Emergent Behaviors over Signed Random Dynamical Networks: State-Flipping Model
Recent studies from social, biological, and engineering network systems have drawn attention to the dynamics over signed networks, where each link is associated with a positive/negative sign indicating trustful/mistrustful, activator/inhibitor, or secure/malicious interactions. We study asymptotic dynamical patterns that emerge among a set of nodes that interact in a dynamically evolving signed random network. Node interactions take place at random on a sequence of deterministic signed graphs. Each node receives positive or negative recommendations from its neighbors depending on the sign of the interaction arcs, and updates its state accordingly. Recommendations along a positive arc follow the standard consensus update. As in the work by Altafini, negative recommendations use an update where the sign of the neighbor state is flipped. Nodes may weight positive and negative recommendations differently, and random processes are introduced to model the time-varying attention that nodes pay to these recommendations. Conditions for almost sure convergence and divergence of the node states are established. We show that under this so-called state-flipping model, all links contribute to a consensus of the absolute values of the nodes, even under switching sign patterns and dynamically changing environment. A no-survivor property is established, indicating that every node state diverges almost surely if the maximum network state diverges.
Keywords. random graphs, signed networks, consensus dynamics
The need to model, analyze and engineer large complex networks appears in a wide spectrum of scientific disciplines, ranging from social sciences and biology to physics and engineering [1, 2, 3]. In many cases, these networks are composed of relatively simple agents that interact locally with their neighbors based on a very limited knowledge about the system state. Despite the simple local interactions, the resulting networks can display a rich set of emergent behaviors, including certain forms of intelligence and learning [4, 5].
Consensus problems, in which the aim is to compute a weighted average of the initial values held by a collection of nodes, play a fundamental role in the study of node dynamics over complex networks. Early work  focused on understanding how opinions evolve in a network of agents, and showed that a simple deterministic opinion update based on the mutual trust and the differences in belief between interacting agents could lead to global convergence of the beliefs. Consensus dynamics has since then been widely adopted for describing opinion dynamics in social networks, e.g., [5, 6, 7]. In engineering sciences, a huge amount of literature has studied these algorithms for distributed averaging, formation forming and load balancing between collaborative agents under fixed or time-varying interaction networks [8, 9, 10, 11, 12, 13, 14, 15]. Randomized consensus seeking has also been widely studied, motivated by the random nature of interactions and updates in real complex networks [16, 17, 18, 19, 20, 21, 23, 24, 25].
Interactions in large-scale networks are not always collaborative since nodes take on different, or even opposing, roles. A convenient framework for modeling different roles and relationships between agents is to use signed graphs. Signed graphs were introduced in the classical work by Heider in 1946  to model the structure of social networks, where a positive link represents a friendly relation between two agents, and a negative link an unfriendly one. In , a dynamic model based on a signed graph with positive links between nodes (representing nations) belonging to the same coalition and negative otherwise, was introduced to study the stability of world politics. In biology, sign patterns have been used to describe activator–inhibitor interactions between pairs of chemicals , neural networks for vision and learning , and gene regulatory networks . In all these examples, the state updates that happen when two nodes interact depend on the sign of the arc between the nodes in the underlying graph. The understanding of the emergent dynamical behaviors in networks with agents having different roles is much more limited than our knowledge about collaborative agents performing consensus algorithms.
It is intriguing to investigate what happens when two types of dynamics are coupled in a single network. Naturally we ask: how should we model the dynamics of positive and negative interactions? When do behaviors such as consensus, swarming and clustering emerge, and how does the structure of the sign patterns influence these behaviors? In this paper, continuing the previous efforts in [36, 37], we answer these questions for a general model of opinion formation in dynamic signed random networks.
