Interconnectivity structure of a general interdependent network.

A general two-layer network consists of two networks G_{1} and G_{2}, whose interconnection pattern is specified by the interconnectivity matrix B. We deduce desirable properties of B from a dynamic process point of view. Many dynamic processes are described by the Laplacian matrix Q. A regular topological structure of the interconnectivity matrix B (constant row and column sum) enables the computation of a nontrivial eigenmode (eigenvector and eigenvalue) of Q. The latter eigenmode is independent from G_{1} and G_{2}. Such a regularity in B, associated to equitable partitions, suggests design rules for the construction of interconnected networks and is deemed crucial for the interconnected network to show intriguing behavior, as discovered earlier for the special case where B=wI refers to an individual node to node interconnection with interconnection strength w. Extensions to a general m-layer network are also discussed.