Phase transitions in Ising models on directed networks

Abstract

We examine Ising models with heat-bath dynamics on directed networks. Our simulations show that Ising models on directed triangular and simple cubic lattices undergo a phase transition that most likely belongs to the Ising universality class. On the directed square lattice the model remains paramagnetic at any positive temperature as already reported in some previous studies. We also examine random directed graphs and show that contrary to undirected ones, percolation of directed bonds does not guarantee ferromagnetic ordering. Only above a certain threshold can a random directed graph support finite-temperature ferromagnetic ordering. Such behavior is found also for out-homogeneous random graphs, but in this case the analysis of magnetic and percolative properties can be done exactly. Directed random graphs also differ from undirected ones with respect to zero-temperature freezing. Only at low connectivity do they remain trapped in a disordered configuration. Above a certain threshold, however, the zero-temperature dynamics quickly drives the model toward a broken symmetry (magnetized) state. Only above this threshold, which is almost twice as large as the percolation threshold, do we expect the Ising model to have a positive critical temperature. With a very good accuracy, the behavior on directed random graphs is reproduced within a certain approximate scheme.

Agreement dynamics on directed random graphs

“…As we have already mentioned, the present studies were motivated by our earlier work on the Ising model on directed graphs [15]. We analysed the heat-bath dynamics version, with a spin variable s i = ±1 (i = 1, .…”

Bipartitioning of directed and mixed random graphs

“…Placing pN directed links is then equivalent to the following method: for each directed pair of vertices (i, j), place a link from i to j with probability p/N . For such a case, an explicit calculation of generating function is straightforward [26] and one obtains that the average relative size of the largest OUT component g O satisfies the equation…”

Prisoner’s dilemma on directed networks

“…We study the model on two kinds of directed random networks where the neighbors of a vertex are connected through outgoing links starting from a vertex. In the directed out-homogeneous networks [49,43] the number of out-links, z, is the same for all vertices. The out-links are generated by selecting randomly, for each vertex of the network, z other (different) vertices.…”

“…Such higher order approximations were applied before to model dynamical processes in regular lattices [40,41,13,3] and networks with an homogeneous degree structure [42]. In directed lattices with a local tree like structure single-site MF approximations may give surprisingly accurate results [43]. In this work, we present an heterogeneous singlesite and pair MF approximation for the PD game on a generic directed network taking into account the degree heterogeneities and degree correlations.…”