Modeling interactions in a dynamic heuristic business network

Abstract

This article presents a novel model for understanding the structure and dynamics of business networks, emphasizing the role of propensities to connect and cooperate as key drivers. The model incorporates behavioral elements and imperfect information updates, departing from traditional rational actor approaches. Starting from the theoretical background, several propositions are outlined, such as dynamism, connection choices, costs, strategy selection, information update, and the update based on experiences. Through simulations, the study successfully demonstrates that the proposed model effectively captures essential characteristics of business networks, including reciprocity, complexity, adaptation, and cooperation. The findings highlight the significance of propensities to connect and cooperate in shaping network structure, evolution, and stability. Particularly, higher propensities to cooperate and connect lead to denser and more cohesive networks, fostering reciprocity, stability, and network performance. The increase only in connection propensities does not have the same result. The lower cooperation propensities result in scale-free networks and asymmetrical distribution of cumulative payoffs. This highlights a crucial insight: different levels of cooperation lead to distinct network properties. Practical implications, increasingly relevant with the rise of digital platforms and metaverse, suggest targeted interventions to enhance network effectiveness, such as incentivizing cooperation, reducing relationship costs, and promoting a culture of trust and collaboration. While providing valuable insights, certain limitations exist, such as not considering the influence of preexisting economic or social structures, equalizing costs and payoffs among actors, and overlooking specific reasons for network connections. Future research should address these refinements and explore their effects on network structure and process characteristics.

Introduction

Regardless of its size, every business actor develops several relationships with the other actors on the market. Those relationships play a role in the businesses’ development and performance, their efficiency and sustainability, as well as resource, knowledge, technology, and innovations development or co-development (Håkansson and Snehota 1995; Heikkilä et al. 2008; Matinheikki et al. 2017; Neghină et al. 2019; Massaro et al. 2019; Anwar and Ali Shah 2020; Khraisha and Mantegna 2020). The lack of such resources and opportunities may contribute to the business’ early failure, which happens globally to more than 50% of new ventures (Anwar and Ali Shah 2020). Therefore, business networks (BNs) capture the attention of both scientists and professionals.

Nevertheless, real-world network exploration bears research challenges (Halinen and Törnroos 2005), such as determining network boundaries, time boundaries, case comparisons, and complexity issues. On the other hand, formal models that aim to describe or prescribe economic and social behavior using networks often impose actors’ rationality and strict economic mechanisms, disregarding the behavioral elements (such as Schweitzer et al. 2009; Izquierdo et al. 2014). While the evolution of cooperation and the role of selective playing, extensively researched using formal models, shed light on the actors’ behavior and presumptions for equilibrium, they still presume actors’ rationality and the behavior governed strictly by optimization given the available information (Izquierdo et al. 2014; Zhang et al. 2016; Hayashi and Yamagishi 1998; Kurokawa et al. 2019; Qu et al. 2019). The empirical approach mainly deals with the macroeconomic level and aggregated data, observing the networks’ structures (for example, Fagiolo et al. 2009; Serrano and Boguna 2003; Miguéns and Mendes 2008; Seok et al. 2021). Though stated approaches contribute to the business network theory, the linking research that incorporates behavioral aspects of the actors revealed by the case studies and behavioral economics literature into formal models is still rare (with a few exceptions, for example, Jiang et al. 2015, Wu and Zhao 2020, Li et al. 2020; as well as experimental results such as Hauk 2003, Rand et al. 2011 and Zhang et al. 2016).

Incorporating behavioral elements increases the model complexity, and so do the non-linear interactions, which can result in chaotic and unpredictable network behavior (Farooqui and Niazi 2016). Therefore, such an approach is often avoided in modeling. However, at this theoretical background accumulation stage, it is necessary to create models that better describe real-world relationships and include or mimic behavioral elements. Recent research has highlighted the importance of incorporating stochastic elements and heterogeneity into models of network interactions. Li et al. (2021) introduced a novel game model with heterogeneously stochastic interactions, demonstrating that such interactions significantly promote cooperation in social dilemmas. By considering the probability distributions of interactions, their work underscores the necessity of accounting for heterogeneity and stochastic processes in understanding network behavior. Recent advances in evolutionary game theory have further expanded our understanding of dynamics in networked environments (Ariful Kabir et al. 2023; Du 2024; Li et al. 2022; Scatà et al. 2016; Wang and Meng 2023; Xiong et al. 2024; Yan 2023; Zhang et al. 2023). Evolutionary game theory is a framework that applies game theory principles to evolving populations, studying how strategic behaviors and traits emerge, persist, or change over time. Evolutionary game theory and adaptive dynamics approaches have made valuable contributions to modeling how gradual evolution leads to adaptation when individuals interact in complex populations (Avila and Mullon 2023), and how adaptive interactions and payoff-driven rewiring of network connections changes the structure of complex networks (Xiong et al. 2024). These frameworks open new avenues for examination how network structure and individual behaviors co-evolve.

BNs can be viewed as a “connectedness of business relationships (which) ties companies into a form of structure with peculiar properties” (Håkansson and Snehota 1995). Another approach observes them as a result of a self-organizing process initiated by business interactions as either emergent or intentional activities (Ritter et al. 2004). The emergent activity approach presumes a relatively passive, non-selective approach to networking. The interaction as a deliberate activity assumes the managing role over a business position in the network. Also, most approaches assume that the business network is generated by interlinking dyadic relationships (Ivanova-Gongne and Torkkeli 2018). The term business networking still means different things to different people, especially given their cultural background, but also individual characteristics such as managers’ sensemaking (Ivanova-Gongne and Torkkeli 2018), which may play a role in interpreting BN and their development.

As a subset of social networks, BNs hold specific structural properties: continuity, complexity, symmetry, and informality (Håkansson and Snehota 1995). Structural properties are usually analyzed using social network analysis (SNA) (Todeva 2006), which is a methodological approach that examines the structure, patterns, and dynamics of relationships between actors, individuals or entities within a network, focusing on how these connections influence behavior, information flow, and overall network properties. Its process characteristics include cooperation and conflict, social interaction, routinization, and adaptation (Håkansson and Snehota 1995). Companies are usually referred to as actors in network modeling.

As we explore the dynamics of traditional business networks, the advent of digital business platforms and the metaverse marks a pivotal shift, significantly magnifying the importance of connectivity, cooperation, and strategic interaction. The idea of applying evolutionary game theory in complex networks in digital systems to foster innovation has already been suggested by Xu et al. (2023), where they focus on resource coordination and value co-creation. In these digital realms, such principles do not merely translate; they evolve to form the backbone of virtual economies and communities, where their impact is exponentially amplified. The transition towards digital and virtual spaces extends the relevance of our research but underscores the urgency to adapt the understanding of business networks. These platforms demand a nuanced exploration of network behaviors, emphasizing the enhanced role of these principles in crafting successful digital ecosystems.

The motivation for creating a dynamic heuristic business network model is to combine previous insights from the distinct approaches to studying this topic: formal models and behavioral aspects of the actors. Our approach aligns with recent trends in theoretical evolutionary game theory that emphasize interdisciplinary approaches and the transfer of insights across fields (Traulsen and Glynatsi 2023). By bridging formal models with behavioral aspects in business networks, we contribute to the cross-fertilization of ideas between behavioral sciences, evolutionary game theory, social network analysis, and business studies. This approach allows for a more nuanced understanding of how cooperation emerges and persists in complex, real-world networks. The goal is to create a business network simulation that better mimics reality than most formal models based strictly on economic mechanisms and payoff maximization. The particular focus of this paper is the examination of cooperation and connectivity. The results of simulations based on the proposed model will be compared to the characteristics of BNs, and whether the goal is achieved will be discussed. The main benefit of a more realistic model is that it begets the examination of different potential scenarios for real-world networks, therefore creating support for examining the interventions’ effectiveness and network management.

Methods

Theoretical framework

The observation of the businesses in a network abandons the classical approach to businesses as resource allocators and opportunity exploiters. It suggests that the development of relationships explains the businesses’ success and that success happens in a dynamic environment due to the interconnections and eventual intertwinement of resources, technologies, and even goals. In a way, each company “bets” on the success of the related businesses, as its own success depends on it (Håkansson and Snehota 1995).

Such commitment is not a result of a single connection; it evolves over a long period with reconnections (it has been reported that business maintains their most essential relationships over 10–20 years (Hallén 1986)), leading to continuity and relative stability, which is a structural property of BNs. The complexity of those connections arises because different individuals involved in those connections create distinct patterns, and the repetitive connections can be different in nature (for example, contracting, delivery, service, co-development of business solutions, etc., (Håkansson and Snehota 1995)).

The symmetry in the BNs emerges from realized more balanced connections between businesses with similar positions in the network and their resources and capabilities (also leading to the positive assortativity), except for the asymmetric relationships between, for example, the startups and business angels or venture capitalists.

While some of those relationships are contractual, many relationships between businesses rely on personal bonds and informal mechanisms, such as trust built on previous experience (Janneck et al. 2008; Massaro et al. 2019; Li et al. 2020). The positive experience from the last connection leads to trust and reconnection, therefore showing the additional intertwinement between the structural characteristics of the BNs. Those structural characteristics play a role in network development and are usually explored using SNA.

On the other hand, process characteristics require a close look at the dyadic relationships. They involve the interplay between the “inherent conflict about the division of benefits from a relationship” (Håkansson and Snehota 1995) and the overall cooperation posture, which creates a basis to analyze them within the game theory framework. Nevertheless, such relationships involve social interaction between individuals, and their personal characteristics play a role in relationship development. Also, those individuals have some social roles (Håkansson and Snehota 1995), each with private social networks bringing about the inequality of the initial position of the actors in the BN, which may play a role in the development of proximity bias (as a form of homophily) in the network (a common occurrence in economic networks, for example, Miguéns and Mendes 2008; Scatà et al. 2016; Seok et al. 2021). While individuals presenting the business goals can be expected to choose optimal strategies, their prior experience, awareness, and personal abilities of assessment and judgment (Camerer et al. 2004; Chugh and Bazerman 2007; Kostelic 2020; Kostelić, 2024) of a situation play a role in the situation sense-making and decision-making, creating a predisposition for a situation misinterpretation. Besides that, actors’ subjective values and preferences, for example, between long-term optimization and short-term temptation (Fudenberg and Levine 2006; Ericson and Laibson 2019), may and do play a role in choosing between the cooperative and defective strategy.

In the case of repetitive cooperative behavior, routinization and adaptation occur, where the first one denotes mechanisms that arise in repetitive relationships that ease the transaction and diminish the costs of connection maintenance. Significant attention has been given to the role of resource constraints and conditional interactions in shaping cooperation within networks. Li et al. (2022) developed a resource-based conditional interaction model highlighting the profound impact of limited resources on cooperation dynamics in spatial social dilemmas. Therefore, supports the role of diminishing costs of connection maintenance in the cooperation dynamics in social networks. The second one represents the coordination of activities that emerge from the connections. The case studies revealed different types of relationships in BN: activity links, resource ties, or actor bonds (Håkansson and Snehota 1995). Whatever the type of the relationship, they are all created by a mutual investment (Håkansson and Ford 2002) and, therefore, inherently carry the cost of connection. The extent of that investment regards the diminished freedom of an actor. While that reduced freedom may implicitly involve realizing the company’s goals, it directly influences the possibility of creating new relationships due to the cost of relationship maintenance. While routinization can be easily incorporated into the models using cost discounting over time, the adaptation requires methods that surpass the general models’ creation goals.

