Equilibrium statistical mechanics of network structures Physica A 334, 583 (2004); PRE 69, (2004); cond-mat ; cond-mat Thanks to: A.-L. Barabási and G. Tusnády Illés FarkasImre Derényi Tamás VicsekGergely Palla Dept. of Biological Physics, Eötvös Univ.

Introduction Growth and restructuring are the two basic phenomena that shape the structure of a network. Although the properties of networks have mostly been analyzed by the tools of statistical physics, only very few recent works have tried to make a connection with equilibrium thermodynamics. Motivations:  Restructuring is often much faster than growth (allowing enough time for equilibration).  By changing the noise level (temperature) of the restructuring, topological phase transition might occur. Equilibrium ensembles are defined as stationary ensembles of networks generated by restructuring processes obeying  ergodicity; and  detailed balance:

Equilibrium ensembles with energy For simplicity, we consider the networks as undirected simple graphs, and their edges (links) are treated as particles. The number of vertices N (which is analogous to the volume) is fixed. Energy can be obtained from  optimization (cost function for the deviations);  the transition rates by reverse engineering (G. Palla, June 8);  trial and error. Micro-canonical ensemble:(for fixed E and M) Canonical ensemble:(for fixed M) Grand-canonical ensemble:

Equilibrium ensembles without energy If the graphs have unequal weights and a fixed number of edges. Micro-canonical ensemble: Canonical ensemble: Grand-canonical ensemble: If every allowed graph has equal weight (e.g. ER graphs). If the graphs have unequal weights and different number of edges. The following energy function can be constructed:

Conditional free energy To be able to monitor topological phase transitions a suitable order parameter  has to be introduced. Possible choices are: The corresponding conditional free energy F( ,T) is defined through: In the thermodynamic limit the most probable value of  is determined by the minimum of F( ,T).

Percolation transition in the Erdős-Rényi graph Even in the micro-canonical ensemble a topological phase transition occurs as a function of the average degree, at.

Global energy If a second order topological phase transition occurs at the critical temperature: For the exponentially decaying probability of large clusters can be compensated by a monotonically decreasing energy function.

Local energy To promote compactification f ( k ) should decrease faster than linear.

f (k)=  k 2 /2 or g (k)=  k/2 If one vertex has already accumulated most edges, then which leads to a parabola with a maximum at  k = (T / M) ln(N). Thus, for T < M / ln(N) the conditional free energy has two minima (at  k = 0 and  k = 1 ), indicating a first order phase transition between compact and disordered topologies.

f (k)=  k ln(k) or g (k)=  ln(k) For this type of local energy, the conditional free energy becomes which is a descending or ascending straight line, depending on whether T 1. The lack of hysteresis indicates that the compact-disordered transition at T = 1 is of second order. To be continued by Illés Farkas...