Complex influence propagation based on trust-aware dynamic linear threshold models

Please note that here we refer to the vast literature on probabilistic models originally designed to explain stochastic processes of information diffusion, which include the classic Independent Cascade(IC) and Linear Threshold (LT) models (Chen et al. 2013), and relating optimization problems, such as influence maximization. By contrast, we will leave out of consideration deterministic models, such as the structural cascades specifically designed to model context/content-sensitive diffusion over an interaction network (e.g., (Krishnan et al. 2016; Das et al. 2016)). Also, it is worth noting that the information diffusion modeling problem we tackle in this work is significantly different from the one addressed by epidemic models, such as SIS, SIR(S), and SEIR(S) (Hethcote 2000), already for the non-competitive scenario. Standard epidemic models are originally defined as compartmental models, since the individuals of a population are divided in compartments that describe an epidemiological state. The parameters used to represent transition rates for changing states are absolute constants, which means that the infection process in compartmental models has a deterministic behavior. Also, standard epidemic models are of mass-action type, since individuals are represented as normalized fraction of a population which randomly interact with each other. As discussed in (Dodds 2018), even social contagion based on stochastic or generalized epidemic models (i.e., there is a probability distribution of rates to govern the infection process) is originally defined on random networks, and its revision to deal with social networks would lead to more complicated models. In this regard, one direction is taken by the stochastic individual-contact, network models, whereby SIS and related models are reformulated by considering a stochastic infection process and a network-based population of individually identifiable elements. In “Comparison with the IC, SIR and SEIR models” section, we present a stage of experimental evaluation devoted to a comparison with such models. However, even if epidemic models have also been used for social influence, they are not the most common approach to such topic (Porter and Gleeson 2016). This is manly due to the fact that identifying and modeling the causal mechanisms of the spread of ideas is more difficult than for the spread of diseases. By constrast, the threshold models for influence propagation (even the simplest ones) have two important features that are not clearly present in epidemic models. First, individuals have different behaviors, being such differences reflected in the distribution of activation thresholds associated with the individuals; by contrast, in stochastic epidemic models, the state-transition probabilities are drawn independently of the individuals’ relations. Second, an individual’s behavior also depends on the behavior of other individuals s/he is linked to: here, it is helpful to think about threshold models as an example of complex contagions, whereby an individual takes an action as a result of the exposure to multiple sources of influence; by contrast, epidemic models are more likely to represent simple contagion, in that a single source of influence (as social contact) may suffice to cause an individual’s action. Moreover, while the transition to the recovered state assumes non-progressivity in stochastic diffusion models such as LT or IC, such a transition in SIR(S) is defined to happen spontaneously, discarding any influence that may result from the interaction with other individuals. For all such reasons, thresholds models are usually considered more appropriate in contexts like the adoption of new technologies or controversial ideas (Chen et al. 2013). And, in our work, we indeed follow this line of research. One further point of divergence adheres to the notion of competitiveness that is somehow found in advanced epidemic models: this refers to the presence of two or multiple groups of individuals (with some distinguishing characteristics) which are however affected by the same, single disease (Han et al. 2003; Ji et al. 2012; Hu et al. 2013), therefore it corresponds to a totally different notion than what is addressed in our work.

In the following, we briefly recall the definition of the LT model, which is at the basis of our proposal; then, we focus on related work that address the aforementioned aspects concerning complex propagation phenomena.

The classic Linear Threshold model. Given a directed graph representing a social network, with estimates of influence probabilities provided as edge weights, nodes can be “activated” (i.e., influenced) through an information cascade starting from an initially selected set of seed nodes (i.e., early-adopters). At the beginning of the information diffusion process, each node is assigned a threshold uniformly at random from [0,1]. The diffusion process unfolds in discrete time steps and follows certain rules: nodes are either active or inactive; once activated, nodes cannot deactivate; an active node may trigger activation of neighboring nodes; a node can be activated at time t+1 by its active neighbors if their total influence weight at time t exceeds the threshold associated to that node. The process runs until no more activations are possible.

Competitive diffusion. A number of studies have been devoted to model competitive diffusion; see, e.g., (Chen et al. 2013) for a general introduction to IC and LT classes of competitive diffusion models. Focusing on competitive diffusion and related optimization problems under the context of misinformation spread limitation, one of the earliest work is (Budak et al. 2011), which proposes a multi-campaign IC model to address the influence limitation problem, i.e., to find a seed set of size k for one, “good” campaign such that the number of nodes influenced by the other, “bad” campaign is minimized. In (Tong et al. 2017), the problem of rumor blocking is addressed under the competitive IC model and a randomized algorithm is developed for the selection of the seed set able to yield the maximum reduction in the number of bad-infected nodes. An influence blocking maximization problem is also addressed in (He et al. 2012), using competitive LT. In (Chen et al. 2011), the two competing cascades correspond to opposite opinions, where the negative one may emerge spontaneously from any user in the network, e.g., a user got disappointed with a purchased item and decides to spread negative opinion among her/his contacts. Lu et al. (2015) also address the aspect of complementarity between two competing campaigns, under the assumption that if the two information items are correlated then the adoption of one item might favor further adoption of the second item over time.

Non-progressive diffusion. While modeling the competitive nature of information cascades, the above works however refer to progressive models. On the contrary, a few studies have been proposed to model non-progressive diffusion. For instance, (Lou et al. 2014) introduces a deactivation function into a continuous non-progressive model, whereas an extension of LT is proposed in (Fazli et al. 2014) to define a non-progressive strict majority model. However, both models are also non-competitive.

Social ties and temporal aspects. All of the aforementioned works discard two important aspects: (i) the nature of social ties and their impact on the influence propagation, and (ii) time aspects concerning the diffusion process. The dichotomy between opposite types of social ties (e.g., friend vs. foe relations) has been widely studied in OSN analysis (e.g., (Leskovec et al. 2010b)), however its incorporation into diffusion models has been relatively little explored so far. For instance, two extensions of competitive LT with negative relations are defined in Talluri et al. (2015), to support positive opinion maximization, and in Weng et al. (2016), to model the adoption of opinions from friends or opposite opinions from foes. All of such models are competitive but do not consider temporal aspects in the activation or propagation processes.

