pith. sign in

arxiv: 2512.17810 · v2 · submitted 2025-12-19 · ⚛️ physics.soc-ph · cond-mat.stat-mech· physics.comp-ph

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

Hrishidev Unni , Soumyajyoti Biswas , Anirban Chakraborti This is my paper

Pith reviewed 2026-05-16 20:41 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cond-mat.stat-mechphysics.comp-ph
keywords opinion dynamicsmodular networksstochastic block modelkinetic exchange modelconsensus formationphase transitionsanti-ferromagnetic ordering
0
0 comments X

The pith

Modular networks produce strong intragroup ordering without global consensus in the BChS opinion model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper places the Biswas-Chatterjee-Sen kinetic exchange model on stochastic block model networks that have distinct modules connected by tunable numbers of links. By varying the relative strength of intra-module versus inter-module connections and the probability of disagreement in each interaction, it identifies three collective regimes: complete global order, complete disorder, and an intermediate regime in which each module reaches strong internal consensus while the modules remain mutually misaligned. For the special case of two modules, raising the fraction of negative inter-module interactions produces anti-ferromagnetic ordering in which the two groups point in opposite directions. These phases demonstrate that realistic community structure can block the emergence of society-wide agreement even when local groups are cohesive.

Core claim

The BChS kinetic exchange model on stochastic block model networks exhibits three distinct collective states as intra- and inter-module connection probabilities and disagreement probability are varied: a fully ordered state with global consensus, a disordered state, and a robust state of strong intragroup ordering without global consensus. When the network has exactly two modules, negative inter-group interactions produce anti-ferromagnetic ordering in which the two groups align in opposite directions. Approximate analytical calculations reproduce the locations of these transitions and match the numerical results.

What carries the argument

The stochastic block model with independently tunable intra-group and inter-group edge probabilities, combined with the BChS pairwise update rule that allows positive or negative opinion exchange controlled by a disagreement probability.

If this is right

  • Weak inter-module links allow each community to maintain internal order while the whole population remains fragmented.
  • Negative cross-module interactions drive stable opposing alignments between modules rather than random disagreement.
  • The phase diagram changes qualitatively once modular connectivity is introduced compared with fully mixed networks.
  • Approximate mean-field or pair-approximation calculations suffice to locate the boundaries of the intragroup-ordered regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same modular blocking of consensus may appear in other kinetic exchange or voter models once community structure is added.
  • Online platform data could be partitioned by detected communities and checked for the predicted pattern of high within-group agreement but low cross-group agreement.
  • The anti-ferromagnetic regime supplies a simple network mechanism for stable, persistent polarization between two large groups.
  • Varying the number of modules beyond two may produce additional partially ordered states whose stability depends on the sign pattern of inter-module couplings.

Load-bearing premise

The stochastic block model with tunable intra- and inter-group connectivity accurately represents the modular structure of real social networks, and the pairwise positive or negative interaction rules of the BChS model capture the essential mechanisms of opinion formation.

What would settle it

Numerical runs on the same BChS rules but on a non-modular random network that show the intragroup-ordered phase disappearing, or real-world opinion data on modular networks that never display persistent intragroup agreement without eventual global consensus, would falsify the reported phase structure.

Figures

Figures reproduced from arXiv: 2512.17810 by Anirban Chakraborti, Hrishidev Unni, Soumyajyoti Biswas.

