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SOE: Fachverband Physik sozio-ökonomischer Systeme

SOE 18: Networks - From Topology to Dynamics II (joint SOE/DY/BP)

SOE 18.1: Vortrag

Donnerstag, 19. März 2020, 17:30–17:45, GÖR 226

Exact Ising partition function computed on networks of low tree-width — •Konstantin Klemm — IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, Palma de Mallorca, Spain

Tree-like approximation is a method commonly used in computing dynamic properties of quenched finite network realizations, including empirical networks. Such properties include expected percolation cluster sizes, epidemic thresholds, and Ising/Potts partition functions. That method is exact only when the network is a tree: removal of one node leaves the network disconnected and this separation recursively holds on the connected components obtained, until reaching the base case of a component with two nodes only. A generalization of this recursive separation is called tree-decompsition of width k, allowing a set of up to k nodes as a separator in each step. In this talk, we show the use of tree-decompositions to obtain exact equilibrium properties for the Ising model and other stochastic processes with detailed balance. On empirical networks of up to 1000 edges, it takes a few seconds to compute the exact value of the Ising/Potts partition function at a given temperature. Computation time is proven linear in size for networks grown by attachment to cliques, such as the Klemm-Eguíluz model [PRE, 2002] and the simplest scale-free network [Dorogovtsev et al, PRE, 2001]. Next to these results, we discuss possibilities and obstacles in generalizing the concept to non-equilibrium processes.