In this paper, we study a scheme of randomized node interaction over a signed network of nodes, and show how the nodes’ states asymptotically evolve under these positive or negative interactions. A sequence of deterministic signed graphs defines the dynamics of the network. Random node interactions take place under independent, but not necessarily identically distributed, random sampling of the environment. Once interaction relations have been realized, each node receives a positive recommendation consistent with the standard consensus algorithm from its positive neighbors. Nodes receive negative recommendations from its negative neighbors. In this paper we investigate a model where neighbors construct negative recommendations by flipping the sign of their true state during the interaction. This definition of negative interaction was introduced in . After receiving these recommendations, each node puts a (deterministic) weight to each recommendation, and then encodes these weighted recommendations in its state update through stochastic attentions defined by two Bernoulli random variables.
Our model is general, and covers many of the existing node interaction models, e.g., consensus over Erdős-Rényi graph , pairwise randomized gossiping , random link failure , etc. We allow the sign of each link to be time-varying as well in a dynamically changing environment. We establish conditions for almost sure convergence and divergence of the node states. We show that under the state-flipping model, all links contribute to a consensus of the absolute values of the nodes, even under switching sign patterns. We also show that strong structural balance  is crucial for belief clustering, which is consistent with the results derived in . In the almost sure divergence analysis, we establish that the deterministic weights nodes put on negative recommendations play a crucial role in driving the divergence of the network. A no-survivor property is established indicating that every node state diverges almost surely given that the maximum network state diverges. Our analysis does not rely on a spectrum analysis as that used in , but instead we study the asymptotic behavior of the node states using a sample-path analysis.
In Section 2, we present the network dynamics and the node update rules. The state-flipping model is defined for the negative recommendations. Section 3 presents our main results on the state-flipping model and the detailed proofs are presented in Section 4. Finally some concluding remarks are drawn in Section 5.
A simple directed graph (digraph) consists of a finite set of nodes and an arc set , where denotes an arc from node to with for all . We say that node is reachable from node if there is a directed path from to , with the additional convention that every node is reachable from itself. A node from which every node in is reachable is called a center node (or a root). A digraph is strongly connected if every two nodes are mutually reachable; has a spanning tree if it has a center node; is weakly connected if a connected undirected graph can be obtained by removing all the directions of the arcs in . A subgraph of , is a graph on the same node set whose arc set is a subset of . The induced graph of on , denoted , is the graph with . A weakly connected component of is a maximal weakly connected induced graph of . If each arc is associated with a sign, either ’’ or ’’, is called a signed graph and the sign of is denoted as . The positive and negative subgraphs containing the positive and negative arcs of , are denoted as and , respectively.
Depending on the argument, stands for the absolute value of a real number, the Euclidean norm of a vector or the cardinality of a set. The -algebra generated by a random variable is denoted as .
2 Random Network Model and Node Updates
We consider a dynamic network where each node holds and updates its belief or state when interacting with other nodes. In this section, we present a general model specifying the network dynamics and the way nodes interact.
2.1 Dynamic Signed Graphs
We consider a network with a set of nodes, with . Time is slotted, and at each slot , each node can interact with its neighbors in a simple directed graph . The graph evolves over time in an arbitrary and deterministic manner. We assume is a signed graph, and we denote by the sign of arc . The sign of arc indicates whether is a friend (), or an enemy () of node . The positive and negative subgraphs containing the positive and negative arcs of , are denoted by and , respectively. We say that the sequence of graphs is sign consistent if the sign of any arc does not evolve over time, i.e., if for any ,
We also define with as the total graph of the network. If is sign consistent, then the sign of each arc never changes and in that case, is a well-defined signed graph.
Note that is defined over directed graphs. The only requirement on and is that they should be disjoint, so the signed graph model under consideration is quite general. In particular, we allow that the two possible edge directions coexist between pair of nodes and that the two directions can have different signs.
Next we introduce the notion of positive cluster in a signed digraph, which will play an important role in the analysis of the belief dynamics (see Fig. 1).
Let be a signed digraph with positive subgraph . A subset of the set of nodes is a positive cluster if constitutes a weakly connected component of . A positive cluster partition of is a partition of into positive clusters , such that .
