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

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.

💡 Research Summary

The paper investigates how modular network structure influences opinion formation when agents follow the Biswas‑Chatterjee‑Sen (BChS) kinetic‑exchange dynamics. Networks are generated with a stochastic block model (SBM) that partitions N agents into c equally sized communities. Within‑community links appear with probability p_in, while between‑community links appear with probability p_out. By fixing a relatively high p_in and varying p_out over several orders of magnitude, the authors create a continuum from strongly segregated modules to almost fully mixed graphs.

The BChS model assigns each agent i a discrete opinion o_i∈{−1,0,+1}. At each elementary step a random edge (i,j) is selected; with probability ½ one endpoint updates its opinion by adding µ o_j, where µ=+1 (attractive interaction) with probability 1−p and µ=−1 (repulsive interaction) with probability p. The “disagreement probability” p therefore controls the balance between conformity and antagonism. After the addition, the opinion is clipped back to the allowed set. Time is measured in single‑edge updates.

Two macroscopic observables are monitored: (i) the global order parameter O=|∑_i o_i|/N, which quantifies overall consensus, and (ii) the intragroup order parameter O_intra = (1/c)∑_g |m_g|, where m_g is the average opinion in community g. Large O and large O_intra indicate global consensus; large O_intra but small O signals a modularly polarized state where each community is internally ordered but communities disagree with each other.

Simulations run for 2×10⁶ transient steps followed by an equally long measurement window; results are averaged over many independent SBM realizations and independent BChS trajectories. The authors explore a three‑dimensional parameter space (p_in, p_out, p) and also vary the number of communities c.

Key findings:

1. Modular Polarization – For low disagreement (p≈0.1) and very weak inter‑community coupling (p_out≈5×10⁻⁴), the system settles into a state with O≈0 but O_intra≈1. Each module becomes almost uniformly +1 or –1, and neighboring modules typically hold opposite majorities. This regime persists over a broad band of p_out values when p is modest.

2. Global Consensus – Increasing p_out (e.g., to 10⁻²) while keeping p low aligns most modules with the same sign, yielding O≈1 and O_intra≈1. The transition from modular polarization to global order occurs sharply as p_out crosses a p‑dependent threshold; the threshold shifts to lower p_out for larger p (more disagreement).

3. Disordered Phase – When the disagreement probability is high (p≈0.6), both O and O_intra remain near zero for all p_out, indicating that repulsive interactions dominate and opinions are finely mixed within each community.

4. Effect of p_in – Scans in the (p_out, p_in) plane show that strong intracommunity connectivity (large p_in) is essential for the modularly polarized phase. With p=0 (no repulsion) the polarized region disappears; any non‑zero p creates a robust zone where O_intra≈1 while O≈0.

5. System‑size and Community‑number Dependence – Increasing the number of modules c sharpens the drop of O from near one to near zero, suggesting a genuine phase transition in the thermodynamic limit (large c). In contrast, O_intra remains high over a much wider range of p, demonstrating that intragroup order is far more resilient than global order.

Overall, the study demonstrates that the mesoscopic architecture of social interactions can qualitatively change collective outcomes. Strong intra‑group cohesion combined with weak inter‑group ties can sustain persistent opinion fragmentation (echo‑chamber‑like states) even when the underlying microscopic dynamics would otherwise lead to consensus on a fully mixed network. These insights have implications for understanding political polarization, the spread of misinformation, and the design of interventions aimed at fostering societal consensus.