A Long-Range Ising Model of a Barabási-Albert Network
Jeyashree Krishnan, Reza Torabi, Edoardo Di Napoli, Carsten Honerkamp, Andreas Schuppert
AA Long-Range Ising Model of a Barab´asi-Albert Network
Jeyashree Krishnan ∗ and Andreas Schuppert † Aachen Institute for advanced study in ComputationalEngineering Science(AICES) Graduate School,RWTH Aachen University, Germany andJoint Research Center for Computational Biomedicine(JRC-Combine),RWTH Aachen University, Germany
Reza Torabi ‡ Department of Physics and Astronomy,University of Calgary, Calgary, Alberta, Canada
Edoardo Di Napoli § Aachen Institute for advanced study in ComputationalEngineering Science(AICES) Graduate School,RWTH Aachen University, Germany andJ¨ulich Supercomputing Center, Forschungszentrum J¨ulich, J¨ulich, Germany
Carsten Honerkamp ¶ Institute for Theoretical Solid State Physics,RWTH Aachen University, Germany andJARA-FIT, Aachen, Germany a r X i v : . [ c ond - m a t . d i s - nn ] M a y bstract Networks that have power-law connectivity, commonly referred to as the scale-free networks, arean important class of complex networks(Albert 2002). A heterogeneous mean-field approximationhas been previously proposed for the Ising model of the Barab´asi-Albert model of scale-free net-works with classical spins on the nodes wherein it was shown that the critical temperature for sucha system scales logarithmically with network size (Bianconi 2002, Aleksiejuk 2002). For finite sizes,there is no criticality for such a system and hence no true phase transition in terms of singularbehavior. Further, in the thermodynamic limit, the mean-field prediction of an infinite criticaltemperature for the system may exclude any true phase transition even then.Nevertheless, with an eye on potential applications of the model on biological systems that aregenerally finite, one may still try to find approximations that describe the relevant observablesquantitatively. Here we present an alternative, approximate formulation for the description ofthe Ising model of a Barab´asi-Albert Network. Using the classical definition of magnetization,defined as the ensemble average of all spins in the network, we show that Ising models on anetwork can be well-approximated by a long-range interacting homogeneous Ising model whereineach node of the network couples to all other spins with a strength determined by the mean degreeof the Barab´asi-Albert Network. In such a effective long-range Ising model of a Barab´asi-AlbertNetwork, the critical temperature is directly proportional to the number of preferentially attachedlinks added to grow the network. This dependence allows us to “control” the critical behavior of aBarab´asi-Albert network by changing the model parameters. The long-range Ising model describesthe magnetization of the majority of the sites with average or smaller than average degree bettercompared to the heterogeneous mean-field approximation. However, the heterogeneous mean-fieldapproximation is better for predicting the onset at higher temperatures.Further, we show that the thermodynamic behavior of a scale-free network is between that ofa lattice and that of a clique. The critical temperatures of lattice and clique form the lower andupper bounds, respectively, of the critical temperature of the Barab´asi-Albert scale-free network.This approximation of an Ising model of a scale-free network to a long-range Ising model allows usto make a direct comparison of a scale-free network to simple graphs such as lattices and cliques ofthe same size. The long-range Ising model is the only homogeneous description of Barab´asi-Albertnetworks that we know of. eywords : Phase transitions, Complex networks, Ising model, Barab´asi-Albert Network,MCMC, Mean-Field approximations . INTRODUCTION The study of disorder and critical phenomena occurring in complex networks has beenan area of extensive study in the last couple of decades (Dorogovstev 2008, Rozenfeld 2008,Strogatz 2001). Owing to the non-trivial topology that is neither regular nor random,complex networks exhibit phase transitions that are markedly different from lattices orcomplete graphs (Ising 1925, Barrat 2000, Ferreira 2010, Herrero 2002, Gitterman 2000,Lopes 2004). Among these, many studies have extensively used the Ising paradigm to modelcriticality in real-world networks owing to its simplicity and broad applicability outside ofstatistical mechanics (Pekalski 2001, Krishnan 2019a, Castellano 2009, Stauffer 2006, Aldana2004, Kumar 2000, Pastor 2015).An important class of complex networks is those that exhibit a power-law distributionand commonly referred to as the scale-free networks. Barab´asi-Albert model is an algorithmthat uses the preferential attachment mechanism to generate such scale-free networks. Inthis model, new nodes are added to existing nodes in the network proportional to the degreeof the existing nodes until the overall network size is generated (Albert 2002).A heterogeneous mean-field approximation has been proposed for the Ising model of aBarab´asi-Albert Network with ± true phase transitionin terms of singular behavior. Further, at the thermodynamic limit ( N → ∞ ), this gives aninfinite critical temperature for the system.This is unlike the well-defined