Random matrix analysis of complex networks

Transcription

1 Random matrix analysis of complex networks Sarika Jalan and Jayendra N. Bandyopadhyay Max-Planck Institute for the Physics of Complex Systems, Nöthnitzerstr. 38, D-87 Dresden, Germany Continuing our random matrix analysis of complex network, in this paper we perform next nearest neighbor spacing distribution analysis and spectral rigidity test to probe long range correlations among the eigenvalues of the adjacency matrix of the various model networks. We show our results for random, scale-free and small-world networks. We find that the spectral rigidity of these networks follows RMT prediction of linear behavior in semi-logarithmic scale with the slope being /π 2. Random and scale-free networks follow RMT prediction for very large scale. Small-world network follows it for the sufficiently large scale, but much less than the random and scale-free networks. PACS numbers: Hc,64.6.Cn,89.2.-a I. INTRODUCTION II. COMPLEX NETWORKS In our previous papers [, 2] we analyzed complex networks under random matrix theory (RMT) framework. We studied nearest-neighbor spacing distribution (NNSD) of eigenvalues spectra of adjacency and Laplacian matrices of various networks studied recently in the literature. We found that the NNSD of these networks follow Gaussian orthogonal ensemble (GOE) statistics, which is one of the most celebrated result of random matrix theory (RMT). We also showed that transition to the small-world and GOE occurs at the same value of random connections in the network. In this paper we make further investigations of the RMT properties of complex networks. NNSD analyzed in [] carries information on the correlation between two adjacent eigenvalues, but it tells nothing about the correlations between two adjacent spacings. To know about longe range correlations among the eigenvalues of the adjacency matrix of networks we perform spectral rigidity test via well known -statistic of RMT. We find that spectral rigidity of these networks follow RMT prediction for very large scale. We also study the next-nearest-neighbor spacing distribution (NNNSD) of the adjacency matrix of various networks. We show our results for various model networks vastly studied in the recent literature, namely, random, scale-free and small-world networks. The paper is organized as follows: after this introductory section, Sec. II explains various aspects of complex networks studies. In Sec. III, we describe some basics of RMT relevant to our studies. In Sec. IV, we analyze next-nearest-neighbor spacing distribution and the - statistic of various networks, namely random, scale-free and small-world networks. Finally, in Sec. V, we summarize and discuss about some possible future directions. Electronic address: sarika@mpipks-dresden.mpg.de Electronic address: jayendra@mpipks-dresden.mpg.de Last years have witnessed a rapid advancement in the studies of complex networks. The main concept of the network theory is to define complex systems in terms of networks of many interacting units. Few examples of such systems are interacting molecules in living cell, nerve cells in brain, computers in Internet communication, social networks of interacting people, airport networks with flight connections, etc [3 5]. Mathematically networks are investigated under the framework of graph theory. In the graph theoretical terminology, units are called nodes and interactions are called edges [6]. Various model networks are introduced to study the behavior of complex systems having underlying network structures. These model networks are based on some simple principles and still capture essential features of the systems. A. Structural properties Erdös and Rényi were the first one to model complex systems by using random graph (ER model). In this model any two nodes are connected with probability p. One of the most interesting characteristics of ER model was the emergence of giant cluster through a phase transition. With the increase in p, while number of nodes in the graph remain constant, a giant cluster emerges through the phase transition. ER model assumed that interaction between the nodes are random [7]. Recently, with the availability of large maps of real world networks, it is observed that the random graph model is not appropriate for studying the behavior of real world networks. Hence many new models are introduced. Watts and Strogatz proposed a model, popularly known as small-world network, which has the properties of small diameter and high clustering [8]. This model shows the small-world transition with the fine tuning of the number of random connections. Moreover, this model network is very sparse : network with a very few number of edges, another property shown by many real-world networks. In addition to above mentioned properties, Barabási and Albert show that degree distributions of many real-world networks

