Interior--Boundary Assortativity Profiles on Networks and Applications to SIS Epidemic Dynamics
Moses Boudourides
TL;DR
The paper tackles the limitation of scalar Newman-style assortativity by introducing interior--boundary assortativity profiles for a fixed partition $\mathcal{P}$, yielding a vector of edge-type components that separate interior/interfacial mixing. It proves a profile collapse theorem showing that a single scalar assortativity is a weighted sum of type-restricted components plus a between-type term, meaning the scalar can mask interface structure. By coupling these profiles with a nonlinear SIS epidemic model, it shows that boundary-dominant endemic equilibria induce a strictly negative boundary-to-interior component $r_{B\to I}(a)$, providing a dynamical mechanism linking conductance bottlenecks to signed assortativity. This framework thus treats assortativity as a dynamical observable tied to partition geometry and flow structure, with implications for SBMs and other nonlinear processes on networks.
Abstract
We introduce interior-boundary assortativity profiles as a structural refinement of Newman's assortativity coefficient and show that they arise naturally from epidemic dynamics on networks. Given a fixed partition of the node set, edges are stratified according to whether their endpoints are interior or boundary nodes relative to the partition, yielding type-restricted assortativity components. We prove an exact decomposition theorem showing how classical scalar assortativity collapses heterogeneous interior-boundary interactions into a single number. We then study a SIS epidemic model and consider equilibrium infection probabilities as node attributes. Under mild connectivity and positivity assumptions, we show that boundary dominance (a dynamical concentration of infection mass on interface nodes) implies a strictly negative boundary-to-interior assortativity component. This establishes a rigorous link between directed conductance, equilibrium flow geometry, and the sign structure of assortative mixing induced by the dynamics. Our results demonstrate that assortativity profiles encode dynamical information invisible to scalar summaries and provide a mathematically grounded bridge between network partition geometry and nonlinear dynamics on graphs.