Several works have studied different types of temporal variables and their impact on spreading processes (e.g., (Liben-Nowell and Kleinberg 2008; Vazquez et al. 2007; Iribarren and Moro 2009)), mainly focusing on lags and delays due to the diverse response-time and heterogeneous susceptibility of users. In (Liu et al. 2012), the authors propose a latency-aware IC model inspired by (Iribarren and Moro 2009), in which an influencing delay is introduced in the activation function. Under this model, a time-constrained IM problem is defined, i.e., to find a seed set of size k such that the expected number of nodes is activated before a given time limit. Another extension of IC is proposed in (Chen et al. 2012), where a notion of meeting probability is introduced to control the activation of neighbors. The models in Liu et al. (2012); Chen et al. (2012) are non-competitive. By contrast, the trust-based latency-aware IC model proposed in (Mohamadi-Baghmolaei et al. 2015) features competitiveness, non-progressivity, temporal delay in propagation, and is also designed to deal with trust/distrust relations.

Dynamic behaviors. All of the previously mentioned works still lack aspects modeling the dynamic behavior of the users. In particular, according to recent studies about polarization of opinion in OSNs (Anagnostopoulos et al. 2015) and related works about misinformation reduction (Kumar and Geethakumari 2014; Lewandowsky et al. 2012), a crucial aspect is to intervene before a competing campaign can reach the users, or at least soon enough, so that a user does not have time to radicalize her/his thoughts. This idea was first captured in (Litou et al. 2016), where a dynamic LT model (DLT) is defined to deal with competitive information cascades. The influence weights temporally decay according to a Poisson distribution, and every node can be either positively or negatively activated at a given time depending on the absolute value of the cumulative influence of its neighbors, while the activation sign depends on the sign of the cumulative influence. Moreover, a dynamic behavior aspect lays on the update of the activation threshold whenever a user switches her/his belief.

The latter work shares with our proposal all features of competitiveness, non-progressivity (although deactivation is not allowed), time-aware propagation, dynamic influence behavior, and incorporation of opposite opinions in the influence probabilities; moreover, it is also based on LT. However, our competitive models differ from DLT in (Litou et al. 2016) since (i) we explicitly model trust and distrust relationships to define the influence probabilities, (ii) our activation function takes into account only the trusted connections while (iii) distinguishing between the two information cascades; (iv) we introduce a quiescence function to model a delay in the information propagation depending on the strength of influence exerted by distrust relations (i.e., foe neighbors); finally, (v) the activation threshold in our models becomes stronger over time as a node is holding a particular belief. In “Comparison with the DLT model” section we shall compare our models with DLT.

In this section we describe our proposed class of Friend-Foe Dynamic Linear Threshold (F²DLT) models, which is comprised of: the Non-Competitive F²DLT (nC-F²DLT), the Semi-Progressive Competitive F²DLT (spC-F²DLT), and the Non-Progressive Competitive F²DLT (npC-F²DLT). We first provide an overview of the framework based on F²DLT. Next, we introduce key features common to all models, then we elaborate on each of them.

Overview

Figure 1 illustrates the conceptual architecture of a framework for information diffusion and influence propagation based on our proposed models. Given a population of OSN users, the framework requires three main inputs: (i) a trust network, which is inferred from the social network of those users to model their trust/distrust relationships; (ii) user behavioral characteristics that are intrinsic to each user (i.e., exogenous to an information diffusion scenario) and oriented to express two aspects: activation-threshold, i.e., the effort needed to activate a user through cumulative influence from her/his neighbors, and quiescence, i.e., the user’s hesitation in being actively committed with the propagation process; and, (iii) one or multiple competing campaigns, i.e., information cascades generated from the agent(s) having viral marketing purposes. Moreover, the information diffusion process has a time horizon, and its temporal unfolding is reflected in the evolution of the information diffusion graph: this also depends on the dynamics of the users’ behaviors in response to the influence chains started by the campaign(s), which admit that users may switch from the adoption of a campaign’s item to that of another one. Putting it all together, our F²DLT based framework embeds all previously discussed aspects that are required to explain complex propagation phenomena, i.e., competitive diffusion, non-progressivity, time-aware activation, delayed propagation, and trust/distrust relations.

Please note that inferring a trust network from a social network is not an objective of this work; rather, we assume that trust relationships between users of an OSN are available and, as we shall describe next in this section, they are exploited as key information to develop our proposed models. Several heuristics have been proposed to infer a trust network from social relations and interactions among users in an OSN. A common approach is to infer trust relationships based on the social influence exerted by users over the network and propagation of trust ratings (Golbeck and Hendler 2006; Overgoor et al. 2012; Hamdi et al. 2013; Jiang et al. 2014). Other studies utilize users’ activities in social media (Gilbert and Karahalios 2009), or users’ attributes and interactions (Liu et al. 2008), or combinations of aspects concerning user affinity, familiarity and reputation (Yin et al. 2012), social influence, social cohesion and the afffective valence expressed by the users in the textual contents they produce (Vedula et al. 2017). The interested reader may refer to (Tang and Liu 2015; Sherchan et al. 2013) for an exhaustive overview on the topic.

Non-competitive model

We introduce the first of the three proposed models, which refers to a single-item propagation scenario. Figure 2 shows the life-cycle of a node in the diffusion graph under this model.

