Collective decision-making with higher-order interactions on $d$-uniform hypergraphs

TL;DR

This work investigates how higher-order group interactions on $d$-uniform hypergraphs shape collective decision-making in a Best-of-$2$ setting with two opinions of qualities $Q_A$ and $Q_B$ and pooling error $α$. It develops a heterogeneous mean-field framework that reduces the dynamics to an ODE for the weighted A-fraction $\langle a\rangle$ and reveals two pooling-threshold bifurcations $α_{\text{crit}}^{(1)}$ and $α_{\text{crit}}^{(2)}$ that demarcate consensus regimes. Crucially, the bifurcation structure depends only on the interaction order $d$ and the quality ratio $Q=Q_B/Q_A$, not on the underlying topology, indicating a form of structural universality. Agent-based simulations on random and scale-free $d$-uniform hypergraphs validate the mean-field predictions and uncover that larger group sizes can, paradoxically, drive the system toward the worst option under certain $α$ and $Q$ conditions. The results set the stage for extensions to Best-of-$n$ problems and temporally evolving hypergraphs, and invite further exploration of how pooling error scales with group size.

Abstract

Understanding how group interactions influence opinion dynamics is fundamental to the study of collective behavior. In this work, we propose and study a model of opinion dynamics on $d$-uniform hypergraphs, where individuals interact through group-based (higher-order) structures rather than simple pairwise connections. Each one of the two opinions $A$ and $B$ is characterized by a quality, $Q_A$ and $Q_B$, and agents update their opinions according to a general mechanism that takes into account the weighted fraction of agents supporting either opinion and the pooling error, $α$, a proxy for the information lost during the interaction. Through bifurcation analysis of the mean-field model, we identify two critical thresholds, $α_{\text{crit}}^{(1)}$ and $α_{\text{crit}}^{(2)}$, which delimit stability regimes for the consensus states. These analytical predictions are validated through extensive agent-based simulations on both random and scale-free hypergraphs. Moreover, the analytical framework demonstrates that the bifurcation structure and critical thresholds are independent of the underlying topology of the higher-order network, depending solely on the parameters $d$, i.e., the size of the interaction groups, and the quality ratio. Finally, we bring to the fore a nontrivial effect: the large sizes of the interaction groups, could drive the system toward the adoption of the worst option.

Figures (8)

• Figure 1: Schematic representation of the probabilities involved in the hypergraph heterogeneous mean-field computations. The focal node $i$ has $k$ incident hyperedges (hyperdegree $k$), denoted by $e_1,\dots,e_r,\dots,e_k$. A generic node $i_r$ in the hyperedge $e_r$ has an excess hyperdegree $j_r$ with probability $q_{j_r}$ and therefore hyperdegree $j_r+1$. With probability $a_{j_r+1}$, she has opinion $A$, and therefore with probability $1-a_{j_r+1}$, she has opinion $B$.
• Figure 2: Equilibria and their stability as a functions of $\alpha$ for different values of $d$ and $Q$: from left to right, the hyperedge size $d$ increases; from bottom to top, the quality ratio $Q$ increases. Green and red dots show the stable and unstable equilibria respectively. Black diamonds illustrate the critical value $\alpha_{crit}^{(1)}$ given in Eq. \ref{['eq:eqalc1']}, and blue stars indicate the critical value $\alpha_{crit}^{(2)}$ given by Eq. \ref{['eq:eqalc2']}.
• Figure 3: Evolution of the critical values: (a) $\alpha^{(1)}_{\text{crit}}$ and (b) $\alpha^{(2)}_{\text{crit}}$ as functions of the parameters $d$ and $Q$. For each pair $(d, Q) \in ([3, 20], [0.1, 1])$, the critical values $\alpha^{(1)}_{\text{crit}}$ (see Eq. \ref{['eq:eqalc1']}) and $\alpha^{(2)}_{\text{crit}}$ (see Eq. \ref{['eq:eqalc2']}) correspond to the thresholds below which the all-B (resp. all-A) solution is stable. These critical values are shown both on the color bar and along the curves.
• Figure 4: We present the bifurcation diagram and the average fraction of agents holding opinion $A$ as a function of the parameter $\alpha$, for both $d$-uniform random and scale-free hypergraphs. The first row (panels a–c) displays the results of the analytical model, while the second row (panels d–f) shows results from the agent-based model (ABM): blue dotted lines correspond to random $d$-uniform hypergraphs with $N = 500$ nodes, and black lines correspond to scale-free $d$-uniform hypergraphs with $N = 1000$ nodes. From left to right, we consider the following parameter settings: $d = 3$ and $Q_B = 0.9$ (panels a, d), $d = 3$ and $Q_B = 0.45$ (panels b, e), and $d = 4$ and $Q_B = 0.45$ (panels c, f). Shaded regions represent the standard deviation. Stable and unstable equilibria are represented by green and red dots, respectively in the first row. The quality of opinion $A$ is fixed at $Q_A = 1$. In the ABM model, the results are averaged over 30 independent simulations, each run for a maximum simulation time of $T_{\text{max}} = 10^5$ iterations.
• Figure 5: Diagram showing the average fraction of agents holding opinion $A$$\bar{\rho}_A$ on a $d$-uniform random hypergraph, as a function of the parameters $\alpha$ and $Q$ (with $Q = Q_B / Q_A$). Panel (a) corresponds to $d = 3$, panel (b) to $d = 4$, panel (c) to $d = 5$, and panel (d) to $d = 20$. Each result is averaged over 30 independent simulations, with a maximum simulation time of $T_{\text{max}} = 10^5$ iterations per experiments.