Interior--Boundary Assortativity Profiles on Networks and Applications to SIS Epidemic Dynamics

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.