Definition 1

Non-Competitive Friend-Foe Dynamic Linear Threshold Model (nC-F²DLT) Let \({\mathcal {G}} = \langle V, E, w, g, q, T \rangle \) be the diffusion graph of Non- Competitive Friend-Foe Dynamic Linear Threshold Model (nC-F ²DLT). The diffusion process under the nC-F ²DLT model unfolds in discrete time steps. At time t=0, an initial set of nodes S₀ is activated. At time t≥1, the following rule applies: for any inactive node \(v \in V \setminus \left (S_{t-1} \cup \widetilde {S}_{t-1}\right)\), if \(\sum \nolimits _{u \in N^{in}_{+}(v) \cap S_{t-1}} w_{uv} \geq g(v,t)\), then v will be added to the set of quiescent nodes \(\widetilde {S}_{t}\), with quiescence time equal to t^∗=q(v,t). Once the quiescence time is expired, v will be removed from \(\widetilde {S}_{t}\) and added to the set of active nodes \(\phantom {\dot {i}\!}{S}_{t^{*}}\). The process continues until T is expired or no more activation attempts can be performed. □

Competitive models

Here we introduce the two competitive F²DLT models. Let us first provide our motivation for developing two different competitive models: through the following example, we illustrate a particular situation that may occur when dealing with two campaigns competitively propagating through a network. Please note that, throughout the rest of this paper, we will consider only two competing campaigns for the sake of simplicity; nevertheless, our proposed models are generalizable to more than two competing campaigns.

Example 1

Figure 3 shows an example activation sequence in a competitive scenario between two information cascades, distinguished by colors red and green. At time t=0, nodes u and z are green-active, and their joint influence causes green-activation of node v as well (since 0.3+0.5≥0.6). At time t=1, as fully influenced by node x, node z has switched its activation in favor of the red campaign. After this switch, at time t=2, it happens that v’s activation state is no more consistent with the (joint or individual) influenced exerted by u and z. In particular, two mutually exclusive events might in principle happen at t=2: either v is deactivated or v maintains its green-activation state. ■

The uncertainty situation depicted in the above example prompted us to the definition of two models, namely semi-progressive and non-progressive F²DLT: the former corresponds to the case of v keeping its current (i.e., green) activation state, whereas the latter corresponds to v returning to the inactive state. Clearly, the two models’ semantics are different from each other: the semi-progressive model assumes that a user, once activated, cannot step aside, unlike the non-progressive one, which instead requires a user to have always the support of her/his in-neighbors to keep activation.

Given two information cascades, or campaigns C^′,C^′′, for every time step t∈T we will use symbols \({S^{\prime }_{t}}\) and \({S^{\prime \prime }_{t}}\) to denote the sets of active nodes, such that \({S^{\prime }_{t}} \cap {S^{\prime \prime }_{t}} = \emptyset \), and analogously symbols \({\widetilde {S}^{\prime }_{t}}\) and \({\widetilde {S}^{\prime \prime }_{t}}\) as the sets of quiescent nodes, for C^′ and C^′′, respectively. Also, \(S_{t} = {S^{\prime }_{t}} \cup {S^{\prime \prime }_{t}}\) and \({\widetilde {S}_{t}} = {\widetilde {S^{\prime }}_{t}} \cup {\widetilde {S^{\prime \prime }}_{t}}\).

It should also be noted that, while sharing the time interval (T) of diffusion, C^′ and C^′′ are not constrained to start at the same time t₀. Nevertheless, for the sake of simplicity, we hereinafter assume that \(t_{0}=t_{0}^{\prime }=t_{0}^{\prime \prime }\) (with t₀∈T), unless otherwise specified (cf. “Results” section).

Definition 2

Semi-Progressive Competitive Friend-Foe Dynamic Linear Threshold Model (spC-F²DLT). Let \({\mathcal {G}} = \langle V, E, w, g,\) q,T〉 be the diffusion graph of Semi-Progressive Competitive Friend-Foe Dynamic Linear Threshold Model (spC-F ²DLT), and C^′,C^″ be two campaigns on \({\mathcal {G}}\). The diffusion process under the spC-F ²DLT model unfolds in discrete time steps. At time t=0, two initial sets of nodes, \({S^{\prime }_{0}}\) and \({S^{\prime \prime }_{0}}\), are activated for each campaign. At every time step t≥1, the following state-transition rules apply:

R1. For any inactive node \(v \in V \setminus \left (S_{t-1} \cup {\widetilde {S}_{t-1}}\right)\), if \(\sum \nolimits _{N^{in}_{+}(v) \cap {S^{\prime }_{t-1}}} w_{uv} \geq g(v,t)\), then v will be added to \({\widetilde {S^{\prime }}_{t}}\); analogously, if \(\sum \nolimits _{N^{in}_{+}(v) \cap {S^{\prime \prime }_{t-1}}} w_{uv} \geq g(v,t)\), then v will be added to \({\widetilde {S^{\prime \prime }}_{t}}\). If both conditions hold, i.e., v can be simultaneously activated by both campaigns, a tie-breaking rule will apply, in order to decide which campaign actually determines the node’s transition in the quiescent state.

R2. When a node v enters the quiescent state corresponding to C^′ (resp. C^′′) for the first time, it will stay in the quiescent node-set \({\widetilde {S^{\prime }}_{t}}\) (resp. \({\widetilde {S^{\prime \prime }}_{t}}\)) until the quiescence time is expired. After that, v will be moved to \({S^{\prime }_{t}}\) (resp. \({S^{\prime \prime }_{t}}\)), i.e., it will become active for C^′ (resp. C^′′).

R3. Given a node v active for C^′′, i.e., \(v \in {S^{\prime \prime }_{t-1}}\), if \(\sum \nolimits _{N^{in}_{+}(v) \cap {S^{\prime }_{t-1}}} w_{uv} \geq g(v,t)\) and \(\sum \nolimits _{N^{in}_{+}(v) \cap {S^{\prime }_{t-1}}} w_{uv} > \sum \nolimits _{N^{in}_{+}(v) \cap {S^{\prime \prime }_{t-1}}} w_{uv}\), then v will be removed from \({S^{\prime \prime }_{t}}\) and added to \({S^{\prime }_{t}}\); analogous rule holds for any node active for the first campaign.

Every node for which none of the above transition-state rules is triggered at time t, it will keep its current state at time t+1. □

The life-cycle of a node in spC-F²DLT is shown in Fig. 4. Note that, once a node becomes active, it cannot turn back to the inactive state, but it can only change the activation campaign. Moreover, switch transitions occur instantly.

