- Research
- Open Access
Replicator equation on networks with degree regular communities
- Daniele Cassese^{1, 2, 3}Email authorView ORCID ID profile
- Received: 8 March 2018
- Accepted: 13 July 2018
- Published: 13 August 2018
Abstract
The replicator equation is one of the fundamental tools to study evolutionary dynamics in well-mixed populations. This paper contributes to the literature on evolutionary graph theory, providing a version of the replicator equation for a family of connected networks with communities, where nodes in the same community have the same degree. This replicator equation is applied to the study of different classes of games, exploring the impact of the graph structure on the equilibria of the evolutionary dynamics.
Keywords
- Replicator equation
- Evolutionary graph theory
- Prisoner’s dilemma
- Hawk-Dove
- Coordination
Introduction
Evolutionary game theory stems from the field of evolutionary biology, as an application of game theory to biological contests, and successively finds applications in many other fields, such as sociology, economics and anthropology. The range of phenomena studied using evolutionary games is quite broad: cultural evolution (Cavalli-Sforza and W 1981), the change of behaviours and institutions over time (Bowles et al. 2003), the evolution of preferences (Bowles 2010) or language (Nowak 2000), the persistence of inferior cultural conventions (Bowles and Belloc 2013). A particularly vaste literature investigates the evolutionary foundations of cooperation (Bowles 2004; Bowles et al. 2004; Doebli et al. 2004) just to name a few. For an inspiring exposition of evolutionary game theory applications to economics and social sciences see (Bowles 2006).
One of the building blocks of evolutionary game theory is that fitness (a measure of reproductive success relative to some baseline level) of a phenotype does not just depend on the quality of the phenotype itself, but on the interactions with other phenotypes in the population: fitness is hence frequency dependent (Nowak 2006a), and as strategies are the manifestation of individuals’ genetic inheritance, individuals are characterised by a fixed strategy throughout their lifetime. The payoffs of the game are in terms of fitness, so if a trait offers an evolutive advantage over another, this means a better fitness for the individual who has inherited that trait. The dynamics resulting from interactions between individuals carrying different traits capture the process of natural selection: the strategy (phenotype, cultural trait) that performs better gives an advantage in term of reproductive success, hence it will reproduce at a higher rate and eventually take over the entire population (Nowak 2006a).
Early models of evolutionary dynamics assume well-mixed population, ignoring the relational structure that constrains interactions between agents. The study of evolutionary dynamics on structured population is the subject of interest of evolutionary graph theory, introduced by (Lieberman et al. 2005). In this framework agents are placed on a network and play the game with their next neighbours, and the least successful (in terms of fitness) are replaced by their most successful neighbours’ offsprings. Evolutionary dynamics on graphs has been applied extensively to the study of cooperation (Santos et al. 2006; Ohtsuki and Nowak 2006; 2008; Allen et al. 2017) showing that there are radical differences with the case of a well-mixed population, and that the success of cooperation depends crucially on the underlying network structure. Analytical results have been derived for evolutionary games on regular networks (Ohtsuki et al. 2005; Ohtsuki and Nowak 2006; Taylor et al. 2007) while more realistic complex networks have been investigated through computer simulations (Maciejewski et al. 2014). This work is an extension of (Cassese 2017), where I studied cooperation on a family of graphs characterised by degree-regular communities, proving that the relation between the structure of the population and the cost of cooperation determines the nature of equilibria for a Prisoner’s dilemma game. In this paper I briefly present the replicator equation for graphs on regular communities, and an algorithm to generate graphs in this family, as well as its application to the Prisoner’s Dilemma as already in (Cassese 2017). In addition to the previous version of this work here I study other classes of games under the replicator dynamics, namely Hawk-Dove and Cooperation games, exploring how the network impacts the equilibria compared to the mean-field case.
