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/*k*β, 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, ...

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