To further understand these complex dynamics and the effects of cooperation, the study of social networks has increasingly drawn from the fields of evolutionary game theory and network science. Evolutionary game theory extends classical game theory by introducing the concept of strategy evolution, where strategies that perform better in terms of payoffs become more prevalent over time (Nowak 2006). This approach is particularly useful for studying cooperation and competition in business networks, as it allows modeling of how businesses adapt their strategies based on past interactions and observed successes or failures. Network science, on the other hand, studies the structure and dynamics of networks, including social and economic networks, by analyzing how nodes (representing entities such as individuals or organizations) and edges (representing interactions or relationships) form and evolve (Jackson 2008). This field has revealed that network topology significantly influences and results from the behavior and outcomes of interactions within the network, with concepts such as small-world networks, scale-free networks, and network centrality playing crucial roles in determining network robustness, efficiency, and resilience (Barabási and Albert 2011; Watts 1999; Xiaong et al. 2024).

Business relationships encompass various elements such as shared alignment, commitment, adaptations, fostering trust, and ongoing social interaction. A reciprocal reliance on outcomes characterizes these relationships, as they cannot be solely controlled by one party (Håkansson and Snehota 1995). This perspective aligns with the application of game theory to represent and analyze such relationships. However, most game theory models assume imposed relationships, where the connection by the “opponent” cannot be stopped (Izquierdo et al. 2014; Zhang et al. 2016; Beranek and Remes 2021), which is not in line with the presumption of mutual orientation. The connection creation should display the mutual preference for the relationship between the two actors.

Nevertheless, if they do connect, the actors proceed to a stage of the strategic interaction, where repeated Prisoner Dilemma (PD) is commonly used (Rand et al. 2011; Jiang et al. 2015; Scatà et al. 2016; Zhang et al. 2016; Kurokawa et al. 2019; Wu and Zhao 2020; Beranek and Remes 2021; Wang and Meng 2023), as it highlights the interplay between the cooperation and conflict in a dyadic relationship. The complex intertwining of cooperation and competition is inherently present in BN: on the one hand, cooperation in BN is a crucial business asset and directly related to its success, while on the other hand, competition exists as seeking an advantage in benefits division at the expense of others, thus playing an important role in new and weak relationships (Ford and Håkansson 2013). The reciprocity in a dyadic relationship was commonly addressed using the tit-for-tat strategy until the out-for-that strategy introduction (Hayashi and Yamagishi 1998). Moreover, the tit-for-tat strategy defies the properties of the cooperation and continuity and relative stability of the relationships and, consequently, the network. The further development of the novel approach usually treats the “out” option not as a part of the game but as a connection choice (possible termination due to the negative experience in the game (Rand et al. 2011)). In such a way, the defective actor is punished, but there is no element of revenge (Rand et al. 2011). The introduction of the “leave” option benefits cooperators and better describes real-world behavior, whether it exists as a connection choice or an in-game strategy (Zhang et al. 2016; Wardil and Amaral 2017; Qu et al. 2019; Kurokawa et al. 2019).

Each relationship is important to an actor, directly through interaction benefits and indirectly through the lens of a position in the network (Ford and Håkansson 2013). Therefore, by introducing the “leave” strategy, the intuition about an actor’s behavior would be reflected in its position in the network. Eventually, the core and periphery polarization occurs (Beranek and Remes 2021). For example, Xiong et al. (2024) discuss how initial network structures can create inherent inequalities among actors. The information on the actor’s position in the network and its type (cooperator or defector) make valuable information for choosing business relationships. Unlike most formal models, real-world networks do not enable the obtainment of complete information.

The available information and its assessment can direct the relationship formation and cooperation (Li et al. 2020), where information deficiency perturbs the development of cooperation (Kurokawa et al. 2019). Hence, the models must define the available information, assuming that not all information is available. If an actor completes a game stage, then it must know its own payoff, which carries the intuition about the strategies played (and the opponent’s type given the chosen strategy). It is generally accepted that the actors continue the relationships with a positive experience and terminate those with a negative experience. Nevertheless, the importance of an actor in the network should also be reflected in the actors’ inclination to connect to more important actors.

However, an actor does not need to (and often cannot) explicitly know the payoff of the other actors and might not even be aware of all the played games in the network. However, the forming network structure can signal the actors’ desirability. Given the information changes in dynamic networks, the actors should update and adjust the choices accordingly (Rand et al. 2011). Each new piece of information requires a recent update, but the update itself should also reflect the errors and biases in individuals’ assessments and judgments. Each person may assess the situation a bit differently or appraise it a bit differently based on their assessment and preferences.

The information in BNs is incomplete, but the actors’ information update therein is also imperfect. The imperfect update and behavior adjustment are in line with the behavioral approaches that claim that people consider only the information that they deem relevant (Hensher et al. 2005; Bazerman and Sezer 2016) and base decision-making and behavior on previous experience, preferences, biases, and heuristics (Camerer et al. 2004; Gigerenzer 2010; Kostelic 2020), introducing an element of randomness and unpredictability at the actors’ level in the modeling. Heuristics present mental shortcuts, one of which is satisficing (Gigerenzer 2010). Satisficing denotes a process of choosing the first “good enough” option that is valued above the person’s notional threshold. In repetitive choosing, a person can discard an option for the next, better one (as in terminating the connection with a defective opponent for the next option above the threshold). Satisficing is a simple heuristic that can be easily implemented in BN modeling. In addition, a negative experience can discourage cooperation, forming new connections, or both. In contrast, positive experiences can reinforce them—meaning that the update process also influences the propensities that govern the choices. Contrary, for an actor with a short-term outcome preference—and thus defective, the positive experience (high payoff) reinforces the exhibited behavior while ignoring the possible benefits of the change of the strategy, whence only repetitive negative experiences (low payoff) lead to the revision of the behavioral propensity towards the cooperation. That may seem like a diluted version of the tit-for-tat strategy. Still, it actually depicts a confirmation bias (Nickerson 1998), where the success due to the exhibited strategy confirms the reasoning for its choice and reinforces the actors’ propensity to use that strategy again. People may but usually do not make drastic changes after only one experience. If the changes occur, it happens gradually with repeated experiences. The subsequent behavior should reflect behavioral elements of the information update and decision-making process, meaning that the modeled behavior can and should change given the experience and available information, as well as the consequential actors’ reflection and revision of the propensities to connect and cooperate.

In sum, observed structural and process characteristics of BNs are not entirely reflected in previously designed models of network interactions. The behavioral point of view demonstrates the necessity for a focused approach to the actor’s behavior in the network and the implementation of behavioral elements. Drawing from the theoretical framework, we conclude that behavioral elements can and should be implemented in the network models of strategic interactions to reflect the characteristics of BNs better.

Model propositions

Based on the theoretical framework, we outline several propositions for the BNs model creation based on heuristics and offer possibilities of their applications, followed by the schematic model outline.

Proposition 1.

The business network development model can be formalized by combining a dynamic network approach and repeated formal PD game.

The repeated interactions modeled as a Prisoner's Dilemma game are a common framework in evolutionary game theory used to study cooperation and defection. This evolutionary approach allows us to simulate how connections and cooperation can emerge and stabilize within a competitive environment. It reflects real-world scenarios where businesses continuously adapt their strategies to maximize their success. We use network science concepts to define the structure and evolution of the business network. Each business is represented as a node, and connections between businesses are represented as edges. The network topology, including measures such as degree distribution, centrality, and clustering coefficient, influences the interactions and the flow of information within the network. By manipulating the network structure, we can study how different topologies affect the emergence and stability of cooperative behaviors.

Application: Dynamism is achieved by repeated connections and interactions with PD games for connected actors. Network boundaries are arbitrarily set on 100 actors. The 100 repetitions bound the time frame. The model is generic, without specifying the connection type or business branch(es) type.

Proposition 2.

One round is composed of three successive parts: choosing the connections, playing a stage of a game between the connected actors, and updating based on the experiences of connecting and playing.

Application: A connection between two actors can occur or not. If the connection occurs between the two actors, they move forward to play one stage of the PD game. At the end of the game, actors gain new experience and update the propensities that govern their behavior, given their predisposition, experience, and the latest information about the resulting actor’s position in the network. The connection and the game can be understood as a mutual arrangement about a joint project, supply chain collaboration, strategic alliances, research and development, etc. The type of connection or business roles is not specified here and can be further developed in future research, in line with (Petrov and Tognazzi 2021) suggestion.

Proposition 3.

The actor’s behavior is not rational but heuristic, governed by the initial propensities to connect and cooperate, with choices based on satisficing and incomplete and imperfect information.

Application: The combination of structural and process approaches enables examining behavioral elements that govern the connections and strategy choices and the resulting network structure and its changes. Therefore, a network structure results from the interaction of actors with individual characteristics, incomplete information, and imperfect available information assessment.

Proposition 4.

The starting actor position must carry an inherent inequality between the actors’ opportunities.

Application: The calculation is governed by the nodes’ (actors’) ID number starting from 1; hence, nodes with smaller numbers „choose “ first, which initially causes the connections of the nodes with closer ID numbers, mimicking: (1) satisficing—choosing first good enough connection; (2) initial inequality of options (lower ID number offers a greater variety of nodes to choose from; by the last nodes, most of the other nodes interesting to connect with could already use their allowed connections and thus, could be unavailable for connection); (3) proximity bias (as a form of homophily). While the size and age of the businesses are not explicitly considered, nor the previous position in the socioeconomic environment and possibly overlapping personal networks, the proposed approach partially mimics the inequality that can be derived from such circumstances.

Proposition 5.

The propensities to connect govern the connections between the actors. The nodes carry different propensities to connect with the other actors and distinct propensities to connect to each of the other actors.

Application: Propensities to connect to any other node are uniformly distributed and distinct. This setting mimics different preferences for connection depending on the actor offered for connection.

Proposition 6.

The actors' propensities to cooperate govern their strategy choices. The actors have different propensities to choose cooperation over defection strategy, which reflects the intuition about their type.

Application: Propensities to cooperate are uniformly distributed and can, but do not have to be equal to propensities to connect (connection and cooperation propensities are treated separately, as a node may want to connect but may not choose cooperative strategy). Propensities to cooperate mimic the underlying preference between short-term and long-term optimization.

Proposition 7.

Each connection carries a symbolic cost. The cost of the retained connections discounts over the rounds and, consequently, allows new connections.

Application: Connections are limited by the cost. The first connection cost is set to $0.5^{1 - 1} = 1,$ and an actor can establish up to 2 connections in the first round, meaning that the allowed amount for the connection maintenance is set to 1 and fixed. Retained connections assume routinization and, therefore, costs of connection maintenance diminish over rounds (a retained connection in the second round costs $0.5^{2 - 1} = 0.5,$ etc.). The cost of connections equals the sum of the first-time connection cost to the power of the number of rounds in which a connection is retained.

