The power of gaps: Multilayer mechanisms and source tracing of complex dynamics in higher-order networks

Higher-order networks, as an emerging tool for describing multi-body interaction relationships, have achieved significant progress in complex propagation research in recent years. While higher-order networks have advanced the modeling of complex contagion, existing studies predominantly focus on densely connected groups, overlooking the critical role of sparsely interacting regions. This study introduces the novel topological concept of "gaps"—sparse, non-higher-order interaction zones—uncovering their profound regulatory influence on propagation dynamics. We partition higher-order networks into three distinct topological domains: higher-order interaction regions governed by strong group coupling, gap regions dominated by weak higher-order structures, and transition regions that bridge the two via hybrid boundary structures. Based on this ternary classification, we ran the SIS complex diffusion model on higher-order networks. Our key findings reveal that the presence of gaps induces striking propagation asymmetry: initial infections originating in interaction regions lead to explosive outbreaks, while those starting in gap regions cause significantly delayed yet ultimately large-scale surges. Crucially, boundary structures within transition regions act as functional conduits, enabling bidirectional signal transfer and giving rise to complex phenomena like discontinuous transitions and bistability. To quantify these dynamics, we develop an innovative probabilistic framework that links topological properties to structure-specific existence probabilities using concise mathematical notation. This framework not only facilitates efficient structural counting from higher dimensions but also enables effective tracking of contagion processes and reverse tracing of the initial infection source ("seed" node). In summary, our work elucidates the multilayer mechanisms governing complex contagion in higher-order networks, highlighting the indispensable role of gaps and boundary structures, and provides both a unified theoretical foundation and novel technical tools for understanding propagation diversity and optimizing early-warning systems.