Biswas-Chatterjee-Sen (BChS) kinetic exchange opinion model on modular networks

TL;DR

The paper addresses how modular interaction topology affects kinetic-exchange opinion dynamics by applying the Biswas-Chatterjee-Sen (BChS) model on stochastic-block networks with tunable intra- and inter-group connectivity and a disagreement parameter $p$. It combines SBM networks with the BChS update rule and monitors global order $O$ and intragroup order $O_{ ext{intra}}$, revealing three steady-state regimes: disordered, globally ordered, and modularly ordered where each group is internally ordered but the whole system is not. A broad modular-polarized phase emerges at low inter-group connectivity and finite $p$, which dissociates from global consensus, and increasing inter-group coupling or reducing $p$ restores global order; these features persist across system sizes and sharpen with more modules. The findings show that modular structure can sustain polarized states and hinder consensus, offering insights into echo-chamber dynamics and informing strategies to promote cross-group alignment.

Abstract

We study opinion formation in a society where agents interact on a modular network generated using a stochastic block model (SBM). Opinion dynamics is modeled through the Biswas-Chatterjee-Sen (BChS) kinetic exchange model, in which agents undergo pairwise interactions that could be positive or negative. By tuning the relative strength of intra- and inter-group connectivity inherent to the SBM, as well as the disagreement probability, we identify distinct collective phases. In particular, we observe a robust regime with strong intragroup ordering but no global consensus, in addition to fully ordered and disordered states. These results demonstrate how modular interaction structure can qualitatively alter collective opinion dynamics and hinder consensus formation.

Figures (5)

