Influence Spread in the Heterogeneous Multiplex Linear Threshold Model

The linear threshold model (LTM) has been used to study spread on single-layer networks defined by one inter-agent sensing modality and agents homogeneous in protocol. We define and analyze the heterogeneous multiplex LTM to study spread on multi-layer networks with each layer representing a different sensing modality and agents heterogeneous in protocol. Protocols are designed to distinguish signals from different layers: an agent becomes active if a sufficient number of its neighbors in each of any a of the m layers is active. We focus on Protocol OR, when a=1, and Protocol AND, when a=m, which model agents that are most and least readily activated, respectively. We develop theory and algorithms to compute the size of the spread at steady state for any set of initially active agents and to analyze the role of distinguished sensing modalities, network structure, and heterogeneity. We show how heterogeneity manages the tension in spreading dynamics between sensitivity to inputs and robustness to disturbances.

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I Introduction

The spread of an activity or innovation across a population of agents that sense or communicate over a network has critical consequences for a wide range of systems from biology to engineering. The adoption of a strategy, such as wearing a face mask during a pandemic, can spread across a social network even when there are only a few early adopters. The observation and response to a threat by one or more vigilant animals can spread through a social animal group. A robot that detects a change in the environment and takes action can spread its behavior across a networked robot team.

To predict and control spread, we present and analyze a new model that captures the realities of multiple inter-agent sensing modalities and heterogeneity in responsiveness of agents to others. We develop and prove the validity of new algorithms that provide the means to systematically determine the spreading influence of a set of agents as a function of multi-layer network structure and agent heterogeneity.

The linear threshold model (LTM), from Granovetter [1] and Schelling [2], describes the spread of an activity as discrete-time, discrete-valued state dynamics where an agent adopts or rejects an activity by comparing the fraction of its neighbors that have adopted the activity to its individual threshold. Kempe et al. [3] used the LTM with random thresholds to investigate spread of an activity over a population on a single-layer network. Lim et al. [4] introduced and analyzed the notion of cascade and contagion centralities in the model of [3]. The LTM on single-layer networks has also been studied in [5, 6, 7, 8, 9] and generalized to continuous-time, real-valued dynamics in [10].

The single-layer network in the LTM represents a single sensing modality or a projection of multiple sensing modalities. Yet, in real-world systems, agents may distinguish the different sensing modalities, rather than project them, in ways that impact spread. For example, someone deciding whether or not to wear a mask may consider as separate signals what they see others doing in the neighborhood and what they hear over social media that others are doing. And, how they act on the signals may differ from person to person. A more readily activated person starts wearing a mask when they observe enough of the first or second group wearing a mask. A less readily activated person starts wearing a mask only when they observe enough of the first and second groups wearing a mask.

In this paper we leverage multiplex (multi-layer) networks to model spread in a population of heterogeneous agents that interact through, and distinguish, multiple sensing modalities. Multiplex networks have been used to study consensus dynamics [11, 12, 13, 14, 15]. Yağan and Gligor [16] studied a multiplex LTM using a weighted average of activity across layers. Other models of spread in the case of multiple sensing modalities are reviewed in [17], but most restrict to homogeneous agents.

In [18], we first introduced the LTM on multiplex networks with homogeneous agents, where the graph for each layer is associated with a different sensing modality, and Protocols OR and AND distinguish signals from different layers to model more and less readily activated agents, respectively. We analyzed the duplex (two-layer) LTM with agents that are homogeneous in protocol and showed how to compute cascade centrality, an agent’s influence on the steady-state size of the cascade. Yang et al. [19] studied the influence minimization problem for the homogeneous model of [18].

Our contributions in the present paper are multifold. First, we define the heterogeneous multiplex LTM to analyze spreading dynamics on an arbitrary number of network layers with agents that employ protocols heterogeneously. Second, we define the heterogeneous multiplex live-edge model (LEM), which generalizes [3]

, and we introduce the live-edge tree to define reachability on this LEM. We prove a key result on equivalence of probabilities for the LTM and LEM.