Note that negative arcs may exist between the nodes of a positive cluster. Therefore, a positive-cluster partition of can be seen as an extension of the classical notion of weak structural balance for which negative links are strictly forbidden inside each positive cluster . From the above definition, it is clear that for any signed graph , there is a unique positive cluster partition , where is the number of maximal positive clusters covering the entire set of nodes.
2.2 Random Interactions
Time is discrete and at time , node may only interact with its neighboring nodes in . We consider a general model for the random node interactions. At time , some pairs of nodes are randomly selected for interaction. We denote by the random subset of arcs corresponding to interacting node pairs at time . More precisely, is sampled from the distribution defined over the set of all subsets of arcs in . We assume that form a sequence of independent sets of arcs. Formally, we introduce the probability space obtained by taking the product of the probability spaces , where is the discrete -algebra on : , is the product of -algebras , , and is the product probability measure of . We denote by the random subgraph of corresponding to the random set of arcs. The disjoint sets and denote the positive and negative arc sets of , respectively. Finally, we split the random set of nodes interacting with node at time depending on the sign of the corresponding arc: for node , the set of positive neighbors is defined as , whereas similarly, the set of negative neighbors is .
The above model is quite general. It includes as special cases the classical Erdős-Rényi random graph , gossiping models where a single pair of nodes is chosen at random for interaction [17, 7], or where all nodes interact with their neighbors at a given time [18, 19, 1, 5]. Independence is the only hard requirement in our random graph process, which is imposed in most existing works on randomized consensus dynamics, e.g., [7, 18, 19, 17]. Non-independent random graph models for randomized consensus were discussed in [23, 24, 41, 22].
2.3 Node updates
Next we explain how nodes update their states. Each node holds a state at . To update its state at time , node considers recommendations received from positive and negative neighbors:
The positive recommendation node receives at time is
The negative recommendation node receives at time is
In the above expressions, we use the convention that summing over empty sets yields a recommendation equal to zero, e.g., when node has no positive neighbors, then .
Now let and be two sequences of independent Bernoulli random variables. We further assume that , , and define independent processes. For any , define and . The processes and represent the attention that node pays to the positive and negative recommendations, respectively.
Node updates its state as
where are two positive constants marking the weight each node put on the positive and negative recommendations.
The role of in (1) is consistent with the classical DeGroot’s social learning model  along trustful interactions. In view of the definition of , in contrast to , the model is referred to as the state-flipping model.
The state-flipping model can be interpreted as a situation where the neighbors connected by a negative link provide false values of their states to each node by flipping their true sign . Under this interpretation it is the head node along each negative arc that knows the sign of that arc. However, the tail node does not see the sign of the arc associated with the recommendations it receive. The weights and attentions of recommendations, represented by and , respectively, are then descriptions of each node’s possible prior knowledge of the signs of its neighbors.
In standard consensus algorithms, nodes communicate relative states. In other words, nodes hold no absolute state information. For the state-flipping model to make sense, there must exist a global origin (state equal to 0) known by each node so that sign flipping is possible in the negative interactions.
Let be the random vector representing the network state at time . The main objective of this paper is to analyze the behavior of the stochastic process . In the following, we denote by the probability measure capturing all random components driving the evolution of the network state.
3 Main Results
In this section, we present our main results. We begin by stating two natural assumptions on the way nodes are selected for updates, and on the graph dynamics. In the first assumption, we impose that at time , any arc is selected with positive probability. The second assumption states that the unions of the graphs over time-windows of fixed duration are strongly connected.
A1. There is a constant such that for all and , if .
A2. There is an integer such that the union graph is strongly connected for all .
The following theorem provides conditions under which the system dynamics converges almost surely. Surprisingly, these conditions are mild: we just require that the sum of the updating parameters and is small enough, and that node updates occur with constant probabilities, i.e., and do not evolve over time. In particular, the state of each node converges almost surely even if the signs of the arcs change over time.
Assume that A1 and A2 hold, and that are such that . Further assume that for any , and for some . Then under the state-flipping model, we have, for all and all initial state ,
In the above theorem, we say that exists if converges to a finite limit as tends to infinity.
