Spectra of Modular Random Graphs
aa r X i v : . [ c ond - m a t . d i s - nn ] A ug Spectra of Modular Random Graphs
G¨uler Erg¨un and Reimer K¨uhn Department of Mathematical Sciences, University of Bath,Claverton Down, Bath BA2 7AY, UK Mathematics Department, King’s College London, Strand, London WC2R 2LS,UK
Abstract.
We compute spectra of symmetric random matrices defined on graphsexhibiting a modular structure. Modules are initially introduced as fully connected sub-units of a graph. By contrast, inter-module connectivity is taken to be incomplete. Twodifferent types of inter-module connectivity are considered, one where the number ofintermodule connections per-node diverges, and one where this number remains finite inthe infinite module-size limit. In the first case, results can be understood as a perturbationof a superposition of semicircular spectral densities one would obtain for uncoupledmodules. In the second case, matters can be more involved, and depend in detail oninter-module connectivities. For suitable parameters we even find near-triangular shapedspectral densities, similar to those observed in certain scale-free networks, in a systemof consisting of just two coupled modules. Analytic results are presented for the infinitemodule-size limit; they are well corroborated by numerical simulations.
1. Introduction
Appreciation has steadily grown in recent years that theories of networked systemsprovide useful paradigms to understand complex processes in various branches of scienceand technology, including the evolution of the internet and the world-wide web, flowsof information, power, or traffic, credit, market, or operational risk, food-webs inecosystems, gene-regulation, protein-protein interactions underlying metabolic processesor cell signalling, immune system response, information processing in neural networks,opinion formation, the adoption of new technologies in societies, or the spread of diseasesor epidemics, and more (see, e.g. [1, 2] for recent reviews).Random matrix theory has long been known to constitute a powerful tool to studytopological properties of the graphs underlying networked systems [3, 4, 5, 1, 6]. E.g.,moments of the spectral density of an adjacency matrix describing a graph, gives completeinformation about the number of walks returning to the originating vertex after a givennumber of steps. pectra of Modular Random Graphs pectra of Modular Random Graphs M -modules case. Results for the extensively and finitely cross-connected two-modules caseare presented and discussed in Secs 4.1 and 4.2, respectively. The paper concludes with asummary and outlook in Sec. 5.
2. A Modular System
We begin by considering the most elementary modular system described by a 2 N × N symmetric block matrix of the form H = M (1) VV t M (2) ! (1)in which the symmetric N × N sub-matrices M (1) , M (2) , and the N × N coupling-sub-matrix V are random. We assume that elements of M (1) and M (2) are independently andidentically Gaussian distributed: M ( µ ) ij ∼ N (0 , J µ /N ), 1 ≤ i ≤ j ≤ N , µ = 1 ,
