Evolution of Scale-Free Random Graphs: Potts Model Formulation

1. Evolution of Scale-Free Random Graphs:Potts Model Formulation STATPHYS22, Bangalore July 5, 2004 DOOCHUL KIM (Seoul National University) Collaborators: Byungnam Kahng (SNU) Kwang-Il Goh (SNU) Deok-Sun Lee (Saarlandes)

2. Plan • Introduction • Static model • Potts model representation • Thermodynamic limit • Finite-size effect • Conclusion

3. Introduction • Scale-free (SF) networks: frequently encountered in Nature. G • Degree of a vertex i: [aij= adjacency matrix element(0,1)] • Degree distribution: • We consider sparse, undirected, non-degenerate graphs only.

4. Introduction Newman et al. PRE (2001)Burda et al. PRE (2001); (2003a,b)Goh et al. PRL (2001)Caldarelli et al. PRL (2002)Chung & Lu Ann. Combinat. (2002)Söderberg PRE (2002)Berg & Lässig PRL (2002)Dorogovtsev et al. Nucl. Phys. B (2003) • Recent references Park & Newman PRE (2003); cond-mat/0405566Farkas et al. cond-mat/0401640 Lee et al. Nucl. Phys. B (2004) • Ensemble of graphs: • Various equilibrium ensembles for SF networks • Microcanonical ensemble with given degree sequence {ki}. • Canonical ensemble with fixed N&L. • Grandcanonical ensemble with fixed N but with fluctuating L. Goh et al. PRL (2001) Lee et al. Nucl. Phys. B (2004)

5. Static Model • Static model of Goh et al.  Evolution of SF random graphs visualization demo • Each site is given a weight (“fitness”) • In each unit time, select one vertex iwith prob. Pi and another vertex j with prob. Pj. • If i=j or aij=1 already, do nothing (fermionic constraint).Otherwise add a link, i.e., set aij=1. • Repeat steps 2,3 NK times (K = time = fugacity = L/N). m = Zipf exponent = 1/(l-1) • “Evolution of Random Graphs” by Erdős-Rényi (ER) • Links are connected with equal probability. • Equivalently, links are randomly attached one by one. • Percolation transition when L=N/2 (appearance of the giant cluster).

6. When m=0  ER case. • Walker algorithm (+Robin Hood method) constructs networks in time O(N). N=107 network in 1 min in a PC. • Monte Carlo simulation with edge addition (deletion) prob. fij (1-fij)  equivalent but inefficient. Static Model Comments • Such algorithm realizes a “grandcanonical ensemble” of graphs G={aij} with weights

7. Static Model 1 fij2KNPiPj 3-l fij1 1 3-l 0 • Recall • When l>3 (0<m<1/2), fij2KNPiPj. • When 2<l<3 (1/2<m<1)fij Comments • Bosonic model (allow multiple links) Prob(aij=n) is Poissonian with aij=2NKPiPj.

8. Potts model representation Hamiltonian: Boltzmann weight: Partition function: Potts model order parameter: ’ Potts model susceptibility mean cluster size • Potts model  Bond percolation through Kasteleyn construction Potts spin: si= 1, 2,  , q.

9. Potts model representation is NOT the Potts model on a scale-free network, but on the complete graph as a tool to generate SF networks. It is NOT the Hamiltonian defining the equilibrium ensemble In our case, Comments

10. Thermodynamic limit • Percolation transition at l>4 l=4.8 3<l<4 l=3.6 2<l<3 l=2.4 Exact analytic evaluation of the Potts free energy: • Vector spin representation  • Integral representation of the partition function • Saddle-point analysis • Explicit evaluations of thermodynamic quantities.

11. Thermodynamic limit Cluster size distributions Branching process approach [cf. Newman et al. 2001] becomes exact (almost no loops in finite clusters). cf. Cohen et al. (2002) for site percolation • Giant cluster size at Kc(N)

12. Finite-size effect 1. Finite size scaling for l>4 (I) and 3<l<4 (II)

13. Finite-size effect 2. Double peaks: 2<l<3 Double peaks in

14. Finite-size effect N N l 2 l 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 3. Clustering coefficient: Clustering coefficient

15. Conclusion • A simple algorithm (“static model”) is introduced to generate SF graphs. • Corresponding equilibrium ensemble is identified and is related to the Potts model with inhomogeneous interactions on complete graph. • Several explicit analytical results are obtained in the thermodynamic limit. • Finite-size scaling near the percolation transition is constructed & tested numerically. • Evolution of the SF random graphs in “time,” i.e., as the # of links increases, shows distinct behaviors for 2<l<3. • Some statistical mechanical problems on the static model can be handled analytically (Work in progress).

16. Conclusion percolating phase • A simple algorithm (“static model”) is introduced to generate SF graphs. • Corresponding equilibrium ensemble is identified and is related to the q=1 limit of the Potts model with inhomogeneous interactions on a complete graph. • Explicit analytical results are obtained in the thermodynamic limit. • Finite-size scaling near the percolation transition is constructed & tested numerically. • Evolution of the SF random graphs in “time,” i.e., as the # of links increases, shows distinct behaviors for 2<l<3. • Some statistical mechanical problems on the static model can be handled analytically (Work in progress). K=L/N l>3 0 scale-free percolating phase K=L/N 2<l<3 0 scale-free