Theorem 1 shows an interesting property of the state-flipping model: negative updates, together with the positive updates, contributes to the convergence of the node states whenever it holds that . The condition guarantees that the absolute values of the node states are non-expansive for all signed graphs, compared to the state non-expansiveness of standard consensus algorithms .
Characterizing the limiting states is in general challenging. There are however scenarios where this can be done, which require the notion of structural balance .
Let be a signed digraph. is strongly balanced if we can divide into two disjoint nonempty subsets and where negative arcs exist only between these two subsets.
To predict the limiting system behavior, we make the following assumption.
A3. is sign consistent.
Recall that denotes the total graph. The following theorem holds.
Assume that A1, A2 and A3 hold, and that are such that . Suppose contains at least one negative arc and that every negative arc in appears infinitely often in . Further assume that for any , and for some . Then under the state-flipping model, we have, for any initial state :
(i) If is strongly balanced, then there is a random variable , with almost surely, such that
(ii) If is not strongly balanced, then
Theorem 2 states that strong structural balance is crucial to ensure convergence to nontrivial clustering states, which is consistent with the result of  derived for fixed graphs under continuous-time node updates. To establish the result, we do not rely on a spectral analysis as in , but rather study the asymptotic behavior for each sample path. From the above theorem, we know that under the strong structural balance condition, the states of nodes in the same positive cluster converge to the same limit, and that the limits of two nodes in different positive clusters are exactly opposite. Using similar arguments as in , the value of can be described as the limit of a random consensus process with the help of a gauge transformation.
Next we are interested in determining whether the states could diverge depending on the values of the updating parameters and . We show that by increasing , i.e., the strength of the negative recommendations, one may observe such divergence. To this aim, we make the following assumptions.
A4. There is an integer such that the union graph is strongly connected for all .
A5. There is an integer such that the union graph is strongly connected for all .
A6. The events , , are independent and there is a constant such that for all and , whenever .
Assume that A1, A4, A5 and A6 hold, and that for any , and for some . Fix . Then under the state-flipping model, there is such that whenever , we have for almost all initial states .
Proposition 1 shows that under appropriate conditions, diverges almost surely if the negative updating parameter is sufficiently large. We can in fact derive an explicit value for .
The main difficulties of establishing Proposition 1 lie in the fact that we need on one hand to establish an absolute bound for the way decreases (which is obtained by a constructive proof), and on the other hand to establish a probabilistic lower bound for the possible increase of (which is obtained combining A4–A6 and by constructing and analyzing sample paths). These constructive derivations are rather conservative since we consider general random graph processes, but they nevertheless establish a positive drift for with an explicit so that almost sure divergence is guaranteed.
We also remark from Proposition 1 that a large deterministic weight on negative recommendations leads to the divergence of the node states. It can also been seen from the forthcoming Lemma 1 that if these weights on the recommendations are sufficiently small, always converges no matter how the random attentions and are selected.
Actually, one may even prove that when grows large when , the state of any node diverges. This result is referred to as the no-survivor property, and is formally stated in the following result.
Assume that A1, A2 and A6 hold, and that for any , and for some . Fix the initial state . Then under the state-flipping model, we have
A similar kind of no-survivor property was first established in  under the model of repulsive negative dynamics for pairwise node interactions. Theorem 3 establishes the same property for the considered state-flipping model, but for general random graph process. From the proof of Theorem 3, it is clear that it is the arc-independence (assumption A6, see ), rather than synchronous or asynchronous node interactions, that directly results in the no-survivor divergence property for dynamics over signed random networks.
In all above results, it can be seen from their proofs that extensions to time-varying and are straightforward under mild assumptions. The resulting expressions are however more involved. We omit those discussions here to shorten the presentation.
In this section, we present the detailed proofs of the results stated in the previous section. We first establish some technical lemmas, and then the proofs of each result.
4.1 Supporting Lemmas
For any , we define and , which will be used throughout the whole paper.
Suppose . Then .
Proof. Observe that . Hence as long as . Now for any ,
which completes the proof.
