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Novel Multidimensional Models of Opinion Dynamics in Social Networks

Sergey E. Parsegov, Anton V. Proskurnikov, Roberto Tempo, Noah E. Friedkin

arXiv:1505.04920v5eess.SYcs.SImath.OC

TL;DR

Social opinion models must represent disagreement and clustering while remaining mathematically analyzable, yet existing multidimensional FJ extensions treat topics independently. This paper introduces an interdependent-topic extension with a coupling matrix, establishes stability and convergence properties, and shows that asynchronous gossip can reach the same final opinions on average.

  • Problem

    Existing multidimensional FJ extensions assume independent topic dimensions, while a model of interpersonal influences generating interdependent belief systems remains unavailable.

  • Method

    The paper modifies FJ dynamics with a constant MiDS coupling matrix that mixes topic-specific opinions after social averaging and prejudice incorporation.

  • Results

    The paper establishes necessary and sufficient stability and convergence conditions and shows that asynchronous gossip reaches the same final opinions on average as synchronous dynamics.

  • Takeaways & Limitations

    Interdependent multi-issue opinions can be modeled within an FJ-based framework while retaining convergence analysis and an equivalent-on-average decentralized protocol.

Abstract

from arXiv · show

Unlike many complex networks studied in the literature, social networks rarely exhibit unanimous behavior, or consensus. This requires a development of mathematical models that are sufficiently simple to be examined and capture, at the same time, the complex behavior of real social groups, where opinions and actions related to them may form clusters of different size. One such model, proposed by Friedkin and Johnsen, extends the idea of conventional consensus algorithm (also referred to as the iterative opinion pooling) to take into account the actors' prejudices, caused by some exogenous factors and leading to disagreement in the final opinions. In this paper, we offer a novel multidimensional extension, describing the evolution of the agents' opinions on several topics. Unlike the existing models, these topics are interdependent, and hence the opinions being formed on these topics are also mutually dependent. We rigorous examine stability properties of the proposed model, in particular, convergence of the agents' opinions. Although our model assumes synchronous communication among the agents, we show that the same final opinions may be reached "on average" via asynchronous gossip-based protocols.

I. INTRODUCTION

Social opinion dynamics require models that preserve mathematically rigorous analysis while representing disagreement, clustering, heterogeneous influence, and multiple interdependent topics. The paper extends the Friedkin–Johnsen framework toward this setting and establishes a foundation for analyzing its dynamics.

  • Motivation: Social networks commonly produce disagreement and irregular opinion clusters rather than unanimous consensus.This motivates models that remain mathematically tractable while capturing complex social behavior.
  • Related work: Homophily and bounded-confidence models explain clustering, but predicting cluster structure from initial conditions remains difficult.Negative ties offer another explanation for polarization, although evidence for ubiquitous negative interpersonal influence is described as unavailable.
  • Related work: The Friedkin–Johnsen model extends DeGroot pooling by incorporating heterogeneous susceptibility and agents’ prejudices.Its scalar dynamics are represented by x(k + 1) = ΛWx(k) + (I −Λ)u, with u containing initial prejudices.
  • Research gap: Existing multidimensional FJ extensions treat topic dimensions independently, whereas this paper targets mutually dependent opinions on several issues.The paper frames the multidimensional opinion vector as containing topic-specific scalar opinions.
  • Contribution: The paper develops stability and convergence conditions for classical and extended FJ models and an asynchronous protocol reaching the same final opinions on average.The contribution extends the scalar framework to interdependent issues and contrasts synchronous deterministic dynamics with randomized gossip communication.

IV. STABILITY AND CONVERGENCE OF THE FJ MODEL

The scalar FJ model is characterized through matrix stability and graph structure. Its convergence is governed by regularity of ΛW, while strong connectivity with at least one stubborn agent guarantees stability.

