A dynamic mechanism for prevalence of triangles in competitive networks

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.

Figures (8)

• Figure 1: Least abundant species vs. coupling strength. The dependence of the minimum-abundance species value $x_{\min}$ at equilibrium on the coupling strength $\tau$ is shown for three graphs with 6 nodes: complete, path, and star. Abrupt jumps in the equilibrium branch appear as discontinuities, occurring at the critical coupling where the equilibrium loses feasibility.
• Figure 2: Bounds for $\tau_{\mathrm{c}}$, the maximum coupling permitting coexistence. Across all graphs $G$, the lower and upper bounds are attained at $G^-$ and $G^+$, respectively, where the definitions of $G^-$ and $G^+$ depend on the constraint imposed on $G$ -- either a fixed number of nodes or a fixed maximum degree, as explained in Theorem \ref{['th:general']}.
• Figure 3: The double star with $\Delta = 3$. Shaded nodes are the hubs where $\mathbf{x^*}$ vanishes first.
• Figure 4: Illustration of the degree-preserving edge-swap (rewiring) operation. Two edges connecting nodes $(1,2)$ and $(3,4)$ are replaced by either crossed or switched configurations, conserving the degree of each node while altering network topology.
• Figure 5: Degree-preserving rewiring that increases clustering also increases the critical coupling. Single network with $n=14$ nodes and $m=20$ edges. The rewiring number counts successive degree-preserving edge swaps relative to the initial network (0); positive values increase the mean clustering coefficient and negative values decrease it. a, Representative networks along the rewiring trajectory. The dashed vertical line marks the initial network. Purple (left) indicates reduced clustering and green (right) increased clustering. b, Critical coupling $\tau_{\text{c}}$ (left axis) and mean clustering coefficient (right axis) versus rewiring number.

Theorems & Definitions (10)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1
• proof
• Remark 1
• Lemma 3
• proof