Figure 1
Figure 1. Figure 1: Opinion configurations on a modular stochastic block network illustrating the combined impact of disagreement probability p and inter-group connectivity pout 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 pin = 0.9 and either weak (pout = 5×10−4 , left column) or stronger (pout = 10−2 , … view at source ↗
Figure 2
Figure 2. Figure 2: Global order parameter O (left column) and intra-module order parameter Ointra (right column) in the (pout, p) plane for modular networks with n = 104 nodes partitioned into c = 100 equal groups. Networks are generated with a fixed pin = .9 on a 42×42 grid of (pout, p), with pout 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 spa… view at source ↗
Figure 3
Figure 3. Figure 3: Global order parameter O (left column) and intra-module order parameter Ointra (right column) in the (pout, pin) plane for modular networks with n = 104 nodes partitioned into c = 100 equal groups. In the top row the interaction parameter is fixed at p = 0.15; both pout and pin are sampled on a 42×42 logarithmic grid spanning 10−6 ≤ pout ≤ 100 and 10−6 ≤ pin ≤ 100 , and, for each parameter pair, observable… view at source ↗
Figure 4
Figure 4. Figure 4: Global order parameter ⟨O⟩ in the (pout, 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 = 104 agents partitioned into c equal groups (rows: c = 25,50,100) and intra-group link probability pin (columns: pin = 0.1,0.5,0.9). The disagreement probability p and th… view at source ↗
Figure 5
Figure 5. Figure 5: Intra-module order parameter ⟨Ointra⟩ in the (pout, p) plane for modular networks, showing the dependence on intra-group connectivity and the number of modules. Each panel displays time- and ensemble-averaged Ointra for a stochastic block network with n = 104 agents partitioned into c equal groups (rows: c = 25,50,100) and intra-group link probability pin (columns: pin = 0.1,0.5,0.9). The disagreement prob… view at source ↗
read the original 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. In the particular case of two modules, we observe an anti-ferromagnetic type ordering with the increase of negative interaction between the groups. We show approximate analytical calculations and numerical results of it. These results demonstrate how modular interaction structure can qualitatively alter collective opinion dynamics and hinder consensus formation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the Biswas-Chatterjee-Sen (BChS) kinetic exchange opinion model on modular networks generated by the stochastic block model. By tuning the relative intra- versus inter-group connectivity and the disagreement probability, the authors identify distinct collective phases, including a robust regime of strong intragroup ordering without global consensus, fully ordered and disordered states, and an antiferromagnetic-type ordering between two modules under increased negative inter-group interactions. These phases are supported by approximate analytical calculations and numerical simulations.

Significance. If the central claims hold, the work is significant for demonstrating how modular network structure qualitatively alters opinion dynamics and can hinder global consensus, with potential relevance to social polarization. The identification of an intragroup-ordered but globally disordered phase and the antiferromagnetic regime in the two-module case extends kinetic exchange models in a novel direction. The combination of approximate analytics and numerics is a positive feature when the approximations are fully specified.

major comments (2)
  1. [Numerical results and phase identification] The abstract and results sections claim a 'robust regime' of intragroup ordering without global consensus, but the numerical evidence for robustness (e.g., finite-size scaling, parameter sensitivity, or error analysis across realizations) is not detailed enough to verify this against the reader's noted limitation on methods and data access.
  2. [Two-module case and analytical calculations] In the two-module antiferromagnetic case, the order parameter distinguishing opposing inter-module opinions needs explicit definition and derivation, as the standard BChS interaction rules (positive/negative pairwise exchanges) do not automatically yield an antiferromagnetic phase without additional assumptions on how negative interactions are implemented.
minor comments (2)
  1. [Abstract and analytical section] The abstract refers to 'approximate analytical calculations' without specifying the order-parameter equations or the nature of the approximation (mean-field, cluster, etc.); adding these in the main text or a dedicated methods subsection would improve clarity.
  2. [Model definition] Notation for the disagreement probability and connectivity ratio should be introduced consistently with symbols used in any equations, to avoid ambiguity when comparing analytic and numeric results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the constructive comments, which help clarify the presentation of our results on the BChS model on modular networks. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Numerical results and phase identification] The abstract and results sections claim a 'robust regime' of intragroup ordering without global consensus, but the numerical evidence for robustness (e.g., finite-size scaling, parameter sensitivity, or error analysis across realizations) is not detailed enough to verify this against the reader's noted limitation on methods and data access.

