Analysis of Complex Networks

2.3 Microscopics: Hamiltonians of Networks – Network Thermodynamics

We now turn to dynamical restrictions on rewirement imposed by a network Hamiltonian. Network Hamiltonians were recently introduced and studied, e.g., in [16, 18–20]. Here we study the Hamiltonian introduced in [29], which was originally motivated by a standard choice of utility functions in the socioeconomic literature [40, 41]. The basic idea is that a node has more utility (less energy) if it has a link to a more “important” node, where the importance of a node is proportional to its degree. The Hamiltonian that we take as a starting point – for a derivation see [29] – reads

where c stands for the adjacency matrix, l is the summation over all links, and Δk is the absolute value of the difference in degree between a pair of nodes, one of them being l. b is a free parameter related to the detailed structure of the log-utility [40, 41] needed to derive this Hamiltonian. Given a Hamiltonian, the canonical ensemble, given by the partition function

using the usual definition of temperature T = 1/kβ, can be simulated, e.g., by the Metropolis algorithm: starting from an adjacency matrix c at time t, a graph ĉ is generated by replacing a randomly chosen edge between nodes i and j with a new edge between randomly chosen, ...