second-order phase transition that is exhibited by a regularlattice. Essentially this difference in critical behavior arises from the connectivity structurethat is fed into the Hamiltonian of the Ising model – short- and long-range connections inthe case of a scale-free network (given by the adjacency matrix)(Albert 2002); purely short-range connections in the case of a lattice (only nearest-neighbor coupling) (Ising 1925). Theheterogeneous mean-field approximation may over-represent the nodes with higher connec-tions. Further, to our knowledge, it does not allow a direct comparison of Ising models ofhomogeneous (such as lattices or cliques) and heterogeneous (any complex network) struc-4ures.Here we present an alternative mean-field approximation of the Ising model of a Barab´asi-Albert Network wherein we choose the classical definition of magnetization defined as theensemble average of all spins in the network. Such a system exhibits up-down symmetrywhen M = 0, and breaking of symmetry when M is non-zero. In this, we approximate theadjacency matrix of the Barab´asi-Albert Network by an effective coupling constant, therebytransforming the network Hamiltonian to the Hamiltonian of a lattice.We show that the Ising model on a Barab´asi-Albert Network can be well-approximated bya homogeneous effective long-range Ising model wherein each node of the network couplesto all other spins with a strength determined to the mean degree of the Barab´asi-AlbertNetwork. This approximation allows us to make a direct comparison of a scale-free networkto simple graphs such as lattices and cliques of the same size. With this, we show that thecritical temperature of these network structures can be directly mapped from one to another,and the classical Ising model. The long-range Ising model describes the magnetization ofthe majority of the sites with average or smaller than average degree better. Preliminaryresults of this work have been presented in the form of a talk and thesis (Krishnan 2019b , c).The paper is organized as follows: in Sec. II we present an approximation of the Isingmodel on a Barab´asi-Albert Network by an effective homogeneous Ising model with long-range interactions and compare the full network with approximation numerically usingMonte Carlo simulations; in Sec. III we analyze the cost of this approximation; followedby Sec. IV where we use the long-range Ising model to compare criticality in regular andscale-free structures. In Sec. V, we compare the proposed approximation with the state-of-the-art and identify the temperature ranges at which each of these models fits best.5 I. APPROXIMATION OF ISING MODEL ON A BARAB ´ASI-ALBERT NET-WORK
Consider the Hamiltonian of the Ising model with spins s i = ± N nodes and m preferentially attached links, H = − N (cid:88) i,j =1 J ij s i s j − h N (cid:88) i =1 s i J ij = J A ij (1)where h is uniform magnetic field; J is the coupling constant; A ij is the adjacency matrix.Since the adjacency matrix A ij is symmetric, the pre-factor is included to not count anypairs twice.The elements of the adjacency matrix A ij are equal to one if there is a link between nodes i and j and zero otherwise. Unlike grid structures, different realizations of a Barab´asi-AlbertNetwork would result in a different adjacency matrix (however with similar connectivity).Therefore mean over multiple realizations of the adjacency matrix is a better estimate forEq. 1. The mean over many copies of the network then has a tensor structure (Bianconi2002) (cf. Appendix VII for summary of this method),[A ij ] = p ij = m √ t i t j = 12 mN k i k j (2)where m is the number of preferentially attached links to construct Barab´asi-Albert Network; N is the network size; and k i is the node degree i . Substituting Eq. 2 in Eq. 1 we have, H = − J mN N (cid:88) i,j =1 k i k j s i s j − h N (cid:88) i =1 s i (3)We can see that the Hamiltonian has non-zero mutual couplings between all pairs i, j of spinswhich however still vary in strength by the factors k i k j . Since k follows a power law degreedistribution it makes it challenging to evaluate Eq. 3. Here we make an approximation byconsidering the first statistical moment of the degree distribution i.e. k = ¯ k , N (cid:88) i,j k i k j s i s j = ¯ k N (cid:88) i,j =1 s i s j (4)Setting all k i = ¯ k would mean that the degree of every node in the network k i is equalto the mean degree of the network ¯ k . If we use this approximation after Eq. 2, it simply6omogenizes the coupling constants between the spin pairs. This way we end up with ahomogeneous model where all spins are coupled irrespective of their distance. A detailedanalysis of the outcomes of this approximation is discussed in Sec. III. Re-writing Eq. 3, H ≈ − J ¯ k mN (cid:88) i,j =1 s i s j − h N (cid:88) i =1 s i (5)The mean degree, ¯ k on a Barab´asi-Albert Network can be approximated as (cf. AppendixVII), ¯ k ≈ m (6)From Eqs. 5 and 6, H ≈ − J eff N (cid:88) i,j =1 s i s j − h N (cid:88) i =1 s i (7)where, J eff = 2 mJN (8)Comparing Eqs. 1 and 7, we have approximated Ising model on a Barab´asi-Albert Net-work by a long-range coupled homogeneous Ising model with an effective coupling constantacting between all pairs of spins, J eff = mJN . This is unlike the