Spectral Analysis of Random Graphs with Given Expected Degrees

1 Moment-Based Spectral Analysis of Random Graphs with Given Expected Degrees

Victor M. Preciado, Member, IEEE, and M. Amin Rahimian, Student, IEEE

Abstract—In this paper, we analyze the limiting spectral distribution of the adjacency matrix of a random graph ensemble, proposed by T (n) (n) Chung and Lu, in which a given expected degree sequence wn = (w1 , . . . , wn ) is prescribed on the ensemble. Let ai,j = 1 if there is an edge between the nodes i, j and zero otherwise, and consider the normalized (random) adjacency matrix of the graph n { } n (n) ensemble: An = √nρn[a ] , where ρn = 1/ w . The empirical spectral distribution of An denoted by Fn( ) is the i,j i,j=1 i=1 i · empirical measure putting a mass 1/n at each of the n real eigenvalues of the symmetric matrix A . Under some technical conditions P n on the expected degree sequence, we show that with probability one, Fn( ) converges weakly to a deterministic distribution F ( ). · · Furthermore, we fully characterize this distribution by providing explicit expressions for the moments of F ( ). We apply our results to · well known degree distributions, such as power law and exponential. The asymptotic expressions of the spectral moments in each case provide signiﬁcant insights about the bulk behavior of the eigenvalue spectrum.

Index Terms—Complex Networks, Random Graph Models, Spectral Graph Theory, Random Matrix Theory. !

1 INTRODUCTION

NDERSTANDING the relationship between structural [14], [15]. U and spectral properties of a network is a key question Random graph models, such as the Erdos-R˝ enyi´ [16], in the field of Network Science. Spectral graph methods (see [17], the scale-free [15], and the small-world models [18], are [1]–[3], and references therein) have become a fundamental a versatile tool for investigating the properties of real-world tool in the analysis of large complex networks, and related networks. We find in the literature several random graphs disciplines, with a broad range of applications in machine able to model degree distributions from empirical data. One learning, data mining, web search and ranking, scientific of best known graphs is the configuration model, originally computing, and computer vision. Studying the relationship proposed by Bender and Canfield in [19]. This model is between the structure of a graph and its eigenvalues is able to fit a given degree sequence exactly (under certain the central topic in the field of algebraic graph theory [1]– technical conditions). Although many structural properties [5]. In particular, the eigenvalues of matrices representing of this model have been studied in depth [20], [21], it is the graph structure, such as the adjacency or the Lapla- not specially amenable to spectral analysis. In contrast, the cian matrices, have a direct connection to the behavior of preferential attachment model proposed in [15] provides a several networked dynamical processes, such as spreading justification for the emergence of power-law degree distri- processes [6], synchronization of oscillators [7], random butions in real-world networks. A tractable alternative to walks [8], consensus dynamics [9], and a wide variety of the preferential attachment model was proposed by Chung distributed algorithms [10]. and Lu in [22], and analyzed in [23]–[25]. In this model, The availability of massive databases describing a great which we refer to as the Chung-Lu model, an expected degree variety of real-world networks allows researchers to explore sequence is prescribed onto a random graph ensemble, that their structural properties with great detail. Statistical anal- can be algebraically described using a (random) adjacency ysis of empirical data has unveiled the existence of multiple matrix. arXiv:1512.03489v3 [math.ST] 24 Mar 2017 common patterns in a large variety of network properties, Studies of the statistical properties of the eigenvalues such as power-law degree distributions [11], or the small- of random graphs and networks are prevalent in many world phenomenon [12]. Random graphs models are the applied areas. Examples include the investigations of the tool-of-choice to analyze the connection between structural spacing between nearest eigenvalues in random models and spectral network properties. Aiming to replicate em- [26], [27], as well as real-word networks [28]–[30]. Empir- pirical observations, a variety of synthetic network models ical observations highlight spectral features not observed has been proposed in the literature [11], [12]. The structural in classical random matrix ensembles, such as a triangle- property that have (arguably) attracted the most attention like eigenvalue distribution in power-law networks [31], is the degree distribution. Empirical studies show that the [32] or an exponential decay in the tails of the eigenvalue degree distribution of important real-world networks, such distribution [33], [34]. as the Internet [13], Facebook, or Twitter, are heavy-tailed and can be approximated using power-law distributions 1.1 Main Contributions In this work we offer an exact characterization for the eigen- The authors are with the Department of Electrical and Systems Engineer- • value spectrum of the adjacency matrix of random graphs ing at the University of Pennsylvania, Philadelphia, PA 19104. with a given expected degree sequence. This characteriza- 2 tion is in terms of the moments of the eigenvalue distribu- each random edge is realized independently of all other tions and it hinges upon application of the moments method edges and in accordance with the probability measure P , {·} from random matrix theory [26], [27]. Accordingly, we give specified below. Let E and Var be the expectation and {·} {·} closed-form expressions describing the almost sure limits of variance operators corresponding to P .2 Following [38] the spectral moments. We then use these moments to draw we allow for self-loops. To each random{·} graph, we associate important conclusions about the graph spectrum and to a (random) adjacency matrix, which is a zero-one matrix bound several spectral quantities of interest. Analyzing the with the (i, j)-th entry being one if, and only if, there is limiting spectral distribution is important when designing an edge connecting nodes i and j. The number of edges statistical tests to investigate the structure of a network incident to a vertex is the degree of that vertex, and by from the observed interconnection data. Such scenarios the volume of a graph we mean the sum of the degrees arise frequently in community detection, where stochastic of its vertices. In this paper, our primary interest is in block models (SBM) find widespread use. SBM captures characterizing the asymptotic behavior of the eigenvalues of the different edge probabilities for different communities the random adjacency matrix as the graph size n increases. but fails to account for variations of node degrees [35]. We consider the distribution of these eigenvalues over Subsequently, the degree-corrected stochastic block model the real line and characterize this distribution through its offers an improved and more realistic null (as compared to moments sequence. Accordingly, the probability of having (n) (n) the Erdos-R˝ enyi´ model) when testing different hypothesis an edge between nodes i and j is equal to ρnwi wj , for community detection [36]. The knowledge of spectrum n (n) where ρn = 1/ i=1 wi is the inverse expected volume. is also important in system-theoretic analysis of networked The adjacency relations in this random graph model are P systems with random interconnections: indeed, we have represented by an n n real-valued, symmetric random applied our methods to study the controllability Gramian ×(n) (n) matrix An = √nρn[aij ], where aij are independent 0- spectra for such systems [37]. (n) (n) (n) 1 random variables with E a = ρnw w . In this The remainder of this paper is organized as follows. Pre- { ij } i j liminaries on the background and motivation of our study, paper, we assume the following sparsity condition on the 3 (n) (n) as well as the random graph model under consideration, are degree sequence : ρnwi wj = o(1) for all i, j, which will presented in Section 2. Our main results on the asymptotic be used to ensure that the distribution of the eigenvalues of spectral moments of the adjacencies of random graphs and the adjacency matrix An converges, almost surely, to a de- the characterization of the limiting spectral distributions terministic distribution that is uniquely characterized by its (for the normalized adjacencies) are presented in Section 3, sequence of moments when n . As a main result of this → ∞ where we also include an outline of the proofs (which are paper, we explicate the technical conditions under which presented in detail in Section 7). In Section 4, we apply our this convergence property holds true (cf. Assumptions A1 results to a case where node degrees are obtained by random to A3 below). Furthermore, we proffer explicit expressions samples from the support of a preset function and show for calculating this moment sequence. These expressions, in how our main results can be applied in analysis of the spec- turn, allow us to upper or lower bound various spectral trum of large random graphs. We consider random graphs metrics of practical interests (cf. Section 4 and discussions with an exponential degree distribution as a special case and therein). derive the asymptotic expressions of its spectral moments. In Section 5 we consider the important case of power-law Characterization of the convergence conditions depend degree distribution, which is known to be a good descriptor critically on the behavior of the extreme values of the for many real-world networks. The asymptotic expressions expected degree sequence as n increases. To that end, we consider two sequences wˆn : n N and wˇn : n N of the spectral moments of power-law networks allow us {(n) ∈ } { (n) ∈ } given by wˆn = maxi [n] wi and wˇn = mini [n] wi for all to analyze the bulk behavior of the eigenvalue spectrum ∈ ∈ n N. Another quantity of interest, whose evolution with in much greater details; in particular, we can quantify and ∈ characterize the similarities with and deviations from the the graph size n plays an important role, is the second-order triangular distribution that is reported in the literature [31], [32]. Section 6 concludes the paper.

