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Wednesday April 29, 2026

PhD Candidacy
Dynamics Shape the Perception of Network Structure
- Callie Reid,
Time: 10:00 am Room: 204A
Abstract/Desc: Dynamical processes can fundamentally reshape how network structure is perceived, with functional organization emerging even when topological modularity is weak or absent. We investigate this phenomenon in nonlinear random walks—mean-field models of diffusion with volume exclusion that are nonlinear on heterogeneous networks and linear otherwise. For connected undirected graphs, we analyze the rate of convergence to equilibrium for n interacting walkers and introduce a localization-based reduction exploiting the spontaneous concentration of Laplacian eigenvectors on specific nodes in large random graphs. We show analytically, and validate numerically, that in scale-free networks the Jacobian spectrum of the nonlinear random walk undergoes a characteristic \emph{spectral deformation}: eigenvalues associated with peripheral, low-degree nodes are compressed toward the origin, while those linked to hubs are systematically displaced outward. This nonlinear spectral reorganization reveals how heterogeneity and crowding jointly redefine the network's effective dynamical structure, offering a principled mechanism for dynamic modularity in complex systems. Building on this perspective, we introduce weighted-node paths in which nodes are weighted according to their crowding capacity, allowing dynamical proximity to be quantified through path-based measures that naturally extend classical closeness and betweenness centrality to account for congestion effects induced by the dynamics.

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This Week in Mathematics