classical Ising model of atwo-dimensional lattice with nearest neighbor coupling only, i.e. has a finite coordinationnumber. In our case, the coordination number is N − H MF = 12 J eff N M − ( J eff N M + h ) N (cid:88) i =1 s i (9)with the thermodynamically configuration-averaged magnetization M = N (cid:104) (cid:80) Ni =1 s i (cid:105) . Usingthis mean-field Hamiltonian we can obtain the partition function which enable us to evaluate M for uniform magnetic field, h as, M = tanh( βh + βJ eff N M ) (10)7he critical temperature T c obtained when h = 0 is, T c = J eff Nk B (11)where k B is the Boltzmann constant. From Eqs. 8 and 11 we have, T c = 2 mJk B (12)From Eq. 12 we note that T c scales linearly with coupling constant, J and number ofpreferentially attached links, m . First, consider magnetization at h = 0. We can ask howthe order parameter decreases as we tend towards the critical point. Just below T = T c , m is small, so we can Taylor expand Eq. 10 and use Eq. 11 to obtain, M ∝ ± (cid:32) T c − TT (cid:33) , T < T c (13)At T = T c and as h →
0, from Eqs. 10 and 12 we have, M ∝ h (14)To compare the quality of this approximation, we performed Monte Carlo simulationsof multiple realizations of the full Barab´asi-Albert Network (Eq. 1 magnetization indicatedas plus marks in Fig. 1) and the approximation i.e. the long-range Ising model (Eq. 7across multiple realizations indicated as cross marks in Fig. 1) using Metropolis local updatealgorithm (Metropolis 1953). For a network of size N = 5 × and magnetic field h = 0,the system is equilibrated for 2 × MC steps and thermodynamic variables sampled over3 × MC steps. Numerical T c is calculated when M ≈ . T > T c ), the effectiveordinary Ising model underestimates magnetization. We also see that there is significantdeviation of the numerical observations of the long-range Ising model (shown by x markers)from the analytical solution (Eq. 10 shown by dotted dashed line). This is due to the smallnetwork size (order of thousands) that we consider here. The mean-field solution is expectedto be exact for very large network sizes (when N → ∞ ).8t very low temperatures, the magnetization values indicated by the Ising model ona Barab´asi-Albert Network are well within the standard deviation of the magnetizationof the long-range Ising model. At intermediate temperatures (and we will note later, at T > T c ), the effective ordinary Ising model underestimates magnetization. At very hightemperatures, the magnetization values indicated by the long-range Ising model agree wellwith the magnetization of the Barab´asi-Albert Network. Further the results Monte Carlosimulations of the full Barab´asi-Albert Network are in reasonable agreement with the mean-field approximation of Eq. 10 (Fig. 1).As the coupling constant and the number of preferentially attached links increases, thesystem takes longer to reach a paramagnetic state. The numerical observations agree withthe linear scaling of critical temperature with network parameters (Fig. 2). The slightdeviation of numerical observations from expected scaling for high m and J could possiblybe an effect of importance sampling in the Monte Carlo scheme. The model that comes outof such an approximation is interchangeably referred to as the long-range Ising model or¯ k -clique model in this paper. 9 Temperature . . . . . . M a g n e t i z a t i o n Long-Range MFScale-free NetworkLong-Range Model
FIG. 1. Comparison of the Ising model of Barab´asi-Albert Network and long-range Ising model ofBarab´asi-Albert Network: The plus markers show magnetization from the Monte Carlo simulationsof the full ferromagnetically coupled Barab´asi-Albert Network (given by Eq. 1). The cross markersindicate the magnetization from the Monte Carlo simulations of the long-range Ising model (givenby Eq. 7). These data come from n = 20 realizations of the Barab´asi-Albert Network for thesame choice of network parameters. The dotted-dashed line shows the mean-field solution of thelong-range Ising model (given by Eq. 10). Simulation parameters: network size, N = 5 × ,preferential links added to grow the network, m = 5, coupling constant, J = 1 and magnetic field, h = 0. A) Number of preferentially attached links, m C r i t i c a l t e m pe r a t u r e , T c (B) Coupling constant, J C r i t i c a l t e m pe r a t u r e , T c FIG. 2. Comparison of numerical and analytical results for scaling of critical temperature, T c with (A) preferentially attached links, m and (B) coupling constant, J . The lines indicate theanalytical approximation of T c from Eq. 12 and the dots indicate the T c calculated from MonteCarlo simulations of the Ising model of the Barab´asi-Albert Network from Eq. 1. Numerical T c iscalculated when M ≈ . II. APPROXIMATION ERROR
The approximation proposed in Sec. II deviates from the expected trend, particularlywhen the system is at intermediate temperatures. Here we analyze the cost of the ap-proximation of the Ising model on a Barab´asi-Albert Network by a long-range Ising modelproposed in Sec. II. In Subsec. III A we do this by analyzing their asymptotic behavior and;in Subsec. III B by numerically evaluating the neglected terms for different temperaturesthat lead to the deviation seen in Fig. 1.
