Topology-Dependent Rivalry Dynamics in Degree- and Spectrum-Controlled Networks

We study how network topology influences frequency–dependent rivalry dynamics in coupled linear oscillator systems, under progressively stronger structural controls. Using degree-matched and Laplacian spectrum-matched null models, we isolate which topological features materially affect rivalry measures beyond trivial degree effects. Networks with fixed mean degree are generated across three canonical families: ring lattices, small-world graphs, and random regular graphs. For each realization, we simulate a linear diffusive network driven at two nearby frequencies and quantify rivalry using phase-locking–based response asymmetry across a controlled frequency detuning grid. We define summary metrics including mean absolute rivalry, collapse depth, and detuning of maximal response. To separate structural mechanisms, we compare real graphs against (i) degree-sequence–preserving nulls and (ii) spectrum-matched nulls constructed via degree-preserving edge swaps optimized to reproduce the lowest non-zero Laplacian eigenvalues. We further measure spectral indicators, including algebraic connectivity and Fiedler vector localization and overlap at driven nodes. Results show that degree matching alone does not remove topology-dependent rivalry effects, while partial spectral matching substantially reduces them in random regular graphs. In contrast, lattice and small-world topologies resist accurate low-eigenvalue spectral matching under degree constraints, indicating that higher-order spectral structure and eigenvector geometry play a functional role. Correlations between rivalry strength, algebraic connectivity, and Fiedler mode localization support this interpretation. These findings demonstrate that rivalry dynamics are sensitive not merely to degree statistics or global spectral density, but to specific low-mode Laplacian structure. The framework establishes a controlled experimental path for linking network topology, spectral geometry, and dynamical competition, providing a foundation for future extensions to nonlinear dynamics and field-theoretic interpretations.