ACKGROUND OTIVATION 2 B &M ¯ 2. Let Pn be the probability measures on the finite product spaces 2.1 Chung-Lu Random Graph Model corresponding{·} to the n(n + 1)/2 independent entries of the n n × real symmetric random adjacency matrix An of the Chung-Lu ran- We consider the Chung-Lu random graph model introduced dom graph ensemble and consider the following product measures: ¯ ¯ ¯ in [22] and analyzed in [23]–[25], in which an expected Pn = Pn Pn 1 ... P1 for all n N. The probability measure P is degree sequence given by the n non-negative entries of the the probability⊗ − measure⊗ on the infinite∈ product space that is the unique{·} T (n) (n) extension (according to Kolmogorov’s existence theorem [47, Section vector w = (w , . . . , wn ) is prescribed over the set of n 1 36]) as n of the consistent probability measures Pn, and E and 1 → ∞ {·} nodes, labeled by [n], of the graph ensemble. In this model, Var are the expectation and variance operators associated with P . We{·} say that an event occurs almost surely if it occurs with probability{·} 1. Throughout this paper, R and N are the set of real and natural one. numbers, N0 = 0 N, n N is a parameter denoting the size of 3. Given three functions f( ) g( ), and h( ) we use the asymptotic the random graph,{ }[n ∪] denotes∈ 1, 2, . . . , n , and card( ) denotes the notations f(n) = O(g(n)) and· f·(n) = o(h· (n)) to signify the rela- { } X cardinality of set . The n n identity matrix is denoted by In, random tions lim supn f(n)/g(n) < and limn f(n)/h(n) = 0, variables are printedX in boldface,× matrices are denoted by capital letters. respectively; in→∞ the| latter case| we also∞ write h(n→∞) =| ω(f(n)). We| use Every vector is marked by a bar over its lower case letter. f(n) (g(n)) to mean that f(n) = (1 + o(1))g(n). 3 average degree dñ, defined in [23] as, spectra for random graphs with independent edges [40]– 2 [42]. In [42], the author shows concentration of the spectral n (n) w n 2 norm for the Laplacian and adjacency matrices around ˜ i=1 i (n) 2 dn = = ρn wi = ρn wn 2. their expectations, under certain technical conditions. These P n w(n) k k i=1 i Xi=1 results are improved by Chung and Radcliffe [40], who use The spectralP moments of the adjacency matrix are derived in a Chernoff-type inequality to approximate the eigenvalues terms of the limiting normalized power-sums of the degree by those of the expected matrix and bound the error with sequence, defined as follows. For each i N, we define high probability. This error bound is O(√wˆn log n) in [40] ∈ the limiting normalized k-th power-sum of the expected and it is later improved to (2 + o(1))√wˆn by [41]. The k n (n) authors in [43] use free probability techniques to give a char- degree sequence as: Λk := limn 1/n i=1 nρnwi . →∞ acterization of the empirical spectral distribution of random For our main results to hold true, we need the expected T P graphs with arbitrary expected degrees in terms of the free degree sequence wn to satisfy the following assumptions: multiplicative convolution of their degree distribution with A1 (Sparse & graphical): ρ wˆ2 < 1, n, and ρ wˆ2 = o(1). the semi-circular density; however, the techniques come n n ∀ n n A2 (All finite moments): Λk exists (and are finite) for all k. short of a closed-form expression for the moments of the A3 (Controlled growth of degrees): (i) wˆn/wˇn = O(log n), limiting spectral distribution, and no attempt is made to 2 (ii) ρnwˆn = ω(1/n), (iii) nρn = o(1), (iv) nρnwˆn = O(1), (v) determine the exact conditions that the degree distribution wˆn = O(log n), and (vi) dñ = o(n/ log n). should satisfy for the techniques to be applicable. Note that Assumption A1, in particular, implies that the edge probabilities are all less than one and, indeed, 3 MAIN RESULT they are asymptotically vanishing (sparsity). Furthermore, In this paper, we offer a moment-based characterization of under Assumption A3 (iv), with nρnωˆn = O(1), the limiting the limiting distribution of the eigenvalues of the Chung- averages Λk are guaranteed to be O(1) and Assumption A2 Lu random graph model under Assumptions A1 to A3. The ensures that they indeed exist. moment sequence provides a versatile tool in the spectral The Assumptions A1 to A3 are satisfied by several analysis of complex networks [44]–[46]. It is worth high- 2 practical degree distributions of interest such as power lighting that, in the sparsity regime ρnwˆn = o(1) and law and exponential distributions, as we will discuss in under the √nρn normalization, the largest eigenvalue of the the examples and case studies later in the paper. In order random adjacency matrix may escape to infinity as n .4 to illustrate Assumption A3, let us consider an Erdos-˝ When investigating the limiting distribution of a sequence→ ∞ Renyi´ random graph with edge probability pn, so that of distributions, it may be the case that some mass escapes (n) ˜ 2 wi =w ˆn =w ˇn = dn = npn and ρn = 1/(n pn) for all n. to infinity, which can cause the limit distribution to not 2 Assumptions A1 and A3 (ii) require pn = ρnwˆn = o(1) and be a probability distribution (does not integrate to one); ω(1/n), respectively (i.e. pn 0 and npn ), agree in such cases, the underlying sequence of distributions is with the assumptions made→ in the most recent→ ∞ analyses not “tight” (cf. [47, Section 25]). However, as we show in concerning Erdos-R˝ enyi´ random graphs under the 1/√npn this paper, the tightness property holds for the sequence normalization [39, Theorem 1.3]. of spectral distributions in the Chung-Lu random graph model when the growth of the degrees is controlled as in 2.2 Spectrum of the Chung-Lu Random Graph Assumption A3. This is because the mass that is escaping to infinity (finitely many largest eigenvalues) is asymptotically In 2003, by a series of results, Chung, Lu and Vu estab- vanishing itself: as n , the contribution of finitely many lished important asymptotic properties of the spectra of eigenvalues to the continuous→ ∞ limiting spectral distribution the adjacency matrices of random graph models with given 2 in the ρnwˆ = o(1) sparse regime and under the √nρn expected degree sequences [23]–[25]. A key result of theirs n normalization is vanishingly small. By the same token, specifies the almost sure limit of the largest eigenvalue of our results complement the characterization of the largest the adjacency matrix, as follows [23, Theorems 2.1 and 2.2]. eigenvalue in [23]–[25], which focuses on the growth rate Theorem 1 (Largest Eigenvalue of Random Graphs). If and concentration of the largest eigenvalue, as described in ˜ dn > √wˆn log n, then with probability one the largest Subsection 2.2. eigenvalue of the (unnormalized) adjacency matrix is To describe our main results we need to introduce some ˜ ˜ 2 (1 + o(1))dn; while if √wˆn > dn log n, then the largest terminology. Let λ (A ) λ (A ) ... λ (A ) be n 1 n ≤ 2 n ≤ ≤ n n eigenvalue is almost surely (1 + o(1))√wˆn. real-valued random variables representing the n eigenval- Moreover, for a random graph model whose ex- ues of the random matrix An ordered from the smallest to the largest. We define δx as the probability measure on R pected degree distribution obeys a power law, the {·} largest eigenvalue is with probability one less than assigning unit mass to the point x R and zero elsewhere. We also define = (1/n) n∈ δ as the ran- 7√log n max √wˆn, dñ . In [23], similar conditions are es- n i=1 λi(An) { } dom probabilityL measure{·} on the real line that{·} assigns a mass tablished for the almost sure convergence of the k-th largest P eigenvalues towards the square root of the k-th largest 1/n to each one of the n eigenvalues of the random matrix expected degree. In [25], relevant results concerning the An. The corresponding distribution can be written as Fn(x) spectra of other matrices, such as the Laplacian, were de- 4. This is true, for example, for Erdos-R˝ enyi´ random graphs with rived. More recent results use the machinery of concentra- edge probability pn and √nρn = 1/√npn normalization, as we tion inequalities to investigate the behavior of the graph demonstrate in Section 3.2. 4