Definition 3

Non-Progressive Competitive Friend-Foe Dynamic Linear Threshold Model (npC-F²DLT) Let \({\mathcal {G}} = \langle V, E, w, g,\) q,T〉 be the diffusion graph of Non-Progressive Competitive Friend-Foe Dynamic Linear Threshold Model (npC-F ²DLT), and C^′,C^′′ be two campaigns on \({\mathcal {G}}\). The diffusion process in npC-F ²DLT evolves according to the same rules as in spC-F ²DLT plus the following rule concerning the deactivation process of an active node:

R4. For any active node v at time t−1, if \(\sum \nolimits _{N^{in}_{+}(v) \cap {S^{\prime }_{t-1}}} w_{uv} < \theta _{v}\) and \(\sum \nolimits _{N^{in}_{+}(v) \cap {S^{\prime \prime }_{t-1}}} w_{uv} < \theta _{v}\), then v will turn back to the inactive state at time t.

Every node for which none of the transition-state rules is triggered at time t (including the ones defined for spC-F ²DLT), it will keep its current state at time t+1. □

It should be noted that a node’s deactivation rule depends on θ_v only (rather than on the whole function g(v,t)); otherwise, every node activated at a given time could deactivate itself in the next time step, due to the increase in its activation threshold. This would eventually lead to a configuration in which all nodes in the network, except the initially activated ones, are in the inactive state. The life-cycle of a node in the npC-F²DLT is illustrated in Fig. 4. Note that, unlike in spC-F²DLT, transitions to inactive state are allowed.

Theoretical properties of the models

In this section we provide insights into the proposed models. Our main goal is to understand how the features introduced in each of our LT-based models impact on the models’ spread behavior, particularly on monotonicity and submodularity properties. We organize our analysis into two parts: the first corresponding to non-competitive diffusion, and the second to competitive diffusion.

Non-competitive diffusion

We show that nC-F²DLT can be reduced to LT with quiescence time, hereinafter denoted as LTqt. By proving the equivalence between the two models, we hence claim that both the monotonicity and submodularity properties hold for nC-F²DLT. Note that since we deal with a progressive model, we assume without loss of generality that, for every node v, the activation-threshold function has a constant value for the whole duration of the diffusion process, i.e., g(v,t)=θ_v.

Definition 4

Reduction ofnC-F²DLT toLTqt. Given \({\mathcal {G}} = \langle V, E, w, g, q, T\rangle \) for nC-F²DLT, a diffusion graph \({\mathcal {G}}_{LT}=\langle V_{LT},E_{LT} \rangle \) can be derived, under LTqt, such that V_LT=V and E_LT={(u,v)|(u,v)∈E,w_uv>0}. Every node v∈V_LT is assigned a quiescence time equal to the maximum value of the quiescence function q_v(·), i.e., \(\tau _{v}^{max} = \tau _{v} + \psi (N_{-}^{in}(v))\). □

Definition 4 exploits the fact that the distrust connections are not involved in the activation process, but only in the calculation of the quiescence time. Therefore, we can assume this time to be the maximum possible value, and hence we can study the propagation under LTqt. The reduction of nC-F²DLT to LTqt is meaningful since the two models are proved to be equivalent, as we report in the following theoretical result.

Proposition 1

The Non-Competitive Trust Threshold Model (nC- F²DLT) and the Linear Threshold Model with quiescence time (LTqt) are equivalent. \(\blacktriangleleft \)

Proof

According to the definition of equivalence of two diffusion models in (Kempe et al. 2003; Chen et al. 2013), in order to prove the equivalence of nC-F²DLT and LTqt we need to prove that the distribution of the active sets for any given seed set S₀ is the same under the two models. We provide a proof by induction, hence we consider the evolution of the active sets during the diffusion rounds.

For the LTqt model, the probability of a node to be activated exactly at time t+1 (with t≥1) is given by:

$$ \begin{aligned} \Pr(v \in \widetilde{S_{t+1}} \mid v \notin S_{t}) &= \frac{\Pr\left(v \in \widetilde{S_{t+1}}, v \notin S_{t}\right)}{\Pr(v \notin S_{t})} \\ &= \frac{\Pr\left(\sum\nolimits_{u \in S_{t-1}}w_{uv} < \theta_{v} \leq \sum\nolimits_{u \in S_{t}}w_{uv}\right)}{\Pr\left(\sum\nolimits_{u \in S_{t-1}} w_{uv} < \theta_{v}\right)} \\ &= \frac{\sum\nolimits_{u \in S_{t} \setminus S_{t-1} }w_{uv}}{1-\sum\nolimits_{u \in S_{t-1}}w_{uv}} \end{aligned} $$

(4)

Above, it should be noted that the joint probability \(\Pr \left (v \in \widetilde {S_{t+1}}, v \notin S_{t}\right)\) corresponds to the probability that the threshold associated with node v falls into the interval denoted by the influence received by v until the previous time step and the one received at the current time step. Moreover, Pr(v∉S_t) is just the probability that, at time (t−1), the influence received by v is still below its threshold. Finally, we derive the last equality in Eq. 4, which intuitively denotes that the influence exerted by the nodes in S_t∖S_t−1, i.e., the nodes turning into the active state exactly in the current time step, is decisive to exceed the threshold θ_v.

For the npC-F²DLT model, the conditional probability \(\Pr \left (v \in \widetilde {S_{t-1}} \mid v \notin S_{t}\right)\) can be derived starting from Eq. 4 by constraining w_uv such that \(u \in N^{in}_{+}(v)\), i.e., only trusted relations are considered. This leads to an equivalent definition of conditional probability, which holds for every time step t and seed set S₀. Therefore, we can conclude that the final active sets will be the same for both models. □

It should be noted that, due to the quiescence times, the sets of active nodes in the two models may not be the same at every time step, but the two final active sets will match each other.

Since the introduction of quiescence time in LT does not have effect on the distribution of the final active nodes (Chen et al. 2013), we obtain the following equivalence: LT ≡LTqt ≡nC-F²DLT. Therefore, the activation function is still monotone and submodular under nC-F²DLT.