2, and that V ij = c ij K ij with K ij ∼ N (0 , J p /c ), 1 ≤ i, j ≤ N , while p ( c ij ) = (cid:18) − cN (cid:19) δ c ij , + cN δ c ij , . (2)Two different limits will be considered: (i) the so-called extensively cross-connected limit c → ∞ , N → ∞ , with the ratio c/N either remaining finite or approaching zero in the N → ∞ -limit. ‡ (ii) the so-called finitely cross-connected limit, where c is kept constant,as N → ∞ . In this case the distribution of inter-module connectivities is Poissonian inthe thermodynamic limit, with average coordination c .We are interested in the spectral density of H , ρ N ( λ ) = 12 N N X k =1 δ ( λ − λ k ) , (3)more precisely in its average ρ N ( λ ) over the random matrix ensemble introduced above.Here, the λ k are the eigenvalues of H . We shall use ρ ( λ ) to denote the (average) spectraldensity in the thermodaynamic limit, ρ ( λ ) = lim N →∞ ρ N ( λ ) . (4)The spectral density is computed from the resolvent via ρ N ( λ ) = lim ε ց − N π Im ∂∂λ ln det [ λ ε − H ] − / , (5) ‡ The second alternative could more appropriately be referred to as sub-extensive, a distinction we arenot going to make here for simplicity, as it does not affect the nature of results pectra of Modular Random Graphs λ ε ≡ λ − iε , and the inverse square root of the determinant is obtained as aGaussian integral. Using u (1) and u (2) to denote N component vectors, and u = ( u (1) , u (2) )to denote their concatenation, we get ρ N ( λ ) = lim ε ց − N π Im ∂∂λ * ln "Z d u (1) d u (2) (2 π/i ) N exp (cid:26) − i u · [ λ ε − H ] u (cid:27) , (6)where angled brackets on the r.h.s denote an average over connectivities { c ij } and weights { M ( µ ) ij } and { K ij } of the non-vanishing matrix elements.The average of the logarithm is evaluated using replica. ρ N ( λ ) = lim ε ց − N π Im ∂∂λ lim n → n ln h Z nN i , (7)with Z nN = Z Y a d u (1) a d u (2) a (2 π/i ) N exp ( − i n X a =1 u a · [ λ ε − H ] u a ) . (8)Here a = 1 , . . . , n enumerates the replica. The average h Z nN i is easily performed [18, 19]. We have h Z nN i = Z Y a d u (1) a d u (2) a (2 π/i ) N exp ( − i X a λ ε ( u (1) a · u (1) a + u (2) a · u (2) a ) ) × * exp ( i X a u (1) a · M (1) u (1) a )+ × * exp ( i X a u (2) a · M (2) u (2) a )+ × * exp ( i X a u (1) a · V u (2) a )+ . Up to subdominant corrections from diagonal matrix elements this gives h Z nN i ≃ Z Y a d u (1) a d u (2) a (2 π/i ) N exp ( − i X a λ ε ( u (1) a · u (1) a + u (2) a · u (2) a ) − J N X a,b ( u (1) a · u (1) b ) − J N X a,b ( u (2) a · u (2) b ) × Y ij cN exp − J p c X a,b u (1) ia u (1) ib u (2) ja u (2) jb − (9)Expanding the exponential in the product (for large c ) and re-exponentiating one obtains h Z nN i ≃ Z Y a d u (1) a d u (2) a (2 π/i ) N exp − i X a λ ε ( u (1) a · u (1) a + u (2) a · u (2) a ) − J N X a,b ( u (1) a · u (1) b ) − J N X a,b ( u (2) a · u (2) b ) − J p N X a,b ( u (1) a · u (1) b )( u (2) a · u (2) b ) (10) pectra of Modular Random Graphs q ( µ ) ab = 1 N X i u ( µ ) ia u ( µ )) ib , µ = 1 , , (11)as order parameters and by enforcing their definition via δ -functions. This leads to h Z nN i ≃ Z Y µ,a,b d q ( µ ) ab dˆ q ( µ ) ab π/N exp { N [ G + G + G ] } (12)with G = − J X ab ( q (1) ab ) − J X ab ( q (2) ab ) − J p X ab q (1) ab q (2) ab (13) G = − i X ab (ˆ q (1) ab q (1) ab + ˆ q (2) ab q (2) ab ) (14) G = ln "Z Y a d u (1) a d u (2) a π/i exp ( − i X ab ( λ ε δ a,b − q (1) ab ) u (1) a u (1) b − i X ab ( λ ε δ a,b − q (2) ab ) u (2) a u (2) b ) = −
12 ln det( λ ε − q (1) ) −
12 ln det( λ ε − q (2) ) (15) In the large N limit the density ofstates is dominated by the saddle-point contribution to Eq. (12). Adopting the by nowwell-established assumptions of replica-symmetry, and invariance under rotation in thespace of replica at the relevant saddle-point [18] q ( µ ) ab = q ( µ ) d δ ab , ˆ q ( µ ) ab = ˆ q ( µ ) d δ ab , (16)one has G = G + G + G ≃ n ( − J q (1) d ) − J q (2) d ) − J p q (1) d q (2) d − i ˆ q (1) d q (1) d − i ˆ q (2) d q (2) d − h ln( λ ε − q (1) d ) + ln( λ ε − q (2) d ) i(cid:27) (17)where terms of order n are omitted.Stationarity of G requires that the RS order parameters solve the following fixed pointequations − i ˆ q (1) d = 12 J q (1) d + 12 J p q (2) d − i ˆ q (2) d = 12 J q (2) d + 12 J p q (1) d pectra of Modular Random Graphs q (1) d = 1 iλ ε − i ˆ q (1) d q (2) d = 1 iλ ε − i ˆ q (2) d These can be combined by eliminating the conjugate variables to give, q (1) d = 1 iλ ε + J q (1) d + J p q (2) d , q (2) d = 1 iλ ε + J q (2) d + J p q (1) d (18) c → ∞ Case