Replicator equation on regular graphs
where b_{ij} depends on the degree of the network, k, the payoff matrix Π and the updating rule. (Ohtsuki and Nowak 2006) derive b_{ij} under three updating rules: Birth-Death: An individual is chosen for reproduction with probability proportional to fitness. The offspring replaces one of the k neighbour chosen at random. Death-Birth: An individual is randomly chosen to die. One of the k neighbours replaces it with probability proportional to their fitness. Imitation: An individual is randomly chosen to update her strategy. She imitates one of her k neighbours proportional to their fitness.
Hence b_{ij} captures local competition on a graph taking account of the gain of ith strategy from i and j players and the gains of jth strategy from i and j players (Nowak et al. 2010). The derived equation is a very good approximation for infinitely large regular graphs with negligible clustering (absence of clustering is the basic assumption behind the moment closure in pair approximation) and provides an easy-to-deal-with differential equation that can be computed at least numerically.
Replicator equation on networks with degree regular communities
In this section I present the extension of the replicator equation to a more complex family of graphs, where nodes can have different degrees. First I define a family of connected graphs (which I call multi-regular graphs) where nodes are clustered in degree-homogeneous communities, such that most of the connections are between same-degree nodes, and few edges connect communities with different degrees. Hence an algorithm to create such networks is proposed, and finally the replicator equation for these networks is introduced.
The definition of the class of multi-regular graphs is motivated by the necessity to have more realistic network structures and at the same time preserving analytical tractability. The homogeneous structure of regular graphs, where all nodes have the same number of neighbours, makes them poorly representative of real world heterogeneous networks (Strogatz 2001). Real world networks are typically characterised by small-world properties (Watts and Strogatz 1998) and scale-free distributions (Barabasi AL 1999), and regular networks fail to satisfy both characteristics: they may have a high clustering coefficient, but usually have large number of hops between pairs of nodes (so they are not small-world), and they trivially are not scale-free, as every node has the same degree. These differences are not without consequences for the dynamics, hence predictions made on regular network models result incorrect if applied to real networks. A standard example can be found in epidemic models: while on regular networks an infection persists if the transmission rate is beyond a finite epidemic threshold, on scale-free networks there is no epidemic threshold, hence infections can spread and persist independently of their transmission rate (Pastor-Satorras and Vespignani 2001). Degree heterogeneity also impacts evolutionary dynamics, and higher heterogeneity has been shown to favour cooperation over defection (Santos et al. 2006). The family of multi-regular graphs is a better representation of real world networks than regular graphs because it allows degree heterogeneity, and at the same time, their local homogeneity allows to derive an analytic expression for the replicator dynamics. Moreover the numerical simulations suggest (but we have no proof) that even if the real population is not structured in degree-regular communities, the replicator dynamics on a multi-regular graph with the same degree distribution of the real population is not far from the dynamics on the real population most of the times.
Multi-regular graphs
Definition 1
A multi-regular graph G is a connected graph partitioned into m degree-homogeneous communities \(C^{i}_{k}\), i={1,…,m}, where each node in community \(C^{i}_{k}\) has degree k, and k≥3. In each community \(C^{i}_{k}\) the number of nodes n_{i} is at least k+1, and n_{i}k must be even. Moreover, the number of connections between different communities must be even.
Definition 2
For each community \(C^{i}_{k}\), call interior those nodes which neighbourhood is entirely contained in the community, and frontier those which have at least one neighbour in a different community.
Notice that we require n_{i}≥k+1 to ensure the existence of a regular graph of degree k on n_{i} nodes, and that we require an even number of edges between nodes in \(C^{i}_{k}\) and nodes outside said community to guarantee that each node in \(C^{i}_{k}\) has degree k. To provide intuition, consider we want a multi-regular graph with two communities of degree k_{1} and k_{2} respectively, and we start with two disconnected regular components of degree k_{1} and k_{2}. If we connect the two components by adding an edge between them, then the two frontier nodes will have degree k_{1}+1 and k_{2}+1 respectively, violating the condition for being in a degree-homogeneous community. If for each of the two frontier vertices we erase one edge other than the one connecting them, then there will be two other nodes (one for each community) violating that condition, as those will now have degree k_{1}−1 and k_{2}−1 respectively. If we connect these two nodes then regularity condition is restored. Notice also that the definition of multi-regular graph implies that the minimal community size is 4, but we are never going to consider such small communities in this work, as the replicator equation provided is a good approximation for large graphs (with at least 10^{5} nodes).