Assume that . Let and assume that for some . Then
Proof. We have:
The lemma is then obtained by applying a simple induction argument.
Assume that . Let and assume that for some . Let . Then conditioned on if , if , we have
Proof. Suppose with . Then we have
where in the last inequality we have used the fact that . It is straightforward to see that (4.1) continues to hold with if . Plugging in the assumption that into (4.1), one gets the desired inequality. This proves the lemma.
Assume that A1 and A2 hold, , and . Further assume that for any , and for some . Then for any initial state , we have .
Proof. We prove this lemma using sample path arguments by contradiction. Let us assume that:
H1. There exist and such that
Let . Define
Note that is a stopping time, and the monotonicity of guarantees that is bounded almost surely . Moreover, is also a stopping time, and it is bounded with probability at least in view of H1. Next, we use Lemmas 2 and 3 to get a contradiction. Plugging in the fact that and invoking Lemma 2, conditioned on , we have that for all :
Now consider the time interval . The independence of , , and guarantees that are independent random variables, and they are independent of (cf., Theorem 4.1.3 in ). From their definitions we also know that , , are i.i.d. with the same distribution as , and Assumption A2 guarantees that is strongly connected. Therefore, there exists a node and such that (note that and are random variables, but they are independent with since is a stopping time). Hence we can apply Lemma 3 and conclude that
with a probability at least . Taking for the introduced in Lemma 2, we have
We can repeat the same argument over time intervals . Assuming that the node set is selected, it follows from the strong connectivity assumption A2 that there exists an arc from to in the union graph of the corresponding interval. In this way we add the tail node of such arc into and obtain for . We can thus recursively find with and bound the absolute values of their states. Finally, we get:
which is impossible and hence, H1 is not true. We have proved that:
The claim then follows easily from Lemma 1.
It is easy to see from the proof that Lemma 4 continues to hold if we relax the requirement of to and for some . Lemma 4 indicates that with sufficient connectivity on the graphs defining the dynamical environment (Assumption A2), the absolute values of the nodes states, will eventually converge to a consensus with probability one under quite general conditions on how the random interactions take place in the environment. Noting that A2 is imposed on the overall underlying graph, this concludes that both the positive and negative links contribute to the node states’ consensus in absolute value.
Let and . Then defines a sure event.
Proof. Let us first assume that . Let such that . Then with , we have
Now assume that . We first prove the following claim.
Claim. Suppose there exits such that with and . Then is a sure event, where
To prove this claim, we distinguish three cases:
Let and assume that there exists such that and . Then and for all . Thus, taking out the term in from (1), some simple algebra leads to
where in the last inequality we have used the assumption that .
Let and assume that for all , and, more generally, for all , which implies that . Observing that , we obtain
Let and satisfy
respectively. Without loss of generality we assume the existence of such and since otherwise the desired conclusions immediately falls to Case (i) and (ii).
Now without loss of generality suppose . From Case (i) and (ii), the desired claim can possibly be violated only when there exists with for some (otherwise we can bound from Case (ii)). While due to the choice of it holds that
We can therefore denote and obtain that and We can thus establish bound for , again applying Case (ii).
From the above three cases, we deduce that if does not hold, then must be true. This proves the claim.
Finally, we complete the proof of the lemma using the claim we just established. Take and . We proceeds in steps.
S1) Let with . Applying the claim with and , we deduce that either the lemma holds or there is another node such that .
S2) If in the first step, we could not conclude that the lemma holds, we can apply the claim to and then obtain that either the lemma holds, or there is a node such that .
The argument can be repeated for applying the claim adapting the value of and . Since there are a total of nodes, the above repeated procedure necessarily ends, so the lemma holds.
The purpose of Lemma 5 is to establish an absolute lower bound regarding the possible decreasing of . This lower bound is absolute in the sense that it does not depend on the random graph processes, and require the constructive conditions and to hold. These conditions are certainly rather conservative for a particular node interaction process, e.g., the pairwise gossiping model , or the i.i.d. link failure model .
4.2 Proof of Theorem 1
From Lemma 4, we know that for any