The average spectral density is obtained bydifferentiation w.r.t λ via Eq. (7). Only terms with explicit λ -dependence in G of Eq. (17)contribute at the saddle point. This gives ρ ( λ ) = 12 π Re h q (1) d + q (2) d i (19)For the limiting cases J p = 0, describing two uncoupled systems, and J = J = J ,describing a coupling of two systems with identical statistics of intra-system couplings weobtain the following explicit analytical results.In the uncoupled case J p = 0 the solution of Eqs. (18) is q ( µ ) d = − iλ ε J µ ± J µ q J µ − λ ε (20)giving ρ ( λ ) = 12 π " Θ(4 J − λ )2 J q J − λ + Θ(4 J − λ )2 J q J − λ , (21)i.e. a superposition of two independent Wigner semi-circular densities as expected.In the case of two coupled systems which are statistically identical, with J = J = J ,the solution of Eqs. (18) is q (1) d = q (2) d = q d with q d = − iλ ε J + J p ) ± J + J p ) q J + J p ) − λ ε (22)leading to ρ ( λ ) = Θ(4( J + J p ) − λ )2 π ( J + J p ) q J + J p ) − λ , (23)i.e. a Wigner semi-circular density, with the radius of the semi-circle given by r =2 q J + J p In the asymmetric case J = J we solve the fixed point equations numerically. A non-zerocoupling leads to a smoothing of the superposition of the two independent semi-circles asanticipated; see Sec. 4 below. pectra of Modular Random Graphs We now consider a system with the same basic setup, except that we assume the averagenumber of cross-connections c to be finite (also in the thermodynamic limit). In thiscase, the analysis is considerably more involved. It combines techniques used in [18] forconnected and of [19] for sparse random matrices, and more specifically the reformulation[14] of the sparse case that allows to proceed to explicit results. In this case the calculationscan be carried out without restricting the distribution of the non-zero matrix elements ofthe inter-module connections to be Gaussian, and we will in developing the theory notmake that restriction.For the average of the replicated partition function we get h Z nN i ≃ Z Y a d u (1) a d u (2) a (2 π/i ) N exp − i X a λ ε ( u (1) a · u (1) a + u (2) a · u (2) a ) − J N X a,b ( u (1) a · u (1) b ) − J N X a,b ( u (2) a · u (2) b ) + cN X ij * exp ( iK X a u (1) ia u (2) ja )+ K − ! (24)in analogy to Eq. (9), where h . . . i K represents an average over the K ij distribution,which is as yet left open. As in Eq. (9), subdominant contributions coming from diagonalmatrix elements are omitted.Decoupling of sites is achieved by introducing order parameters q ( µ ) ab = 1 N X i u ( µ ) ia u ( µ ) ib , µ = 1 , , (25)as in the extensively cross-connected case before, but in addition also the replicateddensities ρ ( µ ) ( u ) = 1 N X i Y a δ (cid:16) u a − u ( µ ) ia (cid:17) , µ = 1 , , (26)as well as the corresponding conjugate quantities. This allows to express Eq. (24) asan integral over the order parameters and their conjugates, combined with a functionalintegral over the replicated densities and their conjugates, h Z nN i = Z Y µ {D ρ ( µ ) D ˆ ρ ( µ ) } Z Y µ,a,b d q ( µ ) ab dˆ q ( µ ) ab π/N exp { N [ G + G + G ] } , (27)with