Detecting Communities in a Gossip Model with Stubborn Agents

Abstract

We consider a community detection problem in a gossip model, where agents randomly interact pairwise, with stubborn agents never changing their states. It is assumed that the agents can be divided into two communities based on their interaction probability with others. Such a model can illustrate how disagreement and opinion fluctuation arise in a social network. The considered problem is twofold: to infer which community each agent belongs to, and to estimate interaction probabilities between agents, by only observing their state evolution. First, stability and limit theorems of the model are derived. An integrated detection and estimation algorithm is then proposed to infer the two communities and to estimate the interaction probabilities, based on agent states. It is verified that the community detector of the algorithm converges in finite time, and the interaction estimator converges almost surely. In addition, non-asymptotic property is obtained for the former, and convergence rate is analyzed for the latter. Simulations are presented to illustrate the performance of the proposed algorithm.

Transient Behavior of Gossip Opinion Dynamics with Community Structure

“…As τ 1 in Theorem 1, the parameter τ 2 can be arbitrarily small (hence the beginning of the transient interval can be arbitrarily close to Θ(1/λ 3 )), but n needs to be large enough in this case. Theorem 2 implies that the phase transition of transient behavior of the gossip model occurs when l Similar to Theorem 1, the requirement of Theorem 2 for the relative size of weights between regular agents and those between regular and stubborn agents is hidden in (7). The next corollary provides an explicit condition for this relative size and characterizes how edge weights influence the duration of the transient interval.…”

“…Given an estimate of the runtime for a process, we may determine whether the current clusters are steady or not. The correspondence between community structure and transient behavior can also inspire development of community detection methods based on state observations [6,7]. Suppose that the network is unknown but several snapshots of an opinion dynamic are available.…”

“…It is known that gossip model (1) converges to a unique stationary distribution π (that is, if X(0) has distribution π, then X(t) has the same distribution for all t ∈ N + ), if at least one regular agent is connected to some stubborn agent and the network is connected [7,25]. Moreover, the stationary distribution π is the same as the distribution of the random series…”

“…So it is necessary to investigate how the opinion formation process, especially during its initial phase, corresponds to its community structure. Such results can also provide insights into design of community detection algorithms based on state observations [6,7] and design of model reduction tools for large-scale networks [8,9].…”

When Less Is More: Systematic Analysis of Cascade-Based Community Detection

“…(2) We consider cascades instead of chains and, in particular, we do not know who infected whom. Recent work [64] studies analytically the problem of detecting two communities from interactions in a gossip model, where some agents never change their state. In this work, we also leverage the (binary) states of the nodes, but the gossiping protocol fundamentally differs from cascades, and the assumption of two communities is too restrictive for our purposes.…”