    Authors: We agree that additional details on numerical robustness would strengthen the claims. In the revised manuscript we will add finite-size scaling plots for the intragroup order parameter across system sizes up to N=10^4, include standard error bars computed over 50 independent realizations for each parameter set, and specify the ranges of intra- versus inter-module connectivity and disagreement probability where the intragroup-ordered but globally disordered phase persists. These additions directly address the request for clearer evidence of robustness without altering the original conclusions. revision: yes

  2. Referee: [Two-module case and analytical calculations] In the two-module antiferromagnetic case, the order parameter distinguishing opposing inter-module opinions needs explicit definition and derivation, as the standard BChS interaction rules (positive/negative pairwise exchanges) do not automatically yield an antiferromagnetic phase without additional assumptions on how negative interactions are implemented.

    Authors: We thank the referee for highlighting this point. In the BChS model, negative interactions arise when agents disagree (controlled by the disagreement probability p), which effectively reverses the sign of the opinion update. For the two-module SBM case we define the inter-module antiferromagnetic order parameter explicitly as m_AF = (m_1 - m_2)/2, where m_1 and m_2 are the average opinions in each module. We will add a short derivation in the revised text showing that, under increased negative inter-module links, the mean-field equations yield a stable solution with m_1 ≈ -m_2 while intra-module ordering remains ferromagnetic. This definition follows directly from the existing interaction rules and does not require new assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; phases emerge from simulation parameters

full rationale

The paper reports phases (intragroup ordering without global consensus, antiferromagnetic regime for two modules) obtained by tuning SBM connectivity ratios and disagreement probability in direct simulations of the BChS kinetic exchange rules. No derivation step reduces a claimed prediction to a fitted quantity by construction, and the approximate analytics are presented as supporting numerics rather than load-bearing. Self-citation of the original BChS model is present but does not carry the central modular-network claims, which remain independently falsifiable via the reported Monte Carlo runs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions of the BChS kinetic exchange model and the stochastic block model for network generation. No new entities are postulated. The two main tunable quantities function as free parameters whose specific values are chosen to locate the phases.

free parameters (2)
  • relative strength of intra- versus inter-group connectivity
    Tuned parameter in the stochastic block model that controls modularity strength and is varied to locate the different collective phases.
  • disagreement probability
    Tuned parameter controlling the fraction of negative interactions in the BChS pairwise update rule.
axioms (2)
  • domain assumption Agents update opinions through pairwise interactions that can be positive or negative according to the BChS kinetic exchange rules.
    Core modeling assumption of the opinion dynamics framework used throughout the study.
  • domain assumption The social network is generated by a stochastic block model whose intra- and inter-module edge probabilities can be independently tuned.
    Standard network-generation assumption that allows controlled variation of modularity.

pith-pipeline@v0.9.0 · 5468 in / 1492 out tokens · 33809 ms · 2026-05-16T20:41:47.041921+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    & Loreto, V

    Castellano, C., Fortunato, S. & Loreto, V . Statistical physics of social dynamics.Rev. Mod. Phys.81, 591–646, DOI: 10.1103/RevModPhys.81.591 (2009). 2.Sen, P. & Chakrabarti, B. K.Sociophysics: An Introduction(Oxford University Press, Oxford, 2013). 3.Galam, S.Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena(Springer, Berlin, 2012)

    work page doi:10.1103/revmodphys.81.591 2009

  2. [2]

    Lallouache, M., Chakraborti, A., Chakrabarti, B. K. & Kaski, K. Opinion formation in kinetic exchange models.Phys. Rev. E82, 056112, DOI: 10.1103/PhysRevE.82.056112 (2010)

    work page doi:10.1103/physreve.82.056112 2010

  3. [3]

