International Workshop on Complex Networks, Seoul (23-24 June 2005) Vertex Correlations, Self-Avoiding Walks and Critical Phenomena on the Static Model of Scale-Free Networks DOOCHUL KIM (Seoul National University) Collaborators: Byungnam Kahng (SNU), Kwang-Il Goh (SNU/Notre Dame), Deok-Sun Lee (Saarlandes), Jae- Sung Lee (SNU), G. J. Rodgers (Brunel) D.H. Kim (SNU)

International Workshop on Complex Networks, Seoul (23-24 June 2005) Outline I.Static model of scale-free networks II.Vertex correlation functions III.Number of self-avoiding walks and circuits IV.Critical phenomena of spin models defined on the static model V.Conclusion

International Workshop on Complex Networks, Seoul (23-24 June 2005) I.Static model of scale-free networks static model

International Workshop on Complex Networks, Seoul (23-24 June 2005) static model  We consider sparse, undirected, non-degenerate graphs only. 1.Degree of a vertex i : 2.Degree distribution: = adjacency matrix element (0,1)

International Workshop on Complex Networks, Seoul (23-24 June 2005) static model  For theoretical treatment, one needs to take averages over an ensemble of graphs G.  Grandcanonical ensemble of graphs:  Static model: Goh et al PRL (2001), Lee et al NPB (2004), Pramana (2005, Statpys 22 proceedings), DH Kim et al PRE(2005 to appear) Precursor of the “hidden variable” model [Caldarelli et al PRL (2002), Soederberg PRE (2002), Boguna and Pastor-Satorras PRE (2003)]

International Workshop on Complex Networks, Seoul (23-24 June 2005) static model 1.Each site is given a weight (“fitness”) 2.In each unit time, select one vertex i with prob. P i and another vertex j with prob. P j. 3.If i=j or a ij =1 already, do nothing (fermionic constraint). Otherwise add a link, i.e., set a ij =1. 4.Repeat steps 2,3 NK times ( K = time = fugacity =  L  /N ).  Construction of the static model m = Zipf exponent When m =0  ER case. Walker algorithm (+Robin Hood method) constructs networks in time O(N).  N=10 7 network in 1 min on a PC. Comments

International Workshop on Complex Networks, Seoul (23-24 June 2005) static model  Such algorithm realizes a “grandcanonical ensemble” of graphs G ={ a ij } with weights   Each link is attached independently but with inhomegeous probability f i,j.

International Workshop on Complex Networks, Seoul (23-24 June 2005) static model Degree distribution Percolation Transition

International Workshop on Complex Networks, Seoul (23-24 June 2005) static model Recall When l>3 ( 0<m<1/2 ), When 2<l<3 ( 1/2<m<1 ) f ij  Comments l3-l 3-l3-l f ij  2KNP i P j f ij  1 Strictly uncorrelated in links, but vertex correlation enters (for finite N) when 2< l <3 due to the “fermionic constraint” (no self- loops and no multiple edges).

International Workshop on Complex Networks, Seoul (23-24 June 2005) II. Vertex correlation functions Vertex correlations Related work: Catanzaro and Pastor-Satoras, EPJ (2005)

International Workshop on Complex Networks, Seoul (23-24 June 2005) Vertex correlations Simulation results of k nn (k) and C(k) on Static Model

International Workshop on Complex Networks, Seoul (23-24 June 2005) Vertex correlations Our method of analytical evaluations: For a monotone decreasing function F(x), Use this to approximate the first sum as Similarly for the second sum.

International Workshop on Complex Networks, Seoul (23-24 June 2005) Vertex correlations Result (1) for

International Workshop on Complex Networks, Seoul (23-24 June 2005) Vertex correlations Result (2) for

International Workshop on Complex Networks, Seoul (23-24 June 2005) N l Result (3) Finite size effect of the clustering coefficient for 2< l <3: Vertex correlations

International Workshop on Complex Networks, Seoul (23-24 June 2005) III.Number of self-avoiding walks and circuits Number of SAWs and circuits The number of self-avoiding walks and circuits (self-avoiding loops) are of basic interest in graph theory. Some related works are: Bianconi and Capocci, PRL (2003), Herrero, cond-mat (2004), Bianconi and Marsili, cond-mat (2005) etc. Issue: How does the vertex correlation work on the statistics for 2< l <3 ?

International Workshop on Complex Networks, Seoul (23-24 June 2005) Number of SAWs and Circuits The number of L-step self-avoiding walks on a graph is where the sum is over distinct ordered set of (L+1) vertices, We consider finite L only. The number of circuits or self-avoiding loops of size L on a graph is with the first and the last nodes coinciding.

International Workshop on Complex Networks, Seoul (23-24 June 2005) Number of SAWs and Circuits Strategy for 2< l <3: For upper bounds, we use repeatedly. Similarly for lower bounds with The leading powers of N in both bounds are the same. Note: The “surface terms” are of the same order as the “bulk terms”. and

International Workshop on Complex Networks, Seoul (23-24 June 2005) Number of SAWs and Circuits Result(3): Number of L-step self-avoiding walks For, straightforward in the static model For, the leading order terms in N are obtained.

International Workshop on Complex Networks, Seoul (23-24 June 2005) Number of SAWs and Circuits Typical configurations of SAWs (2<λ<3) L=2 L=3 L=4 H HH SS BB BB BB SS

International Workshop on Complex Networks, Seoul (23-24 June 2005) Number of SAWs and Circuits Results(4): Number of circuits of size

International Workshop on Complex Networks, Seoul (23-24 June 2005) IV.Critical phenomena of spin models defined on the static model Spin models on SM

International Workshop on Complex Networks, Seoul (23-24 June 2005) Spin models on SM Spin models defined on the static model network can be analyzed by the replica method.

International Workshop on Complex Networks, Seoul (23-24 June 2005) Spin models on SM The effective Hamiltonian reduces to a mean-field type one with infinite number of order parameters, When J i,j are also quenched random variables, do additional averages on each J i,j.

International Workshop on Complex Networks, Seoul (23-24 June 2005) Spin models on SM We applied this formalism to the Ising spin-glass [DH Kim et al PRE (2005)] Phase diagrams in T-r plane for l > 3 and l <3

International Workshop on Complex Networks, Seoul (23-24 June 2005) Spin models on SM Critical behavior of the spin-glass order parameter in the replica symmetric solution: To be compared with the ferromagnetic behavior for 2< l <3;

International Workshop on Complex Networks, Seoul (23-24 June 2005) V. Conclusion 1.The static model of scale-free network allows detailed analytical calculation of various graph properties and free-energy of statistical models defined on such network. 2.The constraint that there is no self-loops and multiple links introduces local vertex correlations when l, the degree exponent, is less than 3. 3.Two node and three node correlation functions, and the number of self-avoiding walks and circuits are obtained for 2< l <3. The walk statistics depend on the even-odd parity. 4.The replica method is used to obtain the critical behavior of the spin-glass order parameters in the replica symmetry solution.

International Workshop on Complex Networks, Seoul (23-24 June 2005) Static Model N=3 1 Efficient method for selecting intergers 1, 2, , N with probabilities P 1, P 2, , P N. Walker algorithm ( )