= n ( , x] = (1/n)card( i [n]: λi(An) x ), from the smallest to the largest as λ1(Aˆ n) λ2(Aˆ n) L { −∞ } { ∈ ≤ } ≤ (n) ≤ where card( ) is the cardinality function, and is referred to ... λ (Aˆ ) and define the random variable ˆm = · n n k as the empirical spectral distribution (ESD) for the random ≤ ˆ k (1/n)trace(An) to be its k-th spectral moment. Also let matrix An. For each x R, Fn(x) is a random variable. (n) (n) ∈ m¯ k = E ˆmk be the expected spectral moments for all Moreover, we define the k-th spectral moment of the ran- k, n. The proof{ of} our main result proceeds as follows. dom matrix An as the following real-valued random vari- Lemma 2, included in Section 7.1, ensures that under (n) k + k (n) (n) able m = (1/n)trace(A ) = ∞ x d Fn(x). Our main k n Assumptions A3 (i), (iii), (v), and (vi) ˆmk mk , almost results establish the almost sure−∞ and weak convergence R surely for each k N. Therefore, the effect of centralization of the empirical spectral distribution of the (normalized) on the spectral moments∈ is asymptotically vanishing, and adjacency matrix A to a deterministic distribution F ( ). We ˆ n · both An and its centralized version An have the same limit- call this distribution the limiting spectral distribution (LSD) ing spectral moments. Concentration results for functionals and we characterize it through its moments sequence, mk of random matrices with independent entries imply that the + k = ∞ x dF (x). According to our main result, the spectral spectral moments concentrate around their expected values; moments−∞ of the limiting spectral distribution F ( ) are spec- R hence, it suffices to calculate the limiting values of the ified in terms of the limiting power-sums of the normalized· (n) expected spectral moments (limn m¯ ). Finally, Lemma →∞ k degree sequence Λk, as follows: 3 (included in Section 7.2) provides the asymptotically exact expressions for the expected spectral moments under Assumptions A1 to A3, completing the proof following the Theorem 2 (Spectral Moments of An). Consider a random graph with a given expected degree sequence method of moments. In the next section, we use the special (n) (n) case of Erdos-R˝ enyi´ random graphs to elaborate on these wn = (w , . . . , wn ), satisfying Assumptions A1 to 1 steps in more details. A3. With probability one, Fn( ) converges weakly to a deterministic distribution F·( ), which is uniquely · + k 3.2 The Case of Erdos-R˝ enyi´ Random Graphs determined by its moments mk = ∞ x dF (x). The −∞ moments of F ( ) are given by: In Erdos-R˝ enyi´ random graphs, denoted by Gn,p, each edge · R is realized with a probability p, independently of other 2 s + 1 r1 r2 rs edges. This is a special case of the Chung-Lu random graph m2s = Λ1 Λ2 ...Λs , (1) s + 1 r1, ..., rs model when the expected degree sequence is given by wn = r s ! X∈R (np, np, . . . , np). Ever since its introduction in the late 1950s m2s+1 = 0, for all s N0, ∈ by Erdos˝ and Renyi´ [16], [17], properties of this well-known s s s with s = r N : rj = s + 1, j rj = 2s class of random graphs have been extensively studied [48], R { ∈ 0 j=1 j=1 } n (n) k [49]. Indeed, the seminal work of Furedi¨ and Komlos´ [50] and Λk = limn 1/n i=1(nρnwi ) . →∞ P P can be used to derive asymptotic properties of the spectra of P Erdos-R˝ enyi´ random graphs; with probability one, putting its largest eigenvalue at (1 + o(1))np and upper-bounding 3.1 Proof Outline the absolute values of the rest by (2 + o(1)) np(1 p). The detailed proof is presented in Section 7.2. Here, we More recently, Feige and Ofek [51] have shown that under− p provide an outline of the proof steps. The crux of the mild conditions on p, the largest eigenvalue of the adjacency argument is in showing that for each k N, the k-th spectral matrix is almost surely pn + O(√pn), and all other eigen- (n) ∈ moments mk converge almost surely to mk; thence con- values are almost surely O(√pn). cluding by the method of moments that with probability one Fig. 1(a) depicts the histogram of eigenvalues for a par- the empirical spectral distributions F ( ) converge weakly ticular realization of the (unnormalized) adjacency matrix n · to F ( ). To begin, we consider the centralized version of the of an Erdos˝ and Renyi´ graph with n = 1000 nodes and · adjacency An, given by p = 0.01. In particular, we can observe that the largest eigenvalue λ (A ) np = 10 is located away from the ˆ (n) 1 n An = An E An = â . (2) − { } ij remaining eigenvalues, which are asymptotically located in h i the [ 2√np, +2√np] [ 6.325, +6.325] interval. Note that â(n) − ≈ − Note that the entries ij have zero mean and, asymptoti- under the √nρn = 1/√np normalization, the largest eigen- cally, a rank-one pattern of variances, given by: value grows as √np, whereas the bulk will be compactly (n) (n) 2 supported in the [ 2, 2] interval. Let us also consider the Var â = â − ij E ij centralized adjacency matrix Aˆ n defined in (2). We plot n o (n) (n) (n) (n) a typical realization of the eigenvalue histogram of the = ρnw w (1 ρnw w ) ˆ i j − i j unnormalized, centralized adjacency matrix An in Fig. 1(b). (n) (n) σ σ , (3) Notice that, as pointed out in Lemma 2 (proved in Section i j 7.1), the effect of centralizing the adjacency matrix An (w) (n) (n) since, by Assumption A1, ρnwi wj = o(1). This rank- is to cancel out the largest eigenvalue (moving it to zero), one pattern is a key property that allows us to calculate while the bulk of eigenvalues remains (almost) unperturbed. closed-form expressions of the limiting spectral moments in Subsequently, the effect of normalization on the limiting Theorem 2. spectral distribution in the √nρn-normalization regime is To proceed, we introduce some necessary notations. negligible. This can be also noticed in Fig. 1(c), where Similarly to An, we consider the eigenvalues of Aˆ n ordered we plot the empirical spectral distributions Fn (x) and 5

Hist(Spec(A )) Hist(Spec(A )) F (x) n n n

Fn(x) CDF Frequency Frequency λ (A ) 1 n λ1(An) x

(a) Eigenvalues of An (b) Eigenvalues of An (c) Eigenvalues

Fig. 1: (a) Eigenvalue histogram of the Erdos-R˝ enyi´ random graph; (b) eigenvalue histogram of the centralized adjacency matrix ; (c) spectral distributions of the normalized adjacency matrix An (blue solid) and the centralized version Aˆ n (red dashed).