Third, we derive Algorithms 1 and 2 to compute influence spread for the heterogeneous multiplex LTM. Algorithm 1 is provably correct and useful for small networks. Algorithm 2

maps the influence spread calculation to an inference problem in a Bayesian network and is efficient for large networks. We prove that calculating influence spread is #P-complete.

Fourth, we derive analytical expressions for influence spread in classes of multiplex networks. We show how ORs enhance and ANDs diminish spreading relative to the projected network. Fifth, we investigate heterogeneity in spreading and show how it can be used to manage the tradeoff between sensitivity to a real input and robustness to a spurious signal.

Section II describes multiplex networks. Sections III and IV introduce the heterogeneous multiplex LTM and LEM, respectively. We prove their equivalence in Section V. Sections VI and VII present Algorithms 1 and 2. Section VIII presents analytical expressions for influence spread. Heterogeneity is studied in Section IX. We conclude in Section X.

Ii Multiplex Networks

A multiplex network is a family of directed weighted graphs . Each graph , , is a layer of the multiplex network. The agent set is the same in all layers. The edge set of layer is and can be different in different layers. Each edge , pointing from to in layer , is assigned a weight . Here we adopt the “sensing" convention for edges: edge exists if agent can sense agent in layer . If edge exists, agent is an out-neighbor of agent in layer . We denote the set of out-neighbors of in layer as . We say that the weight of agent ’s out-neighbor in layer is the weight . We assume the weights of all out-neighbors for every agent sum up to , i.e., for every agent . A monoplex network is a multiplex network with , i.e., with only a single layer.

For undirected graphs, every edge is modeled with two opposing directed edges. For unweighted graphs, every edge can be assigned a weight , where is the out-degree of node in layer and equal to the number of out-neighbors of node in layer . A projection network of is the graph where .

Iii The Heterogeneous Multiplex LTM

The linear threshold model (LTM) is described by a discrete-time dynamical system in which the state of each agent at iteration is inactive with or active with . The LTM protocol determines how the active state spreads through the network. Our focus is on which agents will be active at steady state as a function of which agents are active initially. We define .

In Section III-A, we recall the LTM for monoplex networks. In Section III-B, we generalize the LTM to multiplex networks by defining protocols for how the active state spreads when signals from different layers are distinguished. Our definition allows for heterogeneity among agents in protocol.

Let be the set of agents that are active by the end of iteration . Once active, an agent remains active so that . At , all agents are inactive except the initially active set . Every agent in is called a seed. The LTM protocol determines when inactive agents at iteration become active at iteration . A steady state is reached when .

Iii-a Monoplex LTM

The LTM protocol on a monoplex network is defined as follows (e.g., [3]). Each agent chooses a threshold

randomly and independently from a uniform distribution

. An inactive agent at iteration becomes active at iteration if the sum of weights of its active out-neighbors at exceeds , that is, if . For agents, steady state is reached by .

Iii-B Multiplex LTM

We introduce the LTM on a multiplex network with layers by defining a family of protocols as follows. Each agent chooses a threshold in each layer for and . Each is randomly and independently drawn from the uniform distribution . In general, each agent has different neighbors in different layers. If the sum of weights of active out-neighbors of agent in layer at exceeds , that is, , we say agent receives a positive input from layer at . Otherwise, agent receives a neutral input .

The protocols that determine whether or not an inactive agent at becomes active at account for the possibility that the inputs it receives at from the different layers may be conflicting. Let the average input agent receives at be .

Definition 1 (Multiplex LTM Protocol).

Given multiplex network with seed set , the multiplex LTM protocol for agent is parametrized by as follows:

(1)
(2)
(3)

We identify two protocols for the limiting values of :

Protocol OR: . Inactive agent at iteration becomes active at iteration if it receives a positive input from any layer at ;

Protocol AND: . Inactive agent at iteration becomes active at iteration if it receives positive inputs from all layers at .

The multiplex LTM protocol specifies that inactive agent at iteration becomes active at iteration if it receives a positive input from any of the layers, where . Asymmetric sensitivity to layers can be modelled with a convex combination of .

In this paper, we examine the two limiting cases: Protocol OR, where , and Protocol AND, where . Analysis in these cases is sufficient for understanding heterogeneity and spreading dynamics on multi-layer networks. Our theory can be extended to protocols for .

