Statistical-mechanical iterative algorithms on complex networks

Abstract

The Ising models have been applied for various problems on information sciences, social sciences, and so on. In many cases, solving these problems corresponds to minimizing the Bethe free energy. To minimize the Bethe free energy, a statistical-mechanical iterative algorithm is often used. We study the statistical-mechanical iterative algorithm on complex networks. To investigate effects of heterogeneous structures on the iterative algorithm, we introduce an iterative algorithm based on information of heterogeneity of complex networks, in which higher-degree nodes are likely to be updated more frequently than lower-degree ones. Numerical experiments clarified that the usage of the information of heterogeneity affects the algorithm in Barabási and Albert networks, but does not influence that in Erdös and Rényi networks. It is revealed that information of the whole system propagates rapidly through such high-degree nodes in the case of Barabási-Albert's scale-free networks.

A statistical mechanics approach for scale-free networks and finite-scale networks

“…From the derivation of our model it is evident the connection with "ball in the box" problems [23,24,30]. In our approach the "boxes" map to the degree of the nodes and the "balls" map to the edges of the graph.…”

“…The model again map to the urn model [24] with the probability p(k) ∼ k −γ . Finally another urn ("ball in the box" model ) is presented in [23] in which the same mapping between the network and the "ball/boxes" is made but the probability p(k) = (k!) β is chosen.…”

Free Energy of Disordered Urn Models in the Canonical Ensemble

“…Furthermore, the urn models have been used in research fields of complex networks. 7,8 Recently, the analytical treatment for disordered versions of the urn models have been developed. 5, 8, 9 Leuzzi and Ritort have analyzed a disordered urn model (disordered backgammon model) in the scheme of the grand canonical ensemble.…”

“…5,8,9 Leuzzi and Ritort have analyzed a disordered urn model (disordered backgammon model) in the scheme of the grand canonical ensemble. On the other hand, in the scheme of the canonical ensemble, the replica method has been used in order to calculate the occupation distribution of the preferential urn model.…”

Condensation phase transition in nonlinear fitness networks

“…-Condensation phenomena emerge in various physical contexts, to name a few, the wellknown Bose-Einstein condensation (BEC) in dilute atomic gases [1][2][3], jamming in traffic flow [4,5], wealth condensation in macroeconomies [6], and condensation in zerorange process (ZRP, see e.g., recent review [7] and references therein). Since the pioneered research on complex networks [8][9][10][11][12][13][14][15][16], in the last decade, condensation phenomena, i.e., condensation of links (or edges) in complex networks has also been widely discussed [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. In the context of complex networks, the condensation phase corresponds to the situation that a single node captures a macroscopic finite fraction of total links/edges.…”