Fˆn (x). Indeed, under the √np-normalization, the largest degree of vertex i and rd for number of vertices with degree eigenvalue of An grows as √n, escaping almost surely to d. Notice that the maximum degree is at most s; hence, infinity; every other eigenvalue is almost surely O(1), being rd = 0 for all d > s. For any graph G with s edges, the s asymptotically compactly supported. degree distribution (r1, . . . , rs) satisfies d=1 d rd = 2s and s rd = s+1. In particular, since a connected graph with d=1 P Indeed, as a validity check, it is possible to rederive the s edges and s + 1 vertices is always a tree, we have that G is limiting spectral distribution of the Erdos-R˝ enyi´ random P s s a tree if and only if d=1 rd(G) = s+1 and d=1 d rd = 2s; graph ensemble from Theorem 2 under Assumptions A1 to i.e. the set s defined in Theorem 2 denotes the set of integer ˜ R P P A3. In particular, in this case we have that dn =w ˆn =w ˇn sequences that are valid degree distributions for trees with 2 1 = npn, ρn = n− pn− and Λk = 1 for all k; therefore, for s edges. p = C log n/n the random graph ensemble satisfies As- n We use s+1(r) to denote the set of all rooted ordered sumptions A1 to A3. Theorem 2 implies that the asymptotic trees on s +T 1 nodes whose degree distribution is r. As spectral moments satisfy part of the proof of the main result in (35) we show that card( (r)), i.e. the total number of rooted ordered trees 2 s + 1 1 2s Ts+1 m2s = = =: Cs, (4) with degree distribution r, is given by: s + 1 r1, . . . , rs s + 1 s r s ! ! X∈R 2 s + 1 These moments correspond to that of a semicircular dis- card ( s+1(r)) = . (5) T s + 1 r , . . . , r tribution supported over [ 2, +2], [52], [53]. Note that to 1 s! − obtain a non-trivial support for the bulk of the spectrum, we = (r) Next, if we use the partition s+1 r s s+1 , we can need to investigate the LSD of the adjacency matrix under express the total number of orderedT trees∈R on Ts+1 vertices as, S the normalization √npn = √nρn regime. The quantity Cs C = ( ) = ( (r)) s card s+1 r s card s+1 , and by replacing defined in (4) is the s-th Catalan number. The Catalan num- from (5), weT obtain the right-hand∈R T side equality in (4). bers have great significance in combinatorics: they count the P total number of Dyck paths of length 2s, cf. [26], among many other combinatorial structures. Catalan numbers can also be used to count non-crossing partitions of an ordered 4 DEGREES SPECIFIED BY RANDOM UNIFORM set [26], as well as many other combinatorial structures [54]. SAMPLING Specially relevant is the relationship between the Catalan numbers and rooted ordered trees. A rooted ordered tree T is We now consider for Chung-Lu random graphs for which a tree in which one vertex is designated as the root, and the (n) children of each vertex are ordered (see [54], page 221); i.e. the expected degree of each node i is specified as wi = f (x ), where f ( ) are given functions with a common sup- there is a total order on the vertex set of T , respecting n i n port normalized to· be the unit interval [0, 1], and x , i the partial order - defined as follows: for all j, k (T ), i N { } ⊂ V is a random sample, uniformly and independently{ drawn∈ } j - k iff j belongs to the unique path on T that connects from the unit interval.5 To illustrate our results, let us as- k to the root. There is a bijection between Dyck paths of αx length 2s and ordered trees with s edges (see [26, Lemma sume that fn(x) = ∆ne− , where ∆n, αn > 0 for all n, and 2.1.6]); hence, the number of ordered trees with s edges is 0 x 1. Then, consider a Chung-Lu random graph with ≤ ≤ (n) αxi equal to Cs. We use s+1 to denote the set of all rooted the expected degree sequence specified as wi = ∆ne− , T i [n]. In other words, the degrees of the resulting graph ordered trees on s + 1 vertices that are chosen without ∈ replacement from the set [n]. To understand the significance follow an exponential pattern, which have been observed of the summation over the set s that appears in (4), as in practical scenarios, such as in structural brain networks well as (1), we introduce some additionalR notation pertaing built from diffusion imaging techniques [55], [56]. to trees and their degree sequences. Given a tree T with s The almost sure asymptotic expression for the second- edges and s + 1 vertices labeled 1, . . . , s + 1 , the degree order average degree dñ can be obtained from a Monte- distribution of T is defined as the{ sequence of integers} r(T ) s = (r1, . . . , rs) N0, where rd = rd(T ) is the number of ∈ 5. Given a desired degree distribution, the function fn( ) here plays vertices with degree d in T . We drop the tree argument the same role as the quantile function [47, Section 14] associated· with (T ) when there is no danger of confusion: using di for the the cumulative function of the desired degree distribution. 6

60 100 Carlo average [57, Section XVI.3], resulting in the following 50 expression: 80 40 2 60 n (n) 30 1 2 i=1 wi 40 f (x)dx Frequency ˜ 0 n Frequency 20 dn = (n) 1 n 20 P i=1 wi R0 fn(x)dx 10 1 2 2αx 2α 0 0 0P∆ne− dx R∆n(1 e− ) -6 -4 -2 0 2 4 6 -0.4 -0.2 0 0.2 0.4 0.6 = = − . (6) Eigenvalue Eigenvalue 1 αx 2(1 e α) R 0 ∆ne− dx − − Fig. 2: On the left we have the eigenvalue histogram of a sample 2 We know fromR Theorem 1 that, if ∆n > log n, then realization from the random graph model with exponential dñ > √∆n log n, and the largest eigenvalue of the adjacency degree sequence and n = 1000 vertices. On the right we have the eigenvalue histogram for a power-law random graph. (without √nρn normalization) is almost surely given by dñ. We now use Theorem 2 to write closed-form expressions for the asymptotic spectral moments of A , the normalized n which results in: adjacency matrix of the random graph whose expected (n) (n) k 1 α − kα degree sequence is given by wn = (w1 ,..., wn ). We Λk = α k 1 e− . begin by calculating the almost sure asymptotic expression k(1 e− ) − − of the inverse expected volume of the graph, as follows, Under these conditions, we can apply Theorem 2 to obtain n 1 the following closed-form expression for the asymptotic 1 (n) = w n f (x)dx (7) spectral moments of A : ρ i n n n i=1 Z0 X s (k 1)rk kα rk 1 n∆ 2 s + 1 α − (1 e− ) αnx n α m2s − = n ∆ne− dx = (1 e− ). rk α krk s + 1 r1, ..., rs! k (1 e− ) 0 α − r s k=1 − Z X∈R Y r s 1 s kα k We now proceed to verify the qualifying conditions for α − 2 s + 1 1 e− = − , applying Theorem 2. First, note the following almost sure (1 e α)2s s + 1 r , ..., r k − r 1 s! k=1 ! asymptotic identities for the maximum and minimum de- − X∈Rs Y (9) grees, (n) (n) where in the second equality we have used the identities α s s s ˆwn = max wi ∆n, wˇ n = min wi ∆ne− , (8) i [n] i [n] k=1 rk = s + 1 , k=1 krk = 2s, and k=1(k 1)rk = ∈ ∈ s 1. The histogram of the eigenvalues for the normalized− both of which can be easily verified from the corresponding P− P P adjacency matrix An of a particular realization with param- order statistics for the uniform distribution on the unit eters ∆ = 10, α = 1, and n = 1000 is plotted in Fig. 2 (left). 6 interval. If we set ∆n = log n, then from the set of almost The largest eigenvalue of the centralized adjacency matrix sure asymptotic identities in (6), (7) and (8) we get (n) (n) Aˆ = A [ρ w w ]n in this realization is given n n − n i j i,j=1 dñ = O(∆n) = o(n/ log n), nρn = O(1/∆n) = o(1), by λn(Aˆ n) = 5.6214. We can upper and lower bound this eigenvalue using the k-th spectral moment, for any k, as ˆwn α e = O(log n), ˆwn ∆n = O(log n), follows [27, Equation (2.66)]: wˇ n α 1/k nρn ˆwn = = O(1), ˆ k 1/k ˆ ˆ k 1 e α trace(An) λn(An) n trace(An) . (10) − − ≤ ≤ · 2 ∆nα log n ρn ˆwn = = o(1) and ω(1/n). If we consider k = 20 in (10) and use the asymptotic spectral n n ˆ 20 moment m20 available from (9) to replace for trace(An ), Hence, Assumptions A1 to A3 are all satisfied, and we can then the lower and upper bounds on λn(Aˆ n) are given 1/20 1/20 apply Theorem 2 to obtain the closed-form expressions for by: (m20) = 4.3193 and (n m20) = 6.1011. These the limiting spectral moments of the normalized adjacency values compare reasonably with· the empirically observed matrix A . First note that the k-th order limiting averages n value λn(Aˆ n) = 5.6214. Furthermore, using the techniques Λk are almost surely given by proposed in [45], we can formulate semi-definite programs 1 1 that improve the bounds in (10) by taking into account the k α x k Λ (nρ f (x)) dx = nρ ∆ e− n dx, k n n n n knowledge of all spectral moments up to a fixed order, as Z0 Z0 described in [59, Section 3].