2 2 have power-law, i.e. degree distribution p(k), fraction of nodes that have k number of connections with other nodes, decays as p(k) k γ, where γ depends on the topology of the networks. The scale-free nature of networks implies that some nodes are much more connected than the others are [9]. Barabási-Albert s scale-free (SF) model [9] and Watts- Strogatz s small-world (SW) model have contributed immensely in understanding the evolution and behavior of the real systems having network structures. Following these two new models came an outbreak in the field of networks. These studies have revealed that apart from power law degree distributions and small diameter, real world networks also have modular structures [, ]. Modules are the division of network nodes into various groups within which the network connections are dense, but between which they are sparser. The modularity concept assumes that system functionality can be partitioned into a collection of modules and each module is a discrete entity of several components and performs an identifiable task, separable from the functions of other modules. Studies of real world networks have also given clues to solve the network coloring problem, scale-free networks turned out to be difficult and small-world easier [2]. B. Spectral properties Apart from the above mentioned investigations which focus on direct measurements of the structural properties of the networks, there exists a vast literature demonstrating that properties of networks or graphs could be well characterized by the spectrum of associated adjacency (A) and Laplacian matrix [3, 5]. For a unweighted graph, adjacency matrix is defined in the following way : A ij =, if i and j nodes are connected and zero otherwise. For undirected networks, this matrix is symmetric and consequently have real eigenvalues. Eigenvalues of graph are called graph spectra and they give information about some basic topological properties of the underlying network [3 6]. Spectral properties of networks are also used to understand some of the dynamical properties of interacting chaotic units on networks, for example largest eigenvalue of the adjacency matrix determines the transition to the synchronized state [7]. III. RANDOM MATRIX STATISTICS RMT was initially proposed to explain the statistical properties of nuclear spectra [8]. Later this theory was successfully applied in the study of the spectra of different complex systems such as disordered systems, quantum chaotic systems, large complex atoms, etc [9]. Recently, RMT is also shown to be useful in understanding the statistical properties of the empirical cross-correlation matrices appearing in the study of multivariate time series of followings: the price fluctuations in the stock market [2], EEG data of brain [22], variation of various atmospheric parameters [23], etc. We showed that random matrix theory can also be applied to study complex networks [, 2]. In the random matrix study of eigenvalues spectra, one has to consider two kinds of properties : () global properties, like spectral density or distribution of eigenvalues ρ(λ), and (2) local properties, like eigenvalue fluctuations around ρ(λ). Among these, the eigenvalue fluctuations is the most popular one. The eigenvalue fluctuations are generally obtained from the NNSD of the eigenvalues. The NNSD follows two universal properties depending upon the underlying correlations among the eigenvalues. For correlated eigenvalues, the NNSD follows Wigner-Dyson formula of Gaussian orthogonal ensemble (GOE) statistics of RMT; whereas, it follows Poisson statistics of RMT for uncorrelated eigenvalues. We denote the eigenvalues of network by λ i, i =,...,N, where N is the size of the network. In order to get universal properties of the eigenvalues fluctuations, one has to remove the spurious effects due to variations of the spectral density and to work at constant spectral density on the average. Thereby, it is customary in RMT to unfold the eigenvalues by a transformation λ i = N(λ i ), where N(λ) = λ λ min ρ(λ )dλ is the averaged integrated eigenvalue density [8]. Since we do not have any analytical form for N, we numerically unfold the spectrum by polynomial curve fitting. Using the unfolded spectra, we calculate the nearest-neighbor spacings as s (i) = λ i+ λ i ; and due to the above unfolding, the average nearest-neighbor spacings s becomes unity, being independent of the system. The NNSD P(s ) is πs2 4 defined as the probability distribution of these s (i) s. In case of Poisson statistics, P(s ( ) = ) exp( s ); whereas for GOE P(s ) = π 2 s exp. For the intermediate cases, NNSD( is described ) by Brody formula [24]: P β (s ) = As β exp, where A and B are de- Bs β+ termined by the parameter β. This is a semiempirical formula characterized by the single parameter β. As β goes from to, the Brody formula smoothly changes from Poisson to GOE. Apart from NNSD, the next-nearest-neighbor spacings distribution (NNNSD) is also studied in RMT. We calculate the distribution P(s 2 ) of next-nearest-neighbor spacings s (i) 2 = (λ i+2 λ i )/2 between the unfolded eigenvalues. Here we put a factor two at the denominator to make the average of the next-nearest-neighbor spacings s 2 unity. According to Ref. [8], the NNNSD of GOE matrices is identical to the NNSD of Gaussian symplectic ensemble (GSE) matrices, i.e., P(s 2 ) = π 3 s4 2 exp ( 64 ) 9π s2 2. () The NNSD and NNNSD reflect only local correlations among the eigenvalues. The spectral rigidity, measured

