A framework for continuum modeling of opinion dynamics on a network based on probability of connections

Abstract

We propose a modeling framework to develop a continuum description of opinion dynamics on networks as an alternative to other models, like the ones based on graphons. In a nutshell, the continuum model that we propose aims at approximating the distribution of opinions as well as the probability that two given opinions are connected. To illustrate our framework, we focus on a simple model of consensus dynamics on a network and derive a continuum description using techniques inspired by mean-field limits. We also discuss the limitations of this approach and suggest extensions to account for dynamic networks with evolving connections, stochastic effects, and directional interactions.

Figures (7)

• Figure 1: Diagram of the modeling framework. The discrete system (left) is presented from two perspectives: one describes the evolution of individual opinions $(\omega_i)_{i=1}^N$ over time, where interactions occur on a network represented by the adjacency matrix $(A_{i,j})_{i,j=1}^N$ (see Sec. \ref{['sec:iba']}). The other perspective considers the distributions of opinions, $f^N$, and edges, $g^N$, as described in Sec. \ref{['sec:network description']}. Both perspectives yield equivalent sets of ODEs for short times, provided the "distinguishability" assumption \ref{['as:initial_time']} holds (corresponding to Lemma \ref{['lem:ODE_approx']}). From $f^N$ and $g^N$, the continuum model is formally derived by taking the limit as $N \to \infty$ (Sec. \ref{['sec:limit']}).
• Figure 2: Illustration of the example in Remark \ref{['rem:example irrecoverability f from g']}. On the right, braces correspond to an edge in the network, and a $\bullet$ represents a Dirac delta. These come in pairs, as the network is not directed.
• Figure 3: Comparison between the discrete dynamics (histograms), the continuum model without group labeling (dash line) and the continuum model with group labeling (solid lines). The histogram represents the evolution of the discrete solution, while the dashed line is the solution to the continuous model without group labeling. Here, $N=1000$. See Remark \ref{['rem:validity lemma']} for details.
• Figure 4: Typical LFR graphs for selected values of the mixing parameter $\mu$. We did not include the case $\mu = 1$ as it is visually similar to that of $\mu = 0.5$.
• Figure 5: Initial opinion distribution, assuming three communities of identical size. Each color corresponds to a community.

Theorems & Definitions (26)

• Lemma 2.3: Conserved quantity
• proof
• Proposition 2.4: Convergence to consensus
• Remark 2.5
• proof : Proof of Prop. \ref{['pr:long_time_behaviour']}
• Proposition 2.6
• Lemma 3.2
• Remark 3.3: Example
• Lemma 3.4
• proof : Proof of Lemma \ref{['lem:ODE_approx']}