G = − J X ab ( q (1) ab ) − J X ab ( q (2) ab ) + c Z d ρ (1) ( u )d ρ (2) ( v ) * exp ( iK X a u a v a )+ K − ! G = − i X ab (cid:16) ˆ q (1) ab q (1) ab + ˆ q (2) ab q (2) ab (cid:17) − i Z d u (cid:16) ˆ ρ (1) ( u ) ρ (1) ( u ) + ˆ ρ (2) ( u ) ρ (2) ( u ) (cid:17) pectra of Modular Random Graphs G = ln Z Y a d u a exp ( i ˆ ρ (1) ( u ) − i X ab (cid:16) λ ε δ ab − q (1) ab (cid:17) u a u b ) + ln Z Y a d u a exp ( i ˆ ρ (2) ( u ) − i X ab (cid:16) λ ε δ ab − q (2) ab (cid:17) u a u b ) Here we have introduced abbreviations of the form d ρ ( µ ) ( u ) ≡ d u ρ ( µ ) ( u ) for integralsover densities where appropriate to simplify notation. Eq. (27) is evaluated by thesaddle point method. The saddle point for this problem is expected to be replica-symmetric and symmetric under rotation in the space of replica as in the extensivelycross-connected case. In the present context this translates to an ansatz of the form q ( µ ) ab = q ( µ ) d δ ab , ˆ q ( µ ) ab = ˆ q ( µ ) d δ ab , (28)for the Edwards Anderson type order parameters [18] and ρ ( µ ) ( u ) = Z d π ( µ ) ( ω ) Y a exp [ − ω u a ] Z ( ω ) ,i ˆ ρ ( µ ) ( u ) = ˆ c ( µ ) Z dˆ π ( µ ) (ˆ ω ) Y a exp [ − ˆ ω u a ] Z (ˆ ω ) , (29)i.e. an uncountably infinite superposition of complex Gaussians (with Re ω ≥ ω ≥
0) for the replicated densities and their conjugates [14]. Here we have introducedthe shorthand Z ( ω ) = Z d u exp (cid:20) − ω u (cid:21) = q π/ω . (30)The ˆ c ( µ ) in the expressions for ˆ ρ ( µ ) are to be determined such that the densities ˆ π ( µ ) arenormalised.This ansatz translates path-integrals over the replicated densities ρ and ˆ ρ into path-integrals over the densities π and ˆ π , giving h Z nN i = Z Y µ {D π ( µ ) D ˆ π ( µ ) } Z Y µ d q ( µ ) d dˆ q ( µ ) d π/N exp { N [ G + G + G ] } (31)with now G ≃ n " − J q (1) d ) − J q (2) d ) + c Z d π (1) ( ω )d π (2) ( ω ′ ) * ln Z ( ω, ω ′ , K ) Z ( ω ) Z ( ω ′ ) + K , (32) G ≃ − n h i ˆ q (1) d q (1) d + i ˆ q (2) d q (2) d i − X µ =1 " ˆ c ( µ ) + n ˆ c ( µ ) Z dˆ π ( µ ) (ˆ ω )d π ( µ ) ( ω ) ln Z (ˆ ω + ω ) Z (ˆ ω ) Z ( ω ) , (33) pectra of Modular Random Graphs G ≃ X µ =1 " ˆ c ( µ ) + n ∞ X k =0 p ˆ c ( µ ) ( k ) Z { dˆ π ( µ ) } k ln Z ( µ ) ( { ˆ ω } k ) Q kℓ =1 Z (ˆ ω ℓ ) . (34)Here { dˆ π ( µ ) } k ≡ Q kℓ =1 dˆ π ( µ ) (ˆ ω ℓ ), while { ˆ ω } k ≡ P kℓ =1 ˆ ω ℓ , and p ˆ c ( µ ) ( k ) = ˆ c ( µ ) k k ! exp[ − ˆ c ( µ ) ] (35)is a Poissonian distribution with average h k i = ˆ c ( µ ) , and we have introduced the partitionfunctions Z ( µ ) ( { ˆ ω } k ) = Z d u q π/i exp (cid:20) − (cid:18) iλ ε − i ˆ q ( µ ) d + { ˆ ω } k (cid:19) u (cid:21) = iiλ ε − i ˆ q ( µ ) d + { ˆ ω } k ! / , (36) Z ( ω, ω ′ , K ) = Z d u d v exp (cid:20) − (cid:18) ωu + ω ′ v − iKuv (cid:19)(cid:21) = 2 π √ ωω ′ + K . (37)The stationarity conditions for π (1) ( ω ) and π (2) ( ω ) then readˆ c (1) Z dˆ π (1) (ˆ ω ) ln Z (ˆ ω + ω ) Z (ˆ ω ) Z ( ω ) = c Z d π (2) ( ω ′ ) * ln Z ( ω, ω ′ , K ) Z ( ω ) Z ( ω ′ ) + K + µ (38)and ˆ c (2) Z dˆ π (2) (ˆ ω ) ln Z (ˆ ω + ω ) Z (ˆ ω ) Z ( ω ) = c Z d π (1) ( ω ′ ) * ln Z ( ω ′ , ω, K ) Z ( ω ) Z ( ω ′ ) + K + µ , (39)with µ and µ Lagrange multipliers to take the normalisation of π (1) and π (2) into account.The stationarity conditions for the ˆ π ( µ ) (ˆ ω ) are c Z d π ( µ ) ( ω ) ln Z (ˆ ω + ω ) Z (ˆ ω ) Z ( ω ) = X k ≥ k p ˆ c ( µ ) ( k ) Z { dˆ π ( µ ) } k − ln Z ( µ ) (ˆ ω + { ˆ ω } k − ) Z (ˆ ω ) Q k − ℓ =1 Z (ˆ ω ℓ ) + σ µ (40)where the σ µ are once more Lagrange multipliers to take the normalisation of the ˆ π ( µ ) (ˆ ω )into account.The stationarity