5 THE CASEOF POWER-LAW RANDOM GRAPHS In this section, we study the eigenvalue distribution of 6. More speciﬁcally, if we note that mini [n] xi and maxi [n] xi the random power-law graph proposed by Chung et al. are respectively distributed as Beta(1, n) and∈ Beta(n, 1) variables∈ [58, Chapter 2], then the claimed almost sure limits follow by in [23]. This random graph presents an expected degree (n) (n) (n) the Borel-Cantelli lemma. To see how, consider their expected val- sequence given by wn = (w1 , w2 , . . . , wn ) such that ues: E mini [n] xi = 1/(n + 1) and E maxi [n] xi = n/(n + (n) 1/β 1 1), and{ apply∈ the} Chebyshev inequality{ to their∈ quadratically} de- wi = c i− − , for i = i0 + 1, ..., i0 + n, where β is caying common variance, Var maxi [n] xi = Var mini [n] xi = the exponent of the power-law degree distribution; i.e. the 2 { ∈ } { ∈ } β n/ (n + 1) (n + 2) . number of nodes with degree k is proportional to k− . 7 In this model, we can prescribe a maximum and average Order 2 4 6 8 expected degrees, denoted by ∆ and d, respectively, by Theoretical 9.56e-3 3.10e-4 1.81e-5 1.46e-6 choosing the following values of c and i0 [23]: Empirical 9.5e-3 2.89e-4 1.62e-5 1.26e-6 β 1 β 2 1 d (β 2) − β 1 TABLE 1: Theoretical versus empirical spectral moments (from c = − d n − , i = n − . β 1 0 ∆ (β 1) one sample only) for the power-law network in Example 1. − − For power-law degree distributions, we can asymptotically 100 60 50 50 evaluate the averaged k-th power-sums of the expected 80 40 40 60 30 degrees for n , as follows: 30 → ∞ 40 20 Frequency Frequency 20 Frequency Frequency Frequency Frequency n i0+n 1 i0+n -0.245 0.245 1 1 − 1 k 1 1 k 20 10 10 k β 1 β 1 wi = c i− − cx− − dx 0 0 0 -0.2 0 0.2 -0.2 0 0.2 -0.2 0 0.2 n n n i0 (a)(a) Eigenvalues Eigenvalue (b)(b) Eigenvalues Eigenvalue (c) Eigenvalues i=1 i=i0 Z (c) Eigenvalue X X (b) Eigenvalue Fig. 3: Histograms of the eigenvalues for power-law networks 1 β 1 β 1 k i0+n d k −β −1 k with n = 1000, ∆ = 100, d = 10, and the following values of β: = c − x − = d f ; β, k , n β 1 k i0 ∆ (a) β = 3, (b) β = β∆ 4.44, and (c) β = 6. − − ≈

where follows, we want to compare the spectral density of the d β 2 k β 1 power-law graph with the triangular density function, given f ; β, k = − − ... (11) ∆ β 1 β 1 k × by: − β −1 k − β 1 −β −1 β 1 k 2 4 d (β 2) − − d (β 2) − − b + b2 x, for x [ b/2, 0] , − + 1 − . 2 4 ∈ −  ∆ (β 1) − ∆ (β 1)  t (x; b) =  b b2 x, for x (0, b/2] , (12) ! − ∈ − − 0, otherwise.    Notice that the moments of the power-law distribution are The moments of this density function can be calculated as well-defined only for k < β 1 and the moments diverge follows: − b/2 for k β 1, [60, Section 8.4.2]. Moreover, when the largest k ≥ − mk (b) = x t (x; b) dx (13) degree is much larger than the average degree, i.e., ∆ = b/2 ω (d), the expression inside the square brackets in (11) tends Z− (b/2)k to one; hence (11) simplifies as follows: b = 1 + ( 1)k . − (k + 1) (k + 2) d β 2 k β 1 f ; β, k = (1 + o (1)) − − = f˜(β, k) . In what follows, we want to find conditions under which ∆ β 1 β 1 k − − − the spectral density resembles a triangular distribution. We d measure the similarity in terms of the moments, in particu- Therefore, for k = 1, we have that f ∆ ; β, k 1 and n 7 1 w d, the expected average degree. Furthermore, lar, in terms of the kurtosis of the distribution. From (13), n i=1 i the above expressions can be used to compute the normal- the second and fourth moments of the triangular law are P b2 b4 ized power-sums Λk in Theorem 2, which can then be used given by m2 (b) = 24 and m4 (b) = 240 . The kurtosis of the to compute closed-form values for the asymptotic expected triangular distribution is then given by spectral moments using (1). b b m4 (b) 12 Example 1. Numerical verification of the asymptotic spectral κ = 2 = . (14) (m2 (b)) 5 moments for the power-law degree distributions. In the fol- b lowing numerical simulations, we verify the validity On the other hand,b from Theorem 2, we can compute the of Theorem 2 by computing the first five even-order kurtosis of the power-lawb network: spectral moments of a power-law random graph with 2 m4 2Λ Λ2 2Λ2 n = 1000, β = 3, ∆ = 100, and d = 10. The eigenvalue κ = = 1 = = 2nρd.˜ (15) m2 2 2 Λ2 histogram of one particular realization is plotted in Fig. 2 (Λ1) 1 2 (right). In Table 1, we compare the theoretical values of In our analysis, we use the difference between kurtoses the even spectral moments of the centralized adjacency as a measure of how far the spectral distribution is from matrix with the empirical values of a single random sam- the triangular law. Therefore, the spectral distribution that ple. We would like to remark how, as reported in [31], is closest to the triangular law is the one for which κ = κ. [32], the empirical spectral distribution of the power- According to (14) and (15), κ = κ is satisfied when nρd˜ = law graph under consideration resembles a “triangular” 6/5. Furthermore, if κ < 12/5 (respectively, κ > 12/5), thenb law. In the sequel, we use Theorem 2 to study if this the tail of the spectral distributionb is “fatter” (respectively, distribution is in fact a triangle. “thiner”) than the tail of the triangular law. Moreover, since 2 1 β 2 β 1 ρ nd and d d β−1 β−3 (when ∆ = ω (d)), we have − − 5.1 The Triangular Spectrum 4 7. The kurtosise of a distribution is defined as κ = µ4/σ , where µ4 Many empirical studies of real-world networks have re- is the fourth moment about the mean and σ is the standard deviation. ported triangle-like eigenvalue spectra [31], [32]. In what This ratio is a measurements of how heavy-tailed the distribution is. 8

