1. Equilibrium statistical mechanics of network structures Physica A 334, 583 (2004); PRE 69, 046117 (2004); cond-mat 0401640; cond-mat 0405399. Thanks to: A.-L. Barabási and G. Tusnády Illés FarkasImre Derényi Tamás VicsekGergely Palla Dept. of Biological Physics, Eötvös Univ.

2. 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:

3. 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:

4. 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:

5. 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).

6. 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.

7. 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.

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

9. 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.

10. 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...