conditions for the q ( µ ) d are i ˆ q ( µ ) d = − J µ q ( µ ) d , (41)the corresponding ones for the conjugate variables give q ( µ ) d = ∞ X k =0 p ˆ c ( µ ) ( k ) Z { dˆ π ( µ ) } k D u E ( µ ) { ˆ ω } k = ∞ X k =0 p ˆ c ( µ ) ( k ) Z { dˆ π ( µ ) } k iλ ε + J µ q ( µ ) d + { ˆ ω } k (42) pectra of Modular Random Graphs h . . . i ( µ ) { ˆ ω } k is defined as an average w.r.t. the Gaussian weight in terms of which Z ( µ ) ( { ˆ ω } k ) is defined. We have used Eq. (41) to express the i ˆ q ( µ ) d in terms of the q ( µ ) d .Following [20, 21], the stationarity conditions for π (1) ( ω ) and π (2) ( ω ) are rewritten in aform that suggests solving them via a population based algorithm. In the present case weget [22, 14] ˆ π (1) (ˆ ω ) = Z d π (2) ( ω ′ ) D δ (cid:16) ˆ ω − ˆΩ( ω ′ , K ) (cid:17)E K (43)and ˆ π (2) (ˆ ω ) = Z d π (1) ( ω ′ ) D δ (cid:16) ˆ ω − ˆΩ( ω ′ , K ) (cid:17)E K , (44)in which ˆΩ = K ω ′ , (45)while π ( µ ) ( ω ) = X k ≥ kc p c ( k ) Z { dˆ π ( µ ) } k − δ (cid:16) ω − Ω ( µ ) k − (cid:17) (46)with Ω ( µ ) k − = iλ ε + J µ q ( µ ) d + k − X ℓ =1 ˆ ω ℓ . (47)In Eqs. (43), (44), and (46), we have already invested ˆ c ( µ ) = c to enforce that the ˆ π ( µ ) are normalized.For the spectral density we obtain the same formal result as for the extensively cross-connected system before, ρ ( λ ) = 12 π Re h q (1) d + q (2) d i . (48)The solution of these coupled sets of equations is considerably more involved than for theextensively cross-connected systems considered earlier, as it involves solving equations forthe coupled macroscopic order parameters q (1) d and q (2) d , which are themselves expressedin terms of averages over self-consistent solutions of a pair of non-linear integral equationsparameterised by these order parameters. However, we have found that a populationdynamics in which values of q (1) d and q (2) d are iteratively updated using Eq. (42) by samplingfrom the corresponding populations rapidly converges to the correct solution. pectra of Modular Random Graphs
3. The Multi-Modular Case
Finally, we consider systems with M modules, each of size N , mutually cross-connectedwith finite connectivity. Inside modules we may or may not have all-to-all Gaussiancouplings of variances J µ /N . In addition there are finitely many O (1) module-to-modulecouplings for each vertex, with average connectivities c µν for couplings between nodes inmodules µ and ν (possibly including the case µ = ν ). Evaluating this case requires a fairlystraightforward generalisation of the setup developed earlier. Here we only produce thefixed point equations and the final expression for the average spectral density. We getˆ π ( µ ) (ˆ ω ) = X ν c µν c µ Z d π ( ν ) ( ω ) D δ (cid:16) ˆ ω − ˆΩ( ω, K ) (cid:17)E µν (49)and π ( µ ) ( ω ) = X k ≥ kc µ p c µ ( k ) Z { dˆ π ( µ ) } k − δ (cid:16) ω − Ω ( µ ) k − (cid:17) , (50)as well as q ( µ ) d = ∞ X k =0 p c µ ( k ) Z { dˆ π ( µ ) } k (cid:16) iλ ε + J µ q ( µ ) d + P kℓ =1 ˆ ω ℓ (cid:17) (51)with ˆΩ( ω, K ) = K ω , (52)and Ω ( µ ) k − = iλ ε + J µ q ( µ ) d + k − X ℓ =1 ˆ ω ℓ . (53)In Eq. (49), h . . . i µν denotes an average over the distribution of couplings connectingmodules µ and ν , and we have the normalisation P ν c µν = c µ .For the spectral density we obtain the (obvious) generalisation of those obtained for thetwo modules case before, viz. ρ ( λ ) = 1 M π Re M X µ =1 q ( µ ) d . (54)Generalising these to the situation of varying module-sizes N µ = r µ N , with r µ >
0, isstraightforward. pectra of Modular Random Graphs ρ ( λ ) λ ρ ( λ ) λ Figure 1.