A. Asymptotic behavior of the long-range Ising model
Consider the approximation in Eq. 4 in Sec. II. Re-writing the reduced Hamiltonian, H ≈ − J mN N (cid:88) i,j =1 k i k j s i s j − h N (cid:88) i =1 s i (15)To compare an Ising model on a Barab´asi-Albert Network to a long-range Ising model,we substitute, k i = ¯ k + δk i k j = ¯ k + δk j (16)From Eqs. 15 and 16 we have, H ≈ − J mN N (cid:88) i,j =1 (cid:32) ¯ k + 2¯ kδk i + δk i δk j (cid:33) s i s j − h N (cid:88) i =1 s i ≈ (cid:32) − J ¯ k mN N (cid:88) i,j =1 s i s j − h N (cid:88) i =1 s i (cid:33) − J ¯ k mN N (cid:88) i,j =1 δk i s i s j − J mN N (cid:88) i =1 δk i δk j s i s j (17)The first terms in brackets are the Hamiltonian for a long-range Ising model with aneffective coupling constant J eff = J ¯ k mN . The additional terms are first and second-order errorterms, respectively arising due to the approximation. We know from the simulation of anIsing model on Barab´asi-Albert Network that the system is ordered at T →
0, and all thespins are either +1 or −
1. As T → ∞ the spins are randomly distributed and the system isdisordered (Fig. 1). Consider their asymptotic behavior:12 . Ordered phase: T → In this limit the contribution of the error terms are zero: − J ¯ k mN N (cid:88) i =1 δk i s i s j = − J ¯ k mN N (cid:88) i =1 N (cid:88) i =1 δk i = 0 (18) − J mN N (cid:88) i,j δk i δk j s i s j = − J mN N (cid:88) i =1 δk i s i N (cid:88) j =1 δk j s j = 0 (19)since (cid:80) Ni =1 δk i = (cid:80) Nj =1 δk j = 0 (cf. Appendix VII). Therefore the contribution of the twoextra terms are zero at T →
0. This means that at T →
0, we can map the system to along-range Ising model with an effective coupling constant J eff = J ¯ k mN . This encourages usto use a long-range Ising model with an effective coupling constant and we can estimate T c as, mJk B .In this temperature range (at T ≈ T c with respect to m and J (cf. Figures1 and 2). This is also reflected in the numerical evaluation of the approximation error atlow temperatures (cf. T ≈
2. Disordered phase: T → ∞ In this limit the contribution of the second term is again zero: − J mN N (cid:88) i,j δk i s i s j = − J mN (cid:32) N (cid:88) j =1 s j (cid:33) N (cid:88) i,j δk i s i = 0 (20)Because (cid:80) Nj =1 s j = 0 in this case (when all spins are random). However, the contributionof the third term is not zero in this limit: 13 J mN N (cid:88) i,j δk i δk j s i s j = − J mN (cid:32) N (cid:88) i =1 δk i s i (cid:33)(cid:32) N (cid:88) j =1 δk j s j (cid:33) = − J mN (cid:32) N (cid:88) j =1 δk j s j (cid:33) < < T < T c between the Ising model of aBarab´asi-Albert Network and the proposed approximation as can be seen in Fig. 1. This isalso reflected in the numerical evaluation of the approximation error at high temperatures(cf. T < T c , T ≈ T c and T ≈ T c in Table III B). B. Numerical Evaluation of the Approximation Error