Remark 1.

Protocol OR models agents that are readily activated: there only needs to be sufficient activity among neighbors in one layer at in order for agents to become active at . Protocol AND models agents that are conservatively activated: there needs to be sufficient activity among neighbors in every layer at in order for agents to become active at . Thus, agents with Protocol OR enhance spreading and agents with Protocol AND diminish spreading.

We study heterogeneous networks in which some agents use Protocol OR while the others use Protocol AND.

Definition 2 (Sequence of Protocols).

Let be the protocol used by agent . We define the sequence of protocols to be the protocols used by the agents ordered from agent 1 to agent .

Lemma 1.

For a multiplex network with agents, the multiplex LTM converges in at most iterations.

Proof.

Assume the multiplex LTM converges in more than iterations. Then at least one agent switches from inactive to active in each of the first iterations and these agents are distinct. There is at least one agent in the initial active set that is not one of those agents. This implies at least agents in the network, which is a contradiction. ∎

Iv The Heterogeneous Multiplex LEM

Our approach to analyzing the multiplex LTM generalizes the approach in [3], which uses the live-edge model (LEM) for monoplex networks to analyze the monoplex LTM. In this section we define the multiplex LEM. In Section IV-A, we recall the LEM proposed in [3] for monoplex networks. In Section IV-B, we generalize the LEM to multiplex networks and introduce the notion of reachability- on the multiplex LEM. Unlike in our earlier work [18], the reachability we propose here allows for heterogeneous protocols among agents.

Iv-a Monoplex LEM and Reachability

The LEM for a monoplex network is defined as follows [3]. Let be the set of seeds. Each unseeded agent randomly selects one of its outgoing edges with probability given by the edge weight. The selected edge is labeled as “live", while the unselected edges are labeled as “blocked". The seeds block all of their outgoing edges. Every directed edge will thus be either live or blocked. The choice of edges that are live is called a selection of live edges.

Let be the set of all possible selections of live edges. The probability of selection is the product of the weights of the live edges in selection . Because the selection of live edges can be done at the same time for every node, the LEM can be viewed as a static model. The LEM can alternatively be viewed as an iterative process in the case the live edges are selected sequentially.

A live-edge path [3] is a directed path that consists only of live edges. Let be the set of all possible distinct live-edge paths from agent to . The probability of live-edge path is the product of the edge weights along the path. We say is reachable from by live-edge path with probability , and is reachable from with probability , where .

Alternatively, we can compute in terms of selections of live-edges. Let be the set of all selections of live edges that contain a live-edge path from to . Then, . Likewise, let be the set of all selections of live edges that contain a live-edge path from to at least one node . Then, is reachable from with probability , where .

Iv-B Multiplex LEM and Reachability

We introduce the LEM on a multiplex network as follows.

Definition 3 (Multiplex LEM).

Consider a multiplex network with seed set . In each layer , each unseeded agent randomly selects one of its outgoing edges with probability . The selected edges are labeled as “live", while the unselected edges are labeled as “blocked". The seeds block all of their outgoing edges in every layer. The choice of edges that are live is a multiplex selection of live edges. Let be the set of all possible multiplex selections of live edges. The probability of selection is the product of the weights of all live edges in selection .

The challenge in generalizing the LEM to multiplex networks is in properly defining reachability. Here we introduce the live-edge tree, which we use to define reachability.

Definition 4 (Live-edge Tree).
111To highlight key differences between multiplex and monoplex networks, we assume each has at least one neighbor in each layer. If not, with a slight modification of Defs. 4- 5, the theory and computation are still valid.

Given a set of seeds and a multiplex selection of live edges , the live-edge tree associated with agent is constructed as follows with agent as the root node. Let be the live edge of agent in layer , . Then the children of the root node are agents , and the root node is connected to each child with the live edge in the corresponding layer. The tree is constructed recursively in this way for each child that itself has at least one child. Any agent in the network may appear multiple times as a node in the tree.

Fig. 1 provides an example of a three-layer multiplex network with five agents. The network has only one possible multiplex selection of live edges, given in Fig. 2. Fig. 3 shows the corresponding live-edge tree associated with agent 5.

