A new adaptive two-layer model for opinion spread in hypergraphs: parameter sensitivity and estimation

TL;DR

The paper develops an adaptive two-layer hypergraph model for opinion spread, combining deterministic households with randomly formed workplaces to capture higher-order peer-pressure effects. Dynamics follow a discrete-time majority rule with parameters $(\beta,q,r_1,r_2,\lambda)$ and a weighted majority $a=\frac{h+\lambda w}{1+\lambda}$, enabling analysis via Markov-chain absorbing states and extensive simulations. It then compares three parameter-estimation approaches—linear regression, XGBoost, and 1D CNNs—under partial observables, showing that estimation performance hinges on the strength of peer pressure and the available data, with CNNs excelling in nonlinear regimes. The work provides insights into how adaptive, higher-order interactions shape polarization and offers practical guidance for inferring model parameters from partial time-series data in complex social networks.

Abstract

When opinion spread is studied, peer pressure is often modeled by interactions of more than two individuals (higher-order interactions). In our work, we introduce a two-layer random hypergraph model, in which hyperedges represent households and workplaces. Within this overlapping, adaptive structure, individuals react if their opinion is in majority in their groups. The process evolves through random steps: individuals can either change their opinion, or quit their workplace and join another one in which their opinion belongs to the majority. Based on computer simulations, our first goal is to describe the effect of the parameters responsible for the probability of changing opinion and quitting workplace on the homophily and speed of polarization. We also analyze the model as a Markov chain, and study the frequency of the absorbing states. Then, we quantitatively compare how different statistical and machine learning methods, in particular, linear regression, xgboost and a convolutional neural network perform for estimating these probabilities, based on partial information from the process, for example, the distribution of opinion configurations within households and workplaces. Among other observations, we conclude that all methods can achieve the best results under appropriate circumstances, and that the amount of information that is necessary to provide good results depends on the strength of the peer pressure effect.

Figures (12)

• Figure 1: Different absorbing states configurations with household and workplace structures in the model for $n=10$. Node color represents opinion; the two colors can be switched respectively to gain homomorphic states. The linear model has only two kinds of absorbing states: (a),(b). The nonlinear model has 8 kinds of absorbing states: (a)-(h). For example, for state $(e)$, the value $a$ is $0.6$ for blue vertices and $0.73$ for red ones (with $\lambda=0.5$); while in state $(h)$, the value of $a$ is $1$ for vertices in the first household; $0.73$ for blue vertices in the second household, and $0.6$ for the red vertices. Notice that $0.6$ is the smallest value of $a$ which is larger than $0.5$ and can lead to a stable state. State $(a)$ has modifications with an arbitrary workplace structure. Recall that we have strict inequality both in Step 1 and Step 2; otherwise only homogeneous workplaces would remain.
• Figure 2: Absorption rates of the states shown in Figure \ref{['fig:smallexample']} for the small-scale system with $n = 10$. (i) In the linear model, the probability of reaching a fully homogeneous state decreases with increasing $q$ and increases with higher values of $\beta$. (ii) In the nonlinear model, the absorption rates show less variation across $q$ and $\beta$, although the same qualitative trend for homogeneous states is observed as in the linear case. Absorbing states containing both opinions can be divided into two main categories: (b–d) configurations with two homogeneous households holding opposite opinions, whose absorption rates, shown in (iii), remain nearly constant across the parameter range; and (e–h) configurations with homogeneous workplaces but mixed households, whose absorption rates, shown in (iv), increase with $q$ and decrease with $\beta$.
• Figure 3: Standard deviation of the number of nodes holding opinion $A$ at the stopping time. Results are based on 500 simulations with $n=1000$, each starting from a balanced initial condition of 500 individuals with opinion $A$ and 500 with opinion $B$, chosen randomly, independently of the hypergraph structure. Each run lasted $10^6$ steps or stopped in an absorbing state before, so that on average every individual was updated about 1000 times. (a) The linear model (green solid lines) is compared with the nonlinear model (blue dotted lines). (b) Same as (a), but showing only the nonlinear model for clarity. In both models, $\sigma(N_A)$ increases with $\beta$ and decreases with larger $q$. However, the nonlinear model consistently produces smaller deviations than the linear model, highlighting a fundamental difference between the two dynamics—a distinction that will also be reflected in the polarization results.
• Figure 4:
• Figure 5: We calculated how long it takes for the process to reach a homophily index above $0.4$. All figures show the average of the first timestep $\tau_{hh},\tau_{wp}$ where the homophily index is greater than $0.4$ amongst the household hyperedges and the workplace hyperedges, respectively. When there is no value shown, that means the homophily did not reach $0.4$ in the first $10^6$ steps, all y-axes are in logging time, so 1 log step means 1000 steps in the model.