Generalized mean-field approximation for the Deffuant opinion dynamics model on networks

When the interactions of agents on a network are assumed to follow the Deffuant opinion dynamics model, the outcomes are known to depend on the structure of the underlying network. This behavior cannot be captured by existing mean-field approximations for the Deffuant model. In this paper, a generalized mean-field approximation is derived that accounts for the effects of network topology on Deffuant dynamics through the degree distribution or community structure of the network. The accuracy of the approximation is examined by comparison with large-scale Monte Carlo simulations on both synthetic and real-world networks.

Figure 1

Time evolution of the opinion distribution, averaged over simulations, on configuration model networks ($N={10}^4$) where $90\%$ of nodes have degree $k_{\text{maj}}$ and $10\%$ of nodes have degree 100. Snapshots are shown at the times labeled on the vertical axis (earliest time on top). Two cases are considered: $k_{\text{maj}}=25$ and $k_{\text{maj}}=5$; the latter case has lower connectivity. For comparison, the results on a complete graph (corresponding to the original MF approximation of Ref. [9]) are also shown. One central cluster forms for $k_{\text{maj}}=25$ (a), while for $k_{\text{maj}}=5$ smaller clusters also persist at the boundaries of the opinion space (b).

Figure 2

The distribution of opinions within each degree class on a configuration model network with $90\%$ degree-5 nodes and $10\%$ degree-100 nodes.

Figure 3

The temporal network for one simulation on a configuration model network with $90\%$ degree 5 nodes and $10\%$ degree 100 nodes ($N=1000$) once the system has converged. Links exist between pairs of nodes who can interact, i.e., are connected in the network and have opinion difference less than $\epsilon$. $80\%$ of nodes form one cluster with opinions close to 0.5 (gray fill). The rest, in this case all degree 5 nodes (dark blue boundary), form small clusters closer to the boundaries of the opinion space (black is opinion 0, white is opinion 1).

Figure 4

The distribution of opinions on a configuration model network in which $10\%$ of nodes have degree 100 and $90\%$ of nodes have degree 5.

Figure 5

The initial distribution of opinions within each community and over the whole network for the simulations on SBM networks.

Figure 6

Opinion distribution, averaged over simulations, on a SBM network with $p_w=0.1$ and $p_b=0.01$ and on a complete graph, together with the distributions from the two MF approximations.

Figure 7

Opinion distribution on a SBM network with $p_w=0.1$ and $p_b=0.1$.

Figure 8

Opinion distributions on SBM networks for $p_b=0.02,0.04,$ and 0.06 with $p_w=0.1$, all at $t=25$.

Figure 9

Opinion distributions on SBM networks with $p_w=0.1$ and $p_b=0.01$ for different network sizes and from the class-based MF approximation.

Figure 10

Opinion distributions on annealed SBM networks with $p_w=0.1$ and $p_b=0.01$ for different network sizes and from the class-based MF approximation.

Figure 11

Distribution of opinions on Zachary's Karate Club network.

Figure 12

Distribution of opinions on the politics books network.

Figure 13

Covoting networks in the 91st and 108th U.S. House of Representatives. Edges exist between members who voted the same way on at least $50\%$ of bills.

Figure 14

Distribution of opinions on the 108th House of Representatives covoting network.

Figure 15

Distribution of opinions on the 91st House of Representatives covoting network.

Figure 16

Root-mean-square error between the density from the simulations and the density from the MF approximation.

Figure 17

The distribution of opinions on a configuration model network with a proportion $q_{\text{maj}}$ of nodes having degree $k_{\text{maj}}$ and a proportion $1-q_{\text{maj}}$ of nodes with degree 100 ($\epsilon =0.3$).

Figure 18

The proportion of the population that remains in the two clusters near the boundaries of the opinion space on a configuration model network with a proportion $q_{\text{maj}}$ of nodes having degree $k_{\text{maj}}$ and a proportion $1-q_{\text{maj}}$ of nodes with degree 100 ($\epsilon =0.3$).

Figure 19

The distribution of opinions on a configuration model network with a proportion $q_{\text{maj}}$ of nodes having degree $k_{\text{maj}}$ and a proportion $1-q_{\text{maj}}$ of nodes with degree 100 ($\epsilon =0.25$).