Multi-topic belief formation through bifurcations over signed social networks

TL;DR

The paper develops a nonlinear, multidimensional belief dynamics model on signed social networks, coupling agent interactions with a structured belief system. A neutral equilibrium loses stability at a critical attention value $u^* = \frac{d}{\alpha + \gamma \operatorname{Re}(\lambda) + \beta \operatorname{Re}(\mu) + \delta \operatorname{Re}(\lambda \mu)}$, with $(\lambda,\mu)$ drawn from the spectra of the agent and belief-system graphs; the resulting low-dimensional center manifold governs post-bifurcation behavior. Depending on graph structure and parameters, the system exhibits multi-stable equilibria via pitchfork bifurcations or sustained oscillations via Hopf bifurcations, and external dissonance can shift the dominant eigenstructure from Λ1 to Λ2, drastically altering belief dynamics. The authors connect their deterministic model to Networks of Beliefs theory, elucidating how self-appraisal, internal biases, and cognitive dissonance shape social belief formation and enabling design of decentralized decision-making on engineered networks. Overall, the work provides a principled, graph-analytic framework linking network topology to qualitative belief dynamics with clear implications for polarization, coherence, and oscillatory belief patterns in complex social systems.

Abstract

We propose and analyze a nonlinear dynamic model of continuous-time multi-dimensional belief formation over signed social networks. Our model accounts for the effects of a structured belief system, self-appraisal, internal biases, and various sources of cognitive dissonance posited by recent theories in social psychology. We prove that agents become opinionated as a consequence of a bifurcation. We analyze how the balance of social network effects in the model controls the nature of the bifurcation and, therefore, the belief-forming limit-set solutions. Our analysis provides constructive conditions on how multi-stable network belief equilibria and belief oscillations emerging at a belief-forming bifurcation depend on the communication network graph and belief system network graph. Our model and analysis provide new theoretical insights on the dynamics of social systems and a new principled framework for designing decentralized decision-making on engineered networks in the presence of structured relationships among alternatives.

Figures (7)

• Figure 1: Four types of communication arrows associated to four different cognitive effects on belief formation in model \ref{['eq:dynamics_with_observation']}. Arrow direction follows sensing convention.
• Figure 2: (a) Communication graph $\mathcal{G}_{{\color{black} \rm a}}$ for five agents and (b) belief system graph $\mathcal{G}_{{\color{black} \rm o}}$ for three options with signed adjacency matrices \ref{['eq:adj_mat_ex']}. Solid blue edges are positive connections and red dashed edges are negative connections. Arrow direction follows sensing convention.
• Figure 3: Bifurcation diagrams for Example \ref{['ex:pitchfork']}, generated with MatCont numerical continuation package matcont. Red (blue) lines represent unstable (stable) equilibria. $S_1(y) = \tanh(y + \varepsilon_1 \tanh(y^2))$, $S_2(y) = \frac{1}{2}\tanh(2y + 2\varepsilon_2 \tanh(y^2))$. (a) Symmetric pitchfork bifurcation at $u = u^*$ with$\varepsilon_1 = \varepsilon_2 = 0$, $\langle \mathbf{w}_{{\color{black} \rm a}}\otimes \mathbf{w}_{{\color{black} \rm o}}, \mathbf{b}\rangle = 0$; (b) unfolding of symmetric pitchfork bifurcation with$\varepsilon_1 = \varepsilon_2 = 0$, $\langle \mathbf{w}_{{\color{black} \rm a}}\otimes \mathbf{w}_{{\color{black} \rm o}}, \mathbf{b}\rangle > 0$; (c) unfolding of pitchfork at $u = u^*$ with$\varepsilon_1 = \varepsilon_2 = 0.1$, $\langle \mathbf{w}_{ {\color{black} {\color{black} \rm a}}}\otimes \mathbf{w}_{{\color{black} \rm o}}, \mathbf{b}\rangle = 0$; (d) unfolding of pithfork with$\varepsilon_1 = \varepsilon_2 = 0.1$, $\langle \mathbf{w}_{{\color{black} \rm a}}\otimes \mathbf{w}_{{\color{black} \rm o}}, \mathbf{b}\rangle > 0$. For (b) and (d), $b_{11} = 0.001$, $b_{22} = 0.003$, $b_{53} = -0.002$ and $b_{ij} = 0$ for all other $i\in \mathcal{V}_{{\color{black} \rm a}}, j\in \mathcal{V}_{{\color{black} \rm o}}$. Vertical gray line indicates the bifurcation point $u = u^* \approx 2.742$. Other parameters: $d = 1$, $\alpha = \gamma = \beta = \delta = 0.1$.
• Figure 4: Trajectories $z_{ij}(t)$ for belief formation dynamics \ref{['EQ:value_dynamics']} for 5 agents evaluating 3 options on $\mathcal{G}_{{\color{black} \rm a}}$, $\mathcal{G}_{{\color{black} \rm o}}$ of Fig. \ref{['fig:graphs']} from random initial conditions. The figure illustrates the distribution of agents' states $z_{ij}$ along each option dimension as the network settles to (a) EQ1 from Figure \ref{['fig:pitchfork']}(c); (b) EQ2 from Figure \ref{['fig:pitchfork']}(c); (c) shows same trajectories as (a), grouped by agent instead of option. Belief trajectories on options 1 and 3 overlap for most of the simulation for all agents. Parameters: $u = u^* + 0.05 \approx 2.792$, all other parameters as in Fig. \ref{['fig:pitchfork']}(c).
• Figure 5: (a) Strongly connected, structurally balanced communication graph with node partitions $\mathcal{V}_{{\color{black} \rm a 1}} = \{ 1, 2, 3, 4\}$, $\mathcal{V}_{{\color{black} \rm a 2}} = \{ 5,6,7,8,9,10,11\}$; (b) Strongly connected, structurally balanced belief system graph with node partitions $\mathcal{V}_{{\color{black} \rm o}1} = \{1,2\}$, $\mathcal{V}_{{\color{black} \rm o}2} = \{3,4,5\}$; (c) belief trajectories of all 11 agents on option 1; (d) belief trajectories of agent 7 on all 5 options; belief trajectories on options 4 and 5 overlap for most of the simulation. Parameters: $u = u^* + 0.05 \approx 0.7627$, $\alpha = \gamma = \beta = \delta = 0.1$, $d = 1$, $b_{ij} = 0$ for all $i \in \mathcal{V}_{{\color{black} \rm a}}$, $j \in \mathcal{V}_{{\color{black} \rm o}}$, $S_1(\cdot) = \tanh(\cdot)$, $S_2(\cdot) = \frac{1}{2} \tanh(2 \cdot)$.

Theorems & Definitions (20)

• Proposition II.1: matrices with Perron-Frobenius property
• Definition 1
• Example II.1
• Lemma II.2: Graphs with simple dominant eigenvalue
• Lemma IV.1: Jacobian spectrum bizyaeva2022sustained
• proof
• Theorem IV.2: Belief-forming bifurcation
• Lemma IV.3: Graph conditions
• Proposition IV.4
• Proposition IV.5