Fig. 1: An example of a three-layer multiplex network with five agents. Agent 1 is the seed, which is denoted by the black circle.
Fig. 2: The unique multiplex selection of live edges for the network in Fig. 1.
Fig. 3: The live-edge tree associated with agent 5 for the example three-layer multiplex network of Fig. 1 and the unique selection of live edges of Fig. 2. is the set of distinct branches that end with a seed.

We next define reachability from of an unseeded agent in a multiplex network under a sequence of protocols .

Definition 5 (-Reachability).

Consider multiplex network with seed set and multiplex selection of live edges . Let be the live-edge tree associated with agent . Suppose there are distinct branches in indexed by and of the form: , where , , , and each agent in appears at most once in . We call each a distinct branch that ends in a seed. Denote the set of these branches as . For any subset , let the set of agents in be and the set of edges in be .

Given a sequence of protocols , we say that branch subset is -feasible for if and for every for which , all of ’s live edges belong to . Then, is -reachable from by the selection of live edges with probability if there exists at least one that is -feasible for . Let be the set of all selections of live edges by which is -reachable from . Then, is -reachable from with probability , where .

Remark 2.

The condition for -feasibility for of a branch subset does not make explicit a condition on any for which . This follows since for any such agent , the condition is that at least one of its live edges must be in and this is always true by definition.

To illustrate -reachability, consider the live-edge tree associated with agent 5 in Fig. 3 for the unique selection of live edges in Fig. 2 for the multiplex network of Fig. 1 with seed set . Because the selection of live edges in Fig. 2 is unique, it is chosen with probability . Therefore, agent is -reachable from with probability 1 if there exists at least one that is -feasible for . For agent 5, there are 12 distinct branches that end in a seed, as shown in Fig. 3; thus, . For example, .

We compute -reachability from for agent 5 for each the following three sequences of protocols used by the five agents in the three-layer multiplex network:

(4)
(5)
(6)
  1. Let . Consider . Then, and . Since 5 is the only unseeded node in and , is -feasible for 5. Thus, agent 5 is -reachable from with probability 1.

  2. Let . Consider . Then, . The unseeded nodes for which are . From Fig. 3, observe that all the live edges of nodes 3, 4, and 5, belong to , the edge set of . Thus, is -feasible for 5, and agent 5 is -reachable from with probability 1.

  3. Let . In this case there is no -feasible subset , since and agent 2 has a live edge , which is not in the edge set of any branch in . Thus, agent 5 is not -reachable from .

In the next section we prove the equivalence of the probability that agent is -reachable from for the multiplex LEM and the probability that agent is active at steady state for the multiplex LTM with seed set . For our example, this implies that under or , agent 5 will become active at steady state with probability 1 and under , agent 5 will remain inactive at steady state.

V Equivalence of LTM and LEM

The monoplex LEM was introduced in [3] and proved to be equivalent to the monoplex LTM in the sense that the probabilities of agents being reachable from a set in the LEM are equal to the probabilities of agents being active at steady state given seed set

in the LTM. Computing these probabilities for the LTM is challenging because it requires solving over temporal iterations. However, leveraging the equivalence, the probability distributions can be computed without temporal iteration using the LEM as a static model.

The equivalence is recalled in Section V-A for monoplex networks and proved in Section V-B for multiplex networks.

V-a Equivalence for Monoplex Networks

The LTM and LEM were proved to be equivalent in [3] in the following sense. For a given monoplex network with seed set , the probabilities of the following two events for arbitrary agent are the same:

  1. is active at steady state for the LTM with random thresholds and initial active set ;

  2. is reachable from set under the random selection of live edges in the LEM.

V-B Equivalence for Multiplex Networks

We generalize the equivalence of LTM and LEM to multiplex networks in this section. First, we prove the following lemma that infers an agent’s -reachability from the -reachability of its children in the live-edge tree. We then leverage this lemma to prove the equivalence in Theorem 1.

Lemma 2.

Given a multiplex network with seed set , multiplex selection of live edges and sequence of protocols , consider agent and its associated live-edge tree . Assume ’s live edge in layer connects to agent , . Then the -reachability of from by selection can be inferred from the reachability of its children and its protocol as follows:

  1. Let . Then, is -reachable from by selection of live edges if and only if at least one child is -reachable from by selection of live edges .