3 3 P(s 2 ) P(s 2 ) P(s 2 ) (a) (b) (c) ρ(λ) ρ(λ) ρ(λ) s λ - λ 4 5 FIG. : (Color online) Next-nearest-neighbor spacings distribution (NNNSD) P(s 2) of the adjacency matrices of different networks [(a) random network, (b) scale-free network, and (c) small-world network] is compared with the nearest-neighbor spacings distribution (NNSD) of GSE matrices. Figures are plotted for average over realization of the networks. All networks have N = 2 nodes and an average degree k = 2 per node. by the -statistic of RMT, gives information about the long-range correlations among the eigenvalues and is more sensitive test for RMT properties of the matrix under investigation [8, 2]. Following we describe the procedure to calculate this quantity. The -statistic measures the least-square deviation of the spectral staircase function representing the cumulative density N(λ) from the best straight line fitting for a finite interval L of the spectrum, i.e., (L; x) = x+l L min [ ] 2 N(λ) aλ b dλ (2) a,b x where a and b are obtained from a least-square fit. Average over several choices of x gives the spectral rigidity. The most rigid spectrum is the picket fence with all spacings equal (e.g., D harmonic oscillator), therefore maximally correlated with constant (= /2). At another extreme, for the uncorrelated eigenvalues, = L/5, reflecting strong fluctuations around the spectral density ρ(λ). The GOE case is intermediate of these two extremes. Here depends logarithmically on L, i.e., lnl. (3) π2 λ Slope = ln L 2 3 L FIG. 2: (Color online) statistic for eigenvalues spectra of the random network. The circles are numerical results and the solid curve is GOE prediction of RMT. Inset shows the in semi-logarithmic scale. Figure is plotted for average over realizations of the networks. All networks have N = 2 nodes and an average degree k = 2 per node. IV. RESULTS Following we present the results of the ensemble averaged NNNSD and statistic of random, scale-free and small-world networks. A. Random network First we consider random network generated by using Erdös and Rényi algorithm. We take N = 2 nodes and with probability p =. we make random connections between the pairs of nodes. This method yields a connected network with average degree p N = 2. Note that for very small value of p one gets several unconnected component. Our choice of p is such that it should be high enough to give large connected component typically spanning all the nodes. Spectral density and NNSD of the adjacency matrix of this network follow semicircular and GOE distribution, respectively []. We calculate the NNNSD of the adjacency matrix of this network. We plot NNNSD in the Fig.(a). As expected the NNNSD of the adjacency matrix of this network agrees well with the NNSD of GSE matrices. For completeness, we plot the spectral density of this network in the inset of this figure. Fig. 2 shows the statistic for the spectrum of the adjacency matrix of this network. Here we see that statistic for the random network agrees very good with the RMT predictions for very large L, i.e., L 3. Inset of this figure shows the same in semi-logarithmic scale. Here we see expected linear behavior of with slope.978 which is very close to

4 Slope = ln L L Slope = ln L L FIG. 3: (Color online) statistic for eigenvalues spectra of the scale-free network. The circles are numerical results and the solid curve is the GOE prediction of RMT. Inset plots the in semi-logarithmic scale, in this scale it has the slope.975. Figure is plotted for the average over realizations of the networks. All networks have N = 2 nodes and an average degree k = 2 per node. FIG. 4: (Color online) statistic for the eigenvalues spectra of the small-world network. The circles are numerical results and the solid curve is GOE prediction of RMT. Inset plots the upto L = 3 in semi-logarithmic scale, in this scale has the slope.24. Figure is plotted for the average over realizations of the networks. All networks have N = 2 nodes and an average degree k = 2 per node. the RMT predicted value /π 2.3 [see Eq.(3)]. B. Scale-free network Scale-free network is generated by using the model of Barabási and Albert [9]. Starting with a small number, m of the nodes, a new node with m m connections is added at each time step. This new node connects with a already existing node i with the probability π(k i ) k i (preferential attachment), where k i is the degree of the node i. After τ time steps the model leads to a network with N = τ + m nodes and mτ connections. This model leads to a scale-free network, i.e., the probability P(k) that a node has degree k decays as a power law, P(k) k λ, where λ is a constant and for the type of probability law π(k) that we have used λ = 3. Other forms for the probability π(k) are possible which give different values of λ. The results reported here are independent of the value of λ. Density distribution of the network has the triangular distribution with a peak at ρ(), and NNSD follows GOE statistics []. In Fig. (b), we show that the NNNSD of the adjacency matrix of this network agrees well with the NNSD of the GSE matrices. In the inset of this figure, we plot the well-known triangular distribution of the spectral density of scale-free network. Fig. 3 shows the statistic for the adjacency matrix of the scale-free network. Here we see that statistic for the scale-free network agrees very well with the RMT predictions for very large L, i.e., L 5, and deviations begin to be seen after L = 5. This in- dicates that the scale-free network is very much random but not as much as ER random networks, it has some specific features that cannot be modeled by RMT. Inset of this figure shows the expected linear behavior of in semi-logarithmic scale for L 5 with slope.975, a value very close to the RMT predicted value /π 2. C. Small-world network Small-world networks are constructed using the following algorithm of Watts and Strogatz [8]. Starting with a one-dimension ring lattice of N nodes in which every node is connected to its k/2 nearest neighbors, we randomly rewire each connection of the lattice with the probability p such that self-loop and multiple connections are excluded. Thus p = gives a regular network and p = gives a completely random network. The typical smallworld behavior is observed around p =.5 []. We take N = 2 and average degree k = 2. Spectral density of this network is complicated with several peaks. One peak is at λ =. For different values of k the exact positions of other peaks may vary but overall form of spectral density remains similar []. The spacing distribution again follows GOE statistics with Brody parameter β []. We plot NNNSD of the adjacency matrix of the small-world network in the Fig. (c). We see that the NNNSD agrees well with the NNSD of GSE matrices. Here again in the inset of this figure, we plot the spectral density of this network showing typical multi-peak behavior of smallworld networks. Fig. 4 shows the statistic for the