Generating a random multi-regular graph
- 1
generate \(\sum _{k} n_{k} k \) points.
- 2divide the points in n_{k} buckets in this way:
- (a)
take n_{k} points and put each in a different bucket.
- (b)
add k−1 points to each of these buckets.
- (c)
repeat the procedure for all different k, such that for all degrees k there will be \(n_{k} \mathbb {P}(k)\) buckets with k points each.
- (a)
- 3
take a random point, say it is in a bucket with k points
- 4
join it with probability r to a random point in one of the \(n_{k} \mathbb {P}(k)\) buckets with k points, and with probability 1−r to any of the other points at random.
- 5
continue until a perfect matching is reached.
- 6
collapse the points, so that each bucket maps onto a single node and all edges between points map onto edges of the corresponding nodes.
- 7
check if the obtained graph is simple (e.g. it has no loops or multiple edges).
Replicator equation on multi-regular graphs
where k_{i} is the degree of nodes inside community i and \(\mathbb {P}\left [C_{k_{i}}\right ] \) is the probability that a node is in a community with degree k_{i}, or the fraction of nodes in a community with degree k_{i}, so that the global dynamic is a weighted average of the local dynamics on each community (Cassese 2017).
Prisoner’s dilemma
Prisoner’s dilemma
C | D | |
---|---|---|
C | b−c | −c |
D | b | 0 |
Hawk-Dove game
Hawk-Dove
H | D | |
---|---|---|
H | (b−c)/2 | b |
D | 0 | b/2 |
This game has a similar structure to the Prisoner’s Dilemma, as both parties have incentive to defect and fight to obtain a higher payoff, but a reciprocal aggressive behaviour is detrimental (in expectation) for both. While the Prisoner’s dilemma has a unique dominant strategy, which is mutual defection, Hawk-Dove has two Nash equilibria in pure strategies, namely (Hawk, Dove) and (Dove, Hawk), and one equilibrium in mixed strategies, (Hawk, Dove) = (b/c, 1−b/c). The mixed strategy corresponds to the Evolutionary Stable Strategy in a mean-field evolutionary game, where the equilibrium frequency of hawks is equal to b/c. The equilibrium where everybody in the population is a dove is unstable as long as b>0, so cooperation will never prevail in the mean-field case.
Let us first study the game on a regular graph of degree k under the three different updating mechanisms. The stable equilibrium under death-birth is \(x_{d}^{*} = \left (bk^{2} - bk -ck^{2} +c\right)/c\left (-k^{2}+k+2\right)\), where \(x_{d}^{*}<1\) when c/b<k(k−1)/(k+1). It is easy to check that the equilibrium level of cooperation on a regular graph is greater than the equilibrium in the mean-field case when c/b>2/(k+1), which means that a regular graph always favours cooperation over defection, and the same holds for graphs with regular communities. Computing the equilibria for imitation updating, we can see that the stable equilibrium is [b(−k^{2}−k)+c(k^{2}+2k−3)]/[c(k^{2}+k−6)], which is a non-degenerate mixed equilibrium when c/b<k(k+1)/(k+3) and it is greater than the mean-field when c/b>6/(k+3) which again always holds for k≥3 on both regular graphs, and graphs with degree regular communities.
is a sufficient condition for doves to prevail.
In conclusion reaching cooperation in a Hawk-Dove game on graphs with regular communities is easier than in a corresponding graph with disconnected regular components, in the sense that cooperation is sustainable with a lower relative cost of the aggressive behaviour. Moreover numerical simulations show that, if we compare the distance between the bounds and the true thresholds, we can see that this distance is always greater for imitation, meaning that imitation promotes cooperation more than the other two mechanisms, as it is the case for Prisoner’s dilemma as well.