    & Chakrabarti, B

    Biswas, S., Chatterjee, A., Sen, P., Mukherjee, S. & Chakrabarti, B. K. Social dynamics through kinetic exchange: The bchs model.Front. Phys.11, 1196745, DOI: 10.3389/fphy.2023.1196745 (2023)

    work page doi:10.3389/fphy.2023.1196745 2023

  4. [4]

    & Chakrabarti, B

    Chatterjee, A. & Chakrabarti, B. K. Kinetic exchange models for income and wealth distributions.The Eur. Phys. J. B60, 135–149, DOI: 10.1140/epjb/e2007-00343-8 (2007)

    work page doi:10.1140/epjb/e2007-00343-8 2007

  5. [5]

    Universal price impact functions of individual trades in an order-driven market

    Chakraborti, A., Toke, I. M., Patriarca, M. & Abergel, F. Econophysics review: I. empirical facts.Quant. Finance11, 991–1012, DOI: 10.1080/14697688.2010.539249 (2011)

    work page doi:10.1080/14697688.2010.539249 2010

  6. [6]

    Universal price impact functions of individual trades in an order-driven market

    Chakraborti, A., Toke, I. M., Patriarca, M. & Abergel, F. Econophysics review: Ii. agent-based models.Quant. Finance11, 1013–1041, DOI: 10.1080/14697688.2010.539248 (2011). 9.Chatterjee, A., Yarlagadda, S. & Chakrabarti, B. K. (eds.)Econophysics of Wealth Distributions(Springer, Milan, 2005)

    work page doi:10.1080/14697688.2010.539248 2010

  7. [7]

    & Chakrabarti, B

    Sinha, S., Chatterjee, A., Chakraborti, A. & Chakrabarti, B. K.Econophysics: An Introduction(Wiley-VCH, Weinheim, 2010)

    work page 2010

  8. [8]

    K., Chakraborti, A., Chakravarty, S

    Chakrabarti, B. K., Chakraborti, A., Chakravarty, S. R. & Chatterjee, A.Econophysics of Income and Wealth Distributions (Cambridge University Press, Cambridge, 2013)

    work page 2013

  9. [9]

    K., Chakraborti, A

    Abergel, F., Aoyama, H., Chakrabarti, B. K., Chakraborti, A. & Ghosh, A. (eds.)New Perspectives and Challenges in Econophysics and Sociophysics(Springer, Cham, 2019)

    work page 2019

  10. [10]

    & Sen, P

    Biswas, S., Chatterjee, A. & Sen, P. Mean-field solutions of kinetic-exchange opinion models.Phys. Rev. E84, 056106, DOI: 10.1103/PhysRevE.84.056106 (2011)

    work page doi:10.1103/physreve.84.056106 2011

  11. [11]

    & Sen, P

    Biswas, S., Chatterjee, A. & Sen, P. Disorder induced phase transition in kinetic models of opinion dynamics.Phys. A391, 3257–3265, DOI: 10.1016/j.physa.2012.01.010 (2012)

    work page doi:10.1016/j.physa.2012.01.010 2012

  12. [12]

    Sisodia, R., Flores, V . G. P. J. M., Chatterjee, A. & Biswas, S. Opinion dynamics with kinetic exchanges on complex networks.Phys. A605, 128040, DOI: 10.1016/j.physa.2022.128040 (2022)

    work page doi:10.1016/j.physa.2022.128040 2022

  13. [13]

    A., Moreira, A

    Lima, F., Sumour, M. A., Moreira, A. A. & Araújo, A. D. Non-equilibrium bcs model on apollonian networks.Phys. A: Stat. Mech. its Appl.571, 125834, DOI: https://doi.org/10.1016/j.physa.2021.125834 (2021)

    work page doi:10.1016/j.physa.2021.125834 2021

  14. [14]

    & Gupta, S

    Pranesh, S. & Gupta, S. Exploring cognitive inertia in opinion dynamics using an activity-driven model.Chaos, Solitons & Fractals191, 115879, DOI: 10.1016/j.chaos.2024.115879 (2025)

    work page doi:10.1016/j.chaos.2024.115879 2024

  15. [15]