5 5 spectrum of the adjacency matrix corresponding to the small-world network with p =.5. Here we see that statistic for the small-world network agree very good with the RMT predictions for sufficiently large L, i.e., L 3, but much less than the same for random and scale-free networks. This again implies that, besides randomness, small-world network has some specific features. Inset of this figure shows the expected linear behavior of in semi-logarithmic scale for L 3 with slope.24, a value very close to the RMT predicted value /π 2. V. DISCUSSIONS In our previous work [], following RMT, we introduce a new tool to study complex networks. We showed that inspite of spectral density of the adjacency matrices A being different for different model networks and realworld networks, their eigenvalue fluctuations are same and follow GOE statistics of RMT. We attributed this universality to the existence of the minimal amount of randomness in all these networks. We also showed that randomness in the network connections can be quantified by the Brody parameter coming from the RMT. This is a very interesting fact which directs us to make further investigations of the properties of complex networks under the framework of RMT. We showed that there exists one to one correlation between the diameter of the network and the eigenvalue fluctuations of the adjacency matrix. By changing number of connections in the network we get transition to the GOE distribution. As Erdös and Rényi observed that with the fine tuning of network parameter all nodes get connected with a sudden transition; under the RMT framework our analysis suggests transition to some kind of spreading of randomness over the whole network. In this paper we study spectral rigidity via -statistic of RMT of the model networks studied extensively in the literature. We show that the spectral rigidity of these networks follow universal GOE statistics. -statistic of the random matrices following GOE statistics also have long-range correlations among the eigenvalues. From RMT analogy it tells that there exists long range correlations among the eigenvalues. Moreover, we find that the slope of the spectral rigidity function comes /π 2 as calculated for the GOE statistics theoretically [Eq.(3)]. Additionally we show that NNNSD of the eigenvalues of these model networks are identical to the NNSD of GSE matrices. Note that NNNSD of the matrices whose NNSD follow GOE statistics also follow GSE statistics. Above findings show that the statistics of the bulk of the eigenvalues of the model networks is consistent with those of a real symmetric random matrix and deviation from this could be understood as a system dependent part. analysis show that the random network follows RMT prediction for very long range of L, which is not very surprising as random network follows RMT at each level starting from semi-circular density distributions. However interestingly scale-free and small-world networks also follows RMT for sufficiently large value of L. Beyond this value of L deviation in the spectral rigidity is seen, indicating a possible breakdown of universality. This is quite understandable as small-world network is highly clustered and scale-free network also has specific features like hubs, so it is natural that they are not as random as the random network. particularly small-world network we consider here is generated using Strogatz and Watts algorithm and we take the SW networks exactly at the small-world transition which generates networks with very high clustering coefficient and very less number of random connection. Our results show that these very small number of random connections make network sufficiently random to introduce the correlations among the eigenvalues for the sufficiently long range. According to the many recent studies, randomness in the connection is one of the most important and desirable ingredients for the proper functionality or the efficient performance of the systems having underlying network structures. For instance, information processing in the brain is considered to be because of random connections among different modular structure [25]. We feel that we can study the role of random connections, and behavior and evolution of such systems better under the RMT framework. Also this RMT approach may be used to detect the connections most responsible to increase the complexity of networks. For example effect of oxygen molecule on biochemical network of the metabolic network is recently studied and is shown to increase the complexity of networks leading to a major transition in the history of life [26]. In summary, we introduce RMT analysis of complex networks and we show that these networks follow RMT prediction with the universal GOE statistics. These results tell that we can apply random matrix theory to study the behavior and properties of complex networks. 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