Consensus and Disagreement of Heterogeneous Belief Systems in Influence Networks

TL;DR

This article attributes for the first time, a strong diversity of limiting opinions to heterogeneity of belief systems in influence networks, in addition to the more typical explanation that strong diversity arises from individual stubbornness.

Abstract

Recently, an opinion dynamics model has been proposed to describe a network of individuals discussing a set of logically interdependent topics. For each individual, the set of topics and the logical interdependencies between the topics (captured by a logic matrix) form a belief system. We investigate the role the logic matrix and its structure plays in determining the final opinions, including existence of the limiting opinions, of a strongly connected network of individuals. We provide a set of results that, given a set of individuals' belief systems, allow a systematic determination of which topics will reach a consensus, and which topics will disagreement in arise. For irreducible logic matrices, each topic reaches a consensus. For reducible logic matrices, which indicates a cascade interdependence relationship, conditions are given on whether a topic will reach a consensus or not. It turns out that heterogeneity among the individuals' logic matrices, including especially differences in the signs of the off-diagonal entries, can be a key determining factor. This paper thus attributes, for the first time, a strong diversity of limiting opinions to heterogeneity of belief systems in influence networks, in addition to the more typical explanation that strong diversity arises from individual stubbornness.

Figures (4)

• Figure 1: An illustrative network with 2 individuals discussing 3 topics, with only selected edges drawn for clarity. Each node represents the opinion of an individual for a topic, with red and blue nodes associated with individuals 1 and 2, respectively. The black edges represent interpersonal influence via the weight $w_{ij}$, while the coloured edges represent logical interdependencies between topics. In $\mathcal{G}[\boldsymbol{B}]$, nodes are grouped and ordered by individual in node subset $\tilde{\mathcal{V}}_q$ (as illustrated by the dotted green ellipse groupings) leading to Eq. (\ref{['eq:x_network_system']}). In $\mathcal{G}[\boldsymbol{A}]$, the nodes are grouped and ordered by topic in node subset $\mathcal{V}_p$ (as illustrated by the dotted green ellipses) leading to Eq. (\ref{['eq:y_network_system']}).
• Figure 2: An illustrative example of $\mathcal{G}[\boldsymbol{C}_i]$, with each node representing a topic, and edges representing logical interdependencies between topics (self-loops are hidden for clarity). One can divide the nodes into strongly connected components (each dotted coloured circle denotes a strongly connected component). The results of this paper allow one to progressively analyse each component to establish which topics will have opinions reaching a consensus and which topics will have opinions reaching a persistent disagreement.
• Figure 3: Temporal evolution of opinions for $5$ topics coupled with $\boldsymbol{C}_i$ given in Eq. (\ref{['eq:sim_C_01']}). The solid and dotted lines correspond to individuals with $\boldsymbol{C}_i = \hat{\boldsymbol{C}}$ and $\boldsymbol{C}_i = \bar{\boldsymbol{C}}$, respectively.
• Figure 4: Temporal evolution of opinions for $5$ topics coupled with $\widehat{\boldsymbol{C}}$ replaced by $\widetilde{\boldsymbol{C}}$ given in Eq. (\ref{['eq:sim_C_02']}). The solid and dotted lines correspond to individuals with $\boldsymbol{C}_i = \widetilde{\boldsymbol{C}}$ and $\boldsymbol{C}_i = \bar{\boldsymbol{C}}$, respectively.

Theorems & Definitions (21)

• Lemma 1: bullo2009distributed
• Theorem 1: parsegov2017_multiissue
• Definition 1: Competing Logical Interdependence
• Remark 1
• Remark 2
• Theorem 2
• Theorem 3
• Corollary 1
• Theorem 4
• Corollary 2