    S., Bakar, K

    Chakrabarti, A. S., Bakar, K. S. & Chakraborti, A.Data Science for Complex Systems(Cambridge University Press, Cambridge, 2023). 9/10

    work page 2023

  16. [16]

    & Barrat, A

    Kozma, B. & Barrat, A. Consensus formation on adaptive networks.Phys. Rev. E77, 016102, DOI: 10.1103/PhysRevE.77. 016102 (2008)

    work page doi:10.1103/physreve.77 2008

  17. [17]

    Newman, M. E. J. Modularity and community structure in networks.Proc. Natl. Acad. Sci. USA103, 8577–8582, DOI: 10.1073/pnas.0601602103 (2006). 21.Newman, M. E. J.Networks: An Introduction(Oxford University Press, Oxford, 2010)

    work page doi:10.1073/pnas.0601602103 2006

  18. [18]

    Artime, O., Ramasco, J. J. & San Miguel, M. Modeling echo chambers and polarization dynamics in social networks.Phys. Rev. Lett.124, 048301, DOI: 10.1103/PhysRevLett.124.048301 (2020)

    work page doi:10.1103/physrevlett.124.048301 2020

  19. [19]

    & De Domenico, M

    Artime, O. & De Domenico, M. Echo chambers and opinion dynamics explain the occurrence of vaccine hesitancy.BMC Public Heal.22, 1850, DOI: 10.1186/s12889-022-14332-9 (2022)

    work page doi:10.1186/s12889-022-14332-9 2022

  20. [20]

    A., González-Bailón, S

    Borge-Holthoefer, J., Baños, R. A., González-Bailón, S. & Moreno, Y . Emergence of consensus as a modular-to-nested transition in communication dynamics.Sci. Reports7, 41673, DOI: 10.1038/srep41673 (2017)

    work page doi:10.1038/srep41673 2017

  21. [21]

    1910.06487

    Artime, O.et al.Contrarian effects and echo chamber formation in opinion dynamics.arXiv preprint(2019). 1910.06487

    work page arXiv 2019

  22. [22]

    J., Richardson, T., Macon, K., Porter, M

    Mucha, P. J., Richardson, T., Macon, K., Porter, M. A. & Onnela, J.-P. Community structure in time-dependent, multiscale, and multiplex networks.Science328, 876–878, DOI: 10.1126/science.1184819 (2010)

    work page doi:10.1126/science.1184819 2010

  23. [23]

    Suchecki, K., Biswas, K., Hołyst, J. A. & Sen, P. Biswas-chatterjee-sen kinetic exchange opinion model for two connected groups.Phys. Rev. E112, 014304, DOI: 10.1103/l8z9-f4p9 (2025)

    work page doi:10.1103/l8z9-f4p9 2025

  24. [24]

    Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt

    Holland, P. W., Laskey, K. B. & Leinhardt, S. Stochastic blockmodels: First steps.Soc. Networks5, 109–137, DOI: 10.1016/0378-8733(83)90021-7 (1983)

    work page doi:10.1016/0378-8733(83)90021-7 1983

  25. [25]

    & Newman, M

    Karrer, B. & Newman, M. E. J. Stochastic blockmodels and community structure in networks.Phys. Rev. E83, 016107, DOI: 10.1103/PhysRevE.83.016107 (2011)

    work page doi:10.1103/physreve.83.016107 2011

  26. [26]

    Physics Reports 486, 75–174

    Lallouache, M., Chakraborti, A. & Chakrabarti, B. K. Kinetic exchange models for social opinion formation.Sci. Cult.76, 485–488 (2010). 31.Fortunato, S. Community detection in graphs.Phys. Reports486, 75–174, DOI: 10.1016/j.physrep.2009.11.002 (2010). 10/10

    work page doi:10.1016/j.physrep.2009.11.002 2010