Proposition 8.

The connection is established only if both actors mutually prefer a connection (connection cannot be imposed) and if they can afford the relationship (the connection cost can be covered only from the assigned means for connections).

Application: The connection is established if actors’ multiplied propensities exceed 0.5, indicating a mutually high propensity to connect, and the actor has not yet explored the maximum of allowed connections.

Proposition 9.

The actors will choose the strategies according to their propensity to cooperate. Each business's strategy evolves over time based on the payoffs received from interactions with other businesses.

Application: The game occurs if the two actors establish a connection. The game outcome is governed by the propensity to cooperate rather than maximize the payoff. The payoffs are based on the PD game, with strategies of cooperation and defect, $S = \left\{ {C,D} \right\}$ and game payoffs $\pi = \left\{ {\left( {3,3} \right), \left( {0,4} \right), \left( {4,0} \right), \left( {1,1} \right)} \right\}$. The round outcome for an actor is defined with the payoff diminished by the cost of connections.

Proposition 10.

At the end of a round, the update occurs. The position of an actor in the network is considered the general knowledge for the active actors, while the result of a game played is known only to the players who played a game. Thus, the information that an actor can obtain is incomplete.

Application: An update occurs at the end of each round. Propensities to connect and cooperate change depending on the initial settings (or their values in a previous round), the experience in that round, and the resulting actors’ positions in the network.

Proposition 11.

The actors’ updates are imperfect due to their preferences, perceptions, assessments, and judgments about the event that occurred.

Application: The actor who did not establish a connection in the observed round does not update the propensity to connect, as it has no new experience. It is assumed that the propensities change differently for each individual, given their preferences, perception, assessment (error), and judgment (error) of the event, which is accounted for by including random numbers in the update. The role of the random numbers is to mimic the imperfect update. The exact source of the error is not pinpointed, but it is assumed that an amount of random error exists and appears differently over the actors and the events.

Proposition 12.

The positive experience from connecting to an actor ensures the retainment of the connection in the subsequent round. The negative experience of playing with an actor guarantees disconnection in the next round. The positive experience of connecting to other actors increases the general propensity to connect by an amount. Still, propensities to connect to other actors will also depend on the other actor’s position in the network (measured by a centrality measure).

Application: It is assumed that if a cooperative connection occurred in the first round, the actors would have the highest propensity to keep such a connection (1). It will be terminated if an uncooperative connection occurs, and the propensity to continue such connection is 0, as both cooperative and uncooperative actors prefer to play with a cooperative actor (4 > 1; 3 > 0). The relationship between the gain and the cost of the established connections governs the change in the propensity to connect. While the gain exceeds the cost, the propensity to connect is reinforced and otherwise declines. It is assumed that the propensity to connect to an actor is additionally enhanced by the actor's position in the network at the end of the previous round, measured by normalized eigencentrality. So, the connection depends on the initial propensity (or the propensity of the prior round) times the eigencentrality of the actor corrected for the experience from the previous round. The resulting value of the propensity to connect must be in the interval of [0,1]. While the favorable connections will undoubtedly be retained in the future rounds, the propensity to connect to an uncooperative actor might change over time due to the overall propensity to connect and the uncooperative actor’s position in the network, thus mimicking forgiving or giving a second chance.

Proposition 13.

The positive experience will reinforce the propensity to cooperate, while the negative experience will change the propensity to cooperate in the opposite direction.

Application: If an actor did not establish a connection and hence did not play a game, its propensity for cooperation will not change (no new experience was gained). For any other actor, the propensity to cooperate will vary by the calculated propensity change but only within the interval of [0,1]. If the actor chooses an uncooperative strategy and the share of the earned amount is 0, the propensity to cooperate would increase by a random number. If the player earned more than 0.5 of the max payoff by not cooperating, their propensity is reinforced and diminished by the percentage in the payoff times the random number. If an actor achieves the maximum payoff by cooperating, the propensity to cooperate escalates to 1. Suppose the uncooperative player gained less than half of the max payoff. In that case, their propensity to cooperate will increase for the share times a random number. In the case of a cooperative player who achieved less than half of the max payoff, its propensity to cooperate diminishes for the share times a random number.

The model development is based on an overarching framework outlined in Propositions, where the propensities for connection and cooperation among businesses vary and guide actors’ behavior. This general model allows us to explore how different configurations of these propensities influence network dynamics. The propensities for connection and cooperation are represented within a range that reflects the likelihood of these interactions occurring. Specifically:

• $p_{ij, n}$ denotes the propensity for connection of the node $i$ to node $j$ in the round $n$, with $i \ne j$ and $p_{ij} \ne p_{ji}$, ranging between $\left[ {0, 1} \right]$.

• $p_{i,n}^{C}$ denotes the propensity for the cooperation of the node $i$ in the round $n$, ranging between $\left[ {0, 1} \right]$.

Moreover, the propensities govern the creation of the connection as follows:

• If $p_{ij, n} \cdot p_{ji,n} > t_{cc}$, meaning that both nodes have a sufficient propensity to connect, where $t_{cc}$ denotes a threshold for mutual desirable connectivity,

• And $c_{ij,n} < C_{i,n}^{N}$ and $c_{ij,n} < C_{j,n}^{N}$, meaning that establishing the new connection $c_{ij,n}$ does not exceed the number of allowed connections in a round $n$, which is $C_{i,n}^{N}$ for node $i$ and $C_{j,n}^{N}$ for node $j$ (and reflect the cost of maintaining relationships given available resources),

• Then a new connection is established, $c_{ij,n} = 1$, where $c_{ij,n}$ is a binary variable that measures whether the connection was established or not.

If a connection is established, the node's interaction is modeled as a PD game. The propensity for cooperation, $p_{i,n}^{C}$, determines how likely a node is to cooperate during an interaction. The model defines the cooperation outcome based on the propensities of the connected nodes. For a game between nodes $i$ and $j$ in round $n$:

If $c_{ij,n} = 1$ and $p_{i,n}^{C} > t_{co}$ and $p_{j,n}^{C} > t_{co}$, then $g_{ij,n} = b$.

If $c_{ij,n} = 1$ and $p_{i,n}^{C} > t_{co}$ and $p_{j,n}^{C} < t_{co}$, then $g_{ij,n} = d$.

If $c_{ij,n} = 1$ and $p_{i,n}^{C} < t_{co}$ and $p_{j,n}^{C} > t_{co}$, then $g_{ij,n} = a$.

If $c_{ij,n} = 1$ and $p_{i,n}^{C} < t_{co}$ and $p_{j,n}^{C} < t_{co}$, then $g_{ij,n} = c$.

If $c_{ij,n} = 0$, then $g_{ij,n} = 0$,

where $g_{ij,n}$ represents the payoff of node $i$ when playing with node $j$ in round $n$ and $a > b > c > d, c \ge \frac{1}{2}\left( {a + d} \right)$, and $t_{co}$ represents a threshold between cooperative and uncooperative behavior.

Based on the experience in the round $n$, actors revise their propensities and update the payoff, cost of maintained relationships, and network statistics. These revisions and updates reinforce the propensities that have led to better outcomes and shift the propensities that have led to undesirable outcomes. The connections and cooperation in the next round are governed by the reflection of the experience, namely the cost of maintaining connections, network characteristics, and the payoff (as an indicator of other actors’ cooperativeness).

The cost associated with maintaining connections is updated in each round. The cost for node $i$ in round $n$ is given by:

$$c_{i,n}^{C} = c^{c} \cdot \mathop \sum \limits_{j = 1}^{100} c_{ij,n} .$$

For connections retained from previous rounds:

$$c_{ij,n}^{C} = c^{c} \cdot c_{ij,n} + \mathop \sum \limits_{k = 1}^{n - 1} c_{ij,k} \cdot c^{cn - k + 1} ,$$

where $c_{ij,n}^{C}$ is the cumulative cost of connections, $c_{ij,n}$ is established connection in round n, $c_{ij,k}$ are connections retained for $k$ rounds, and $c^{c}$ is a cost of the new connection. This cost moderates the allowed number of connections in the round n,

$C_{i,n} , C_{j,n} , i,j \in v, i \ne j$ is.

$$C_{i,n} = c_{i,k} + C_{i,n}^{N} ,$$

where each node has been allocated an amount of $C_{i}$ which they get to spend in a round, so $C_{i,n} \le C_{i}$; $c_{i,k}$ denotes retained connections, and $C_{i,n}^{N}$ denotes allowed new connections, and is calculated as:

$$C_{i,n}^{N} = INT\left( {\frac{{c_{{i,\left( {n - 1} \right)}} }}{0,5}} \right).$$

The aggregate payoff for node $i$ at the end of round $n$ is calculated as:

$$g_{i,n}^{A} = \mathop \sum \limits_{j = 1}^{100} g_{ij,n} ,$$

that is, a sum of the payoffs gained from each played game, $g_{ij,n}$ in the round $n$.

The share of the possible maximum payoff, given the strategy choice, is:

$$s_{i,n} = \frac{{g_{i,n}^{A} }}{{g_{max,i,n} }}.$$

where $g_{max,i,n}$ is the maximum possible payoff for node $i$ in round $n$, based on the strategy choices.

For each round $n$, network statistics are updated to include measures such as degree, centrality (harmonic, closeness, betweenness, eigenvector), clustering coefficient, number of triangles, hubs, and authorities. These statistics help analyze the overall network dynamics and structure. Network measures are general knowledge, and actors may use them to revise their connection preferences.

The propensities for cooperation and connection are updated based on the outcomes and interactions in each round. For each, a change is calculated based on the relevant experience and information, and then the propensities are updated for the change.

Change in cooperation depends on their previous propensity to cooperate and the share in payoff:

If $c_{ij,n} = 0$, then $p_{i,n}^{cc} = 0$.

If $p_{i,n}^{C} \le t_{co}$ and $s_{i,n} = 0$, then $p_{i,n}^{CC} = r$.

If $p_{i,n}^{C} > t_{co}$ and $s_{i,n} = 0$, then $p_{i,n}^{CC} = - r$.

If $p_{i,n}^{C} \le t_{co}$ and $s_{i,n} \ge t_{s}$, then $p_{i,n}^{CC} = - r \cdot s_{i,n}$

If $p_{i,n}^{C} \le t_{co}$ and $s_{i,n} < t_{s}$, then $p_{i,n}^{CC} = r \cdot s_{i,n}$.

If $p_{i,n}^{C} > t_{co}$ and $s_{i,n} \ge t_{s}$, then $p_{i,n}^{CC} = r \cdot s_{i,n}$.

If $p_{i,n}^{C} > t_{co}$ and $s_{i,n} < t_{s}$, then $p_{i,n}^{CC} = - r \cdot s_{i,n}$.

where $p_{i,n}^{C}$ denotes the existing propensity to connect, $s_{i,n}$ is the share of the maximum possible payoff given the strategy choice, $r$ is a small random number, and $p_{i,n}^{CC}$ is a resulting change in the propensities, given the experience in the round $n$, $t_{co}$ represents a threshold between cooperative and uncooperative behavior, and $t_{s}$ is a threshold for perceiving the gain as sufficient.