Consider the degree distribution term in Eq. 5, N (cid:88) i,j k i k j s i s j (22)which is approximated by the mean degree. Re-writing Eq. 16 we have, k i = ¯ k + δk i k j = ¯ k + δk j (23)Plugging Eq. 16 in Eq. 5, N (cid:88) i,j k i k j s i s j (cid:124) (cid:123)(cid:122) (cid:125) LHS = ¯ k N (cid:88) i,j s i s j (cid:124) (cid:123)(cid:122) (cid:125) t +2 ¯ k N (cid:88) i,j δk i s i s j (cid:124) (cid:123)(cid:122) (cid:125) t + N (cid:88) i,j δk i δk j s i s j (cid:124) (cid:123)(cid:122) (cid:125) t (24)The approximation presented in Sec. II truncates the contribution of the degree distri-bution in Eq. 22 to t . This approximation is hence a zero-order approximation in δk . Are-scaling of the square of the mean degree term in t , ¯ k allows us to map the Ising modelon a Barab´asi-Albert Network to the long-range Ising model. t includes fluctuation termsarising from the deviation of node degree from mean degree. For nodes with a very high14egree, depending on the temperature, these terms may be high. t is the second-order con-tribution of these fluctuation terms, and hence its contribution can be very high dependingon the temperature.Table III B summarize the contribution of each term at different temperature rangesevaluated numerically. The sum of the contribution of all terms (LHS) is very high at lowtemperatures ( T ≈
1) for all choice of network parameters, given that all spins take +1 or − T ≈ T c ) where the spin configuration can beeither +1 or −
1. At critical temperature ( T ≈ T c ), LHS is another order of magnitude lowersince the effect of most spins cancels each other.Let us now consider observations for Barab´asi-Albert Network of N = 5 × (Table III B(A)(B)(C)). At T ≈ T (ordered phase in Subsubsec. III A 1), t contributes most. In otherwords, the network can be well-approximated by the ¯ k − clique model at this temperatureas can be verified from numerical simulations in Fig. 1 (cf. limit cases Subsec. III A). Attemperatures close to critical temperature 0 < T < T c (disordered phase in Subsubsec.III A 2) the deviation from mean degree owing to high degree nodes causes t to explodecausing the deviation observed in numerical simulations in Fig. 1. At T ≈ T c and 2 T c ,close to paramagnetic state, t contribution is high in most cases (not in (C)). The netcontribution (LHS) may be negative since most spins may take values − N = 10 .Overall, at ordered phases (very low temperatures or temperatures higher than criticaltemperatures), the proposed approximation is reasonable. At intermediate temperatures,there is a significant deviation between the approximation and the Barab´asi-Albert Networkowing to the dominance of the fluctuation term t . Overall, the Monte Carlo simulationsof the long-range Ising model proposed here (Eq. 7) is a modest approximation of the fullBarab´asi-Albert Network (Eq. 1). 15 A) Temperature,
T t t t t LHS T ≈ T . . × − . × − − .
88 498912 . T < T < T c .
77 275 .
93 275 .
93 85392 .
04 86556 . T ≈ T c . − . − . − .
07 103 . T ≈ T c .
74 7 . × − . × − − .
54 3 . (B) Temperature,
T t t t t LHS T ≈ T .
06 1 . × − . × − − .
32 179164 . T < T < T c − . − . − .
05 30406 .
41 29397 . T ≈ T c . − . − .
45 97 . − . T ≈ T c − .
94 1 . × − . × − − . − . (C) Temperature,
T t t t t LHS T ≈ T . − . × − − . × − .
23 20203 . T < T < T c − .
55 0 .
03 0 .
03 3631 .
96 3607 . T ≈ T c . − . − .
98 10 . − . T ≈ T c .
07 2 . × − . × − − . − . (D) Temperature,
T t t t t LHS T ≈ T . . × − . × − − .
90 98565 . T < T < T c .
86 77 .
67 77 .
67 16339 .
75 16805 . T ≈ T c . − . − .
69 4 . − . T ≈ T c .