2 3/2 β∆ 2 β∆ 1 6 6 Λ2 = − − = , and Λ3 = 2 , we β∆ 1 β∆ 3 5 1+√6 2+√6 − − ( ) (− ) have that 6 m (β ) = 2 27 + √6 14.14, 6 ∆ 25 ≈ which does not coincide with m (b ) 7.71. 6 ∆ ≈ b 6 CONCLUSIONS Fig. 4: Power-law degree distribution (left) and eigenvalue histogram (right) of the air traffic control network. In this work, we have investigated the asymptotic behavior of the bulk of eigenvalues of the adjacency matrix of random graphs with given expected degree sequences. We have showed that, in the nρ normalization regime and under that κ = κ when √ n some technical assumptions on the expected degrees se- 2 (β 2) 6 quence, the empirical spectral distribution of the adjacency − = . b (β 1) (β 3) 5 matrix converges weakly to a deterministic distribution, − − which we have characterized by providing closed-form Solving the above equation, we obtain the following critical expressions for its limiting spectral moments. value of β for which κ matches κ: β = 2 + √6 4.44. In ∆ We have illustrated the application of our results by other words, the kurtosis of the spectral distribution≈ is equal analyzing the spectral distribution of large-scale networks to that of the triangular law only when β = β . b ∆ with exponential degree distributions, which appear in Remark 1. In practice, the values of β in real world networks structural brain networks obtained from diffusion imaging. are below the threshold value β∗ 4.44 (cf. [61]). We have also applied our results to analyze the spectrum Therefore, the spectral tails of real networks≈ are mostly of power-law random graphs, which are of great practi- “fat”, which is in accordance with empirical observations cal importance. Using the closed-form expressions for the reported in [13]. For example, we include in Fig. 4 the asymptotic spectral moments in Theorem 2, we have in- power-law degree distribution of the (symmetric and vestigated the triangle-like spectrum of power-law random unweighted) adjacency matrix of the air traffic con- graphs. In particular, we have provided quantitative rela- trol network constructed from the USA’s FAA (Federal tionships to show how the parameters of the power-law Aviation Administration) National Flight Data Center degree distribution affect the shape and properties of the (NFDC), Preferred Routes Database [62]. Nodes in this graph spectrum. Furthermore, closed-form expressions of network represent airports and links are created from the asymptotic spectral moments allow us to bound spectral strings of preferred routes recommended by the NFDC. properties of practical interest, such as the support of the This network has a total of 1,226 nodes, 2,615 edges, spectral bulk. and a power-law exponent of 3.7 (which is below the β threshold value ∆). As predicted, the corresponding 7 PROOFOF MAIN RESULTS eigenvalue distribution, plotted in Fig. 4, presents a ‘fat’ tail. The argument leading to the proof of our main convergence result in Theorem 2 is based on the method of moments, and it is executed in two steps. We begin by When the spectral distribution resembles a triangular showing (in Section 7.1, below) that with probability one law, it is possible to approximate the support of the spectral (n) (n) limn m = limn ˆm , i.e., the adjacency matrix bulk of eigenvalues by computing the value of b in (12) →∞ k →∞ k A and its centralized version Aˆ share the same almost such that the second moment of the triangular distribution n n sure limits for their spectral moments. Next, in Section 7.2, matches the second moment of the theoretical spectral dis- we prove that these common almost sure limits are in fact tribution. Since the second moment of the spectral distribu- 2 given by mk in closed-form, as claimed in Theorem 2 (main tion is given by m2 = Λ1 = 1 and the second moment of the b2 result). triangular distribution is m2 = 24 , the value of b for which these moments match is given by: b∆ = √24. In Fig. 3(b), 7.1 Effect of Centralization on the Spectral Moments we indicate this value of bb∆ in the x-axis, which in this case is given by n√ρb∆ 0.245. The following set of results measures the effect of the cen- ≈ One may ask whether the spectral distribution of a tralization in (2) by comparing the spectral moments of Aˆ n random power-law network does indeed follow a triangular and A as n . Indeed, centralization by subtracting the n → ∞ density when β = β∆. The answer is no. This can be mean E An from the adjacency matrix An has the effect { } verified by comparing the sixth moments of the triangular of shifting the largest eigenvalue towards zero. Example 1 distribution with b = b∆ with the sixth moment of the demonstrates this shifting; however, in the sequel we shall power-law spectral density when β = β∆. In particu- show that the subsequent effect on the spectral moments is lar, the sixth moment of the triangular law is given by asymptotically vanishing, under certain mild assumptions m (b ) = 54/7 7.71. On the other hand, from Theo- on the degree sequences. 6 ∆ ≈ rem 2, the sixth spectral moment of the power-law network We know from Theorem 1 that, if dñ > √wˆn log n, then is given by m (β ) = 2Λ3Λ + 3Λ2Λ2. Since Λ = 1, λ (A ) d˜ with probability one. We use a variation of this b 6 ∆ 1 3 1 2 1 1 n n 9 result to prove that the column vector of expected degrees, surely identical, and therefore the effect of centralization on T (n) (n) T denoted by wn = (w1 , . . . , wn ) , is asymptotically al- spectral moments is asymptotically vanishing. most surely an eigenvector of An associated with its largest Lemma 2 (Vanishing Effect of Centralization). Under As- eigenvalue. In particular, as λn(An) concentrates around dñ, (n) sumptions A3 (i), (iii), (v), and (vi), it is true that ˆmk the vector (1/n)Anwn also concentrates around the vector (n) 1 ˜ mk , almost surely, for each k N. n dn wn. This is important when characterizing the effect of ∈ the centralization in (2) in light of the fact that Proof: To begin, consider the k = 1 case. From (2), we 3/2 (n) (n) 3/2 T have E An = √nρn wi wj = √nρn wnwn { } i,j [n] ∈ (n) 1 1 h i ˆm = trace(Aˆ n) = trace(An An ). √nρndñ T 1 E = w w . (16) n n − { } w 2 n n k nk2 From (16) we know that Lemma 1 (Eigenvector Concentration). Under Assumption nρ d˜ T 1 √ n n ˜ A3 (vi), it is true that A w ρ /nd˜ w almost trace(E An ) = trace(wnw ) = √nρndn, n n n n n n { } w 2 n surely. n 2 p k k wherefrom it follows that trace(Aˆ n) = trace(An) √nρndñ − Proof: The i-th component of Anwn is a random is true for all n, and in particular with probability one as variable given by: n . For general k N, we have → ∞ ∈ n 1 1 (n) (n) (n) ˆ k k [A w ] = nρ a w . ˆm = trace(An) = trace(An E An ) . n n i √ n ij j (17) k n n − { } j=1 X To proceed, consider the binomial expansion of (An (n) (n) k − Since aij is a Bernoulli random variable with P aij = E An ) consisting of a sum of the product of non- (n) (n) { commutative{ } elements, as follows: 1 = ρnwi wj , we have that } k k 1 k 2 n 2 An + An− ( E An ) + An− ( E An )An + ... (n) (n) ˜ (n) − { } − { } E [Anwn] = √nρnwi ρn wj = √nρndnwi , k 2 2 k 3 { i} + An− ( E An ) + An− ( E An )An( E An ) + ... j=1 − { } − { } − { } X Consider any product term of the form, ˜ and E Anwn = √nρndnwn. Next, note that each of the { } k1 k2 k3 kp (n) (n) Π(k) = An ( E An ) An ... ( E An ) , (18) summands √nρnaij wj in (17) are independent bounded − { } − { } (n) (n) (n) random variables satisfying √nρna w [0, √nρnw ] where k = (k , . . . , k ) is an integer partition of k, consist- ij j ∈ j 1 p almost surely. Hence, we can apply Hoeffding’s inequality ing of p 1 positive integers satisfying k1 + ... + kp 1 = k. ≥ − [63, Theorem 2 ] to obtain that, for each i and any ε > 0, Let k˜ be the sum of evenly indexed integers, i.e., k˜ = k2 + k4 + ... + k2 p/2 . We claim that as n , 1 ˜ (n) ·b c → ∞ P [Anwn] dnw ε n i − i ≥ k/2 ˜k 1 (nρn) dn k˜ trace(Π(k)) ( 1) , (19) 2 2 n → n − 2 2n ε 2nε /dñ 2 exp  − 2  = 2e− . almost surely. To see why, first note that for any ki, ≤ n (n) T T k 2(ki 1) nρn i=1 wi (w w ) i = w − w w , so taking the k -th power   n n k nk2 n n i  P  of both sides in (16), we get Next, note that given dñ = o(n/ log n) per Assumption A3 ki ( nρ d˜ ) T (vi), and for any α > 1, we get that when n is large enough ki √ n n 2 ˜ ( A ) = − w w , ˜ 2 2nε /dn α E n 2 n n dn < 2nε /(α log n). Hence, 2e− < 2/n forms a − { } wn 2 summable series in n, and by the Borel-Cantelli lemma [47, k k and replacing in (18) yields Theorem 4.3], we get that T T k˜ wnw wnw ρ d˜ ˜ k1 n k3 n 1 √ n n (n) Π(k) = ( √nρndn) An 2 An ... 2 . (20) P [Anwn]i wi ε, infinitely often = 0, − wn 2 wn 2 ( n − √n ≥ ) k k k k Under Assumption A3 (vi), from Lemma 1 we know that

(1/n)A w ρ /nd˜ w almost surely. Multiplying both n n n n n which holds true for any ε > 0. Therefore, we have sides by Aki 1 for any integer k yields n − p i ˜ 1 ρ d ki √ n n (n) 1 (√nρnd˜n) lim [Anwn] = lim w = 1. ki P n i n i An wn wn, ( →∞ n →∞ √n ) n n