• Figure 1: Opinion configurations on a modular stochastic block network illustrating the combined impact of disagreement probability $p$ and inter-group connectivity $p_{\text{out}}$ on global and intragroup ordering. Each panel shows a network with $N=400$ agents arranged in $c=10$ equal-sized modules, generated with strong intragroup connectivity $p_{\text{in}} = 0.9$ and either weak ($p_{\text{out}} = 5\times 10^{-4}$, left column) or stronger ($p_{\text{out}} = 10^{-2}$, right column) inter-module coupling. Node colors encode individual opinions after $10^7$ asynchronous BCS update steps (with $p$ the probability of repulsive interaction): blue for $-1$, light gray for $0$, and red for $+1$. The top row corresponds to high disagreement ($p = 0.6$), where opinions are disordered within and across modules for both values of $p_{\text{out}}$, while the bottom row shows low disagreement ($p = 0.1$), where small $p_{\text{out}}$ produces a modular-polarized state with internally ordered but mutually opposed modules, and larger $p_{\text{out}}$ drives the system toward a more globally aligned configuration.
• Figure 2: Global order parameter $O$ (left column) and intra-module order parameter $O_{\mathrm{intra}}$ (right column) in the $(p_{\mathrm{out}},p)$ plane for modular networks with $n = 10^4$ nodes partitioned into $c = 100$ equal groups. Networks are generated with a fixed $p_{\mathrm{in}}=.9$ on a $42 \times 42$ grid of $(p_{\mathrm{out}},p)$, with $p_{\mathrm{out}}$ logarithmically spaced between $10^{-7}$ and $10^{-2}$ in the bottom row and between $10^{-3}$ and $10^{-1}$ in the top row, while $p$ is linearly spaced between $0$ and $0.5$ in all panels. For each parameter pair in the bottom row, observables are averaged over $n_I = 6$ independent network realizations and over a stationary time window following transients of $T_s = 2 \times 10^{6}$ steps and a measurement period $T_m = 2 \times 10^{6}$ steps; the top row shows the same quantities for a narrower $p_{\mathrm{out}}$ range with $n_I = 50$ realizations per point, providing a higher-precision estimate of the phase-diagram structure. Color encodes the averaged values of the corresponding order parameter (see color bars), illustrating how increasing inter-group connectivity $p_{\mathrm{out}}$ and interaction parameter $p$ drive the system from disordered to globally and modularly ordered regimes.
• Figure 3: Global order parameter $O$ (left column) and intra-module order parameter $O_{\mathrm{intra}}$ (right column) in the $(p_{\mathrm{out}},p_{\mathrm{in}})$ plane for modular networks with $n = 10^{4}$ nodes partitioned into $c = 100$ equal groups. In the top row the interaction parameter is fixed at $p = 0.15$; both $p_{\mathrm{out}}$ and $p_{\mathrm{in}}$ are sampled on a $42 \times 42$ logarithmic grid spanning $10^{-6} \le p_{\mathrm{out}} \le 10^{0}$ and $10^{-6} \le p_{\mathrm{in}} \le 10^{0}$, and, for each parameter pair, observables are averaged over $n_I = 30$ network realizations after a transient of $T_s = 2\times 10^{6}$ steps and a measurement window of $T_m = 2\times 10^{6}$ steps. In the bottom row the same procedure is repeated for $p = 0$ on a $20 \times 20$ logarithmic grid with $10^{-5} \le p_{\mathrm{out}} \le 10^{0}$ and $10^{-5} \le p_{\mathrm{in}} \le 10^{0}$, again using $n_I = 30$ realizations per parameter pair. Color encodes the time-averaged values of the corresponding order parameter (see color bars), showing how the presence or absence of the interaction $p$ qualitatively changes the phase diagram in terms of intra-group connectivity $p_{\mathrm{in}}$ and inter-group connectivity $p_{\mathrm{out}}$.
• Figure 4: Global order parameter $\langle O \rangle$ in the $(p_{\mathrm{out}},p)$ plane for modular networks, showing the dependence on intra-group connectivity and the number of modules. Each panel displays time- and ensemble-averaged $O$ for a stochastic block network with $n = 10^{4}$ agents partitioned into $c$ equal groups (rows: $c = 25, 50, 100$) and intra-group link probability $p_{\mathrm{in}}$ (columns: $p_{\mathrm{in}} = 0.1, 0.5, 0.9$). The disagreement probability $p$ and the inter-group link probability $p_{\mathrm{out}}$ are sampled on a $30 \times 30$ grid, with $p$ ranging from $0$ to $0.5$ and $p_{\mathrm{out}}$ logarithmically spaced between $10^{-7}$ and $10^{0}$. For each parameter quadruple $(p_{\mathrm{in}},p_{\mathrm{out}},p,c)$, the BChS dynamics is run for a transient of $T_{\mathrm{ss}} = 2\times10^{6}$ steps followed by a measurement window of $T_{\mathrm{m}} = 2\times10^{6}$ steps, and results are averaged over $n_I = 20$ independent realizations. Color encodes the resulting mean global order parameter $\langle O \rangle$ (see color bar), highlighting how increasing inter-group connectivity and disagreement probability jointly drive the system from disordered to globally ordered regimes, and how the location and sharpness of this transition depend on $p_{\mathrm{in}}$ and $c$.
• Figure 5: Intra-module order parameter $\langle O_{\mathrm{intra}} \rangle$ in the $(p_{\mathrm{out}},p)$ plane for modular networks, showing the dependence on intra-group connectivity and the number of modules. Each panel displays time- and ensemble-averaged $O_{\mathrm{intra}}$ for a stochastic block network with $n = 10^{4}$ agents partitioned into $c$ equal groups (rows: $c = 25, 50, 100$) and intra-group link probability $p_{\mathrm{in}}$ (columns: $p_{\mathrm{in}} = 0.1, 0.5, 0.9$). The disagreement probability $p$ and the inter-group link probability $p_{\mathrm{out}}$ are sampled on a $30 \times 30$ grid, with $p$ ranging from $0$ to $0.5$ and $p_{\mathrm{out}}$ logarithmically spaced between $10^{-7}$ and $10^{0}$. For each parameter quadruple $(p_{\mathrm{in}},p_{\mathrm{out}},p,c)$, the BChS dynamics is run for a transient of $T_{\mathrm{ss}} = 2\times10^{6}$ steps followed by a measurement window of $T_{\mathrm{m}} = 2\times10^{6}$ steps, and results are averaged over $n_I = 20$ independent realizations. Color encodes the resulting mean intra-module order parameter $\langle O_{\mathrm{intra}} \rangle$ (see color bar), highlighting how increasing inter-group connectivity and disagreement probability jointly affect the persistence of strong ordering within modules for different $p_{\mathrm{in}}$ and $c$.