Coordination game
Coordination game
A | B | |
---|---|---|
A | a | b |
B | c | d |
Discussion
In this paper I presented an extension of my previous work (Cassese 2017), providing a version of the replicator equation for a family of graphs characterised by degree-regular communities. As examples of possible application of this equation, here I study the evolutionary dynamics of three game classes: Prisoner’s dilemma, Hawk-Dove and Coordination games. It is shown that graphs with degree-regular communities promote cooperation both in the Prisoner’s dilemma and in the Hawk-Dove game for imitation and death-birth updating, and that imitation updating in both cases is more favourable to cooperation than death-birth. The results confirm that higher degree heterogeneity favours cooperation, and this can be better understood by comparing the dynamics on a multi-regular graph with the dynamics on a graph with disconnected regular components. In the case of the Prisoner’s dilemma with birth-death updating, in all those components where the degree is such that b/c>k_{i} cooperators will prevail, viceversa in the other components defectors will prevail (and in some of them we could also have a mixed equilibrium). So the only way to have cooperation prevailing globally is b/c>k_{max}+2, where k_{max} is the largest degree of the graph. Adding a few connections between these regular components, as we do in a multi-regular graph, changes the picture completely, and cooperation prevails if b/c is greater than the average degree, which is a much easier condition to meet. The same is true for imitation updating, where we would have that each disconnected component may reach a different equilibrium depending on their degree, with cooperation prevailing locally where b/c>k_{i}+2, and globally only if b/c>k_{max}+2, while on a multi-regular graph we have the milder condition \(b/c>\sum _{i} (k_{i}+2) \mathbb {P}\left [C_{k_{i}}\right ]\). Analogously, for the Hawk-Dove game on a graph with regular disconnected components, cooperation prevails globally if c/b>k_{max} for birth-death, c/b>k_{max}(k_{max}−1)/(k_{max}+1) for death-birth and c/b>k_{max}(k_{max}+1)/(k_{max}+3) for imitation, and each of these conditions is stronger than the corresponding condition on multi-regular graphs as in Eqs. (10), (14), (17) respectively. If these conditions are not met, each disconnected component will be in a different equilibrium depending on its degree, with some components where doves prevail, others where the two strategies coexist.
In the Coordination game on graphs with regular disconnected components, the Pareto-efficient strategy needs to yield a higher payoff than the one needed on a multi-regular graph in order to be both Pareto-efficient and risk-dominant globally, so we can say that graphs in this family promote Pareto-efficiency over risk-dominance. Moreover, on a graph with disconnected components we may have that the Pareto-efficient strategy is also risk-dominant on some components and only Pareto-efficient on others, depending on their degree.
In conclusion the results show that multi-regular graphs enhance cooperation and favour Pareto-efficiency compared to both the complete graph (well-mixed population) and the regular graph.
The replicator equation provided can be applied to any game on such graphs, so further research directions include the study of other game classes, in particular games with more than two strategies.
Declarations
Acknowledgements
The author thanks an anonymous reviewer and Hisashi Ohtsuki for useful comments.
Funding
The author wishes to thank support from FNRS (Belgium).
