Multilayer network

A multilayer network is a network in which nodes participate in multiple interacting layers simultaneously, each with its own topology and dynamics. The framework generalizes single-layer network theory to capture systems like the global financial system, where institutions are linked through lending, derivatives, equity, and payment networks with different contagion properties. A shock absorbed in one layer may propagate catastrophically in another, making multilayer analysis essential for understanding systems whose connectivity cannot be reduced to a single graph.

The mathematical formalism of multilayer networks extends adjacency matrices to tensor representations that encode both intralayer connections (within a layer) and interlayer connections (between the same node in different layers). This added complexity introduces new phenomena — such as interlayer coupling-dependent percolation and layer-specific cascade dynamics — that have no single-layer analog. The theory remains underdeveloped: while single-layer networks have mature results for robustness, epidemic spreading, and community detection, multilayer counterparts often lack analytical solutions and require computational exploration.