The update stage serves to revise the propensities. Updated propensities are then utilized in the next round:

If $c_{ij,n} = 0$, then $p_{i, n + 1}^{C} = p_{i,n}^{C}$.

If $p_{i,n}^{C} + { }p_{i,n}^{CC} \ge 0$ and $p_{i,n}^{C} + { }p_{i,n}^{CC} \le 1$ then $p_{i,n + 1}^{C} = p_{i,n}^{C} + { }p_{i,n}^{CC}$.

If $p_{i,n}^{C} + { }p_{i,n}^{CC} < 0$, then $p_{i,n + 1}^{C} = 0$.

If $p_{i,n}^{C} + { }p_{i,n}^{CC} > 1$, then $p_{i,n + 1}^{C} = 1$,

where $p_{i,n}^{C}$ is a propensity to connect of an actor $i$ in the round $n$, $c_{ij,n}$ is a connection between actors $i$ and $j$ in the round $n$, $p_{i,n}^{CC}$ is the calculated change, and $p_{i,n + 1}^{C}$ is a propensity to connect of an actor $i$ in the next round.

Unlike cooperative behavior, the propensity to connect differs between the experience with a certain node but may affect overall propensity. First, the change is calculated, which affects the actor’s general propensity to connect:

If $c_{ij,n} = 0$, then $p_{ij,n}^{co} = 0$.

If $g_{i,n}^{A} > c_{i,n}^{c}$, then $p_{ij,n}^{co} = s_{i,n} \cdot r$.

If $g_{i,n}^{A} \le c_{i,n}^{c}$ and $s_{in} = 0$, then $p_{ij,n}^{co} = - r$.

If $g_{i,n}^{A} \le c_{i,n}^{c}$, then $p_{ij,n}^{co} = - s_{i,n} \cdot r$.

If $c_{ij,n} = 0$ and $g_{i,n}^{A} > 0$, then $p_{ij,n}^{co} = r$

Based on the previous propensity to connect, calculated change, and other nodes’ positions in the network, propensities are updated:

If $c_{ij,n} = 1$ and $p_{j}^{c} > 0.5$, then $p_{{ij, \left( {n + 1} \right)}} = 1$.

If $c_{ij,n} = 1$ and $p_{j}^{co} \le 0.5$, then $p_{{ij, \left( {n + 1} \right)}} = 0$.

If $p_{ij,n} \cdot e_{j,n} + p_{ij,n}^{co} \ge 0$ and $p_{ij,n} \cdot e_{j,n} + p_{ij,n}^{co} \le 1$, then $p_{{ij,\left( {n + 1} \right)}} = p_{ij,n} \cdot e_{j,n} + p_{ij,n}^{co}$.

If $p_{ij,n} \cdot e_{j,n} + p_{ij,n}^{co} < 0$, then $p_{{ij,\left( {n + 1} \right)}} = 0$.

If $p_{ij,n} \cdot e_{j,n} + p_{ij,n}^{co} > 1$, then $p_{{ij,\left( {n + 1} \right)}} = 1$,

where $c_{ij,n}$ measures whether the connection was established, $p_{ij,n}$ is a propensity to connect of an actor $i$ in the round $n$, $p_{ij,n}^{co}$ is calculated change, $g_{i,n}^{A}$ is an aggregate payoff of the games played in round $n$, $r$ is a small random number, $s_{i,n}$ is a share of payoff in a round $n$, and $e_{j,n}$ is eigenvector centrality of node $j$ at the end of round $n$.

The condition in connection creation is to maintain existing cooperative relationships. In the case of an established cooperative relationship, $cond_{{ij,\left( {n + 1} \right)}} = 1$.

If $cond_{{ij,\left( {n + 1} \right)}} = 1$, then $c_{{ij,\left( {n + 1} \right)}} = 1$.

If $p_{{ij, \left( {n + 1} \right)}} \cdot p_{{ji, \left( {n + 1} \right)}} > t_{cc}$ and $\mathop \sum \limits_{k = 1}^{i} c_{ij,n} < C_{i,n}$ and $\mathop \sum \limits_{l = 1}^{j} c_{ij,n} < C_{j} ,n$, then $c_{{ij,\left( {n + 1} \right)}} = 1$, else $c_{{ij, \left( {n + 1} \right)}} = 0$,

where $c_{{ij,\left( {n + 1} \right)}}$ is a connection between actor $i$ and $j$ in the next round. If the connection has not been previously established and maintained, then the new connection may be established if the connection conditions are satisfied.

Due to the dynamic nature of the model and a multitude of variables and parameters, the model requires a simulation approach for validation. For operationalization of this approach, values have been attributed to the fixed parameters: $t_{cc} = 0.5$, $t_{co} = 0.5$, $t_{s} = 0.5$, $a = 4$, $b = 3$, $c = 1$, $d = 0$, $C_{i} = 1$, and $c^{c} = 0.5$. The model's schematic representation (Fig. 1) links the offered model propositions to applications and allows examination of the effects of different propensities. One hundred rounds will be simulated to assess the model's reflection of key business network (BN) characteristics.

Fig. 1

The model of dynamic network development based on heuristic decision-making and imperfect information update with incomplete information

Despite the theoretical framework supporting all the propositions, the proposed applications need to be thoroughly examined through simulations. To assess whether the model accurately reflects important characteristics of business networks, 100 rounds of simulations were conducted. These simulations explored various combinations of initial propensities to connect and cooperate. A network size of 100 nodes was chosen to balance computational efficiency and the ability to observe significant interactions and emergent behaviors. In real-world business networks, not all companies will develop more than 100 connections. Most small or new companies fail within the first year, during which they establish fewer connections (Anwar and Ali Shah 2020). Typically, only larger companies evolve to maintain connections with more than 100 other businesses. While the specific effects of larger network sizes will not be explored in this study, we believe that the 100-node network provides a reasonable approximation for studying business network dynamics. In addition, the previous research shows that most significant trends occur in the initial stages, so the 100 rounds represent a lengthy period, more than enough to examine this phenomenon. While we acknowledge that some systems might not reach a steady state within 100 rounds, this still provides ample interactions to draw meaningful insights into the relevance and consequences of business network dynamics.

Manipulating the distribution of propensities to connect and cooperate will create different scenarios, allowing for a closer observation of the business network (BN) properties. The variations in the lower boundary of the propensities to connect distribution and the different ranges of uniform distributions create distinct scenarios that can help assess the network's adaptation. By comparing the BN properties across these different cases, we can examine how the network adapts and whether specific distributions lead to convergence, diversification, or stable states.

The six cases will be more closely observed through the BN properties:

• Simulation 1 (S1): the propensities to connect and cooperate are uniformly distributed within the range of [0, 1]:

  • $$p_{ij, n} \sim U\left( {0,1} \right)$$
  • $$p_{i,n}^{C} \sim U\left( {0,1} \right)$$

• Simulation 2 (S2): the propensity to connect is uniformly distributed in the range [0.3, 1], and the propensity to cooperate is uniformly distributed within the range of [0, 1]:

  • $$p_{ij, n} \sim U\left( {0.3,1} \right)$$
  • $$p_{i,n}^{C} \sim U\left( {0,1} \right)$$

• Simulation 3 (S3): the propensity to connect is uniformly distributed in the range [0.5, 1], and the propensity to cooperate is uniformly distributed within the range of [0, 1]:

  • $$p_{ij, n} \sim U\left( {0.5,1} \right)$$
  • $$p_{i,n}^{C} \sim U\left( {0,1} \right)$$

• Simulation 4 (S4): the propensity to connect is uniformly distributed in the range [0, 1], and the propensity to cooperate is uniformly distributed within the range of [0.3, 1]:

  • $$p_{ij, n} \sim U\left( {0,1} \right)$$
  • $$p_{i,n}^{C} \sim U\left( {0.3,1} \right)$$

• Simulation 5 (S5): the propensity to connect is uniformly distributed in the range [0, 1], and the propensity to cooperate is uniformly distributed within the range of [0.5, 1]:

  • $$p_{ij, n} \sim U\left( {0,1} \right)$$
  • $$p_{i,n}^{C} \sim U\left( {0.5,1} \right)$$

• Simulation 6 (S6): the propensity to connect is uniformly distributed in the range [0.5, 1], and the propensity to cooperate is uniformly distributed within the range of [0.5, 1]:

  • $$p_{ij, n} \sim U\left( {0.5,1} \right)$$
  • $$p_{i,n}^{C} \sim U\left( {0.5,1} \right)$$

The simulation results are scrutinized using social network analysis (SNA). The network differences and evolutions are analyzed through degrees, densities, betweenness centrality, hub scores, coreness, eccentricity, transitivity, and assortativity using the R package igraph (Csardi and Nepusz 2006). The script used to create the network, generated data, and visualizations are available in the Supplementary files.

It is expected that manipulating propensities to connect influences the network's connectivity and structure. By observing the BN properties, such as connectivity patterns, density, or degrees, for each scenario, we can gain insights into how the different distributions impact the network's connectivity and overall structure. This analysis will help understand how the propensities to connect affect the network's behavior and adaptation. By comparing the outcomes of each scenario, we can determine which distribution of propensities to connect leads to desirable network properties, such as improved stability, higher gains, or enhanced collaboration. This evaluation helps identify distributions that promote optimal network performance.

Results and discussion

The different initial settings of propensities to cooperate and to connect lead to different network evolution. This is evident in Fig. 2.

Fig. 2

Networks’ graphs in a round 100

In each of the simulations, there were 100 rounds. However, the developed network structures differ from highly compartmentalized in S1 to a completely connected network in S6. While the first insights into the results shed light on the role of the propensities, the intricacies of dynamic development and the differences that arise from different settings yet remain to be investigated. In subsequent sections, we explore the dynamics and evolution of network topology. The results show how various levels of propensities to connect and cooperate govern the development of business networks, which we compare to previous findings on business networks.

Average degrees

In the fifth and sixth simulations (S5, S6), the average degrees rapidly increase over time, which indicates that, on average, each node has a larger number of connections to other nodes and that connectivity within the network is growing (Fig. 3). In S4, only the propensity to cooperate is enhanced, leading to a moderate rise in average degrees.

Fig. 3

Average degrees over rounds with different initial propensities settings

Increasing average degree measures may signify an expanding network where actors are forming more connections with one another. This can signal increased collaboration, information sharing, and resource exchange within the business network. It may indicate a network that is becoming more interconnected and fostering greater opportunities for partnerships and interactions. Given that the higher growth of average degrees occurs in S5 and S6 or lower growth in S4, it looks like the initial propensity to cooperate plays a more pronounced role in network development than the initial propensity to connect.

However, for S1–S3, the average degree measures stay small and oscillate within a small range, suggesting a relatively stable network connectivity level over time. This indicates that, on average, actors maintain a similar number of connections with other actors, and there are minimal changes in the overall level of connectivity. Lower average degree values indicate that, on average, each actor has fewer connections to other actors. This can also result from bad experiences from interactions with noncooperative actors, which decrease both propensities to connect and cooperate. This paints a picture of a network that has reached a relatively stable state with a small but consistent level of collaboration and information exchange among actors.

