A dynamic mechanism for prevalence of triangles in competitive networks

TL;DR

This paper proposes that the prevalence of triangles in competitive networks may naturally result from stability requirements in ecological systems, linking network structure to species coexistence and stability.

Contribution

It introduces a novel hypothesis that triangles emerge from stability constraints in competitive interactions, supported by theoretical bounds and network optimization analyses.

Findings

Coexistence guaranteed below a certain coupling strength related to maximum degree

Higher clustering correlates with stronger stability in network models

Real-world plant networks show more clustering and stability than random models

Abstract

Triangles are abundant in real-world networks but rare in standard null models for sparse graphs. Existing explanations typically rely on explicit triadic closure mechanisms or geometry-based connection rules. We propose an alternative hypothesis: the frequent appearance of triangles may arise naturally from the requirement of dynamic stability that maintains coexistence of species in Lotka-Volterra systems with competitive interactions. To evaluate this idea, we show that, across all possible interaction graphs, coexistence is guaranteed whenever the coupling strength is below the reciprocal of the graph's maximum degree. We also show that coexistence can persist up to a critical coupling strength of 1, which leaves a large gap that is unexplained by the graph degrees alone. These lower and upper bounds are achieved for star and complete graphs respectively. To investigate what structural properties of the interaction graph control the critical coupling within the gap, we optimise networks algorithmically while keeping the degree sequence fixed. We find that networks supporting stronger interaction strengths consistently exhibit higher clustering coefficients in several network models. Moreover, in real-world grassland plant networks, we observe higher clustering and stronger stability than those expected from a configuration model with the same degree sequence. Our result suggests that triangles, and clustering in general, may emerge as a structural signature of stabilising competition.