  2. Let . Then, is -reachable from by selection of live edges if and only if every child is -reachable from by selection of live edges .

Proof.

Let be the corresponding live-edge tree associated with agent ’s child for .

1) Let and suppose is -reachable from by selection for some . By Definition 5, the set of distinct branches that end with a seed in is nonempty and there exists a subset that is -feasible for such that for every for which , all of ’s live edges belong to , where and are the sets of agents and edges in , respectively.

For every branch , there exists a branch in . Let be the set of all branches in that correspond to branches in . Then, is the set of agents in and is the set of edges in . Thus, since and is -feasible for , by Definition 5, must be -feasible for . Agent is therefore -reachable from by selection . This proves the “if” part of the statement.

If no child is -reachable from by selection , then there exists no nonempty in that is -feasible for . This implies there is no nonempty in that is -feasible for . Therefore, agent cannot be -reachable from by selection . This proves the “only if” part of the statement.

2) Let and suppose is -reachable from by selection , for every . By Definition 5, for every , there exists a subset in that is -feasible for . For every let be the set of all branches in that correspond to branches in as defined in the proof of 1). Let with agent set and edge set . By construction, and all of agent ’s live edges belong to . It follows that is -feasible for and thus agent is -reachable from by selection . This proves the “if” part of the statement.

If there is one child that is not -reachable from by selection , then there exists no set in that is -feasible for . Suppose there is a set in that is -feasible for . Since , by Definition 5, it follows that edge . For every branch in that starts with and edge , i.e., , we denote the set of all corresponding branches as . It follows that is -feasible for in and is reachable from . This is a contradiction. Thus, there is no set that is -feasible for and agent cannot be -reachable from by selection . This proves the “only if” part of the statement. ∎

While Definition 5 uses the LEM to define -reachability in a static way, the LEM can also be used to reveal the -reachability of agents as an iterative process over time [3] as follows. First, using Lemma 2, determine the -reachability of the agents with at least one edge coming from initial set . If an agent is determined to be -reachable from , add it to to get a new reachable set . In the next iteration, follow the same procedure and get a sequence of reachable sets . The process ends at iteration if , where is the set of agents that are -reachable from . The mapping between static and temporal determinations of reachability for the LEM is the key to proving the equivalence of the multiplex LTM and multiplex LEM. We also make use of the following definitions.

Definition 6 (LTM-related events).

We define () to be the event that the sum of weights of active out-neighbors of in layer does (does not) exceed at : and . Let

Definition 7 (LTM-related probabilities).

For agent that is inactive at , we define the probability that becomes active at as if uses Protocol OR, and as if uses Protocol AND.

Definition 8 (LEM-related events).

We define () to be the event that agent ’s live edge in layer does (does not) connect to the reachable set at . Let

Definition 9 (LEM-related probabilities).

Consider the LEM as an iterative process. If agent , then we define the probability that as if uses Protocol OR, and as if uses Protocol AND.

We state the equivalence of multiplex LTM and multiplex LEM in the following theorem.

Theorem 1.

For a multiplex network with seed set , multiplex selection of live edges and sequence of protocols , the probabilities of the following two events regarding an arbitrary agent are the same:

  1. is active at steady state for the multiplex LTM under with random thresholds and initial active set ;

  2. is -reachable from the set under random selection of live edges in the multiplex LEM.

Proof.

We prove by mathematical induction.

We use , which we have from [3]. We show i) and ii) , so by induction over the iterations, the probabilities of the two events in the statement of the theorem are the same.

i) Proving (when uses Protocol OR)

In the LTM, being inactive at means none of ’s thresholds is exceeded at and being active at means ’s thresholds in at least one layer is exceeded at . In this case, the probability that is exceeded at is

. The last equality holds because random variables

, , …, are independent. Then is the complement of the probability that its threshold in none of the layers are exceeded, i.e.,

In the LEM, by Lemma 2, means all of ’s live edges are not connected to and means at least one live edge of is connected to . In this case, the probability that ’s live edge in layer connects to is . Then is the complement of the probability that none of ’s live edges connects to , i.e., .