The comparison shows that the increased propensities to cooperate facilitate accelerated network growth and expansion as actors also become more receptive to establishing new connections, thereby driving the continuous evolution of the network. Cooperative behavior fosters enduring and stable relationships, promoting the development of trust-based networks but also encouraging actors to engage in repeated interactions, thereby contributing to the network's gradual and sustainable advancement. However, given Fig. 3, it can also be concluded that if only the propensity to connect rises, there is no corresponding increase in the average degrees (S2, S3).

The node degree distributions in S5–S6 are bimodal, suggesting two distinct groups within the network, with one group having limited connectivity and the other being more interconnected (Fig. 4). It is similar for S4, which is also bimodal and very positively skewed, with one mode in 0 indicating a group with limited (or none) connectivity and the other with a higher number of connections.

Fig. 4

The degrees distribution in the round 100

The node degree distributions in S1–S3 are highly skewed with a mode of 0, indicating a large number of actors with no connections. The distributions are positively skewed, meaning there are fewer actors with a high number of connections than the other networks. This suggests a less interconnected and less dense network structure with a few prominent actors.

One usually referred property of social networks is the power-law distribution of degrees, meaning the network is scale-free. A hypothesis testing using the poweRlaw package (Gillespie 2015), where H₀: data is generated from a power law distribution, yielded the following p-values: 0.82, 0.34, 0.54, 0.02, 0, 0 for S1–S6, respectively. The null hypothesis that the data follows power-law distribution cannot be rejected for S1–S3 at a 5% significance level. However, the null hypothesis is rejected for all simulations with enhanced cooperation.

While some research suggests the prevalence of scale-free networks in many social and biological networks, the distribution of node degrees does not always follow a power-law distribution. While degree distributions are not often in research focus, there are instances where deviations from the power-law distribution occur. For example, international trade networks have not been scale-free for most years and have become decentralized (Li et al. 2020). Tourism networks in 2018 did not exhibit power-law degree distributions (Kostelić and Turk 2021), and degrees in supply chain networks do not seem to follow power-law distribution (Wiedmer and Griffis 2021). On the other hand, the research devoted to examining networks’ degrees revealed that they are not, in fact, so frequent in the real world (Broido and Clauset 2019).

The hypothesis results indicate that different levels of cooperation lead to distinct network properties, while the manipulations with only propensities to connect lead to power-law degree distributions. Furthermore, conflict, rather than cooperation, could be the key driver for the scale-free networks observed in this study. In this research design, the conflict stems from the actor's emphasis on short-term temptation over long-term optimization. This opens up opportunities for further exploration and understanding of how cooperative behaviors impact business networks’ overall structure and dynamics.

Network density

Figure 5 depicts the evolution of shares of established connections of all possible connections (which is 4950 connections for an undirected network with 100 nodes). In other words, it shows the evolution of network density. In the last round, there were 160 connections in S1, 201 in S2, 185 in S3, 610 in S4, 1401 in S5, and 1493 in S6, resulting in densities of 0.032, 0.04, 0.037, 0.123, 0.283, and 0.302, respectively.

Fig. 5

Share of established connections of all possible connections

The World Trade Network is one of the oldest networks, with almost all interconnected actors and a high density of over 0.9 (Antonietti et al. 2022). In comparison, a relatively novel global network of industrial robot trade reached a density of 0.65 in 2017 (Li et al. 2019). However, other networks that are less developed reveal lower densities, such as almost 0.3 in the global tourism network (Kostelić and Turk 2021), 0.12 in the education network (Barnett et al. 2016), 0.22 in the international coal network in 2008 (Chen et al. 2022), or oscillates between 0.2 and 0.6 in the financial network (Chinazzi et al. 2013). The interregional trade network in Indonesia achieved a density of 0.14 (Santoso et al. 2019). While social networks usually have higher densities, business networks may have lower densities because of strategic and selective connections. Asian-Australasian cruise shipping network demonstrates a variable density from 0.03 to 0.2 over its clusters (Kanrak and Nguyen 2022). Small oscillations over time were observed in the industrial software clusters, with the lowest at 0.028 and the highest at 0.035 (Kim et al. 2014). The network density in the production cluster in the film industry cluster was 0.16, which researchers deemed a relatively high value compared to the Brandenburg timber industry cluster, which had a density of no more than 0.04 (Krätke 2002). Another example is the study of the big business sector in Britain from 1904 to 1976, which revealed a shift in the overall density of the interlock network, rising from 0.013 to 0.017 (Scott 1988). The analysis of affiliation networks of Indian listed companies revealed varying densities over the clusters, from 0 to 0.08 (Prem Sankar et al. 2015). Crop trade networks show evolving network densities of up to 0.075 (Zhang and Zhou 2022). The network density values depend on many factors, starting from how long the network has been developing, the network size, the type of the network, cooperation and connectivity preferences, constraints, and limitations.

S5 and S6 show higher density, indicating more established connections, while S1-S3 have lower density and fewer connections. However, they all show gradual changes over time and different growth rates of network densities. The network density values of S1-S6 are comparable to various real-world networks, reflecting the diversity in interconnectedness among different types of business networks.

Average betweenness centrality

Betweenness centrality quantifies how often a node lies on the shortest path connecting pairs of other nodes. In the case of simulations S1–S3, the average betweenness centrality slowly increases over time, suggesting that certain network actors are gaining prominence in terms of their control over information flow (Fig. 6). The overall increasing betweenness centrality indicates that there are actors that act as critical intermediaries, connecting different parts of the network. In business networks, such actors facilitate efficient communication, information flows, or even resource allocation.

Fig. 6

Average betweenness centrality for networks throughout 100 rounds

The pronounced oscillations (S4) may indicate fluctuations in the influence and importance of specific actors over time, potentially influenced by network dynamics or actor behavior changes. This can imply that certain actors are exchanging and becoming more influential or central in terms of their control over information transfer. It is not uncommon for business networks (aside from trade and tourism) to show oscillations in centrality parameters, so this mimics an evolving network where different actors achieve gains and suffer losses, changing or improving their positions in the network. The overall increasing betweenness measures suggest a growing concentration of power or influence within the network. It might also indicate the network's formation of key hubs or connectors.

Conversely, if the average betweenness measures decrease over time (S5, S6), it implies that actors have less influence in facilitating communication and information flow between other actors. This could suggest a more decentralized, distributed, or balanced network structure, where information flow is not primarily reliant on a few highly central actors. Decreasing betweenness could also suggest a network evolving towards greater autonomy or independence of individual actors, as in the case of declining centrality measures in the global industrial robot trade (Li et al. 2019).

Betweenness centrality may be crucial for new ventures seeking supplier partners (Bloodgood et al. 2017). Pursuing organizations in high betweenness centrality networks may require an early start, while less dense networks allow for a delayed approach. New ventures may target networks with high or low betweenness centrality depending on the legitimacy diffusion stage to accelerate or enhance diffusion. Low-density and low-betweenness centrality networks offer a less contentious environment, facilitating faster acceptance and improved performance. However, ventures requiring extensive diffusion may benefit from focusing on dense or high-betweenness centrality networks for optimal outcomes. Initially, legitimacy may be harder to establish in high-betweenness centrality networks, but once achieved, diffusion is faster and more comprehensive compared to low-density or low-betweenness centrality networks (Bloodgood et al. 2017).

In addition, the study (Huang and Hsueh 2023) provides evidence that betweenness centrality plays a significant role in shaping exploratory and exploitative innovation within ego networks or personal networks. Specifically, the findings reveal that ego-network density acts as a positive moderator for the impact of betweenness centrality and structural holes on exploratory innovation. Another research (Telles et al. 2020) assumes the role of betweenness in innovation diffusion. The combination of insights creates a basis for a line of future research worth examining after role assignment to the actors and the extension of this simulation to a multi-layered network (Petrov & Tognazzi 2021).

Average hub scores

A hub score is calculated by assessing the number of incoming links (in-degree) of a node in a network, indicating its significance as a central point for receiving connections from other actors. The increases in average hub scores with the oscillations (which can be observed in the initial rounds, Fig. 7) suggest that certain actors in the network are temporarily becoming more central and influential as hubs. Higher hub scores indicate that more actors have a greater number of connections to other actors, making them important for information flow, resource exchange, and collaboration within the network. The increasing hub scores could indicate the emergence or strengthening of key players within the business network. It suggests a temporary concentration of power, expertise, or influence in a few actors, which could shape decision-making, resource allocation, and overall network dynamics. S1–S3 demonstrates a slower growth trend with expressed oscillations, while S4–S6 exhibit higher growth rates with fewer oscillations. The networks S4 and S5 have higher propensities to cooperate, while propensities to connect stay uniformly distributed from 0 to 1. That means the actors are initially less willing to connect but have a lower probability of encountering uncooperative actors, which fosters further connectivity.

Fig. 7

Average hub scores for networks throughout 100 rounds

On the contrary, in S1–S3, the actors are initially more willing to connect, with a higher probability of encountering uncooperative actors. This hinders fast connection growth and diminishes propensities to connect to uncooperative actors, thus maintaining alliances with a few trusted connections. That is also reflected in average degrees and network density.

The international coal trade network has exhibited changes in top hub actors over the years (Chen et al. 2022). In that case, they are governed by changes in the supply and demand for coal. The changes in leading actors are common in other networks, such as tourism networks (Miguéns and Mendes 2008; Kostelić and Turk 2021). They reflect the network dynamism and results of interactions.

Average coreness

The average coreness measures increase over time in the cases of S4–S6, which suggests that the structural connectedness of actors within the network is growing (Fig. 8). Higher average coreness values indicate that, on average, actors are more central and play more crucial roles in the network's connectivity. It may indicate a network becoming more cohesive and reliant on a core set of influential actors.

Fig. 8

Average coreness for networks throughout 100 rounds

However, that does not occur for S1–S3, where the average coreness measures are low, with very slow growth over time. That suggests that the structural importance and centrality of actors within the network are relatively limited. The slow growth in average coreness measures suggests minimal changes in the importance of individual actors over time. It may indicate a network with a more distributed structure where influence and centrality are spread across a larger number of actors without significant concentration in a core subset.

For example, the industrial robot trade exhibited a gradual increase in the nodes belonging to the core, followed by stabilization (Li et al. 2019). However, upon closer inspection, they revealed that the trade in the core cluster is more diversified than peripheral nodes, showing that the high coreness does not necessarily mean a lack of variety or complexity.

Transitivity

The network becomes more clustered or cohesive if the average transitivity measures increase over time, most expressed in S4–S6 (Fig. 9). They seem to converge to a transitivity value of 0.8. Higher average transitivity values indicate that actors are more likely to form triangles or clusters, where connections tend to exist between already connected actors. Increasing average transitivity measures may signify the emergence of more localized interactions and strengthening relationships within specific clusters or communities in the network. It indicates a network evolving towards a more cohesive and tightly connected structure.

