The soft consensus model: dynamics, complexity, and modularity

We consider the network dynamics of the soft consensus model and we investigate the dynamical behavior of two important measures of network structure, complexity and modularity. The dynamical mechanism of the soft consensus model is driven by the minimization of a cost function combining a collective measure of dissensus with an individual mechanism of opinion changing aversion. The dissensus measure plays a key role in the model and induces a network of pairwise interactions between the individual opinions. The pairwise interaction coefficients take values in the unit interval and define the adjacency matrix of an undirected weighted graph. The measures of complexity and modularity express in two different ways how far the network structure is from some appropriate, transitive partitioning. We illustrate their dynamical behavior in the context of the soft consensus model by means of a number of computer simulations. 1 The soft consensus model Let {i = 1, . . . ,n} be an index set for a group of agents. The individual opinions of these agents are denoted by ri ∈ [0,1] with i = 1, . . . ,n. The soft consensus network model of group decision making [11,12,13,6,5,3,4] is based on a graph representation in which each node i encodes the individual opinion value ri, and each edge (i, j) encodes the interaction coefficient (strength of interaction) between agents i and j, expressed in the (0,1) interval: larger interaction values correspond to stronger interactions. Consider the decreasing sigmoid function σ(x) = 1 1 + eβ(x−α) ∈ (0,1) (1) with two free parameters 0 < α < 1 and β > 0. The former is a threshold parameter, σ(α) = 2 , and the latter controls the polarization of the sigmoidal response around the threshold value, σ (α) = − 4 . The interaction coefficients vi j (α,β) ∈ (0,1) defining the adjacency matrix of an undirected weighted graph are computed by applying the non-linear σ filter to the square distance between the two opinion values ri and r j , vi j = v ji = σ (