Group adaptation drives opinion dynamics in higher-order networks

TL;DR

This work proposes a bounded confidence model that incorporates both mechanisms: group discussions can lead to global agreement among members, while strong internal disagreement causes groups to split, with resulting subgroups merging with others.

Abstract

In modern interconnected societies, opinions and beliefs can quickly spread across large populations, giving rise to collective behaviors such as the adoption of social norms or polarization. These phenomena have motivated many models aimed at reproducing emergent properties from simple interaction mechanisms. In particular, opinion dynamics models describe how individual opinions evolve through interactions and study the conditions for global consensus or polarization. Most models assume that these interactions occur between pairs of agents, typically on a fixed network structure. However, opinion changes can occur in groups, which may also undergo adaptive changes if disagreement arises. Here, we propose a bounded confidence model that incorporates both mechanisms: group discussions can lead to global agreement among members, while strong internal disagreement causes groups to split, with resulting subgroups merging with others. We systematically study the model outcomes as a function of agents' tolerance for agreement. Strikingly, adaptivity suppresses key effects of group interactions, restoring a phenomenology close to that of pairwise interactions. In particular, adaptivity enables the formation of large groups and prevents fragmentation at small tolerance. It also restores a phase transition from polarization to consensus, which would otherwise disappear in a non-adaptive group-based model. Overall, our work shows that both adaptivity and group interactions shape the structure of social ties and global opinion dynamics in a population.

Figures (5)

• Figure 1: Illustration of the two phases composing one time step of the model. Nodes (agents) are colored according to their opinion and each hyperedge (group) is identified by a number and a different color. (a) Initial configuration composed by four groups. (b) Agents forming group 1 and group 2 converge respectively to a common opinion. Group 4 instead splits into two overlapping subgroups (4a and 4b), while group 3 divides into two disjoint subgroups (3a and 3b, with 3a being an isolated node). (c) Several of the subgroups resulting from the splits of groups 3 and 4 merge with other hyperedges, generating new groups (3a+4a and 1+3b).
• Figure 2: Main features of the model in the steady state. Each row shows the behavior of a specific quantity as a function of the confidence parameter $\varepsilon$; columns correspond to different initial group size $M$. $\mathcal{H}_0$ is an $M$-uniform ER hypergraph and opinions are initially uniformly distributed in $[0,1]$. Top row: number of connected components ($N_{cc}$); second row: relative size of the two largest connected components ($S_{cc}/N$) and variance of the largest one (inset); third row: relative number of hyperedges, with respect to the initial value ($E/E_0$); fourth (bottom) row: maximum (dashed line) and average (solid line) hyperedge size ($|e|$). Inside the panels, curves are colored according to the size of the system $N$, from $N=500$ to $N=5000$ (see legend). For each value of $\varepsilon$, results are averaged over $20$ independent realizations of the model, starting from different instances of $\mathcal{H}_0$ and $x_i(0)$.
• Figure 3: Distributions within the steady state, for different values of the confidence parameter $\varepsilon$. First row: distribution of relative sizes of connected components $s_{cc}$. Second row: distribution of agents' opinions $x$. Third row: number of connected components with opinions $y$. The starting structure of interactions is a 4-uniform ER hypergraph (see also central column of Fig. \ref{['fig: ER_max_min']}) with $\langle k_4 \rangle = 10$, and the initial opinions are uniformly distributed in $[0,1]$. Each color corresponds to a specific size of the system (see legend). For $P(x)$ and $P(y)$ we show only the values related to one single run with $N=1000$ and one single run with $N=5000$, to improve readability. The values of $P(s_{cc})$ are instead averaged over 50 independent realizations.
• Figure 4: Distributions of structural features within the steady state, for four values of $\varepsilon$. First row: distribution of hyperedge (i.e. group) sizes. Second row: hyperdegree distribution of the initial (dashed lines) and final (continuous line) hypergraphs. The starting structure is a 4-uniform ER hypergraph with average hyperdegree $\langle k_4 \rangle = 10$; initial opinions are uniformly distributed in $[0,1]$. Each line color corresponds to a specific size of the system (see legend). The results are averaged over 50 independent realizations of the model.
• Figure 5: Temporal evolution of structural and dynamical features of the system for four values of $\varepsilon$. First row: number of different opinions present in the system ($N_{op}$). Second row: relative number of hyperedges ($E/E_0$, w.r.t. the initial value $E_0$). Third row: average (continuous line) and maximum (dashed line) group size ($|e|$). The insets contain the early evolution of the corresponding quantities. The starting structure is a 4-uniform ER hypergraph with average hyperdegree $\langle k_4 \rangle = 10$; the initial opinions are uniformly distributed in $[0,1]$. The results are averaged over 50 independent runs of the model.