Fig. 9

Transitivity for networks throughout 100 rounds

In the case of S1–S3, the measures are lower with very slow growth over time but exhibit high oscillations. That points to a network with inconsistent clustering behavior. This scenario suggests that the network experiences varying local connections or community formations but lacks stability in its clustering patterns. The reason may be the uncooperative actors that cause a loss for the cooperative ones, which discontinues those connections. If uncooperative actors become involved in triangles or clusters, their presence can be disruptive. This disruption may manifest through the cooperative actors’ loss and termination of established connections with these actors. Additionally, the negative experience stemming from their behavior can influence the cooperative actors’ propensities to connect and cooperate, impacting their willingness to connect and cooperate with other actors. That would lead to a sparse or scattered network structure, with a relatively low tendency for actors to form triangles or clusters.

The biggest cluster size

The biggest cluster size reaches almost all 100 actors in round 50 for S5 and S6 and has been minimally oscillating since (Fig. 10). That suggests a network with a growing and stable cohesive structure. The increasing cluster size indicates the formation and expansion of a dominant cluster in the network. As the cluster encompasses a larger portion of the actors, it implies a higher level of interconnectivity and potential collaboration among businesses. The minimal oscillations in cluster size in the latest rounds suggest a relatively stable structure, with the dominant cluster maintaining its size and cohesion.

Fig. 10

The biggest cluster size for networks throughout 100 rounds

In the case of S4, the biggest cluster is smaller (about 60 actors) with expressed oscillations in the initial rounds, which indicates a network with fragmented and fluctuating cluster structures. The consistent biggest cluster size of around 60 actors after the first 20 rounds suggests continuity and possible presence of multiple small clusters or disconnected components in the network. This scenario implies a network where businesses tend to form smaller, localized collaborations or relationships rather than a single dominant cluster.

The biggest cluster sizes of around 50 actors, with oscillations, occur for S1-S3. That can indicate a network with relatively small clusters, possibly with changes in structure or occasional dissolution. The lower and slowly growing cluster size suggests the presence of multiple smaller clusters in the network, indicating a less cohesive structure. The oscillations in cluster size imply a dynamic and evolving network with frequent changes in the formation and decline of clusters. This scenario suggests a network where businesses may form connections with uncooperative actors that change over time.

It is common for social, economic, and business networks to have clusters. Examples are the Asian-Australasian cruise shipping network (Kanrak and Nguyen 2022), the timber industry cluster (Krätke 2002), and affiliation networks of Indian listed companies (Prem Sankar et al. 2015). The analysis of the network of business angels indicates a possibility of clustering based on geographical proximity (Gvetadze et al. 2020). A similar is true for trade and tourism networks (Fagiolo et al. 2009; Serrano and Boguna 2003; Seok et al. 2021; Kostelić and Turk 2021), and their analyses revealed shifts in clusters and communities over time regarding their participants and size. Community structures are evident in supply chain networks, and the number of communities can differ among firms and across various industries (Wiedmer and Griffis 2021). Changes regarding the size are depicted in the oscillations of the biggest cluster size, which, combined with the conclusions about transitivity and assortativity, lead to the conclusion that the changes also occur in clusters’ participants.

Higher propensities to cooperate can lead to network convergence, where actors form coherent and stable clusters or communities with common goals and attributes. Conversely, lower cooperation propensities can lead to network diversification, with actors forming distinct and disconnected groups. Diminishing costs may facilitate convergence and diversification, as actors can maintain existing relationships while exploring new connections.

Modularity

For S5 and S6, the average modularities decrease more rapidly over the first 20–30 rounds and then oscillate with a slow declining trend (Fig. 11). That shows networks that initially had a higher degree of discernable communities. Still, the communities gradually merged or became less distinct over time. The decreasing modularity indicates a decrease in the strength of the community structure, with more connections occurring between actors from different communities. The insight into clusters shows that, in this case, the communities merged into one giant component.

Fig. 11

Average modularity for networks throughout 100 rounds

In the case of S1–S3, the average modularity decreases harshly in the first rounds, with a slow decrease and oscillations. This indicates a network with a weaker and less pronounced community structure almost from the beginning. The significant reduction in modularity at the beginning suggests a disruption or breakdown of distinct communities within the network. These networks have higher shares of uncooperative actors, which cause structural breaks.

S4 has a steadily declining modularity, indicating continuous fragmentation in the structure. This resulted in the lowest modularity value in the last round, which pointed to a fragmented structure with limited community formation. This scenario suggests a network where businesses may have weaker tendencies to form cohesive groups or communities; connected nodes create a homogeneous group, which also means they are neutral when forming connections.

Assortativity

Apart from S4, where assortativity remains stable, the actors show a diminishing tendency to form connections with similar actors regarding specific attributes or characteristics (Fig. 12). Higher average assortativity values (S4–S6) in the beginning indicate a network where actors with similar attributes, such as degree, size, or other relevant features, are more likely to connect. A positive assortativity was expected, as propensities to connect are, among other parameters, also governed by the actors’ eigencentralies. However, when combined with other measures, it can be noticed that these networks are more cohesive and more balanced in terms of differences between actors’ degrees. So, high assortativity does not have to indicate preferential attachment in this case but a network with similarly successful actors.

Fig. 12

Assortativity throughout 100 rounds

The oscillating assortativity values in S1–S3 suggest networks with inconsistent assortative mixing patterns. The low average assortativity values (such as in S2, in rounds 17–20, or S3 in the last rounds) indicate a state of a network where actors do not exhibit strong preferences for connecting with others of similar attributes. The high oscillations in assortativity measures indicate fluctuations in the level of assortative mixing over time. This scenario suggests that the network experiences varying patterns of connectivity based on attribute similarity but lacks stable assortative tendencies. For example, in the case of the crop network, assortativity oscillates over the years (Zhang and Zhou 2022).

While the examined networks in S1–S6 are primarily assortative, given the degrees, real-world business networks can be assortative, disassortative, or both. For example, the examination of the Asian-Australasian cruise shipping network (Kanrak and Nguyen 2022) revealed that assortativity could be observed if location, rating score, or degrees were observed. On the other hand, if actor properties such as population, visitors, popularity, or attractions are examined, the network shows disassortative tendencies.

Continuity

Continuity refers to the long-term and ongoing nature of relationships within a business network. It suggests relative stability within the network, indicating a persistent connection between actors. Routinization, on the other hand, relates to the development of routine mechanisms, established patterns of behavior within repeated relationships, and diminished costs. Despite their distinct definitions, continuity and routinization share some similarities. Both properties arise from repeated interactions and relationships within a business network. Continuity is achieved by maintaining long-term connections, while routinization emerges due to the familiarity and repetition of interactions. Both continuity and routinization contribute to the overall structure and stability of the network, facilitating smoother transactions and enhancing the efficiency of business relationships (Håkansson and Snehota 1995).

In this network, the costs of maintaining relationships decrease for retained relationships. This cost reduction allows for greater availability of resources that can be allocated toward forming new connections. This flexibility promotes network growth, facilitates the formation of additional relationships, and potentially enhances collaboration, resource sharing, and performance within the network. As a result, the number of node degrees, representing the number of connections an actor has, increases over successive rounds. In other words, the network is characterized by a growing number of connections if relationships are maintained and new connections are established.

If the cost of maintaining existing connections decreases sufficiently (< 1), it allows for allocating additional resources to afford a new connection in the next round. This increase in available resources enables the actor to raise the maximum number of allowed connections by one, particularly when the current round has already reached its maximum capacity.

Figure 13 illustrates the maximum number of connections distribution in round 100 for different network scenarios. In the case of S5 and S6, the distribution is bimodal with a long right tail. The distributions suggest that these networks have a balanced and relatively higher connection capacity than S1–S4. This indicates that the actors in these networks have sufficient resources and capabilities to establish and maintain multiple connections, leading to a more extensive network structure.

Fig. 13

The density of maximum allowed connections in round 100

However, the two increases in the frequencies, at about 20 and 50 nodes, show that two groups with different levels of connectivity exist within the structure. While this cannot result from the propensities to connect or cooperate, especially in the case of S6, this may be related to the node positioning. Given the satisficing principle, nodes are connected to the first available nodes that they can afford, given the connections’ costs. However, that may have left some nodes unconnected or at peripheral positions in the initial stages of the game, which later resulted in differences in retained and maximal allowed connections.

For S4, the density curve reveals a bimodal distribution, with approximately 3 and 35 nodes appearing more frequently in round 100. This suggests that some actors in this network have the capacity to establish a larger number of connections, while others (more of them) are limited to a smaller number. This could reflect varying resources, capabilities, or strategic choices among the actors in S4, resulting in a more heterogeneous network. The actors with a higher number of connections may have more opportunities for reciprocity and cooperative interactions, contributing to a higher level of reciprocity within their subset of the network. However, the actors with fewer connections may face limitations in establishing reciprocal relationships, which can affect the overall reciprocity dynamics of the network. The interplay between these two subsets may influence the network performance, as the more connected actors can potentially compensate for the limitations of the less connected ones. Still, the overall effectiveness and efficiency of the network may be compromised due to the unequal distribution of connections.

On the other hand, for S1–S3, the density curves indicate a positively skewed distribution, with 2 or 3 nodes being the most common occurrence in round 100. The actors in these networks may have fewer resources or constraints that restrict their ability to establish multiple connections. The restricted capacity for connections can limit the opportunities for reciprocity and cooperation among actors. This may hinder the development of strong collaborative relationships and impede the flow of resources and information within the network, resulting in lower levels of reciprocity and potentially suboptimal network performance. This can also explain the revealed simpler and more localized structure with fewer overall connections.

The capacity for connections in the networks influences the consequences for reciprocity and network performance. This capacity is inherently related to propensities to cooperate and connect. Still, it can also result from the initial node's position or interaction with cooperative vs. uncooperative actors in the initial stages. Networks with a higher capacity tend to foster stronger reciprocity and exhibit better performance. In contrast, networks with a more limited capacity may face challenges to reciprocity and overall network effectiveness. The specific distribution of the maximum number of connections, whether positively skewed or bimodal, further shapes the dynamics of reciprocity and performance within the network.

Reciprocity

Given that the network is undirected, all connections are reciprocal and mutually established. In this case, we observe reciprocity regarding the chosen strategy in the game. It is calculated separately for cooperative and uncooperative strategies for each round. To calculate the share of reciprocal relationships, the number of relationships where both players chose to cooperate is divided by all possible connections. Similarly, the share of reciprocal uncooperative relationships has a number of connections where both players select uncooperative strategies in the numerator.

Based on Fig. 14, we can conclude that reciprocity in business networks increases over time, particularly in the case of cooperative behavior in S4, S5, and S6, as expected. For S4, the share of reciprocal relationships grows to slightly over 0.02 over 100 rounds. This indicates a noticeable increase in reciprocal interactions within the network. While the share is still relatively low compared to S5 and S6, it suggests increasing reciprocity in cooperation.

Fig. 14

Share of reciprocal relationships in all possible relationships over 100 rounds

In the case of S5 and S6, the share of reciprocal relationships shows significant growth, reaching almost 0.3 for S5 and S6. This indicates a substantial increase in reciprocal interactions over time in these networks. The higher share of reciprocal relationships suggests a greater emphasis on mutual cooperation and exchange within the network.

