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Note updated: 9-30-2024

Statistical Physics Models of Belief Dynamics: Theory and Empirical Tests

Mirta Galesic, Daniel L. Stein   (2017)

Summary

  • Describes models for representing belief dynamics in a social network using a simple discrete-spin statistical physics framework
  • Demonstrates how the parameters of these models can be determined from empirical data
  • Presents the predictive value of the models using real-world studies of belief change in social networks

Belief models

The framework for these models describes a collection of autonomous "agents", a set of possible states for each agent, and the dynamics by which an agent updates their state over time

Ising model

Each agent's belief at time tt can be represented by an Ising spin σi(t)=±1\sigma_i(t) = \pm 1; so, each agent can only have one state (agree/disagree, Democrat/Republican, etc.). The "satisfaction" of agent ii can be represented by a Hamiltonian-like function Hi\mathcal{H}_i; the lower the value of Hi\mathcal{H}_i, the higher the satisfaction, and vice-versa:

Hi=hieffσi=hisocσihiσi\mathcal{H}_i=-h_i^{\text{eff}}\sigma_i=-h_i^{\text{soc}}\sigma_i-h_i\sigma_i

hieffh_i^{\text{eff}} describes the total effective field (Note: like influence) on a node, including its internal field, hih_i, and "social field", hisoch_i^\text{soc}

Initial conditions

The spin value, σi\sigma_i for each node ii must be set to some initial value; it's often approximated using survey results or inferring them from secondary data sources, like social media behaviors

Social interaction term

The social interaction term, hisoch_i^\text{soc}, describes the "social field", or how the beliefs of node i's neighbors influence the belief of node i. The paper considers three social interaction rules:

  1. Voter model (unbiased random copying): each node chooses a single neighbor, uniformly at random, and adopts that neighbor's belief state
  2. Majority rule model (conformism): each node adopts the majority belief state amongst its neighbors
  3. Expert rule model: each node adopts the belief state of one specific neighbor considered to be an expert, or somehow favored, on the issue

Internal field term

The internal field term, hih_i describes how likely a belief is to be accepted by node ii. The paper argues that the relative importance of "intrinsic predispositions vs. social influence" differs per individual, so they introduce 0α10 \leq \alpha \leq 1, which describes the relative weight between these two factors:

Hi=ασihisoc(1α)hiσi\mathcal{H}_i=-\alpha \sigma_i h_i^{\text{soc}} - (1-\alpha)h_i\sigma_i

Note: I'm not really sure what the basis is behind the introduction of α\alpha, and why it's zero-sum.

Network structure

In order to test the dynamics of beliefs within a social network, the network needs a structure (the total number of nodes, and the edges between each node). The paper uses five different network structures (below).

network-structures

The ring lattice and fully connected graph approximate some real-world networks (Note: really?) and enable analytical solutions of a system's dynamics. The more realistic structures, like small-world and stochastic block models, are not typically analytically solvable.

Deterministic vs. stochastic updating

The reliability of information transfer, β0\beta \geq 0, describes the accuracy of a node's perception of the beliefs of their neighbors. If you're familiar with statistical mechanics, β\beta behaves as an inverse "temperature" of the system: when β\beta is small, the reliability of information transfer is low (weakly held beliefs with random fluctuations, not easily observable social information, etc.).

β\beta is used while updating the state of an agent. If an agent can lower its energy at a given timestep (ΔHi<0\Delta \mathcal{H}_i < 0), it does so with probability one. If changing its state raises its energy (ΔHi>0\Delta \mathcal{H}_i > 0), it does so with probability [1+exp(βΔHi)]1[1 + \exp(\beta \Delta \mathcal{H}_i)]^{-1}