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Opinion Dynamics with Multi-Body Interactions
Leonie Neuhäuser, Michael T. Schaub, Andrew Mellor, Renaud Lambiotte
TL;DR
Pairwise models may be insufficient to describe social dynamics shaped by group influence and reinforcement. The paper introduces a three-body consensus model for opinion formation on hypergraphs, and finds that nonlinear multi-body interactions can shift average opinions and allow one group to dominate another. These results highlight the role of reinforcing group effects within the studied model.
Problem
Pairwise interaction models may be insufficient for phenomena involving group influence and reinforcement in social dynamics.
Source: §1 · p. 2Method
The paper introduces a three-body consensus model for opinion formation on hypergraphs, with triplet interactions and an interaction function assumed identical across interacting triplets.
Source: §2 · p. 2Results
Nonlinear multi-body interactions can shift average opinion and, in two connected polarized groups, lead one group's opinion to clearly dominate the other.
Source: §4 · p. 10Takeaways & Limitations
The findings support the relevance of reinforcing group effects for understanding processes governed by such effects in society.
Source: §4 · p. 10With a Heaviside modulation, which is not positive-definite, the dynamics need not converge to global consensus and can instead settle into a polarized state.
Source: §3.5 · p. 10
Abstract
from arXiv · showhide
We introduce and analyse a three-body consensus model (3CM) for non-linear consensus dynamics on hypergraphs. Our model incorporates reinforcing group effects, which can cause shifts in the average state of the system even in if the underlying graph is complete (corresponding to a mean-field interaction), a phenomena that may be interpreted as a type of peer pressure. We further demonstrate that for systems with two clustered groups, already a small asymmetry in our dynamics can lead to the opinion of one group becoming clearly dominant. We show that the nonlinearity in the model is the essential ingredient to make such group dynamics appear, and demonstrate how our system can otherwise be written as a linear, pairwise interaction system on a rescaled network.
1 Introduction
Network models commonly represent dynamics through pairwise interactions, but such models may miss phenomena produced by group influence and reinforcement. These higher-order effects are relevant to social dynamics and may help explain polarization and echo chambers.
1 Introduction
- Networked dynamics are often described by differential equations for pairwise interactions, with node states and an adjacency matrix specifying the network.The interaction function describes how two nodes affect one another.Source: §1 · p. 1
- Opinion formation models studied on networks include the de Groot model, bounded-confidence models, and threshold models.Diffusion and oscillator dynamics are also examples of networked dynamics.Source: §1 · p. 2
- Pairwise interaction models may not capture phenomena involving group interactions, where the basic unit of interaction includes more than two nodes.Examples discussed include collaborations and neuronal activity,.Source: §1 · p. 2
- Group influence and reinforcement may be important in social dynamics and may contribute to higher-order effects such as hate communities, echo chambers, and polarization.These dynamics are described as not merely based on pairwise interactions.Source: §1 · p. 2
2 A Multi-body interaction model for non-linear consensus
The paper introduces a three-body consensus model in which node triplets interact through a modulated influence function. The formulation allows reinforcing effects while preserving specified state symmetries, under a simplifying assumption of identical interactions across triplets.
2 A Multi-body interaction model for non-linear consensus
- The three-body consensus model (3CM) describes opinion formation through interactions among triplets of nodes.The triplet interactions are governed by differential equations.Source: §2 · p. 2
- Each triplet's influence combines the joint effect of two nodes on a third with a modulation based on the difference between the first two nodes' states.The interaction function is assumed to be the same for every interacting triplet.Source: §2 · p. 2
- When the modulation function decreases with state difference, two nodes with similar states reinforce their influence, while dissimilar states diminish it.This behavior is reminiscent of nonlinear voter models.Source: §2 · pp. 2–3
- For scalar node states, the model is invariant to a global shift or sign change, corresponding to its translation and rotation invariance.The interaction function is described as quasi-linear despite its modulation.Source: §2 · p. 3
3 Results
The 3CM can be rewritten as a weighted pairwise system, but nonlinear reinforcement creates higher-order effects and can shift consensus. In clustered systems, interaction asymmetry and the modulation function shape both the consensus value and its timescale.
3.1 Reduction to network model and higher-order effects
- The model can be expressed using a time-varying, state- and topology-dependent weighted adjacency matrix, but constant modulation reduces it to a rescaled linear pairwise system.Thus, multi-body effects beyond rescaled pairwise interactions require nonlinear modulation.Source: §3.1 · pp. 3–4
3.2 Convergence to consensus and average-opinion dynamics
- For positive modulation, global consensus is a fixed point and convergence is guaranteed; the average opinion is not generally conserved, but is conserved when the induced Laplacian is balanced.Constant modulation yields linear dynamics with a symmetric Laplacian.Source: §3.2 · p. 4
3.3 Shifts towards majority opinions on complete graphs
- With decreasing modulation on a complete hypergraph, an initial imbalance drifts further toward the majority opinion, while linear dynamics conserve the average.The shift may be interpreted as peer pressure.Source: §3.3 · p. 5, Figure 1 · p. 5
3.4 Opinion dynamics in clustered systems
- In clustered systems, triplet orientation changes relative cluster influence and can shift consensus toward either cluster; even small topology imbalances can produce large consensus differences.A minority cluster can dominate under nonlinear reinforcement, whereas linear coupling leaves the majority opinion dominant.Source: §3.4 · pp. 6–7
3.5 Time-scale separations in clustered systems and multi-body interactions
- Clustered systems show fast within-cluster convergence followed by slower inter-cluster dynamics; with a Heaviside modulation, inter-cluster dynamics become linear and conserve the average once cluster means are closer than the threshold.The Heaviside function is not positive-definite, so global consensus is not guaranteed; separated clusters can remain polarized.Source: §3.5 · pp. 8, 10, Figure 5 · p. 9
4 Conclusion
The paper concludes that nonlinear, reinforcing multi-body interactions can shift average opinion and allow one of two polarized groups to dominate. Unlike linear pairwise models, outcomes depend on both network structure and initial states; the authors identify control strategies as future work.
4 Conclusion
- Reinforcing nonlinear multi-body interactions can shift average opinion and make one of two connected, polarized groups clearly dominant.The authors state that these phenomena do not appear in corresponding pairwise systems.Source: §4 · p. 10
- Unlike standard linear two-body models, the 3CM's asymptotic behavior depends on initial states as well as network structure, allowing relatively small coherent groups to dominate.The authors suggest that seeding initial states or changing topology could steer opinion toward a desirable outcome.Source: §4 · p. 10
- Future work includes developing computable control heuristics and higher-order centralities for identifying influential actors and connections.The proposed matrix representation W is a possible starting point, though the authors note it assumes a known fixed initialization.Source: §4 · p. 11