On the other hand, for S1–S3, the share of reciprocal cooperative relationships remains low, reaching up to 0.05 over 100 rounds. This implies that reciprocity is not a prominent characteristic in these networks, and there is a relatively lower emphasis on cooperation and exchange compared to S4, S5, and S6.

If we calculate reciprocity differently and replace the denominator with the number of established relationships, then in the last round, shares of cooperative reciprocal relationships reach 0.71, 0.82, 0.81, 0.95, 1, and 1 for S1–S6, respectively.

Since all reciprocal uncooperative relationships in S1–S6 remain low, reaching the most up to 0.05 over 100 rounds, it further supports the conclusion that cooperative reciprocity is an important characteristic in the business networks. When the share of uncooperative reciprocal relationships remains low, it means that the majority of connections formed in the networks are cooperative. This means that actors within the network are more likely to engage in mutually beneficial and cooperative interactions where both parties reciprocate and maintain a positive relationship over time. The governing mechanisms also exclude uncooperative actors from the network, leading to the larger connected networks where there were few or no uncooperative actors in the initial stages. The contrast between the low share of uncooperative and the increasing share of cooperative reciprocal relationships in S4, S5, and S6 reinforces the notion that reciprocity is a significant factor in these networks’ development.

The increasing share of reciprocal relationships in S4–S6, compared to S1–S3, suggests that reciprocity becomes more prevalent and significant over time in these networks. This indicates a stronger emphasis on mutual cooperation and exchange, which can contribute to the development and performance of the businesses involved. The fact that the share of nonreciprocal relationships remains low across all simulations supports the conclusion that reciprocity is an important characteristic of business networks. It suggests that actors within these networks prioritize and value cooperation and exchange, leading to a higher prevalence of cooperative relationships.

However, this also shows that the actors adapt to their environment based on previous experiences. Actors in S1–S3 are more likely to experience uncooperative behavior, which can lead to a bad experience and shape their future propensities to connect and cooperate. Moreover, a cooperative actor experiencing connections with uncooperative actors might suffer such a loss that will lead to their exclusion from further participation. That mimics the business’s early failure, which can be caused by the lack of resources and opportunities (Anwar and Ali Shah 2020), but also by wrong partnership choices.

Network performance

Based on the average cumulative gain for each round increasing over 100 rounds in S1-S6, it can be concluded that network performance improves over time (Fig. 15). Higher cumulative gains indicate that the network actors are experiencing greater success and profitability as the rounds progress. This suggests that the business networks facilitate efficient resource allocation and collaboration among the actors, improving overall performance.

Fig. 15

Average cumulative gain over 100 rounds

The increase in cumulative gains can be connected to other characteristics of the networks, such as reciprocity. The fact that the share of reciprocal relationships also increases in S4, S5, and S6 further supports the connection between reciprocity and network performance. Reciprocal relationships, where actors engage in mutually beneficial interactions, can contribute to higher cumulative gains by fostering trust, cooperation, and long-term commitment between the actors. As a result, the actors are more likely to exchange resources, knowledge, and support, leading to improved performance and cumulative gains over time.

Additionally, the higher cumulative gains in S5 and S6 (over 3000) compared to S1–S3 (up to over 100) and S4 (over 1000 in the last round) indicate that these networks are particularly effective in generating positive outcomes for the actors involved. This could be attributed to factors such as the complexity of relationships, the symmetry in resource distribution, or the informality of connections. These characteristics might contribute to a more favorable business environment that enhances performance and facilitates higher cumulative gains. The higher cumulative gains observed in S5 and S6 compared to S1–S4 indicate that these networks exhibit particularly strong performance and effectiveness.

Another way to examine network performance is the examination of the Pareto 80/20 rule (Fig. 16). In this case, it is examined whether the top 20% of the actors achieve 80% of the gains. Since this is a property usually related to scale-free networks, it is not unexpected to reveal that S1–S3 seem to converge to the 80/20 rule. Networks with lower levels of cooperation (S1–S3) exhibit characteristics typically associated with scale-free networks. This suggests that a few actors in these networks dominate in terms of resource accumulation or performance outcomes.

Fig. 16

Examination of Pareto 80/20 rule

However, the networks with more cooperative actors achieve a more balanced gain division. In the case of S4, the top 20% of the actors achieve about 60% of the gain. However, in the case of S5 and S6, the top 20% of the actors achieve about 30% of the gain.

This result complements the previous finding that S1–S3 behave as scale-free networks, indicating that the degree distribution in these networks follows a power-law distribution. The observation that networks with higher cooperation levels depart from scale-free networks and exhibit a more balanced gain division suggests that cooperation may shape the distribution of gains within the network. Higher cooperation levels lead to a more equitable distribution of resources and benefits among the actors.

Scenarios overview

We see that manipulating propensities to connect influences network connectivity, structure, and evolution. Increasing propensities to cooperate leads to higher network density, more established connections, overall accelerated network growth, increased cohesion, and improved performance outcomes. Networks with higher initial cooperation levels tend to have a more balanced distribution of connections among actors, resulting in a more equitable sharing of resources and benefits. Increasing assortativity suggests actors preferentially connect to others with similar attributes. However, in balanced networks, where actors have similar degrees, it ceases to signal preferential attachment. Reciprocity in cooperation is more pronounced in networks with higher initial propensities to cooperate, indicating a stronger emphasis on cooperation and exchange. Networks with lower initial cooperation levels may experience slower growth, fragmented structures, and suboptimal performance outcomes.

The propensities to cooperate and connect significantly influence the characteristics of business networks, particularly concerning whether the networks exhibit scale-free properties. In the simulations, the networks with higher levels of cooperation (S4, S5, and S6) displayed characteristics of non-scale-free networks, while networks with lower cooperation levels (S1, S2, and S3) exhibited scale-free properties. The higher cooperation levels contributed to the emergence of non-scale-free networks, where actors had a more equal number of connections and positive outcomes. This suggests that a cooperative environment fostered a more collaborative and interconnected network structure, allowing for smoother resource exchange and mutual support among the actors.

The positive and negative experiences from connecting and playing with other actors influence the retention or termination of connections in subsequent rounds, which is highlighted in the network measures. Actors in S1 have the most opportunities to interact with uncooperative players, which hinders the network evolution, resulting in slower growth rates, lower cohesion, and suboptimal performance outcomes. The network exhibits fragmented and fluctuating community structures, with varying patterns of connectivity based on attribute similarity. Reciprocity in cooperation exists but at lower levels compared to other scenarios. Similarly, the cumulative gains are lower than networks with higher cooperation levels, indicating less effective resource allocation and collaboration. A business network mimicked by S1 could reflect a network where collaborations and information exchange among actors remain low, with significant fluctuations in network connections and slow growth.

S2 shows similar trends to S1 but slightly higher density and centrality measures. More significant growth in cumulative gains, indicating better network performance due to a higher average propensity to connect. The network shows inconsistent clustering behavior, indicating a less cohesive structure. The reciprocity in cooperation remains low, suggesting less emphasis on cooperation and exchange. The cumulative gains are lower than networks with higher cooperation levels, indicating less effective resource allocation and collaboration. In a real-world context, this simulation could represent a business network where there is a moderate level of connectivity between actors. Businesses are not initially but become more selective in forming connections, preferring to collaborate with others who exhibit a certain level of cooperation, causing a less cohesive structure.

S3 has an additionally enhanced propensity to connect and a slightly higher average degree than S1 and S2. The simulation 3 (S3) network still has low average degree measures with very slow growth and few oscillations. The biggest cluster size indicates the presence of small clusters or disconnected components. Cumulative gains show slow growth over time, indicating lower network performance. Still, a low share of reciprocal cooperative relationships, comparable to S2, suggests a limited emphasis on cooperation. In the context of business networks, hubs can represent influential or prominent actors with a high number of connections, while sparsely connected actors represent actors with fewer connections.

In a real-world context, this simulation could represent a business network where actors are initially more willing to connect, but the propensities to cooperate vary across the network. This results in a network with a mix of cooperative and non-cooperative actors, leading to limited growth in connections, fluctuations, and the formation of smaller, localized collaborations. The first three networks (S1–S3) have the lowest densities, observed in some real-world networks and discussed in Chapter 3.2. This may be also explained by the nature of business networking, which often involves more selective and strategic connections rather than having many connections with everyone in the network.

S4 has a higher average propensity to cooperate, and its bimodal actor degree distribution indicates a network with two distinct groups, one with more connections and one with fewer. The modularity values indicate the merging of distinct communities over time, leading to a more unified network structure. The average coreness measures increase over time, suggesting a growing reliance on a core set of influential actors. The network exhibits increasing average degrees, network density, betweenness centrality, and hub scores compared to S1-S3. Also, the reciprocity in cooperation rises over time, indicating a stronger emphasis on cooperation and exchange. The cumulative gains are higher than networks with lower cooperation levels, suggesting more effective resource allocation and collaboration.

In a real-world example, this simulation could mimic a business network with two levels of connectivity among actors. The network has the potential for a growing number of connections. Still, due to a mix of cooperative and non-cooperative actors, the result is limited growth in connections, with fluctuations and the formation of smaller collaborations. The heterogeneity in S4's network suggests a combination of both tightly connected and loosely connected actors, representing a hybrid network structure. This hybrid structure can offer the benefits of both cohesive collaboration within the core group and access to diverse resources and perspectives from the peripheral actors. The heterogeneity usually occurs in a network that brings together diverse actors with distinct characteristics and roles.

In practice, a network like S4 would require management strategies that recognize and leverage the strengths and characteristics of both clusters. It would involve fostering collaboration and knowledge sharing within the tightly connected cluster while also encouraging interactions and engagement with actors from the sparser cluster. This approach can help harness the network's diversity and maximize the potential for innovation, growth, and resilience.

S5 created a network with an enhanced initial propensity to cooperate, whereas, in S6, both propensities have higher averages. A higher network density characterizes these networks; average degree measures indicate more established connections and growing connectivity. Higher betweenness centrality and hub scores suggest the emergence of influential actors. Increasing average coreness and transitivity indicate the network's growing centrality and cohesion. Higher cumulative gains and significant growth in the share of reciprocal cooperative relationships suggest better performance and greater emphasis on cooperation.

S5 reflects a scenario where a high propensity to cooperate governs network development. The average degree measures in S5 show significant growth over time, indicating an expanding network with increased collaboration, information sharing, and resource exchange. The network shows a tendency towards cohesive and tightly connected structures with increasing transitivity and cluster sizes. In a real-world context, S5 could represent a business network that experiences a rapid increase in connections and interconnectivity among actors, fostering partnerships and opportunities for collaboration.

In a real-world context, S6 could mimic a business network where actors are highly willing to connect and have a high propensity for cooperation. This could result in a network with a significant number of connections and a strong culture of collaboration among actors, likely assembled to achieve a shared goal. The network would likely exhibit a high level of interconnectivity, shared resources, and extensive business partnerships. Actors within this network would value cooperation and actively seek out opportunities for collaboration, leading to a cohesive and tightly-knit business community.