  • Stability and convergence: Theorem 1 states that Λ11W11 is Schur stable and characterizes stability by the absence of oblivious agents.With oblivious agents, convergence additionally requires the corresponding W22 block to be regular.
  • Graph conditions: A strongly connected interaction graph with at least one stubborn agent guarantees stability of the FJ model.Strong connectivity ensures every agent is stubborn or connected by a walk to a stubborn agent.
  • Convergence criterion: The FJ model converges if and only if A = ΛW is regular, with the limit of its powers determining the limiting behavior.For the FJ input term, regularity is also the necessary and sufficient convergence condition.
  • Caveat: In unstable but convergent models, the inverse formula for the stationary point can fail because I −ΛW is not invertible.When oblivious agents are present, the same prejudices may correspond to multiple stationary points, and iteration and series limits can differ.
  • Approximation: The stable dynamics can be approximated by models in which all agents are stubborn as α approaches 1.For convergent FJ models, the approximation converges uniformly in u over compact sets.

V. A MULTIDIMENSIONAL EXTENSION OF THE FJ MODEL

The paper extends FJ dynamics from scalar opinions to vector-valued opinions, with each agent holding topic-specific opinions across multiple issues. This extension supplies the representation needed for multidimensional opinion analysis.

  • Vector opinions: The multidimensional extension assigns each agent a vector opinion xi(k) ∈ R^m.Each vector component represents the agent’s opinion on one of m different issues.
  • Vector opinions: The model’s topic dimensions represent distinct issues within a multidimensional opinion state.The supplied formulation introduces vector opinions but does not yet describe inter-topic coupling.

A. Opinions on independent issues

The paper contrasts independent-topic opinion dynamics with a multidimensional FJ model in which topic interdependence couples opinions across issues. The coupling matrix can substantially alter steady opinions while preserving an invariant opinion domain under suitable conditions.

  • Independent issues: Independent topics evolve as separate scalar FJ systems, so each topic-specific opinion can be analyzed and computed independently.
  • Interdependent issues: The MiDS matrix C adds introspective transformations that mix averaged topic-specific opinions before each agent’s update.C captures interrelations between topics; setting C = I_m reduces the model to the usual FJ dynamics.
  • Examples: The coupling matrix substantially changes final opinions: dependent topics prevent some salmon attitudes from becoming negative, unlike the independent case.With C, fish attitudes become more positive and salmon attitudes less positive than initially, while the dependent-topic case differs from C = I_2.
  • Interdependent issues: Interdependent topics create additional couplings between topic-specific opinions, unlike the two uncoupled copies shown in the independent-topic structure.These extra ties are imposed by C and connect opinions associated with different issues and agents.
  • Examples: Positive intertopic couplings bring topic-specific opinions closer together, whereas contrary issues may require negative couplings to represent consistency.
  • Invariance: If C maps a convex set D into itself, then D is invariant: initial opinions in D remain in D throughout the dynamics.This provides a general invariance result, including the hypercube when the corresponding condition on C holds.

C. Convergence of the multidimensional FJ model

The multidimensional FJ model is represented in matrix form and analyzed for stability and convergence. Stability depends on the spectral radii of the social-influence and topic-coupling matrices, while convergence additionally requires regularity of the coupling matrix in relevant cases.

  • The multidimensional dynamics can be written using stacked opinion and prejudice vectors, yielding a matrix representation of the model.
  • Stability: Row-stochastic C preserves the original FJ stability conditions, but non-stochastic C can also yield stability when its spectral radius is sufficiently small.The sufficient condition stated is ρ(C) < ρ(ΛW)^-1, and it is not necessary for stability.
  • Stability: The model is stable if and only if ρ(ΛW)ρ(C) < 1, with the ultimate opinions then determined by the corresponding stable system.
  • Stability: When ρ(C) = 1, stability is equivalent to stability of the scalar FJ model, meaning the absence of oblivious agents.
  • Convergence: Convergence requires C to be regular and, depending on the system decomposition, either C* = 0 or an additional condition specified by the convergence theorem.Regularity means that lim_k→∞ C^k exists; convergence is therefore stricter than stability in the stated setting.
  • Extensions: Allowing interdependencies among initial topic opinions through x(0) = [I_n ⊗ D]u changes neither stability nor convergence conditions.

VI. OPINION DYNAMICS UNDER GOSSIP-BASED COMMUNICATION

The paper extends gossip-based opinion dynamics to multidimensional opinions and shows that suitable asynchronous protocols recover the deterministic model’s steady opinion on average.