Spectral densities for J = 1, J = 0 .
1, and J p = 0 . J p = 0 .
4. Results
The results we obtain for extensively cross-connected systems can be understood in termsof superpositions of Wigner semi-circles one would expect for the spectral densities ofthe modules if they were uncoupled, but smoothed (and broadened) by the interaction.We have checked that our results agree perfectly with simulations, but have not includedresults of simulations in the figures for the extensively cross-connected systems below.In Fig. 1 we explore the effect of the inter-module coupling strength J p on the spectraldensity of a system of two coupled modules with intra-module coupling strengths J µ differing by one order of magnitude. The reader is invited to compare the results withthose expected for in a situation where the two modules were non-interacting, namely asimple superposition of semi-circular densities of radii 2 J and 2 J , respectively.In Fig. 2 we keep the strength of the inter-module couplings but vary the ratio of the twointra-module couplings.Results for uncoupled modules or for coupled modules with identical intra-modulecoupling statistics are not shown; they were found to be in perfect agreement with thesimple analytical results presented in Sec. 2.1.2. The modifications generated by cross-connections that remain finite in number andstrength for each node in each of the blocks are more pronounced than those createdby extensive (infinitesimal) cross-connections. We mention just two fairly drasticmodifications. First, the spectral density of a system of finitely cross-connected modules pectra of Modular Random Graphs ρ ( λ ) λ ρ ( λ ) λ Figure 2.
Spectral densities for J = 1, J = 0 .
25 (left), and J = 0 . J p = 0 . does no longer have sharp edges with square-root singularities of the spectral density atthe edges as in the case of extensively cross-connected systems, where it derives from aperturbed superposition of semi-circular densities, but rather tails with decay laws thatdepend on the nature of the distribution of cross-connections. In the present case ofPoisson distributions of cross-connections we find these tails to exhibit exponential decay.Second, even with identical intra-module coupling statistics the spectral density is nolonger given by a simple semi-circular law as it is for the extensively cross-connected case,but rather exhibits marked deviations from the semi-circular law, with details dependingboth on the number and the strengths of the cross-connections, as shown in Fig. 3.As the numerics in the present finitely cross-connected case is considerably moreinvolved we present results of the analytic theory together with checks against numericaldiagonalization. Figs 3-4 demonstrate that the analytic results are in excellent agreementwith numerical simulations, virtually indistinguishable for the parameters and thestatistics used.Fig 3 exhibits spectral densities of a two-module system with identical intra-modulecoupling statistics, and Gaussian cross-connections with a Poissonian degree statistics, ofaverage cross-coordinations 2 and 5 respectively. In the first case the spectral density hasa shape close to triangular, though more rounded at the tip than what is known fromspectral density of certain scale-free systems.Fig. 4 looks at a system of two modules having intra-module connections of differentstrengths, and average cross-coordinations 2; results resemble the corresponding caseof extensive cross-connectivity, apart from the exponential tails, which are an exclusivefeature of the finitely cross-connected case. pectra of Modular Random Graphs ρ ( λ ) λ ρ ( λ ) λ Figure 3.
Spectral densities for J = J = 1 and J p = 1 / √ c for c = 2 (left) and c = 5 (right). Results of numerical diagonalizations of 500 matrices containing twocoupled blocks, each of dimension 1000, are shown for comparison (dashed lines); theyare virtually indistinguishable from results of the analytic theory. ρ ( λ ) λ Figure 4.
Spectral density for J = 1, J = 0 . J p = 1 / √ c for c = 2 (full line).Results of numerical diagonalizations of 500 matrices containing two coupled blocks, eachof dimension 1000, are shown for comparison (dashed line), and are once more virtuallyindistinguishable from results of the analytic theory.
