Localization and universality of eigenvectors in directed random graphs
LLocalization and universality of eigenvectors in directed random graphs
Fernando Lucas Metz
Physics Institute, Federal University of Rio Grande do Sul, 91501-970 Porto Alegre, Brazil andLondon Mathematical Laboratory, 18 Margravine Gardens, London W6 8RH, United Kingdom
Izaak Neri
Department of Mathematics, Kings College London, Strand, London, WC2R 2LS, UK (Dated: July 28, 2020)Although the spectral properties of random graphs have been a long-standing focus of networktheory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytictreatment. We present a general theory for the statistics of the right eigenvector components indirected random graphs with a prescribed degree distribution and with randomly weighted links. Weobtain exact analytic expressions for the inverse participation ratio and show that right eigenvectorsof directed random graphs with a small average degree are localized. Remarkably, the criticalmean degree for the localization transition is independent of the degree fluctuations. We also showthat the dense limit of the distribution of the right eigenvectors is solely determined by the degreefluctuations, which generalizes standard results from random matrix theory. We put forward aclassification scheme for the universality of the eigenvector statistics in the dense limit, which issupported by an exact calculation of the full eigenvector distributions. More generally, this paperprovides a theoretical formalism to study the eigenvector statistics of sparse non-Hermitian randommatrices.
Introduction.
Complex systems, such as neural net-works [1–3], ecosystems [4], and the World Wide Web[5, 6], consist of components that interact along the edgesof large directed networks. Therefore, a problem of fun-damental importance is how network structure affects theproperties of complex systems.Much insight in the dynamics of a complex systemis gained from the eigenvalues and eigenvectors of theadjacency matrix representing its interaction network.For example, the dynamics in the vicinity of a station-ary state is governed by the eigenvalues and eigenvec-tors of the adjacency matrix [7, 8], which is importantin the study of disease spreading [9–12], synchronizationof coupled oscillators [13, 14], and stability of biologicalsystems, such as, neural networks [15, 16], ecosystems[17, 18], and gene regulatory networks [19, 20]. Eigenval-ues and eigenvectors of sparse matrices are also impor-tant to evaluate spectral algorithms for ranking nodes[21–23], measuring node centrality [24, 25], detectingcommunities [26–28], and for recovering signals [29].Properties of eigenvectors of random graphs have beenmainly studied for undirected graphs [30–43], where animportant feature is the delocalization-localization transi-tion . Localized eigenvectors occupy a few sites, whereasdelocalized eigenvectors are extended over the whole sys-tem. In general, the delocalization-localization transi-tion implies a qualitative change in the properties of asystem. Examples are the metal-insulator phase transi-tion in solid state physics [30, 37], the transition froman algorithmically successful to a failure phase in spec-tral algorithms [29, 44], and the transition from a phasegoverned by a collective mode to a phase governed by alocalized mode in dynamical systems [10]. Besides that,eigenvector localization also impacts the efficiency of net- work centrality measures [25, 45] and the propagation ofperturbations in ecosystems [46].The statistical properties and the localization of eigen-vectors of directed random graphs have so far eluded amathematical analysis. Notable exceptions are modelsdefined on one-dimensional chains, such as the Hatano-Nelson model [47, 48] and the Feinberg-Zee model [49]for the (de)pinning of vortex lines in superconductors.Recently these models have been extended to consider lo-calization in one-dimensional biological systems [16, 50].In this Letter, we develop an exact theory for the sta-tistical properties of the right (or left) eigenvectors ofdirected random graphs with a prescribed degree distri-bution and random couplings. We derive exact analyticexpressions for the inverse participation ratio and forthe critical point of the localization-delocalization tran-sition. Surprisingly, when the moments of the degree dis-tribution are finite, the critical point of the localization-delocalization transition is independent of the degree dis-tribution. Moreover, the right eigenvectors are localizedif the degree distribution has diverging moments. We alsoshow that in the dense limit the statistics of the compo-nents of right eigenvectors are only determined by degreefluctuations. In this limit, we obtain distinct universalityclasses that depend on an exponent that quantifies thedegree fluctuations.
Model set-up.
We consider random matrices A of di-mension n × n with elements A ij = J ij C ij , i, j ∈ { , , . . . , n } , (1)where C ij ∈ { , } are the entries of the adjacency matrix C of a directed random graph with a prescribed degreedistribution p K in ,K out ( k, (cid:96) ) = p K in ( k ) p K out ( (cid:96) ) (2) a r X i v : . [ c ond - m a t . d i s - nn ] J u l of indegrees K in and outdegrees K out . We set C ij = 1when there exists a directed link pointing from i to j ,such that the outdegree (indegree) of the i -th node isgiven by K out i = (cid:80) nj =1 C ij ( K in i = (cid:80) nj =1 C ji ). The J ij are real-valued independent and identically distributedrandom variables drawn from a distribution p J ( x ).Directed random graphs with a prescribed degree dis-tribution [51–56] have been used to model the WorldWide Web [5, 6] and neural networks [1, 3, 57]. In thismodel, the indegrees and outdegrees are drawn indepen-dently from Eq. (2) subject to the constraint (cid:80) nj =1 K in j = (cid:80) nj =1 K out j , and subsequently nodes are randomly con-nected according to the given degree sequences. Sincethe degree distributions are specified at the outset, thismodel provides the ideal setting to explore the influenceof network topology on the spectral properties of A .In what follows, brackets (cid:104)·(cid:105) denote the average withrespect to the distribution of A . In particular, we use c = (cid:104) K out (cid:105) (3)for the mean outdegree, and we denote the variance of arandom variable X by var( X ) = (cid:104) X (cid:105) − (cid:104) X (cid:105) . Spectra of infinitely large matrices A . The spectrumof A has been studied in Refs. [58–61]. For n → ∞ and c >
1, directed random graphs have a giant stronglyconnected component [62] and the spectral distribu-tion ρ A ( λ ) = n − (cid:80) nj =1 δ [ λ − λ j ( A )] of the eigenval-ues { λ j ( A ) } nj =1 is supported on a disk of radius | λ b | = (cid:112) c (cid:104) J (cid:105) centered at the origin of the complex plane. Inaddition, if c > c gap = (cid:104) J (cid:105)(cid:104) J (cid:105) , (4)then there exists an eigenvalue outlier located at λ isol = c (cid:104) J (cid:105) that is separated from the boundary λ b by a finitegap. Figure 1 shows the eigenvalues for an example of adirected random graph, where one clearly identifies theoutlier λ isol and the boundary λ b of ρ A ( λ ) for n → ∞ . Distribution of the right eigenvector components.
Aright eigenvector (cid:126)R ( λ ) associated to an eigenvalue λ of A satisfies A (cid:126)R ( λ ) = λ (cid:126)R ( λ ) . (5)In this paper, we study localization of (cid:126)R ( λ ) with thedistribution p R ( r | λ ) = lim n →∞ n n (cid:88) i =1 δ [ r − R i ( λ )] (6)of the entries of (cid:126)R and we also study universality classesin the dense limit c → ∞ .If λ is an outlier ( λ = λ isol ) or λ is located at theboundary of the spectrum ( λ = λ b ), then p R ( r | λ ) fulfills − − p c h J i λ isol = c h J i Re( λ ) I m ( λ ) Boundary λ b FIG. 1. Eigenvalues of three realizations (circles, triangles,and squares) of the adjacency matrix A of directed randomgraphs with n = 500 (see Eq. (1)). The indegrees and outde-grees follow a Poisson distribution with average c = 5. Theweights J ij are drawn from a Gaussian distribution p J withmean and variance equal to one. the equation [59–61] p R ( r | λ ) = ∞ (cid:88) k =0 p K out ( k ) (cid:90) k (cid:89) j =1 d x j d r j p J ( x j ) p R ( r j | λ ) × δ r − λ k (cid:88) j =1 x j r j , (7)with d r ≡ dRe r dIm r . Equation (7) is exact for in-finitely large directed random graphs with a prescribeddegree distribution [61]. Inverse participation ratio.
The localization of (cid:126)R ( λ )can be characterized in terms of the inverse participationratio (IPR) [36, 63, 64] I ( λ ) ≡ lim n →∞ n (cid:80) ni =1 | R i ( λ ) | ( (cid:80) ni =1 | R i ( λ ) | ) = (cid:104)| R ( λ ) | (cid:105)(cid:104)| R ( λ ) | (cid:105) , (8)where we have used that I is self-averaging [65]. TheIPR is finite if (cid:126)R ( λ ) is delocalized, whereas I ( λ ) divergesif (cid:126)R ( λ ) is localized on a finite number of nodes.From Eq. (7), we derive in the Supplemental Mate-rial [65] exact expressions for the IPR when λ = λ isol or λ = λ b . We find that I ( λ b ) = ( γ + 1) (cid:2) (cid:104) ( K out ) (cid:105) − c (cid:3) c ( c − (cid:104) J (cid:105) / (cid:104) J (cid:105) ) , (9)where γ = 2 when λ b ∈ R and γ = 1 when λ b / ∈ R . FromEq. (9), it follows that I ( λ b ) ≥ γ + 1 and, consequently, { R i ( λ b ) } ni =1 are non-Gaussian random variables if either p K out or p J has nonzero variance. Analogously, the IPRat λ = λ isol reads I ( λ isol ) = 3 β (cid:104) J (cid:105) ( c (cid:104) J (cid:105) − c (cid:104) J (cid:105) ) + β (cid:0) c (cid:104) J (cid:105) − c (cid:104) J (cid:105) (cid:1) β ( c (cid:104) J (cid:105) − c (cid:104) J (cid:105) )+ 12 β (cid:104) J (cid:105)(cid:104) J (cid:105) (cid:0) c (cid:104) J (cid:105) − c (cid:104) J (cid:105) (cid:1) ( c (cid:104) J (cid:105) − c (cid:104) J (cid:105) ) ( c (cid:104) J (cid:105) − c (cid:104) J (cid:105) )+ 4 β (cid:104) J (cid:105) (cid:0) c (cid:104) J (cid:105) − c (cid:104) J (cid:105) (cid:1) β ( c (cid:104) J (cid:105) − c (cid:104) J (cid:105) ) ( c (cid:104) J (cid:105) − c (cid:104) J (cid:105) )+ 6 β (cid:104) J (cid:105) (cid:0) c (cid:104) J (cid:105) − c (cid:104) J (cid:105) (cid:1) β ( c (cid:104) J (cid:105) − c (cid:104) J (cid:105) ) , (10)where β (cid:96) ≡ ∞ (cid:88) k = (cid:96) +1 p K out ( k ) k !( k − (cid:96) − , (cid:96) = 1 , , . (11)Figure 2 illustrates Eqs. (9) and (10) as a function of c for a Gaussian distribution p J and three different outde-gree distributions: Poisson, exponential, and Borel distri-butions (see Supplemental Material [65]). All momentsof these degree distributions are finite and each p K out isparametrized only by c . Figure 2 shows that the IPR isfinite if c is large enough and it diverges for small c , whichproves the existence of a delocalization-localization phasetransition in directed random graphs. The localization phase transition.
There are twomechanisms for localization, one which is governed byfluctuations of J ij , and a second one that is governed bydegree fluctuations.The first mechanism is illustrated in Fig. 2 and it holdsfor an arbitrary p K out with finite moments. In this case,from Eqs. (9) and (10) it follows that right eigenvectorsassociated to λ = λ b and λ = λ isol are localized when c is smaller than c b = (cid:104) J (cid:105)(cid:104) J (cid:105) and c = (cid:104) J (cid:105)(cid:104) J (cid:105) , (12)respectively. Thus, the critical points for the localiza-tion transitions only depend on the lower moments of p J and they are independent of p K out . When the J ij areconstant, then c b = c isol = 1 such that the delocalization-localization transition is governed by the percolationtransition for the strongly connected component [62]. Onthe other hand, when there is disorder in J ij , then c b > c isol > p J is a Gaussian distribution with mean µ and variance σ . In this case, c gap , c b and c isol only depend on theratio σ/µ . A few generic aspects of eigenvector local-ization in directed random graphs, which also hold fornon-Gaussian p J , are illustrated in Fig. 3. First, theeigenvector (cid:126)R ( λ isol ) is delocalized when (cid:104) J (cid:105) > (cid:104) J (cid:105)(cid:104) J (cid:105) because c gap > c isol . Second, the transition lines fulfill c gap < c isol < c b for (cid:104) J (cid:105) < (cid:104) J (cid:105)(cid:104) J (cid:105) . Lastly, we observe h J i = 1 h J i = 2 (a) c I ( λ i s o l ) PoissonExponentialBorel (b) h J i = 0 h J i = 1 c I ( λ b ) FIG. 2. The inverse participation ratio I ( λ ) of the right eigen-vectors associated to the outlier eigenvalue λ isol [Panel (a)]and to an eigenvalue λ b / ∈ R at the boundary of the spec-trum [Panel (b)]. Equations (9) and (10) (different line styles)are shown as a function of the average degree c for differentoutdegree distributions: Poisson, exponential, and Borel (seeSupplemental Material [65]). The weights J ij are drawn froma Gaussian distribution p J with first and second momentsindicated on each panel. The different symbols are resultsobtained from the numerical solution of Eq. (7), while directdiagonalization results for I ( λ ) are presented in the Supple-mental Material [65]. The results for the Borel distributionare rescaled as I ( λ isol ) → I ( λ isol ) /c in panel (a). . . localizeddelocalized ga pp e d un ga pp e d h J i = h J ih J i σ/µ c c isol , Eq. (12) c b , Eq. (12) c gap , Eq. (4) FIG. 3. Phase diagram for localization of right eigenvectorsassociated to the outlier λ isol and to eigenvalue λ b at theboundary of the spectrum. The distribution p J is Gaussianwith mean µ and standard deviation σ . that the critical transitions c gap , c isol and c gap intersectin a common point because of the identity c = c b c .The second mechanism for localization is due to largedegree fluctuations. From Eqs. (9) and (10), it followsthat I ( λ b ) → ∞ if (cid:104) ( K out ) (cid:105) → ∞ and I ( λ isol ) → ∞ if (cid:104) ( K out ) (cid:105) → ∞ , independently of the distribution p J .Hence, localization of (cid:126)R ( λ b ) and (cid:126)R ( λ isol ) also occurs ingraphs with power-law degree distributions. In the se-quel, we show that degree-based localization persists inthe dense limit. Localization and universality in the dense limit.
Letus explore the localization and universality of eigenvec-tors in the dense limit c → ∞ . In Fig. 2, we observethat I ( λ ) flows to different asymptotic values for c (cid:29) c → ∞ , we analyze the moments of the distribution p R .We characterize the dense limit of p R ( r | λ isol ) using therelative variance R c = var[ R ( λ isol )] (cid:104) R ( λ isol ) (cid:105) , (13)while we choose to characterize the dense limit of p R ( r | λ b ) through the kurtosis K c = (cid:68) (Re R ( λ b )) (cid:69)(cid:68) (Re R ( λ b )) (cid:69) = (4 − γ )2 I ( λ b ) , (14)where the second equality in Eq. (14) follows from thefact that odd moments of p R ( r | λ b ) are zero [65]. Setting c → ∞ in Eqs. (13) and (14), we obtain [65] R ∞ = lim c →∞ var[ K out ] c , (15) K ∞ = 3 (cid:18) c →∞ var[ K out ] c (cid:19) , (16)which shows that the dense limit of p R is determined bythe degree distribution. We see that, in general, p R ( r | λ b )and p R ( r | λ isol ) are not Gaussian in the dense limit.With the purpose of classifying the universal behaviorof p R for c → ∞ , let us consider degree distributions thatsatisfy var[ K out ] = Bc α ( c (cid:29) , (17)where α and B depend on the specific choice of p K out ( k ).Equation (17) holds for most degree distributions, includ-ing those addressed in Fig. 2. Plugging this ansatz forvar[ K out ] in Eqs. (15) and (16), we obtain three univer-sality classes for lim c →∞ p R ( r | λ ), which are determinedby the exponent α that controls the degree fluctuations.The results for the universality classes are summarizedin table I. We find that for α ≤ (cid:126)R ( λ b )and (cid:126)R ( λ isol ) are delocalized in the limit c → ∞ , whereasfor α > α < α = 2 α > R ∞ B ∞K ∞ B ) ∞ Example Poisson Exponential BorelTABLE I. The relative variance R c of (cid:126)R ( λ isol ) and the kurto-sis K c of (cid:126)R ( λ b ) in the dense limit c → ∞ (see Eqs. (15) and(16)), together with an example of the outdegree distribution p K out in each regime of α (see Eq. (17)). The eigenvector distributions in the dense limit.
Theresults in Table I indicate that p R ( r | λ ) is universal inthe dense limit. Below we present explicit expressionsfor p R ( r | λ ) when c → ∞ . Henceforth we set (cid:104)| R | (cid:105) = 1without loosing generality.The characteristic function of p R ( r | λ ) is given by [65] g R ( u, v | λ ) = ∞ (cid:88) k =0 p K out ( k ) e k ln F ( u,v | λ ) , (18)where F ( u, v | λ ) = (cid:90) d x p J ( x ) (cid:90) d r p R ( r | λ ) e − xzr λ + xz ∗ r ∗ λ ∗ , (19)and z = u + iv . The symbol ( . . . ) ∗ denotes complex-conjugation. If λ ∈ R , the eigenvector components arereal and F ( u, v | λ ) does not depend on v .Setting λ = λ isol or λ = λ b in Eq. (19), we can expand F ( u, v | λ ) for c (cid:29) O (1 /c ) if α ≤ α >
2, becausethe moments of p R can diverge in this regime. Thus,performing this expansion for α ≤ F ( u, v | λ ) in Eq. (18), we obtain[65] g R ( u, v | λ b ) = ∞ (cid:88) k =0 p K out ( k ) exp (cid:20) − γk c (cid:0) u + (2 − γ ) v (cid:1)(cid:21) , (20) g R ( u, v | λ isol ) = ∞ (cid:88) k =0 p K out ( k ) exp (cid:18) − i ukc √ Bc α − + 1 (cid:19) . (21)Remarkably, the characteristic functions in the denselimit are fully specified by degree fluctuations and areindependent of p J .For degree distributions where lim c →∞ var[ K out ] /c =0 ( α < p K out ( k ) = δ k,c inEqs. (20) and (21), leading to [65] p R ( r | λ b ) = 1 π e −| r | ( λ b / ∈ R ) , (22) p R ( r | λ isol ) = δ [Im( r )] δ [Re( r ) − . (23)Equation (22) yields the well-known Porter-Thomas dis-tribution for the eigenvector components of Gaussianrandom matrices [66, 67]. Thus, standard results fromrandom matrix theory are recovered when α < p K out is an exponential distribution, for which α = 2,we obtain in the limit c → ∞ [65] p R ( r | λ b ) = 2 π K (2 | r | ) ( λ b / ∈ R ) , (24) p R ( r | λ isol ) = √ δ [Im( r )] Θ [Re( r )] e −√ r ) , (25)where Θ( x ) is the Heaviside step function and K ( x ) isa modified Bessel function of the second kind [68]. Fig-ure 4 illustrates the shape of the distributions p R given r ) p R ( R e ( r ) | λ i s o l ) ExponentialPoissonRegular .
51 (b) | r | p | R | ( | r || λ b ) FIG. 4. The dense limit c → ∞ of the distribution p R (Re( r ) | λ isol ) of the real part of the eigenvector componentsat λ isol [Panel (a)], and of the distribution p | R | ( | r || λ b ) of thenorm of the eigenvector components at λ b / ∈ R [Panel (b)].The solid red lines and the dashed black lines are, respectively,the analytic results for regular/Poisson and exponential de-gree distributions (see Eqs. (22-25)), while the symbols aredata obtained from the numerical solutions of Eq. (7) with c = 100. The numerical data for regular/Poisson graphs inpanel (a) is a Gaussian distribution with variance of O (1 /c ),approaching the Dirac delta distribution (vertical arrow) for c → ∞ . by Eqs. (22-25), and compares them with numerical so-lutions of Eq. (7) for c = 100. In the SupplementalMaterial [65], we also derive the analytic expressions forlim c →∞ p R ( r | λ b ∈ R ) when α ≤ Conclusions.
We have shed light on the relationshipbetween graph topology and the localization of righteigenvectors in directed random graphs. If the momentsof the outdegree distribution p K out are finite, then righteigenvectors at the edge of the spectrum are localizedbelow a critical mean outdegree. It is striking that thecritical points for the localization transitions are univer-sal, in the sense they only depend on the lower momentsof the distribution p J of the edge weights, regardless ofthe network topology. Therefore, localization in directedrandom graphs is fundamentally different from localiza-tion in undirected graphs, for which degree fluctuationsare important [34–36, 39, 69–71]. Indeed, eigenvectorsin the tail of the spectrum of undirected random graphsare localized for any p J if the degree distribution hasan unbounded support. Degree-based localization is alsopossible for directed random graphs, but then p K out hasdivergent moments.We have also studied localization and universality ofthe eigenvectors in the dense limit. In this limit, thedistribution p R of the right eigenvector components isonly determined by the graph topology, independentlyof the distribution p J . If the outdegree fluctuations aresmall enough, then eigenvectors are delocalized and p R isgiven by the same universal distribution as in the case ofGaussian random matrices [66, 67]. On the other hand, ifthe outdegree fluctuations are large enough, then eigen-vectors are localized and the distribution p R depends on p K out . More generally, these results indicate that Gaus-sian random matrix theory describes well the spectralproperties of dense graphs only when the degree fluctua-tions are sufficiently small.The techniques developed in the present paper can beused to study localization phenomena in non-Hermitianquantum systems [47–49, 72, 73], neural networks [16,50], ecosystems [17, 18], and real-world networks [42, 74].The relation between the dynamical properties of thesesystems and the localization properties of eigenvectors isan interesting topic of future research.The authors thank Jacopo Grilli for interesting discus-sions. F.L.M. thanks London Mathematical Laboratoryand CNPq/Brazil for financial support. 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