Example 2

Consider Fig. 5 , where the propagation process unfolds according to the LTqt dynamics. Nodes u and z are chosen as initial seeds. Thresholds and weights are set such that θ≤w_uv and max{w_vx,w_zx}<θ_x≤w_vx+w_zx, therefore the combined influence of v and z is required for the activation of node x. The dashed edge denotes a distrust connection removed as a result of the reduction defined in Definition 4. In the initial time step (t=0), u activates v causing its transition from the inactive state to the quiescent state (in yellow). When \(t=\tau _{v}^{max}\), v turns to the active state, and together with z it becomes able to trigger the activation of node x (which will eventually become active by the time-horizon T).

It should be noted that the same dynamics holds for the nC-F ²DLT model, apart from the difference that concerns the quiescence time of node v: this would be less than \(\tau _{v}^{max}\) since y, a foe of v, is not involved in the propagation process. ■

Competitive diffusion

We focus here on spC-F²DLT and npC-F²DLT, and show that both models can be reduced to the Homogeneous Competitive Linear Threshold (H-CLT) with Majority Vote as tie-breaking rule (Chen et al. 2013). This is a competitive, progressive model based on LT, for which it is known that its activation function is monotone but not submodular regardless of the particular tie-breaking rule.

To begin with, we might recall that the non-progressive LT-based diffusion can be reduced to the progressive case, using a particular form of layered graph (Kempe et al. 2003). Given a time interval T and a diffusion graph G=〈V,E〉 for non-progressive LT, a new graph G^T can be derived such that every node v∈V will have a replica v_t in every layer at time t∈T, and for every edge (u,v)∈E there will be an edge (u_t−1,v_t) in G^T.

Unfortunately, this serialization technique cannot be directly applied to our models, since it is not designed to deal with competitive or non-progressive diffusion and it discards activation or delayed propagation aspects. In the following, we define serialization techniques that are suitable for our competitive models and treat one particular configuration at a time. One general requirement is related to the time horizon to bound the unfolding of the diffusion process. In fact, when dealing with competitive models, the termination guarantee is lost. A simple example is provided next to depict such a non-termination scenario.

Example 3

In Fig. 6 , nodes u and z are chosen as seed for the green campaign and the red one, respectively. Nodes v and x become green-active and red-active, respectively, at time t=1. Next, they will constantly switch their activation campaign, causing non-termination of the diffusion process. ■

CONFIGURATION 1: No quiescence time, constant activation-threshold.

We assume that q(v)=0 and g(v,t)=θ_v, for all v∈V,t∈T. For both spC-F²DLT and npC-F²DLT, we claim their reduction to the H-CLT model with majority voting as tie-breaking rule.

Definition 5

spC-F²DLT graph serialization for reduction toH-CLT. Given a time interval T, we define a layered graph G^T=〈V^T,E^T〉 such that, for each layer at time t∈T, every node v∈V will be represented in V^T as a tuple \(\left \langle v^{1}_{t},v^{2}_{t},v^{3}_{t} \right \rangle \). Instances \(v^{1}_{t}\) and \(v^{2}_{t}\) have activation-threshold equal to 0, while \(v^{3}_{t}\) has the same threshold as the original node v∈V. The set of edges is defined as \(E^{T} = \left \{\left (u_{t}^{1},v_{t+1}^{3}\right) \mid (u,v) \in E, t, t+1 \in T \right \} \cup \left \{\left (v_{t}^{3},v_{t}^{2}\right) \mid v \in V, t \in T\right \} \cup \left \{\left (v_{t}^{2},v_{t}^{1}\right) \mid v \in V, t \in T\right \} \cup \left \{\left (v_{t}^{1},v_{t+1}^{2}\right) \mid v \in V, t \in T\right \}\), and the following constraint on edge weights must hold: \( \forall v_{t}^{2} \in V^{T}, \ w\left (v_{t-1}^{1},v_{t}^{2}\right) < w\left (v_{t}^{3},v_{t}^{2}\right).\) □

In the above definition, triples act as connectors between two consecutive time-layers. The role of any connector component is as a sort of “switch” to enable a node choosing between its activation state in a layer and the one in the subsequent layer. In other words, node \(v^{1}_{t}\) is the main instance of node v, since the activation state of \(v^{1}_{t}\) reflects the state of v in the original graph, under spC-F²DLT at time t; node \(v^{3}_{t}\) is the instance of v connected with other nodes from layer at t−1, therefore it reflects the influence received by v in the original graph, at time t−1; if the activation attempt to \(v^{3}_{t}\) fails, node \(v^{2}_{t}\) will be activated with the same state of v; otherwise, according to the edge weight constraint (cf. Definition 5), \(v^{2}_{t}\) will switch to the other campaign, and then will propagate to instance \(v^{1}_{t}\). Recall that \(v^{1}_{t}, v^{2}_{t}\) have zero activation-threshold. Figure 18 in Appendix A shows an example of serialization for a spC-F²DLT diffusion graph with time horizon set to 2.

It should be emphasized that, compared to the serialization method in (Kempe et al. 2003), we require replication of each node in each layer, and additional edges connecting the replica-instances, in order to allow the maintenance of the activation state when no activation event occurs between two time-consecutive layers.

Analogous reduction technique can be defined for the npC-F²DLT model.