5. Summary and Conclusions
We have evaluated spectral densities of symmetric matrices describing modular systems.Modularity is regarded as one of several routes to create heterogeneity in interactingsystems. In some biological systems in fact, modularity of interactions appears to be anatural consequence of compartmentalization; systems with cellular structure, or sub-structures within cells come to mind, where heterogeneity of interaction patterns dueto modularity of the system would seem to enjoy a greater degree of plausibility thanheterogeneity as observed in certain scale free systems.In any case, whenever large systems with different levels of organization are considered, pectra of Modular Random Graphs
References [1] R. Albert and A.-L. Barab´asi. Statistical Mechanics of Complex Networks.
Rev. Mod. Phys. , 74:47–97, 2002.[2] S.N. Dorogovtsev and J.F.F Mendes.
Evolution of Networks: from Biological Networks to the Internetand WWW . Oxford University Press, Oxford, 2003.[3] D. Cvetkovi´c, M. Doob, and H. Sachs.
Spectra of Graphs - Theory and Applications, 3rd Edition .J.A. Barth, Heidelberg, 1995.[4] B. Bollob`as.
Random Graphs . Cambridge Univ. Press, Cambridge, 2001.[5] I. Farkas, I. Der´eny, A.L. Barab´asi, and T. Vicsek. Spectra of Real World Graphs: Beyond theSemi-Circle Law.
Phys. Rev. E , 64:026704, 2001.[6] S. N. Dorogovtsev, A. V. Goltsev, J. F. F. Mendes, and A. N. Samukhin. Spectra of ComplexNetworks.
Phys. Rev. E , 68:046109, 2003.[7] A. L. Barab´asi and R. Albert. Emergence of Scaling in Random Networks.
Science , 286:509–512,1999.[8] D. J. Watts and S. A. Strogatz. Collective Dynamics of “Small-World” Networks.
Nature , 393:440,1998.[9] M. Mitrovi´c and B. Tadi´c. Spectral and Dynamical Properties in Classes of Sparse Networks withMesoscopic Inhomogeneity. arXiv:cond-mat/0809.4850, 2008.[10] K.-I. Goh, B. Kahng, and D. Kim. Spectra and Eigenvectors of Scale-Free Networks.
Phys. Rev.E , 64:051903, 2001.[11] T. Nagao and G. J. Rodgers. Spectral Density of Complex Networks with a Finite Mean Degree.
J. Phys. A , 41:265002, 2008.[12] G. J. Rodgers, K. Austin, B. Kahng, and D. Kim. Eigenvalue Spectra of Complex Networks.
J.Phys. A , 38:9431–9437, 2005.[13] D. Kim and B. Kahng. Spectral densities of scale-free networks.
Chaos , 17:026115, 2007.[14] R. K¨uhn. Spectra of Sparse Random Matrices.
J. Phys. A , 41:295002, 2008.[15] T. Rogers, I. P´erez-Castillo, K. Takeda, and R. K¨uhn. Cavity Approach to the Spectral Density ofSparse Symmetric Random Matrices.
Phys. Rev. E , 78:031116, 2008.[16] T. Rogers and I. P´erez-Castillo. Cavity Approach to the Spectral Density of Non-Hermitean SparseMatrices.
Phys. Rev. E , 79:012101, 2009. pectra of Modular Random Graphs [17] D. S. Dean. An Approximation Scheme for the Density of States of the Laplacian on RandomGraphs. J. Phys. A , 35:L153–L156, 2002.[18] S. F. Edwards and R. C. Jones. The Eigenvalue Spectrum of a Large Symmetric Random Matrix.
J. Phys. A , 9:1595–1603, 1976.[19] G. J. Rodgers and A. J. Bray. Density of States of a Sparse Random Matrix.
Phys. Rev. B ,37:3557–3562, 1988.[20] R. Monasson. Optimization Problems and Replica Symmetry Breaking in Finite Connectivity Spin-Glasses.
J. Phys. A , 31:513–529, 1998.[21] M. M´ezard and G. Parisi. The Bethe Lattice Spin Glass Revisited.
Eur. Phys. J. B , 20:217–233,2001.[22] R. K¨uhn, J. van Mourik, M. Weigt, and A. Zippelius. Finitely Coordinated Models for Low-Temperature Phases of Amorphous Systems.
J. Phys. A , 40:9227–9252, 2007.[23] R. K¨uhn and J. van Mourik.