From we have . So that we conclude that .

ii) Proving (when uses Protocol AND)

In the LTM, being inactive at means at least one of its thresholds is not exceeded at and being active at means all of ’s thresholds are exceeded at . In this case, there are possible events, with probabilities denoted as . We have that . The other probabilities have a similar form but with one or more, but not all, of the replaced by . Since and are mutually exclusive for all , we have .

In the LEM, by Lemma 2, means at least one of ’s live edges are not connected to and means all of ’s live edges are connected to . In this case, there are possible events, with probabilities denoted as . We have that . The other probabilities have similar form but with one or more, but not all, of the replaced by . Since and are mutual exclusive for all , we have .

Since the thresholds are independent of one another, can be separated into the product of terms, each of which involves events associated to one layer only: , where is either or . Since the live edges that each agent uses are independent of one another, similar analysis applies to : , where is either or . It follows from that and . ∎

Theorem 1 generalizes the equivalence of LTM and LEM to multiplex networks. Leveraging Theorem 1, we can calculate an agent’s influence, in terms of spreading information through the network, without needing to simulate the multiplex LTM.

Vi Computing Multiplex Influence Spread

In this section we define multiplex influence spread and multiplex cascade centrality for the LTM. We then derive and prove the validity of an algorithm to compute them.

Vi-a Monoplex Influence Spread and Cascade Centrality

The monoplex influence spread of agents in , denoted , is defined as the expected number of active agents at steady state for the monoplex LTM given the network and initial active set  [3]. The monoplex cascade centrality of agent , denoted , is the influence spread of agent defined in [4] as , the expected number of active agents at steady state for the monoplex LTM given and .

Vi-B Multiplex Influence Spread and Cascade Centrality

Influence spread and cascade centrality are naturally generalized to the multiplex setting as follows.

Definition 10 (Multiplex influence spread).

The multiplex influence spread of agents in , denoted , is defined as the expected number of active agents at steady state for the multiplex LTM given the network , sequence of protocols , and initial active set . Let and be expected value and probability, respectively, conditioned on . Then

(7)
Definition 11 (Multiplex cascade centrality).

The multiplex cascade centrality of agent , denoted , is defined as

(8)

When , we replace with in the superscript. For example, when , we write and . When is understood we drop it from the superscript.

Vi-C Computing Multiplex Influence Spread and Centrality

We can directly compute multiplex influence spread and multiplex cascade centrality by computing probabilities of -reachability for the LEM, which is much easier than computing probabilities of agents being active at steady state for the LTM. We summarize in a corollary to Theorem 1.

Corollary 1.

Given multiplex network and sequence of protocols , multiplex influence spread of agents in and multiplex cascade centrality of agent can be determined as

(9)
Proof.

By Definition 5, is the probability that is -reachable from in the multiplex LEM of Definition 3. By Theorem 1, . Thus, by Definition 10, . In case , by Definition 11, . ∎

Algorithm 1 uses (9) to compute multiplex influence spread and can be specialized to compute cascade centrality.

Algorithm 1 (Compute multiplex influence spread ).

Given multiplex network and sequence of protocols :

  1. Find the set of all possible selections of live edges for multiplex network and initially active set . Calculate the probability of each .

  2. For each agent find , the set of all such that is -reachable from by selection .

  3. Calculate .

  4. Calculate .

Although the algorithm is not efficient, we can use it to accurately calculate multiplex influence spread for multiplex networks with a small number of agents. Next, we propose an efficient approach that sacrifices accuracy to calculate multiplex influence spread for large networks.

Vii A Bayesian Network Approach

In this section, we map the problem of computing multiplex influence spread into a problem of probabilistic inference in Bayesian networks (BN). This means we can compute multiplex influence spread by using an appropriate algorithm for inference in BNs, such as the loopy belief propagation algorithm. We first recall the definition of a BN.

Definition 12 (Bayesian network).

Let , where and , be a directed acyclic graph (DAG). Each node is associated with a random variable . Denote the set of out-neighbors of as . Let be the probability of conditioned on the states of nodes in . Then is a Bayesian network

if the joint distribution of the random variables is factorized into conditional probabilities: