Deterministic Versus Stochastic Consensus Dynamics on Graphs

Abstract

We study two agent based models of opinion formation -one stochastic in nature and one deterministic. Both models are defined in terms of an underlying graph; we study how the structure of the graph affects the long time behavior of the models in all possible cases of graph topology. We are especially interested in the emergence of a consensus among the agents and provide a condition on the graph that is necessary and sufficient for convergence to a consensus in both models. This investigation reveals several contrasts between the models -notably the convergence rates -which are explored through analytical arguments and several numerical experiments.

Exponential convergence towards consensus for non-symmetric linear first-order systems in finite and infinite dimensions

“…Under this strong connectivity assumption, we can now identify the consensus value in terms of an eigenvector v of A * , for which we prove the existence and positivity properties that were only assumed in [40]. We also recover the L 2 -convergence towards consensus obtained in finite dimension in [50], extend it to the infinite-dimensional model, and provide the sharp convergence rate in both cases. We state hereafter our two main results, valid in finite and infinite dimensions, and we start by providing the proper consensus value.…”

“…Many studies of the Hegselmann-Krause model were recently implemented, as in [7,29,31,39,42]. We point out [37,50], where the authors proposed graph-theory related ideas to develop their theory. Second-order models were also studied in [23,25,31,38,45].…”

“…In [31,38], the authors have developed a L ∞ theory, or hybrid theories which are at least partially based on L ∞ tools. Although the latter approach is remarkable, the L 2 framework remains more convenient to investigate stability and robustness properties of the system, or to design asymptotically stabilizing controls, see [37,50].…”

Bounded confidence dynamics and graph control: Enforcing consensus

“…Models of opinion formation often have the feature that agents can only interact if they are connected in an underlying network structure. Therefore, a hallmark of the study of these models is examining how the interplay between the topology of the underlying network and the interaction rules affect the distribution of opinions among the agents [19,24,27]. Of particular interest is how these factors can lead to the emergence of a consensus among the agents.…”

“…One might assume that the attractive nature of the interactions causes the emergence of consensus to be a ubiquitous feature of this class of models, however this is not the case. The manner in which agents are connected in the underlying network has a large effect on the distribution of opinions observed among the agents [19,27]. The interplay between the network structure and the interaction rules can often cause the analysis of these models to be very involved; a popular strategy is to use simplifying assumptions on the network structure such as symmetry of connections or static connections that do not change throughout the evolution of the model [20,24,25,29].…”