The claimed concentration of eigenvector around wn now almost surely. Hence, taking the limits of both sides in (20) follows by the countable intersections of the above almost we get sure events over all i N. k T ∈ 1 ˜ (√nρndñ) w w We can now proceed to give conditions under which the Π(k) ( 1)k n n , ˆ n − n w 2 spectral moments of An and An are asymptotically almost k nk2 10 almost surely. Taking the trace of both sides and using the where each summand in the last term corresponds to a T fact that trace(w w / w 2) = 1, leads to (19) as claimed. sequence of k nodes, which can be interpreted as a closed n n k nk2 The above applies invariably to any of the product terms walk of length k, denoted by w = (i1, i2, . . . , ik 1, ik, i1), k − appearing in the binomial expansion of (An E An ) , in the complete graph with n nodes, denoted by n. Fur- − k { } K with the exception of the leading term, trace(An), which thermore, we associate a weight to each walk equal to does not simplify any further. Next, note that given any 1 the expected value of the product of the adjacency entries ≤ k˜ k, the number of terms Π(k) in the binomial expansion along the walk. In other words, we define the weight (n) (n) (n) (n) ≤ k function ω(w) := â â ... â â . Our task is for which (19) holds true, is exactly k˜ ; wherefore we get E i1i2 i2i3 ik 1ik iki1 { − } that with probability one, to study such weighted sums of closed walks. To this end, we define the set of vertices and edges visited by w as ˆ k 1 k trace(A ) = trace(An E An ) (w) = ij : j [k] and (w) = ij, ij+1 : j [k 1] . n n − { } VFor any {e ∈(w),} we defineE N({{e, w) as} the number∈ − of} k ∈ E 1 k ˜ k k˜ k times that w transverses the edge e in any direction. We trace(An) + (√nρndn) ( 1) . n  − k˜  denote by k the set of all closed walks of length k on ˜ ! kX=1 that startW and end on the vertex labeled by i . It is   Kn 1 useful to partition the set k into subsets k,p defined as ˜ W W k k k k k,p = w k : card( (w)) = p , i.e. the set of closed We next use a trite identity, 0 = (1 1) = 1+ k˜=1( 1) k˜ W { ∈ W V } to get − − walks of length k visiting p vertices. Furthermore, it is P convenient to define the following subset of k,p: 1 ˆ k 1 k ˜ k W trace(An) trace(An) (√nρndn) , n n − k,p = w k,p : N(e, w) 2 for all e (w) , W { ∈ W ≥ ∈ E } almost surely. The claim now follows upon noting that i.e., the set of walks in k,p for which each edge is traversed 1 k c (√nρndñ) = o(1), k . This is true because dñ < W n N at least twice. Note that, any walks that do belong to k,p 2 ∀ k ∈ k k k wˆ /wˇn so that (√nρndñ) < (√nρn) (w ˆn/wˇn) wˆ and we W n n will have a zero weight, because for any w k,pK k,p, known from Assumptions A3 (i) and A3 (v) that (w ˆn/wˇn) there exists an edge i, j (w) such that N∈( Wi, j ,W wc) = and wˆn are both O(log n) while Assumption A3 (iii) gives 1. Hence, by independence,{ } ∈ E { } c that nρn is o(1). Hence, upon division by n we get that 1 k (√nρndñ) is o(1), for all k as claimed. n N( k,l ,w) (n) (n) { } ω(w) = E âij E  âkl  = 0,   7.2 Almost Sure Limits of the Spectral Moments n o  k,l  (w{)YK }∈i,j E {{ }} Having established that the spectral moments of A and Aˆ n n  (n) since E âij = 0; recall from (2) that the entries â have are asymptotically identical, in this section we establish that { } ij the spectral moments of the centralized adjacency matrix zero mean. We can now employ the shorthand notation: ˆ An are indeed given by (1) in Theorem 2, i.e. we have 1 (n) µk,p = ω(w), (22) ˆmk mk, almost surely for each k N. Each spectral n ∈ 2 w moment is a functional that maps the O(n ) independent ∈XWk,p entries of a random matrix to a real value. Concentration to rewrite (21) as follows: c of functionals of random matrices around their expected k/2 +1 b c values have been established in great depth and generality. (n) k/2 Accordingly, the deviation of these functionals (or so-called m¯ k = (nρn) µk,p, (23) p=1 linear spectral statistics) from their expected values are char- X acterized using Gaussian central limit theorems [64, Chapter where the upper limit of p = k/2 + 1 in the summation 9], [65], as well as exponential tail bounds [26, Sections follows by the pigeonhole principle,b c since that every walk 2.3 and 4.4], [66, Section 4.14]; which, in particular, imply w 2k,p with p > k + 1 has at least one edge e (w) (n) ∈ W ∈ E that ˆm converge almost surely to their limiting expected such that N(e, w) = 1; whence, for all p > k+1, 2k,p = ∅. k W values. Hence, to determine the limiting spectral moments A key step in the proof of Wigner’s semicircle law [52], (n) (n) it suffices to calculate limn E ˆm = limn m¯ , a [53] is that for k even, the summation in (23)c is asymp- →∞ { k } →∞ k task which we undertake in our next lemma. totically dominated by those closed walks belonging to Lemma 3 (Limiting Spectral Moments). Under Assumptions k,k/2+1; i.e. those in which every edge is repeated exactly (n) (n) W twice. In order words, for every k as n , µk,k/2+1 A1 to A3, it is true that m¯ 2s m2s and m¯ 2s 1 = o(1) → ∞ − c µ p < k/2 + 1 for all s N. dominates every other k,p with . The reason ∈ is that by repeating an edge more than twice one looses Proof: a factor of order n in the available choices for vertices First, notice that that comprise the closed walk, this causes the number of such walks and their contributions to be asymptotically (n) 1 k m¯ = E trace(Aˆ ) k n n dominated. The same principle applies to the walks that 1 comprise the right-hand side of (23) and forms the base of = â(n) â(n) ... â(n) â(n) , E i1i2 i2i3 ik 1ik iki1 (21) our analysis. n − 1 i ,...,i n ≤ 1X k≤ n o We begin by analyzing (23) for the case k = 2s, i.e. 11

(n) s s+1 m¯ 2s = (nρn) p=1 µ2s,p. We show the desired dominance and (28) to get µ by first lower bounding the term 2s,s+1 and then upper 4 2 s+1 p p 1 2p 2 p P µ2s,p p s − ρ − wˆ − n bounding the terms µ2s,p for p < s + 1, as follows. First, 2 n n µ ≤ 4 ρs wˇ2s(n s)s+1 note that the number of walks in 2s,s+1 can be counted, 2s,s+1 n n W 2s −s as follows (according to Lemma 2.3.15 in [27]): s6 wˆ n 1 n , c ≤ 2ρ wˆ2 wˇ n s n s n! (n s)s+1 2s n n n − − card( ) = C − . (24) W2s,s+1 (n s 1)! s ≥ s + 1 s where in the first inequality we have used that − − ! 2s 2p+2 We nextc lower-bound the contribution of each walk in 2s 2s − 2s 2p+2 2s − , as ω(w) ρs wˇ2s, which holds true because 2p 2 ≤ 2s 2p + 2 ≤ 2s,s+1 n n ! WN(e, w) = 2, e ≥(w), and together with (24) and (22), − − impliesc that ∀ ∈ E and in the second inequality we take into account that the greatest lower bound is achieved for p = s, cf. [50]. The s+1 (n s) 2s s 2s proof now follows upon noting that under Assumptions µ2s,s+1 − ρnwˇn . (25) 2 ≥ n(s + 1) s A1, A3 (i), and A3 (ii), we have ρnwˆn < 1, wˆn/wˇn = ! 2 O(log n) and ρnwˆn = ω(1/n); whence, the rate of growth Note that Assumption A3 implies that ρ = o(1/n) 6 2 2s (iii) n of (s / ρnwˆn) (w ˆn/wˇn) is slower than a poly-logarithmic while wˇn < wˆn = O(log n); hence, √ρnwˇn = o(log n/√n) term times n, and we get that µ2s,p/µ2s,s+1 = o(1), as s 2s (n) and in particular we have that ρnwˇn = o(1) ensuring that s desired. Thus, having shown that m¯ 2s = (nρn) (1 + o(1)) the right hand side does not grow unbounded with increas- µ2s,s+1, we can now proceed to derive the asymptotic ing n and is in fact vanishing with n. Next, to upper-bound expressions of µ2s,s+1 as n ; in particular, to show that µ2s,p for all p < s+1, we make use of the following bound, s → ∞ limn (nρn) µ2s,s+1 = m2s. But before we embark on the | | →∞ which is developed by Furedi¨ and Komlos´ in Section 3.2 proof of the asymptotic expressions for the even moments, of [50], and is subsequently used in Lemma 2 of [67] and we use the upper-bound in (28) to show that the odd-order Equation (5) of [68] as well; for all p [n]: ∈ moments are asymptotically vanishing: First, note from (27) that with k = 2s+1 each of the terms µ2s+1,p, p s + 1 can n! k 2(k 2p+2) 2p 2 ≤ card k,p p − 2 − be upper-bounded, as follows: W ≤ (n p)! 2p 2! − − c p 1 2s + 1 2(2s 2p+3) 2p 2 p 1 2p 2 p k 2(k 2p+2) 2p 2 µ2s+1,p n − p − 2 − ρ − wˆ − n p − 2 − . (26) n n ≤ 2p 2 | | ≤ 2p 2! − ! − 2s + 1 p 1 2(2s+3) 2s 2 − Furthermore, we can bound the contribution of each walk (s + 1) 2 nρnwˆn . (29) p 1 2p 2 ≤ s w k,p as ω(w) ρn− wˆn − because with p distinct ! ∈ W | | ≤ vertices in walk w, there is at least p 1 distinct edges, (n) − 2 Next, using (23) with k = 2s + 1, we have m¯ 2s+1 = for eachc of which we can use the bound ρnwˆn. Indeed, for s+1 any edge (i, j) (w), we have N( i, j , w) 2, while p=1 µ2s,p, and replacing from (29), the odd spectral mo- (n) ∈ E (n) {(n) } ≥(n) (n) ments can now be upper bounded as âij 1 as by definition âij = aij ρnwi wj , so P | |( ≤n) (n) (n) − (n) (n) (n) that â = 1 ρnw w with probability ρnw w m¯ ij − i j i j | 2s+1| (n) (n) (n) (n) (n) s+1 and âij = ρnwi wj with probability 1 ρnwi wj . 2s + 1 p 1 − − √nρ (4nρ )s (s + 1)4s+6 nρ wˆ2 − Using (3), we thus obtain the following bound: n n s n n ≤ ! p=1 N( i,j ,w) 2 X (n) { } (n) 2 s+1 E âij E âij 2s + 1 nρ wˆ 1 ≤ s 4s+6 n n = √nρn(4nρn) (s + 1) 2 − (n) (n) (n) (n) 2 s ! nρnwˆn 1 = ρnw w (1 ρnw w ) ρnwˆ . − i j − i j ≤ n a 2s + 1 4s+6 2s = √nρn (s + 1) (2nρnwˆn) (1 + o(1)) s !

