Localized complex dynamics in higher-order networks: Heterogeneous topological coupling and effective degree models

Higher-order networks have become a vital tool for delineating multi-body interactions in complex systems, offering unique capabilities for modeling contagion dynamics. However, existing research largely focuses on higher-order structures between nodes, frequently neglecting the significant role of non-higher-order areas, or "gap regions" in spreading processes. To address this, we propose a new framework that partitions networks into distinct "higher-order interaction regions" and "gap regions", connected by "transition zones". This approach facilitates a principled distinction between emergent interactive structures and conventional network constructs based on intrinsic properties. We further introduce an innovative effective degree model to refine node classification based on regional contexts and local topology. Building on this, we establish a multi-layered contagion process where infection rates depend on both the scale of neighborhood infection and microscopic transmission rates via distinct pathways. To enhance tractability, a closure-based self-consistent method is developed to transform higher-order traits into feasible closed forms, allowing systematic tracking of complex dynamics. SIS rumor propagation simulations on synthetic and empirical networks, with seeds in various regions, show that the initial source profoundly impacts the process—specifically regarding prevalence, diffusion radius, and diameter. Notably, seeding in transition zones causes more explosive and widespread outbreaks. This study deepens our understanding of dynamic diversity in higher-order networks and offers quantitative tools for strategies in rumor control and epidemic warning.