Mean field solution of the Ising model on a Barabási-Albert network. Ginestra Bianconi arXiv:cond-mat/0204455v4 [cond-mat.stat-mech] 17 Sep 2002 Department of Physics, University of Notre Dame, Notre Dame,Indiana 46556,USA The mean field solution of the Ising model on a Barabási-Albert scale-free network with ferromagnetic coupling between linked spins is presented. The critical temperature Tc for the ferromagnetic to paramagnetic phase transition ( Curie temperature) is infinite and the effective critical temperature for a finite size system increases as the logarithm of the system size in agreement with recent numerical results of Aleksiejuk, Holyst and Stauffer. ——————– PACS numbers: 89.75.-k, 89.75.Hc, 05.40.-a Keywords: Networks, Curie temperature, Mean field approximation, Critical phenomena E-mail Address: gbiancon@nd.edu ——————– Pj sum The Ising model is the starting point for the study of α=1 kα with 2mtj and substituting into (1) the dynamic solution for the connectivity [7] second order phase transitions and cooperative phenomena. Traditionally it has been solved exactly on periodic r t lattices in one and two dimensions [1] or in a Bethe lattice , (2) ki (t) = m t i [2]. Recently attention has been addressed on the complex structure of scale-free networks [3] with power-law we obtain connectivity distribution P (k) ∼ k −γ . These networks m 1 describe for example biological systems where proteins . (3) pi,j = √ 2 ti tj are nodes and physical protein interactions are the links [4]. Moreover they constitute an intriguing substrate for The adjacency elements of the network ǫi,j are equal one the spreading of infectious diseases [5], and for percolaif there is a link between node i and j and zero othertion phenomena [6]. The Ising model has been applied to wise. Consequently their mean over many copies of the the evolving scale-free network with γ = 3 [the BarabásiBA network has a tensor structure: Albert (BA) network [7]] by Aleksiejuk et al. (AHS) [8] [ǫi,j ] = pi,j and to random graph with arbitrary degree distribution [9,10]. By introducing a spin on each node of the netm 1 = √ work, they found that ordering occurs for temperatures 2 ti tj T below the effective critical temperature Tc , and that 1 ki kj . (4) = the nature of the phase transition is very different from 2mN that on periodic lattices. The Ising-type spins si = ±1 placed on the nodes of a In this work I show that a mean field solution of the BA network are subjected to the Hamiltonian Ising model on the BA network can be treated as a MatX X tis model [11] with the substitution ξi → ki and that hi si (5) Ji,j si si − H=− the scaling of the effective critical temperature Tc for feri i,j romagnetic transition increases as the logarithm of the system size in agreement with recent numerical calculawith the local magnetic field hi and the coupling Ji,j non tions [8]. zero only for nodes i and j connected by a link: Let us consider a BA network of N nodes. StartJi,j = Jǫi,j (6) ing from a small number of nodes n0 and links m0 (n0 , m0 << N ), the network is constructed iteratively were J > 0, and ǫi,j is the adjacency matrix of the BA by the constant addition of nodes with m links. The new network. links are attached preferentially to well connected nodes The exact solution of the Hamiltonian is given by in such a way that at time tj the probability pij that the X new node j is linked to node i with connectivity ki (tj ) is Ji,j sj + hi )] > . (7) < si >=< tanh[β( given by j ki (tj ) . pi,j = m Pj α=1 kα The mean field equation for the mean local magnetization < si > is given by (1) If N is large we can approximate the total number of edges present in the network at time tj , given by the < si >= tanh[β(J N X j=1 1 [ǫi,j ] < sj > +hi )] (8) where we approximated < tanh(x) > with tanh(< x >) and we performed the mean of the adjacency matrix [ǫi,j ] over the different realization of the network. If we define S by S= N 1 X ki < si > 2mN i=1 (9) after substituting [ǫi,j ] by (4) we can rewrite Eq.(8) < si >= tanh[β(Jki S + hi )]. (10) Multiplying both sides of the equation by ki /2mN and summing over i we can solve (10) self-consistently for S, N 1 X S= ki tanh[β(Jki S + hi )]. 2mN i=1 FIG. 1. The analytical prediction for the ferromagnetic effective critical temperature (solid line) is compared with the AHS simulations (open circles) with m = 5 as a function of the network size. (11) The effective critical temperature– Using S as the order parameter of the system, and its self-consistent definition Eq.(11), it is easy to find the effective critical temperature (Tc ). As S ∼ 0 we can approximate the tanh(x) with x and β with βc giving, Z N 1 S= dt′ βc JSk 2 (t′ ) 2mN 1 m = Jβc S ln(N ), (16) 2 Thus the BA Ising model looks like a Mattis model [11] with the substitution ξi → ki . (12) This is interesting because the Mattis model is a simple solvable model of spin-glass [12] and suggest that in a spin-glass on a scale-free network there will be a noise term deriving both from the geometry of the system and the random interactions. i.e. Tc m = ln(N ). J 2 This shows that for finite size system there is an effective critical temperature that increases linearly with the interaction J and logarithmically with the number N of the nodes of the network. This result is in agreement with the numerical results of AHS [8] as it is shown in Fig. 1. In Fig. 1 we have compared the analytical results Eq. (17) for the case m = 5 with the numerical results of AHS. However we know that a finite system cannot have a phase transition, therefore the true critical temperature goes to infinite in the thermodynamic limit N → ∞. This shows that in a scale-free network the ordered phase is the only allowed phase in the thermodynamic limit. I am grateful to professor A.-L. Barabási for useful discussion and help. This work was supported by NSF. The susceptibility– The susceptibility of the order parameter S, describing the response to a uniform magnetic field h, is given by ∂S 1 = β ∂h 2mN (1 − 2 i ki (1− < si > ) P 2 1 i ki (1− < si 2mN βJ P >2 )) . (13) If we define the mean magnetization M as M= the susceptibility χ = χ= 1 X < si >, N i ∂M ∂H (17) (14) is given by ∂S 1 X (1− < si >2 )[βJki + β] N i ∂h [1] J. M. Yeomans ’Statistical mechanics of phase transitions ’ (ed. Oxford Science Publications) (1992). (15) 2 [2] M. F. Thorpe, in Excitations in Disordered Systems, ed. M. F. Thorpe (Plenum press, New York & London,1982). [3] R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002). [4] H. Jeong, S. P. Mason, A.-L. Barabási and Z. N. Oltvai Nature 411, 41 (2001). [5] R. Pastor-Satorras and A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001). [6] R. Cohen, K. Erez, D. ben-Avraham and S. Havlin, Phys. Rev. Lett. 85, 4626 (2000); R. Cohen, K. Erez, D. benAvraham and S. Havlin, Phys. Rev. Lett. 86, 3682 (2001). [7] A.-L. Barabási and R. Albert, Science 286, 509 (1999). [8] A. Aleksiejuk, J. A. Holyst and D. Stauffer, Physica A 310, 260 (2002). [9] S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, Phys. Rev. E 66, 016104 (2002). [10] M. Leone, A. Vazquez, A. Vespignani and R. Zecchina, Eur. Phys. J. B 28,191 (2002). [11] D. C. Mattis Phys. Lett. 56 A, 421 (1976). [12] D. Amit, Modeling brain function,(Cambridge University Press, Cambridge, 1989) 3