89 5 . × − . × − − . − . t to t evaluated at (a) T ≈ < T < T c , (c) at analytical T c and (d) T ≈ T c . T c is calculated from Eq. 12 fornetwork parameters: (A) N = 5000 , m = 5, (B) N = 5000 , m = 3, (C) N = 5000 , m = 1, (D) N = 1000 , m = 5. LHS is the sum of all terms on the left hand side of Eq. 24. V. ISING MODEL OF LATTICE, SCALE-FREE NETWORK AND CLIQUE ANDTHEIR RELATIONSHIPS
The approximation of a scale-free network such as the Barab´asi-Albert Network to alattice-like Ising system now allows us to make a direct comparison of complex networks tosimpler structures such as lattices and cliques. Let us consider mean-field approximationsof the Ising model of the three topologies:
1. Classical Ising Model of a Lattice
This is the well-established two-dimensional Ising model of a quadratic lattice whereonly short-range nearest-neighbor interactions are allowed, as illustrated in Fig. 3 (A) . Thenumber of interacting spins on a spin, z , is equal to the number of nearest neighbors, z = 4and critical temperature T c is given by (Ising 1925), T lattice c = 2 . J (25)
2. Ising Model of a Clique
In a k − clique, all nodes interact with each other since it is a complete graph (as illustratedin Fig. 3 (C) ).The number of interacting spins on clique is ( N − N is the totalnumber of spins and the critical temperature T c , T clique c = ( N − Jk B ≈ N Jk B (26)since N >>
3. Ising Model of a Scale-Free Network
However, a scale-free network has connectivity that ranges from nodes that interact short-range only to nodes that may have both short- and long-range interactions (as illustratedin Fig. 3 (B) ). It can be approximated by a network of interacting spins where the averagenumber of interacting spins on a spin is ¯ k = 2 m and the critical temperature T c as proposedin Sec. II as, 17 sfnet c = 2 mJk B (27)Among these three structures, a regular two-dimensional lattice has the lowest connectiondensity. Generally, then, the connectivity structure of a scale-free network is between thetopology of a lattice and a clique. Therefore we expect the Ising model of a scale-free networkto exhibit a critical temperature between lattice and clique.Since mean-field calculations approximate the same critical exponent for all the abovethree topologies, the vital difference is in the critical temperature of T c . For a Barab´asi-Albert Network with preferentially attached links, m >
2, or 3, the critical temperatureis higher than the critical temperature of a two- or three-dimensional lattice with nearest-neighbor interactions. On the other hand, the critical temperature of a Barab´asi-AlbertNetwork is less than that of the critical temperature of a clique because m should be lessthan ( N − to preserve its scale-free structure. Therefore, T lattice c < T sfnet c < T clique c , unless m < J , the Ising simulations of a scale-free network show atrend between lattice and clique where lattice is the lower bound (undergoes phase transitionat low temperatures) and clique is the upper bound (undergoes phase transition at highertemperatures) as can be verified from the Monte Carlo simulations of the Ising models onthe three topologies as shown in Fig. 4. In a regular lattice, the spins are well-connectedand hence exhibits spin flips at low temperatures (indicated as circles in Fig. 4). However, aclique is well-connected; therefore, it requires that the system is heated much longer beforespins flip down (indicated as box markers in Fig. 4).Finally in Sec. II we showed that the approximation of a Ising model on a Barab´asi-Albert Network can be interpreted as a special case of a long-range Ising model with areduced effective coupling J eff (Eq. 8). As a corollary we can make the inference that it canbe intepreted as a special case of Ising model on a clique as well. Hence the model couldbe alternatively referred to as the ¯ k -clique model as mentioned in the previous sections.Consider Eq. 12, 18 c = 2 mJk B = N mJN k B T c = N J eff k B where J eff = 2 mJN (28)Eqs. 12 and 28 show the relationship between the three topologies in terms of the criticaltemperature. Phase transition in a complex structure such as the scale-free network is,in a way, fundamentally an alternative formulation of phase transitions in homogeneousstructures such as a clique and a lattice. FIG. 3. Illustration of networks of nine nodes with different topologies: (A)
Two-dimensionalperiodic square lattice (short-range interactions only) (B)
Scale-Free Network (both short- andlong-range interactions) (C)
Clique (all nodes interact with each other). Temperature − − − M a g n e t i z a t i o n Scale-free networkCliqueLattice
Temperature . . . . . . M a g n e t i z a t i o n T sfapprox c SF-Approximation
Temperature . . . . . . M a g n e t i z a t i o n T clique c Clique