Next, using the deﬁnition (22), together with the upper- b 2s + 1 c p 1 2p 2 4s+6 bounds ω(w) ρ − wˆ − and (26), we obtain: = √nρn (s + 1) O(1) = o(1), | | ≤ n n s !

1 k 2(k 2p+2) p 1 p 1 2p 2 a 2 µk,p 1 p p − 4 − ρn− wˆn − . (27) where in (=) we use Assumption A3 (ii): ρnwˆn = ω(1/n) ≤ n − 2p 2! 2 − to conclude that nρnwˆn >> 1 and simplify the leftmost b Using k = 2s in (27), we get that for all p < s + 1: fraction; in (=) we use Assumption A3 (iv): nρnwˆn = O(1) c and in (=) we use Assumption A3 (iii): √nρn = o(1), which 1 2s p 1 4(s p+1) 2p 2 p 1 together complete the proof of the claim about vanishing µ2s,p 1 p 4 − p − wˆn − ρn− . (28) ≤ n − 2p 2! odd moments. Having thus shown the asymptotic relations − (n) (n) m¯ 2s+1 = o(1) and m¯ 2s = (1 + o(1)) µ2s,s+1, we are now ready to show that µ2s,s+1 are asymptotically equal to m2s, To show the dominance of µ2s,s+1 over µ2s,p for p < completing the proof of the claim about the limiting spectral s + 1, we form the ratio between the two inequalities (25) moments in Lemma 3. 12

Starting from the definition (22) with k = 2s and p = (d1(T ), . . . , ds+1(T )) we have: s + 1, we have that s+1 s+1 1 (n) dk(T ) µ2s,s+1 = ω(w) (30) w (33) n ik 1 i ,...,i n k=1 ! w X2s,s+1 j=1 ≤ 1 Xs+1≤ Y ∈W card( i1,...,is+1 )=j 1 (n) (n) (n) (n) X { } = c ρ w w (1 ρ w w ), n n n d (T ) d (T ) n n i j − n i j (n) 1 (n) s+1 T s+1 i,j (T ) = ... wi1 ... wis+1 ∈TX { }∈EY iX1=1 iX2=1 isX+1=1 s+1 n s+1 s dj (T ) where in the last equality we write the summation in terms (n) rj (T ) = wk = Sn,dj (T ) = Sn,j of the set of rooted ordered trees (using a bijection described j=1 k=1 ! j=1 j=1 below), and use (3) together with the independence of the Y X Y Y entries to express the weight of each tree as the product (n) (n) (n) (n) of the terms ρ w w (1 ρ w w ) corresponding to where we use the partial power-sums notation Sn,k = n i j n i j k − n (n) each edge i, j of the tree. The second equality in (30) is w for all k, n N, and we know that their { } i=1 i ∈ based on a bijection between the set of walks in 2s,s+1 k limiting averages Λk = limn (1/n)(nρn) Sn,k exist and W P →∞ and the set of rooted ordered trees on s + 1 vertices, s+1, are finite by Assumption A2. Next, note that as follows. First, note that the set of edges visitedc byT any nsρ2s walk in 2s,s+1 is always a tree with s + 1 vertices (since lim n W n we need at least s distinct edges for the induced graph →∞ n × s s+1 to be connected).c Second, it is clear that for any walk in (n) dk(T ) wi = 0, 2s,s+1, each edge must be visited exactly twice and the k ! W j=1 1 i1,...,is+1 n k=1 total number of edges in the induced graph is exactly s. X card≤( i ,...,iX ≤)=j Y { 1 s+1} Furthermore,c any walk w 2s,s+1 corresponds to a ∈ W which is true since, for each j [s], the term in the curly depth-first traversal of this induced tree (see [69] for more ∈ details about depth-first traversalsc of a tree) and we can, brackets above can be bounded as s 2s s+1 thus, impose a total order on the vertices of the induced tree n− (nρn) 0 wˆdk(T ) (using the order of the first appearance in the walk w). ≤ n n 1 i1,...,is+1 n k=1 ! card≤( i ,...,iX ≤)=j Y { 1 s+1} j s 1 1 n 2s n − − 2s s+1 (nρnwˆn) (nρnwˆn) = o(1), ≤ n j! ≤ j! where in the last equality we use the fact that Assumption A3 (iv): nρnwˆn = O(1). In other words, as n the summation over j = 1, . . . , s + 1 is dominated→ by its∞ last terms j = s+1. We now take limits of both sides in (32) and use the preceding asymptotic dominance relation together with (33) to get that

(n) s lim m¯ = lim (nρn) µ2s,s+1 (n) (n) n 2s n Next note that since per Assumption A1: ρnw w →∞ →∞ i j → 2s s 0 as n , for any s fixed we obtain (nρn) rj (T ) → ∞ = lim s+1 Sn,j s n n T s+1 (1 o(1))ρ (n) (n) →∞ ∈T j=1 µ = − n w w . (31) X Y 2s,s+1 i j s jrj (T ) n (nρn) rj (T ) T s+1 i,j (T ) ∈TX { }∈EY = lim r (T ) Sn,j n T s+1 n j For any tree T , it is always true that →∞ ∈T j=1 X s Y r (T ) di(T ) j (n) (n) (n) = Λj , (34) w w = w , T s+1 i j i ∈T j=1 i,j (T ) i (T ) X Y { }∈EY ∈VY where in the penultimate equality we have used the facts where di(T ) is the degree of node i in the tree T . Hence, (31) that, for any of the trees T s+1, the degree distribution can be written as: s ∈ T s (r1, . . . , rs) satisfies d=1 d rd = 2s and d=1 rd = s + 1. 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ACKNOWLEDGMENTS This work was supported in part by the United States Na- tional Science Foundation under grants CNS-1302222 and IIS-1447470.