Definition 6

npC-F²DLT graph serialization for reduction to H-CLT. Given a time interval T, we define a layered graph G^T=〈V^T,E^T〉 such that, for each layer at time t∈T, every node v∈V will be represented in V^T as a tuple \(\left \langle v^{1}_{t},v^{2}_{t},v^{3}_{t} \right \rangle \). Instances \(v^{1}_{t}\) and \(v^{2}_{t}\) have activation-threshold equal to 1 and 0, respectively, while \(v^{3}_{t}\) has the same threshold as the original node v∈V. The set of edges is defined as \( E^{T} = \left \{\left (u_{t}^{1},v_{t+1}^{3}\right) \mid (u,v) \in E, t,t+1 \in T \right \} \cup \left \{\left (v_{t}^{3},v_{t}^{2}\right) \mid v \in V, t \in T\right \} \cup \left \{\left (v_{t}^{2},v_{t}^{1}\right) \mid v \in V, t \in T\right \} \cup \left \{\left (v_{t}^{3},v_{t}^{1}\right) \mid v \in V, t \in T\right \} \cup \left \{\left (v_{t}^{1},v_{t+1}^{2}\right) \mid v \in V, t, t+1 \in T \right \}\), and the following constraints on edge weights must hold: \( \forall v_{t}^{2} \in V^{T}, \ w\left (v_{t-1}^{1},v_{t}^{2}\right) < w\left (v_{t}^{3},v_{t}^{2}\right)\), and \( \forall v_{t}^{1} \in V^{T}, \ w\left (v_{t}^{2},v_{t}^{1}\right) + w\left (v_{t}^{3},v_{t}^{1}\right) = 1.\) □

It should be noted that the last condition in Definition 6 imposes nodes \(v_{t}^{2}\) and \(v_{t}^{3}\) to hold the same activation state in order to activate \(v_{t}^{3}\).

Analogously to the reduction of spC-F²DLT to H-CLT, we can conveniently devise a notion of “connector” component between any two consecutive layers, which however in this case should also account for node deactivations. Figure 19 in Appendix A shows an example of connector for the npC-F²DLT model.

Claim 1

For any given diffusion graph \({\mathcal {G}}\) under spC-F²DLT (resp. npC-F²DLT), assuming constant activation-threshold and no quiescence time, every node v in \({\mathcal {G}}\) is active at time t∈T if and only if its corresponding instance \(v^{1}_{t}\) is active in the serialized graph G^T (resp. npC-F²DLT). \(\blacktriangleleft \)

CONFIGURATION 2: Constant quiescence time, constant activation-threshold.

We assume that q(v)=τ_v and g(v,t)=θ_v, for all v∈V. For both spC-F²DLT and npC-F²DLT, we claim their reduction to H-CLT with majority voting as tie-breaking rule.

In this case, we need to consider that, whenever a node is activated, its quiescence time may not expire before the time horizon; for this reason, we will consider only nodes reachable from \(S_{0} = S^{\prime }_{0} \cup S^{\prime \prime }_{0}\) within T, for any two given seed sets \(S^{\prime }_{0}\) and \(S^{\prime \prime }_{0}\). To identify such nodes, we define a quiescence-aware distance measure that accounts for the quiescence times along the path connecting any two nodes. Given nodes u,v, and the set P(u,v) of all paths between u and v, the distance from u to v will be measured as \( d(u,v) = \min _{p \in P(u,v)} \sum \nolimits _{x \in p} \tau _{x} \). Moreover, we denote with d(S₀,v) the minimum distance between nodes u∈S₀ and v. By exploiting this distance, we will discard all nodes that cannot be “contagious” before the end of T, say t_max. Therefore, the node set V^T of the layered graph is defined as:

$$V^{T} = \left\{ \left\langle v_{t}^{1},v_{t}^{2},v_{t}^{3} \right\rangle \mid \forall v \in V, \ t \in T, \ d(S_{0},v) < t_{max}\right\}. $$

Each node v∈V with quiescence time τ_v will have connections from the previous layers according to the following rule: for any layer at time t, if t<d(S₀,v) then v will not have any incoming edges, otherwise all incoming edges of v will be from the layer at time t−τ_v−1.

Using the above settings in the serialization method previously presented, it can easily be demonstrated that both spC-F²DLT and npC-F²DLT can be reduced to an equivalent H-CLT model.

Claim 2

For any given diffusion graph \({\mathcal {G}}\) under spC-F²DLT (resp. npC-F²DLT), assuming constant activation-threshold and constant quiescence time, every node v in \({\mathcal {G}}\) is active at time t∈T if and only if its corresponding instance \(v^{1}_{t}\) is active in the serialized graph G^T (resp. npC-F²DLT). \(\blacktriangleleft \)

CONFIGURATION 3: Variable quiescence time, constant activation-threshold.

We assume that q(v,t) is variable, while g(v,t)=θ_v, for all v∈V,t∈T.

Like in the previous case, we need to specify the seed sets \(S^{\prime }_{0}, S^{\prime \prime }_{0}\). However, note that the quiescence time of a node now depends on the actual activation state of its in-neighborhood (cf. Eq. 3), which makes it unfeasible a direct serialization of the whole diffusion graph.

Starting from the original diffusion graph \({\mathcal {G}}\), we derive an “intermediate” graph \(\widehat {{\mathcal {G}}}\), which is equivalent to \({\mathcal {G}}\) unless each node v∈V is associated with a quiescence time interval \([\tau _{v},\tau _{v}^{max}]\), where \(\tau _{v}^{max}=\tau _{v} + \psi (N_{-}^{in}(v))\). Let us denote with \({\mathcal {G}}^{min}\) the instance of \(\widehat {{\mathcal {G}}}\) such that the quiescence time of every \(v \in \widehat {{\mathcal {G}}}\) is τ_v, and with \({\mathcal {G}}^{max}\) the instance of \(\widehat {{\mathcal {G}}}\) such that the quiescence time of every \(v \in \widehat {{\mathcal {G}}}\) is \(\tau _{v}^{max}\).

Although we cannot assert that spC-F²DLT and npC-F²DLT are equivalent to H-CLT under the layered graph obtained by applying the previously described serialization techniques, an important theoretical result can nonetheless be provided, as reported next.