FIG. 4. Log-log plot of Monte Carlo simulations of Ising model of (A) two-dimensional Latticeof size N = 1024 (indicated by circles) (B) long-range Ising model of a Barab´asi-Albert scale-freenetwork of size N = 1000 and preferentially attached links, m = 5 (indicated by plus markers) (C) Clique of size N = 1000 (indicated by square boxes). Numerical results validate results frommean field calculations (cf. Sec. II): T lattice c = 2 . T sfnet c = 10, T clique c = 999. Critical temperatureof Ising models of clique and lattice form the upper and lower bounds respectively for criticaltemperature of a scale-free network. . COMPARISON OF MEAN FIELD THEORIES FOR ISING MODEL OF BARAB ´ASI-ALBERT NETWORK The long-range Ising model of a Barab´asi-Albert Network proposed in Sec. II is one of themean-field formulations for phase transitions occurring in Barab´asi-Albert Network. long-range Ising model uses a global averaged mean-field with degree distribution approximatedby the mean degree to arrive at the mean-field approximation in Eq. 10. The degree-weightedmean-field approximation, however, uses a node dependent expectation value instead. Tocompare the two models, we performed Monte Carlo simulations of the long-range Isingmodel of Barab´asi-Albert Network (indicated as triangle markers in Fig. 5) and degree-weighted Ising model of Barab´asi-Albert Network (indicated as circle markers in Fig. 5)for selected nodes of the network to infer how well they fit with their respective mean-fieldapproximations (indicated in dashed and dotted-dashed lines respectively). For a networkof size N = 5 × , the system is equilibrated for 2 × MC steps and thermodynamicvariables sampled over 3 × MC steps. We categorize nodes in the network based on theirdegree as follows:- nodes of all degree ( k );- nodes with low degree ( k < ¯ k );- nodes with degree slightly higher than mean degree ( k > k );- nodes with degree significantly higher than average ( k > k ) and- nodes with high degree ( k > k )The total magnetization of long-range Ising model and degree-weighted model of Barab´asi-Albert Network agree reasonably well with the trend predicted by the mean-field theory(first column of Fig. 5). The total magnetization of the long-range Ising model follows thatof nodes with a low degree while the degree-weighted model “underestimates” magnetiza-tion owing to the lower weighting of the nodes (second column of Fig. 5). However, fornodes with a degree slightly higher than the mean degree or above, long-range Ising modelexhibits finite magnetization even at very high temperatures and does not agree with mean-field (third, fourth, and fifth columns of Fig. 5 LRM). These high degree nodes that order21t high temperatures may form an effective magnetic field for the low-degree nodes that areyet to order, thereby not reaching a paramagnetic state. On the other hand, though thedegree-weighted model “overestimates” magnetization above unity, it accurately predictsthe ordering of nodes with a high degree (third, fourth, and fifth columns of Fig. 5 DW). Insummary, the long-range Ising model describes the magnetization of the majority of siteswith mean or smaller than mean degree better, while the degree-weighted theory is betterfor predicting the onset at higher temperatures. FIG. 5. Comparison of the long-range Ising model (first row, indicated as LRM) and degree-weighted theory (second row, indicated as DW): numerical results are shown by triangle and circlemarkers respectively. Mean-field approximation is shown by dashed and dotted lines respectively.The first column shows the total magentization of all nodes in the network; second to fifth columnshows magnetization of selected nodes of the network of degrees k < ¯ k , k > k , k > k and k > k respectively where ¯ k ≈
10 for the chosen simulation parameters ( N = 5 × , m = 5, J = 1).Expected T c for long-range Ising model = 10 (Eq. 12); expected T c for degree-weighted theory= 21 .
29 (Bianconi 2002). (A) (B)(C)
FIG. 6. Comparison of Mean Field Theories: Figure shows the mean field equations for (A)
Total magnetization of all nodes in the network, (B)
Net magnetization of nodes with high degree( k i ≥ k ); (C) Net magnetization of nodes with low degree ( k i < ¯ k ). Dashed lines show globalmean field theory, dashed lines with dots show homogenized long-range mean field theory, dottedlines show degree weighted or heterogeneous mean field theory and straight lines show local meanfield theory in each case. Network parameters: N = 5 × , m = 5; and coupling constant, J = 1. ethod Magnetization Local spin expectation valueLong-range M = N (cid:80) Ni =1 (cid:104) s i (cid:105) (cid:104) s i (cid:105) = tanh( βJ (cid:104) k (cid:105) M )Degree-weighted M = N (cid:80) Ni =1 k i (cid:104) k (cid:105) (cid:104) s i (cid:105) (cid:104) s i (cid:105) = tanh( βJ k i M )Global mean-field M = N (cid:80) Ni =1 (cid:104) s i (cid:105) (cid:104) s i (cid:105) = tanh( βJ k i M )Local mean-field M i = (cid:80) Nj =1 A ij (cid:104) s i (cid:105) (cid:104) s i (cid:105) = tanh( βJ M i )TABLE II. Comparison of Mean Field formulation for phase transitions in occurring in Barab´asi-Albert network I. CONCLUSIONS