  • Protocol motivation: Asynchronous gossip communication is motivated as a practical alternative to simultaneous communication, which is difficult to implement in large-scale social networks.Each interaction updates one agent using its prejudice and a modified neighbor opinion while other opinions remain unchanged.
  • Protocol definition: The multidimensional gossip protocol samples an interaction-graph arc uniformly and updates the communicating agent’s opinion.The extension covers multidimensional opinions and generalizes earlier gossip algorithms beyond Λ = I − diag W.
  • Ergodicity: Under ρ(ΛW) < 1, row-stochastic C, Γ1 = ΛW, and Γ2 = (I − Λ)W, the expected opinion converges to the deterministic FJ model’s final opinion.The theorem states lim k→∞ Ex(k) exists and equals x′.
  • Ergodicity: The Cesàro-Polyak averages converge almost surely and in Lp to the same limiting opinion.The result also remains valid for broader choices of Γ2 satisfying the stated support and coefficient conditions.
  • Scalar special case: The scalar gossip algorithm is a special case of a broader class of protocols satisfying the multidimensional ergodicity theorem.For the scalar case, the random process is almost surely and mean-square ergodic and reaches the FJ final opinion.
  • Limitation: The random opinions themselves do not converge, instead exhibiting nondecaying oscillations around their average limit.Their distribution may converge in probability to a random vector determined by Λ, W, and C.

VII. NUMERICAL EXPERIMENTS

Numerical experiments illustrate convergence, topic-interdependence effects, and the correspondence between synchronous multidimensional dynamics and averaged gossip dynamics.

  • Experimental scope: The experiments illustrate convergence of the synchronous multidimensional FJ model and its lazy gossip version.The numerical section uses simulations to illustrate the theoretical dynamics.
  • Examples 3–5: In Examples 3–5, independent and interdependent topic models produce different opinion trajectories, with topic interdependencies causing substantial drags for agents 1, 2, and 4.Figure 5 compares C = I2, a stochastic C, and the MiDS matrix C.
  • Extended DeGroot model: In the extended DeGroot model, positive topic ties can drive topic-specific opinions toward agreement, whereas negative coupling can produce polarization.The same initial opinions and topic matrices are used while setting Λ = In.
  • Gossip comparison: Cesàro-Polyak averages in the gossip counterparts converge to the deterministic model’s limits, while the random opinions oscillate and do not converge.This correspondence is shown for the models from Examples 3 and 4.
  • Hierarchical network: The hierarchical simulation uses n = 51 agents arranged as one totally stubborn leader and 10 groups of 5 agents.Each subgroup has a local representative, with influence propagating through the hierarchy.

VIII. ESTIMATION OF THE MIDS MATRIX

The paper formulates estimation of the MiDS matrix C from experimental opinion data as convex optimization, while discussing feasibility constraints and solution existence.

  • Motivation: Estimating social influence structure supports network analysis, control, and sensing, and this section focuses on estimating the MiDS matrix C.The matrices W and Λ are assumed known.
  • Finite-horizon procedure: A finite-horizon experiment records initial opinions and opinions after T full conversation rounds, then fits C to the model equations.The resulting data are used in an optimization problem based on the observed trajectory.
  • Constraints: Model feasibility can be enforced by restricting C to a known closed convex set, such as row-stochastic matrices or matrices satisfying the model’s structural conditions.With these constraints, the estimation problem remains a convex quadratic program.
  • Optimization: Alternative convex objectives, including ℓ1 and maximum-error forms, turn the estimation problem into standard linear programs.The stated objectives use sums or maxima of residual norms.
  • Existence: Under the stated compactness, continuity, or strict-convexity conditions, an optimum exists for the corresponding estimation problem.The existence result also covers unbounded closed convex sets when the objective is a strictly convex norm.

B. Infinite-horizon identification procedure

The infinite-horizon procedure estimates C from stabilized opinions rather than full trajectories, but it applies only to stable models and does not provide a predictable convergence time.

