Discrete conservation laws and port-Hamiltonian systems on graphs and complexes

In this paper we present a unifying geometric framework for modeling various sorts of physical network dynamics as port-Hamiltonian systems. Basic idea is to associate with the incidence matrix of the graph a Dirac structure relating the flow and effort variables associated to the edges, internal vertices, and boundary vertices of the graph. This Dirac structure captures the basic conservation/balance laws of the system. Examples from different origins such as consensus algorithms and coordination control strategies for multi-agent systems share the same structure. The framework is extended to k-complexes primarily motivated by the discretization of continuous conservation laws.