In this work, we have shown that the phase transition behavior of an Ising model onBarab´asi-Albert Network, wherein the order parameter of the system is defined as the en-semble average of all spins of the network, is well-approximated by a long-range Ising modelwherein each node has neighbors equal to the mean degree of the Barab´asi-Albert Network.Such a model of Barab´asi-Albert Network works well at low temperatures and close to crit-ical temperatures (0 ≤ T < T c ); but the approximation appears to be limited at criticaltemperature ( T ≥ T c ) owing to the error arising from nodes of very high degree. The crit-ical temperature for the long-range Ising model scales linearly with Barab´asi-Albert modelparameters and coupling constant of the Ising model.This dependence allows us to control the critical behavior of a Barab´asi-Albert networkby changing the model parameters. We infer that the structure (network) and dynamics (theIsing model) are closely interconnected to each other. We can change the control parameterof the Ising model (such as temperature) by changing the network parameters (such as linksadded), which opens up a new window into the study of the critical behavior of Barab´asi-Albert networks and real-world networks that have connectivity close to a power-law degreedistribution. The long-range Ising model describes the magnetization of the majority of thesites with average or smaller than average degrees better compared to the degree-weightedtheory, which predicts the onset at higher temperatures better.Lattices, scale-free networks, and cliques have traditionally been treated as graphs withdifferent structures that exhibit phase transition behavior that is very far from each other.It has, therefore, not been possible to compare the phase transition behavior between thesestructures objectively owing to their unique structural connectivity. We have shown that thescale-free network behaves like a clique or a lattice with a reduced effective coupling. Thisapproximation of an Ising model of a scale-free network to a long-range Ising model allowsus to make a direct comparison of a scale-free network to simple graphs such as lattices andcliques of the same size. With this, we have shown that the critical temperature of thesenetwork structures can be directly mapped from one to another. We infer that the phasetransition behavior of regular and complex structures are not very different, and the criticaltemperature on these structures can be mapped from one topology to another.25 II. APPENDIXA. Approximation of ensemble average of adjacency matrix by network parame-ters
Here we summarize the approach from Bianconi (2002) to reduce mean adjacency matrixover many realization of Barab´asi-Albert Network to network parameters. Let us consider aBarab´asi-Albert Network of N nodes. Starting from a small number of nodes N and links m (where N , m << N ), the network is constructed iteratively by the constant additionof nodes with m links. The new links are preferentially attached to well connected nodes insuch a way that at time t j , the probability p ij that the new node j is linked to node i withconnectivity k i ( t j ) is given by, p ij = m k i ( t j ) (cid:80) jα =1 k α (29)is proportional to the number of links k i at time t j , and number of preferentially attachedlinks m . The dynamic solution of connectivity at time t i is, k i = m (cid:114) tt i (30)From Eqs.29 and 30 we have, p ij = m m (cid:113) tt i (cid:80) jα =1 k α ( t ) (31)If N is large we can approximate the total number of edges in the network at time t j ,given by the sum (cid:80) jα =1 k α as, j (cid:88) α =1 k α = m + 2 mt j ≈ mt j (32)because m << N . The factor 2 comes from the fact that as we create a link which connectstwo nodes, the number of links of each of them increases by 1. Substituting Eq. 32 in 31,26 ij = m (cid:113) t j t mt j = m √ t i t j (33)The adjacency elements of the network A ij are equal to 1 if there is a link between node i and j and 0 otherwise. Consequently the mean over many copies of a Barab´asi-AlbertNetwork (cid:104) A ij (cid:105) = p ij = m √ t i t j (34)From Eq. 30 we can re-write for t = N steps, k i ( t ) = m (cid:114) tt i k i ( N ) = m (cid:114) Nt i t i = m Nk i (35)and similarly, t j = m Nk j (36)From Eqs. 35 and 36, (cid:104) A ij (cid:105) = m (cid:113) m Nk i (cid:113) m Nk j = 12 mN k i k j (37)The average of the adjacency matrix over many realizations can be approximated by thenetwork parameters as, (cid:104) A ij (cid:105) = 12 mN k i k j (38)27he mean degree of the network ¯ k can be approximated from Eq. 32 as,¯ k = 1 N N (cid:88) i =1 k i = 1 N mN ¯ k ≈ m (39) B. Degree Distribution Deviation
Consider the sum of degree distribution deviation summed over i (as in Eq. 18), N (cid:88) i =1 δk i = N (cid:88) i =1 ( k i − ¯ k )= N (cid:88) i =1 k i − N (cid:88) i =1 ¯ k = 0 (40)since ¯ k = N (cid:80) Ni =1 k i . Hence the average over deviation of degree distribution from meandegree is zero. VIII. FUNDING
This work was supported by the Exploratory Research Space (ERS) Seed Fund 2017 inComputational Life Sciences (CLS001). All simulations were performed using the RWTHCompute Cluster under general use category; priority category allocated to AICES andJRC users; and with specific computing resources granted by RWTH Aachen Universityunder project rwth0348. The authors gratefully acknowledge the generous support of theaforementioned funding and computing resources.
IX. ACKNOWLEDGMENTS
JK thanks Richard Polzin for his help with the network illustration in this paper; andAjay Mandyam Rangarajan for the many useful discussions.28 [email protected]; permanent address: MTZ, Pauwelstrasse 19, Level 3, D-52074,Aachen, Germany † [email protected] ‡ [email protected] § [email protected] ¶ [email protected], R., Barabasi, A.-L. Statistical mechanics of complex networks
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