Conclusion

The proposed model bridges insights from network science, business studies, game theory, and behavioral economics, making it relevant to a wide audience. The model is based on the assumptions of continuity, complexity, and informality as BN structural characteristics but also allows for reviewing process characteristics: cooperation and conflict, routinization, and adaptation.

Given that the model is based on behavioral elements followed by heuristics and satisficing in making choices, the model does not consider the rational agents aiming for payoff maximization. Instead, it is assumed that the actors will behave according to their initial preferences and imperfectly update them, given their interaction experience. The connection and strategy choice is governed by propensities to connect and cooperate, resulting from the heuristics with randomness effect. The randomness element illustrates individual differences in preferences, perception, assessment, and judgment of an event, resulting in an imperfect information update and behavioral adjustment.

The study offers a comprehensive approach to understanding business networks by combining network science and game theory. Integrating these two disciplines allows a deeper understanding of how individual actors' decisions to connect and cooperate influence the overall network structure and performance. The simulation shows that the developed business network demonstrates relevant characteristics of BNs, successfully mimicking reality. Therefore, the main theoretical contribution of the established model is the connection between the previous strains of research involving formal models and case studies. Besides that, the model allows for further examination of potential scenarios for real-world networks, exploration of the changes in the network based on the initial settings, review of the interventions in the network, or rule changes (or imposing a new rule).

The study introduces the concept of propensities to connect and cooperate as drivers of network structure and evolution. By incorporating these propensities into the analysis, the research provides a novel perspective on how businesses form and maintain connections within the network. The results yield different network types based on their initial propensities to cooperate. It identifies networks with low cooperation as scale-free networks with dominant actors, while networks with higher cooperation exhibit a more balanced distribution of connections and resources among actors.

The research highlights the significance of reciprocity in fostering network performance and cumulative gains. Understanding how reciprocity evolves over time and its relationship with network effectiveness provides insights into the key factors driving network success.

By exploring the dynamic nature of network properties, such as clustering, density, and average degrees, the study sheds light on how business networks continuously adapt and change over time. This dynamic perspective offers new knowledge on the evolution of business networks and the factors contributing to stability or change. Different initial propensities to cooperate lead to network convergence or diversification. Higher cooperation levels result in more cohesive, stable, and efficient networks, while lower cooperation levels lead to fragmented and less stable networks.

The study compares various network scenarios and their effects on network properties. This comparative approach deepens the understanding of the network's sensitivity to different parameters and provides a comprehensive assessment of the implications for businesses.

While it is outside the research's focus, the study shows the impact of diminishing costs of maintaining relationships (imitating routinization) on network growth and evolution. It demonstrates how reduced costs allow actors to allocate more resources to form new connections, leading to network expansion and potential collaboration. In addition, behavioral elements, imperfect updates, and the inclusion of randomness create uncertainty that mimics human decision-making.

The research bridges insights from network science, business studies, and behavioral theories, offering an interdisciplinary approach to analyzing business networks. Integrating diverse disciplines enriches the research and contributes to a more holistic understanding of business network dynamics.

We find that the proposed framework of heuristic business networks captures the main characteristics of business networks: relationships and interactions, cooperation and reciprocity, complexity and continuity, symmetry, behavioral elements, and dynamic nature.

Theoretical contribution

By varying the propensities to connect and cooperate, we reveal or confirm various properties of business networks. These properties refer to business network structure, evolution, reciprocity, complexity, adaptation, convergence and diversification, and stability. They stem from the interplay of propensity to connect, cooperation, and conflict.

Business network structure

A higher propensity to connect among actors in the network leads to denser connections and larger network structure. Actors are more likely to establish and maintain connections, resulting in a well-connected and cohesive network. A higher propensity to cooperate fosters collaborative relationships among actors. Cooperative actors tend to develop mutually beneficial connections, contributing to a network structure focused on resource sharing and collaboration.

Networks with uniformly distributed propensities to cooperate (from 0 to 1) result in degree distribution that follows the power law. These networks also demonstrate convergence with the Pareto 80/20 rule. Still, the same is not true for instances with higher propensities to cooperate. This highlights a crucial insight: different levels of cooperation lead to distinct network properties. Moreover, the results suggest that conflict, rather than cooperation, can be the key driver for the scale-free networks observed in this study.

It opens opportunities for further exploration and understanding of how cooperative behaviors impact business networks' overall structure and dynamics. Additionally, it sheds light on the potential role of conflict in shaping scale-free network structures, which can affect network resilience, resource allocation, and information flow. Diminishing costs of maintaining relationships allow actors to sustain and expand their connections, leading to a more extensive and interconnected network structure.

Network evolution

Higher propensities to connect can lead to more rapid network growth and expansion, resulting in more cohesive and efficient networks. Actors are more open to forming new connections, contributing to the continuous evolution of the network.

Higher propensities to cooperate promote long-term and stable relationships, leading to the evolution of trust-based networks. Cooperation encourages actors to engage in repeated interactions, contributing to the network's gradual development.

Lower relationship costs allow actors to maintain existing connections and form new ones more easily, promoting network growth and evolution over time.

Reciprocity

Higher propensities to cooperate foster reciprocal relationships within the network. Actors are more likely to engage in mutually beneficial interactions, leading to increased reciprocity and trust-building among actors.

Reciprocity also contributes to symmetry in resource distribution within the network, promoting more equitable resource-sharing among actors. Also, lower reciprocities result in an 80/20 division of outcomes.

Complexity

Higher propensities to connect and cooperate can result in a more complex network structure. The network becomes enriched with various relationships and collaborative interactions among actors, leading to increased complexity. In addition, lower relationship costs allow actors to interact more, increasing network complexity over time.

Adaptation

Cooperative relationships provide a platform for flexible responses and adaptive behavior within the network. Similarly, lower relationship costs allow actors to adjust their connections and adapt to evolving network dynamics more easily. Adaptive actors in a network can respond effectively to challenges and opportunities, leading to the retention of cooperatives and the termination of uncooperative connections.

Cooperation and conflict

Higher propensities to cooperate foster a culture of collaboration and mutual support within the network, reducing the likelihood of conflicts. Lower relationship costs encourage actors to maintain positive and cooperative relationships, minimizing conflicts and enhancing cooperation.

Symmetry

Higher cooperation levels lead to more symmetric network structures, with actors having similar degrees and resource distribution. On the contrary, lower cooperation levels may result in asymmetric network structures, with a few dominant actors having significantly more connections and resources than others.

Convergence and diversification

Higher propensities to connect and cooperate can lead to network convergence, where actors form clusters or communities with common goals and attributes. Conversely, lower propensities to connect and cooperate can result in network diversification, with actors forming distinct and disconnected groups. Diminishing costs may facilitate convergence and diversification, as actors can maintain existing relationships while exploring new connections.

Stability

Higher cooperation levels promote network stability as actors engage in more long-term relationships, reducing connection terminations and disruptions within the network. Lower cooperation levels result in a less stable network, with actors terminating relationships more often, which leads to fluctuating cluster structures. Lower relationship costs enhance network stability by enabling actors to sustain and adapt relationships more easily, reducing the likelihood of disruptions.

To sum up, the simulations reveal that propensities to connect and cooperate and factors like diminishing relationship costs significantly impact business network properties. Higher propensities to cooperate and to connect contribute to denser and more interconnected networks, fostering reciprocity and improved network performance. The interplay of these factors influences the networks' structure and stability, leading to variations in network evolution, convergence, diversification, and stability across different scenarios.

Practical implications

Understanding the impact of propensities to connect and cooperate allows for targeted interventions. Network managers can enhance the willingness of actors to cooperate through incentives, trust-building activities, and relationship-building initiatives.

In addition, managing relationship costs can be an intervention strategy. Reducing transaction costs, administrative burdens, and other barriers can encourage actors to maintain and expand their connections, fostering network growth. This can particularly be effective for new ventures, especially if they had an early encounter with an uncooperative actor.

Nevertheless, interventions can be aimed at shaping the network structure and involve facilitating the formation of clusters or communities through targeted introductions, collaborations, and strategic partnerships. While facilitating interaction, it is necessary to promote cooperative reciprocity. To promote reciprocity, businesses can implement strategies that encourage mutually beneficial interactions, reward cooperative behavior, and build a culture of trust and collaboration. Possibly encouraging cooperation, supporting long-term relationships, offering resources for maintaining connections, and fostering a collaborative culture within the network can lead to increased reciprocity, stability, and overall network effectiveness.

Businesses should consider the propensity to connect and cooperate when selecting strategic partners. Collaborating with actors that align with their willingness to form and sustain connections can lead to more successful partnerships. Long-term relationships generally cost less, and understanding the impact of relationship costs on network growth can help businesses allocate resources effectively. Businesses should take a long-term perspective in their network-building efforts, recognizing that network evolution and performance often require sustained efforts and investments. Investing in relationship maintenance and expansion can lead to improved network performance. Businesses should focus on building stable and resilient networks by nurturing long-term relationships and addressing potential disruptions. In these simulations, disruptions are brought on by uncooperative actors. While such behavior could be externally penalized, the actors should try to balance their experience and future behavior. That leads to the need to recognize the importance of adaptation in business networks; companies should adopt adaptive strategies to respond to changing network dynamics and evolving needs.

The research's practical implications offer valuable guidance for businesses seeking to optimize their network strategies. The identified relationships between propensities, reciprocity, stability, and performance provide actionable insights for companies aiming to strengthen their network positions. By investigating the dynamics of connectivity and cooperation, our study offers critical insights into the strategic design and management of business networks, proving indispensable for thriving in the increasingly digital and interconnected business landscape of the future.

As businesses navigate through the intricate web of digital platforms and venture into the vast potential of the metaverse, the practical implications of our research on the propensities to connect and cooperate become ever more critical. These digital frontiers, characterized by their dynamic interactions and expansive networks, heighten the importance of establishing strategic partnerships and fostering collaborative environments. The insights gained from our study are not only relevant but integral to mastering the digital domain. Implementing targeted strategies for connectivity and cooperation in these virtual spaces is key to unlocking growth, driving innovation, and building resilient digital ecosystems. This expansion of our research's applicability into the realms of digital and metaverse platforms illustrates the universal need for nuanced network management strategies that embrace the complexities and opportunities of our increasingly connected world.

Limitations and future research

The problem of thorough complexity—parallel links and dependence on the previously existing economic, social, political, or market structure is not considered. Also, the initial cost, or the cost of the new connection, is equal over the actors, assuming that maintenance of each new connection requires the same amount of resources, disregarding the position of the actor in the network to which the connection is made. The same approach is valid regarding the payoff, which is held the same, disregarding the position of the actors in the network involved in strategic interaction. In addition, the links are not defined as either activity links, resource ties, or actor bonds, so the reason for the connection remains general and unexplored. Also, as a process characteristic, adaptation is not considered in its usual sense of activities, coordination, knowledge, technology, and innovation co-development. It is examined through the propensities’ updates based on the successful interactions—implicitly allowing for actors’ and networks’ adaptations as a presumed supportive process in the background. Those refinements remain for future research.