Authors’ contributions
DC was responsible for the research in full. The author read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Allen, B, Lippner G, Chen YT, Fotouhi B, Momeni N, Yau ST, Nowak MA (2017) Evolutionary dynamics on any population structure. Nature 544:227–230.ADSView ArticleGoogle Scholar
- Axelrod, R, Hamilton WG (1981) The evolution of cooperation. Science 211:1390–1396.ADSMathSciNetView ArticleMATHGoogle Scholar
- Barabasi AL, AR (1999) Emergence of scaling in random networks. Science 286:509–12.ADSMathSciNetView ArticleMATHGoogle Scholar
- Bowles, S (2004) The evolution of strong reciprocity: Cooperation in heterogeneous populations. Theor Popul Biol 65:17–28.View ArticleMATHGoogle Scholar
- Bowles, S (2006) Microeconomics, Behavior, Institutions, and Evolution. Princeton University Press, Princeton, NJ.Google Scholar
- Bowles, S (2010) Endogenous preferences: The cultural consequences of markets and other economic institutions. J Econ Lit XXXVI:75–111.Google Scholar
- Bowles, S, Belloc M (2013) The persistence of inferior cultural-institutional conventions. Am Econ Rev 103:1–7.Google Scholar
- Bowles, S, et al. (2003) The coevolution of individual behaviors and social institutions. J Theor Biol 223:135–147.MathSciNetView ArticleGoogle Scholar
- Bowles, S, et al. (2004) Explaining altruistic behavior in humans. Evol Hum Behav 24:153–172.Google Scholar
- Cassese, D (2017) Replicator equation and the evolution of cooperation on regular communities. In: Cherifi C, Cherifi H, Karsai M, Musolesi M (eds)Complex Networks & Their Applications VI, 869–880.. Springer, Cham.Google Scholar
- Cavalli-Sforza, L, W FM (1981) Cultural Transmission and Evolution: a Quantitative Approach. Princeton University Press, Princeton, NJ.Google Scholar
- Doebli, M, et al. (2004) The evolutionary origin of cooperators and defectors. Science 306:859–862.ADSView ArticleGoogle Scholar
- Lieberman, E, Hauert C, Nowak MA (2005) Evolutionary dynamics on graphs. Nature 233:312–316.ADSView ArticleGoogle Scholar
- Maciejewski, W, Fu F, Hauert C (2014) Evolutionary game dynamics in populations with heterogenous structures. PLoS Comput Biol 10:1003567.ADSView ArticleGoogle Scholar
- Matsuda, H, et al. (1992) Statistical mechanics of population - the lattice lotka-volterra model. Prog Theor Phys 88:3176–3205.View ArticleGoogle Scholar
- Nowak, MA (2000) Evolutionary biology of language. Philos Trans R Soc Lond B Biol Sci 355:1615–1622.View ArticleGoogle Scholar
- Nowak, MA (2006a) Evolutionary Dynamics: Exploring the Equations of Life. The Belknap Press of Harvard University Press, Cambridge.MATHGoogle Scholar
- Nowak, MA (2006b) Five rules for the evolution of cooperation. Science 314:1560–1563.ADSView ArticleGoogle Scholar
- Nowak, MA, Tarnita CE, Antal T (2010) Evolutionary dynamics in structured populations. Philos Trans R Soc B 365:19–30.View ArticleGoogle Scholar
- Ohtsuki, H, et al. (2005) A simple rule for the evolution of cooperation on graphs and social networks. Nature 441:502–505.ADSView ArticleGoogle Scholar
- Ohtsuki, H, Nowak MA (2006) The replicator equation on graphs. J Theor Biol 243:86–97.MathSciNetView ArticleGoogle Scholar
- Ohtsuki, H, Nowak MA (2008) Evolutionary stability on graphs. J Theor Biol 251:698–707.MathSciNetView ArticleGoogle Scholar
- Pastor-Satorras, R, Vespignani A (2001) Epidemic spreading in scale-free networks. Phys Rev Lett 86:3200–3203.ADSView ArticleGoogle Scholar
- Santos, FC, Pacheco JM, Lenaerts T (2006) Evolutionary dynamics of social dilemmas in structured heterogeneous populations. PNAS 103:3490–3494.ADSView ArticleGoogle Scholar
- Strogatz, SH (2001) Exploring complex networks. Nature 410:268–276.ADSView ArticleMATHGoogle Scholar
- Taylor, PD, Day T, Wild G (2007) Evolution of cooperation in a finite homogeneous graph. Nature 447:469–472.ADSView ArticleGoogle Scholar
- Watts, DJ, Strogatz SH (1998) Collective dynamics of ’small-world’ networks. Nature 393:440–2.ADSView ArticleMATHGoogle Scholar