Claim 3

For any diffusion graph \({\mathcal {G}}\) under spC-F²DLT (resp. npC-F²DLT), with campaigns C^′,C^′′, assuming constant activation-threshold and variable quiescence time, for any seed sets \(S^{\prime }_{0}\) and \(S^{\prime \prime }_{0}\), it holds that:

$$ \sigma^{\prime}_{H{-}CLTmax}(S^{\prime}_{0},S^{\prime\prime}_{0}) \leq \sigma^{\prime}(S^{\prime}_{0},S^{\prime\prime}_{0}) \leq \sigma^{\prime}_{H\text{-}CLTmin}(S^{\prime}_{0},S^{\prime\prime}_{0}), $$

(5)

where σ^′ is the number of nodes activated by C^′ under spC-F²DLT (resp. npC-F²DLT), \(\sigma ^{\prime }_{H\text {-}CLTmax}(S^{\prime }_{0},S^{\prime \prime }_{0})\) and \(\sigma ^{\prime }_{H\text {-}CLTmin}(S^{\prime }_{0},S^{\prime \prime }_{0})\) are the number of nodes activated by C^′ under H-CLT in the layered graph obtained by serialization of spC-F²DLT (resp. npC-F²DLT) on \({\mathcal {G}}^{max}\) and \({\mathcal {G}}^{min}\), respectively. \(\blacktriangleleft \)

Enabling variable quiescence time, i.e., ψ(·), means that the exact time required by each node to make a transition from the quiescent state to the active one cannot be established in advance at the beginning of the propagation process. Since for any node v the quiescent time ranges within \([ \tau _{v}, \tau _{v}^{max}]\), we devise two opposite scenarios. In the first scenario, represented by the rightmost side of Eq. 5, each node is assumed to wait the minimum amount of time, i.e., τ^min, before its activation; this leads to a higher fraction of nodes that could be activated before the time horizon T is reached. The second scenario, represented by the leftmost side of Eq. 5, assumes that each node has to wait the maximum possible quiescence time, i.e., τ^max; as a consequence, a smaller fraction of nodes will be able to complete the activation process before the time limit, thus leading to a lower spread.

CONFIGURATION 4: No quiescence time, variable activation-threshold.

We assume that q(v)=0 and g(v,t)=θ_v+𝜗(θ_v,t), for all v∈V,t∈T. For both spC-F²DLT and npC-F²DLT, we claim their reduction to H−CLT with majority voting as tie-breaking rule. In the following, we refer to the biased activation-threshold function, although it is easy to show analogous considerations for the non-biased activation-threshold function.

Because of the dynamic behavior of the activation-threshold function, we cannot predict its value at any particular time step of the diffusion process; nevertheless, by specifying the value of coefficient δ in Eq. 1, we can derive the value of \(t_{v}^{max}\), which would suggest how many time-layers we have to look back in order to know the actual threshold value of v at a particular time t. In order to capture such dynamic aspect in H-CLT, we define a further serialization technique, built on top of the previously defined. We will restrict to a particular case, afterwards we provide some rules that apply to the general case.

Let us assume to focus on a particular node v, and at any two consecutive time steps of activation for the same campaign its threshold increases by δ. Again, node v will have replicas for any time-layer t, i.e., \(\langle v^{1}_{t},v^{2}_{t},v^{3}_{t} \rangle \), with the first replica, \(v^{1}_{t}\), holding the actual state of v in the corresponding serialized graph for the competitive model. In addition, we introduce further replicas, in number equal to the value \(t_{v}^{max}\); suppose, for the sake of simplicity, \(t_{v}^{max} = 3\), we derive replica nodes \(\left \langle v_{t}^{3,r1},v_{t}^{3,r2},v_{t}^{3,r3} \right \rangle \), such that each of them will have a threshold value in [θ_v,1] with increment of δ. Figure 7 illustrates this new component in the serialized graph.

Because this component is introduced as an extension of the previous techniques, the meaning of the nodes \(v^{1}_{t-1},v^{1}_{t-2},v^{1}_{t-3}\) remains the same as in the previous cases. On the right side of Fig. 7, each of the additional replicas has a different value of threshold and it is connected with nodes coming from the previous layers. Clearly, the overall behavior of this component depends on the weights attached to every edge in the structure. In this regard, we define the following constraints on the edge weights:

$$ \left\{\begin{array}{ll} w^{3,r1} > w_{1}^{13} & \quad (a) \\ \forall i>1 \quad w_{i}^{13}=w^{3,ri} & \quad (b) \\ \forall i \geq 1 \quad w_{i}^{13} > \sum\nolimits_{j>i}^{n} w_{j}^{13} & \quad (c)\\ \forall i \geq 1 \quad w^{3,ri} > \sum\nolimits_{j>i}^{n} w^{3,rj} & \quad (d) \\ w^{3,r1} - w_{1}^{13} < w_{n}^{13} & \quad (e) \end{array}\right. $$

(6)

It should be noted that the activation attempts are performed directly on the replicas. Therefore, the above constraints on the edge weights control whether a node assumes the state derived as the outcome of the most recent activation attempts, or the one consistent with its personal history. as the outcome of the most recent activation attempts or the one consistent with its personal history. Each of the aforementioned inequality contributes to this decision process, following a different purpose. Eq. 6(a) ensures that the state derived from the last activation attempt is always preferred to the one derived from the previous time step. Eq. 6(b) ensures that the information coming from the previous time steps shall be given the same importance as the one derived from the current replicas. Eq. 6(c-d) ensures that the most recent information, i.e., the closest previous time steps, has higher priority than the earliest one. Eq. 6(e) ensures that there is consistency with respect to the state assumed in the closest previous time step and farthest involved time step (e.g., the third previous time step in the addressed scenario).

Moreover, the threshold of the “central” node in the component (\(v^{3}_{t}\)) is set to w^3,r1, to ensure sequentiality of the diffusion. By setting \(\theta _{v_{t}^{3}}\) equal to w^3,r1, we avoid that \(v_{t}^{3}\) can be activated by its own replicas belonging to layers preceding the t−1-th layer.

Figure 8 shows how the above defined connector is integrated into a serialization technique. In the figure, only the connections incident on vertex v are expanded. The red edges are the ones connecting consecutive layers, therefore the replica \(v_{t}^{3,r1}\) is connected with the previous layer, the replica \(v_{t}^{3,r2}\) is connected with the second previous layer and so on. Blue edges represent the new connections due to the introduction of this new component.