  • Applicability: The infinite-horizon experiment applies only to stable models, restricting the admissible MiDS matrices to those satisfying ρ(C) < ρ(ΛW)^−1.The admissible set C is assumed convex and closed, with every element satisfying the stability condition.
  • Procedure: Agents communicate until their opinions stabilize, after which the final opinion x′ is used to fit C through an optimization problem.The procedure does not require agents to record their complete opinion histories.
  • Optimization: For polyhedral constraints and Euclidean loss, estimation is a convex quadratic program; ℓ1 or ℓ∞ loss yields a linear program.The solution exists for any closed and convex admissible set C.
  • Trade-offs: Compared with finite-horizon identification, the infinite-horizon procedure uses less stored data but cannot predict how quickly convergence will occur.Its applicability is limited to stable models, whereas finite-horizon identification does not depend on system convergence.
  • Standard form: Vectorization converts Kronecker-product constraints into standard linear constraints Ax = b.The transformation uses vec identities and represents the unknown matrix C through c = vec C.

D. Numerical examples

The numerical examples demonstrate identification of the multidimensional influence structure from either final opinions or finite conversation trajectories. Perturbations produce estimated matrices that remain close to the underlying model.

  • Two numerical examples illustrate procedures for identifying the multidimensional influence structure.
  • An infinite-horizon experiment estimates the matrix from agents’ final opinions by minimizing a residual.The reported minimal residual is ∥ε∥2 = 0.9322.
  • The estimated steady opinion is C = [35.316, 11.443, 35.092, 9.483, 75, −50, 52.386, 4.915]⊤.
  • A finite-horizon experiment uses T = 3 conversation rounds to estimate a row-stochastic matrix from observed opinions.
  • The examples’ perturbations make the estimated MiDS matrix differ from, but remain close to, the original matrix.

IX. PROOFS

The proofs characterize stability and convergence by decomposing substochastic influence matrices and analyzing their stochastic components. For the multidimensional model, convergence depends jointly on the social influence matrix and the intertopic dependence matrix.

  • A substochastic matrix with spectral radius 1 contains a maximal stochastic subset, enabling an upper-triangular decomposition.The remaining diagonal block is Schur stable, while the stochastic block is row-stochastic.
  • For the FJ model, the maximal stochastic subset of ΛW consists exactly of oblivious agents.
  • The non-oblivious block Λ11W11 is Schur stable, while the oblivious-agent block equals W22.
  • Scalar FJ convergence requires W22 to be regular, meaning its powers converge to a limiting matrix.
  • The multidimensional system is stable exactly when ρ(ΛW)ρ(C) < 1; when ρ(C) = 1, this reduces to ρ(ΛW) < 1.
  • Multidimensional convergence requires regularity of both the social submatrix W22 and the topic-dependence matrix C.
  • The asynchronous gossip process is almost surely ergodic and its expected opinions converge to the deterministic steady opinion.
  • For p = 2, the mean-square convergence error is bounded by χ/(k + 1), with χ depending on ρ(ΛW) and the prejudice vector u.

X. CONCLUSION

The conclusion presents a multidimensional FJ model for interdependent topics in static social networks and establishes its convergence theory. It also identifies future work on empirical validation, robustness, controllability, and heterogeneous topic-dependence matrices.

  • The paper proposes a multidimensional FJ extension for agents whose opinions concern two or more interdependent topics.The model represents an ideological or belief system within a static social-network topology.
  • The model addresses an open problem concerning interpersonal influence mechanisms in the formation of ideological-belief systems.
  • Necessary and sufficient conditions are established for stability and convergence, with convergence meaning finite limits for every initial condition.
  • The paper also addresses identification of the multi-issue interdependence structure.
  • Although the model is synchronous, an asynchronous decentralized gossip protocol can reach the same final opinions.
  • Future research includes large-data experimental validation, robustness and controllability, and stability for heterogeneous MiDS matrices.

APPENDIX

The appendix reviews regularity for row-stochastic matrices and explains its spectral and geometric consequences. Regularity determines convergence of matrix powers toward a projector onto the eigenspace associated with eigenvalue 1.

  • A row-stochastic matrix is regular when every eigenvalue other than 1 lies strictly inside the unit circle.
  • Full regularity additionally requires eigenvalue 1 to be simple, so its eigenspace consists only of scalar multiples of the all-ones vector.
  • For irreducible matrices, regularity and full regularity are equivalent to primitivity.Primitivity means that some matrix power is strictly positive.
  • Under regularity, the invariant subspace for eigenvalue 1 is preserved while components associated with other eigenvalues decay to zero.
  • The limiting operator A* is the projector onto the eigenspace corresponding to eigenvalue 1.
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