Claim 4

For any given diffusion graph \({\mathcal {G}}\) under spC-F²DLT (resp. npC-F²DLT), assuming variable activation-threshold and no quiescence time, every node v in \({\mathcal {G}}\) is active at time t∈T if and only if its corresponding instance \(v^{1}_{t}\) is active in the serialized graph G^T (resp. npC-F²DLT). \(\blacktriangleleft \)

Our theoretical inspection of the proposed models, whose technical details have been presented in “Theoretical properties of the models” section, revealed two important findings:

(F1): The non-competitive, progressive model, nC-F²DLT, is proven to be equivalent to LT with Quiescence Time; therefore, the activation function in nC-F²DLT is monotone and submodular.

(F2): The competitive, non-progressive models, spC-F²DLT and npC- F²DLT, can be reduced, via graph serialization, to Homogeneous Competitive LT (Chen et al. 2013), which is competitive and progressive, and has monotone, non-submodular activation function; therefore, the activation function in spC-F²DLT and npC- F²DLT is monotone but not submodular. It should be emphasized that the basic technique of graph serialization introduced in (Kempe et al. 2003) to reduce the non-progressive LT-based diffusion to the progressive case, cannot be applied to our proposed models, since it is not designed to deal with competitive or non-progressive diffusion and it discards activation or delayed propagation aspects; to overcome this issue, we provided new serialization techniques and relating definitions of layered-graphs that are suitable for our competitive models, focusing on particular settings of the activation-threshold and quiescence functions.

The two findings clearly have different impact on the development, upon our F²DLT models, of approximate solutions to influence maximization, rumor blocking, and related problems. On the other hand, in terms of expressiveness of our competitive F²DLT models, it should be noted that the serialization techniques require the construction of layered graphs whose size easily grows with some of the models’ parameters, making the application of such serialized graphs unfeasible at a large scale. Therefore, using our competitive F²DLT models turns out to be essential in the representation of complex, dynamic propagation phenomena.

Our proposed class of trust-aware, dynamic models for non-competitive and competitive information diffusion offers a versatile solution for enhanced understanding of complex influence-propagation phenomena that occur in real-life network scenarios. Our models are also unique, since they have significantly different behavioral dynamics w.r.t. epidemic models and the dynamic linear threshold model, according to theoretical considerations that were also clearly supported by empirical evidence in our experimental evaluation.

In this regard, it would be interesting to study how to learn the various parameters in our models for the corresponding IM scenarios. Note that learning parameters for IM tasks is a challenging problem, which is still largely open, given the relatively little work done even under basic diffusion models (Saito et al. 2008; Goyal et al. 2010). Major difficulties are in the assumption of availability of past propagation data from which the parameters would be learnt, which is in general difficult to obtain, and the large number of parameters, which poses efficiency issues. To overcome these aspects, the approach in (Vaswani et al. 2017) appears to be particularly promising, as it does not depend on the rules that control how the propagation unfolds over time. Nonetheless, we expect that the learning problem in our setting will easily become much more challenging, given the presence of other parameters than just the diffusion probabilities, and the need for coping with competitive influence scenarios. Therefore, we believe there will be much work to do in such a direction.

Figure 18 shows an example of serialization for a spC-F²DLT diffusion graph with time horizon set to 2. Dashed lines correspond to the edges in the original graph, whereas solid lines correspond to the edges in the resulting serialized graph. Each of the four nodes in the original graph is replicated as a triple on each of the two time-layers. Triples act as “connectors” between two consecutive time-layers.

Analogously to the reduction of spC-F²DLT to H-CLT, we can conveniently devise a notion of “connector” component between any two consecutive layers, shown in Fig. 19, which in the case of npC-F²DLT needs to account for node deactivations.

Example 4 shows a selection of possible configurations for the component utilized in competitive models shown in Fig. 7, in order to prove the correctness of the set of constraints in Eq. 6.

Example 4

In the example of Fig. 20, we assume that node v in the original graph needs three consecutive time steps to reach the unit value for its threshold.

On the right side of each subfigure, there are the additional node replicas:

\(\left \langle v_{t}^{3,r1},v_{t}^{3,r2},v_{t}^{3,r3} \right \rangle \), where \(v_{t}^{3,r3}\) has the maximum value for the activation threshold.

Figure 20a represents the case when the node has already reached the maximum value of its threshold and its in-neighbors are only able to activate the first two replicas, which is not enough for making v change its activation campaign. We need thus to verify that:

Note that the inequality in (9) holds (cf. Eq. 6 (e)).

Figure 20b shows the case when v is active for just two consecutive time steps. Therefore, the configuration on the right side of the figure is enough to activate v in favor of the red-campaign. Therefore, the following inequality must hold:

Above, inequality in (12) holds as given in Eq. 6 (a).

Figure 20c shows the case when the node v switched from one campaign to the opposite in the middle of the three consecutive time steps. In this case the activation of node \(v_{t}^{3,r1}\) must guarantee the change of activation campaign. So the following inequality must hold:

$$\begin{array}{*{20}l} \overbrace{w^{3,r1} + w_{2}^{13}}^{red} > \overbrace{w_{1}^{13}+w_{3}^{13}}^{green} \end{array} $$

(13)

Indeed the validation is straightforward, because all the summations on the left side in 13 are by definition greater than the one on the right side.

Figure 20d shows the case when at time step t none of the in-neighbors of v is able to activate it. In this case, the node will keep its previous state, and it will do it only after the activation of the corresponding replica at the very previous time, this allow us to guarantee the sequentiality of the whole process. ■

Figure 21 provides a focus on the behavior of SIR and SEIR models in terms of varying parameters (i.e., transmission rate β, recovery rate γ, and incubation rate σ) for the analysis discussed in “Comparison with the IC, SIR and SEIR models” section.

We observe that, for both models, most of the infections tend to occur at the early time steps of the propagation as β increases. On the other hand, higher values of γ yield cascades that show a smoother decay over time, and consequently they last longer than those corresponding to smaller γ.