Structural Analysis of Laplacian Spectral Properties of Large-Scale Networks

1 Structural Analysis of Laplacian Spectral Properties of Lar ge-Scale Networks V ictor M. Preciado, Member , IEEE, Ali Jadbabaie, Senior Member , IEEE , and Geor ge C. V erghese, F ellow , IEEE Abstract Using methods from algebraic graph theory and con ve x optimization, we study the relationship between local structural features of a network and spectral properties of its Laplacian matrix. In particular , we deriv e expressions for the so-called spectral moments of the Laplacian matrix of a network in terms of a collection of local structural measurements. Furthermore, we propose a series of semidefinite programs to compute bounds on the spectral radius and the spectral gap of the Laplacian matrix from a truncated sequence of Laplacian spectral moments. Our analysis sho ws that the Laplacian spectral moments and spectral radius are strongly constrained by local structural features of the network. On the other hand, we illustrate how local structural features are usually not enough to estimate the Laplacian spectral gap. I . I N T RO D U C T I O N Understanding the relationship between the structure of a network and the beha vior of dynam- ical processes taking place in it is a central question in the research field of network science [1]. Since the behavior of many networked dynamical processes is closely related with the Laplacian eigen v alues (see [2], [3] and references therein), it is of interest to study the relationship between structural features of the network and its Laplacian eigen v alues. V .M. Preciado and A. Jadbabaie are with the Department of Electrical and Systems Engineering at the University of Pennsylvania, Philadelphia, P A 19104 USA. (e-mail: preciado@seas.upenn.edu; jadbabai@seas.upenn.edu). G.C. V erghese is with the Department of Electrical Engineering and Computer Science at the Massachusetts Institute of T echnology , Cambridge, MA 02139 USA. (e-mail: verghese@mit.edu). This work was supported by ONR MURI “Next Generation Network Science” and AFOSR “T opological and Geometric T ools for Analysis of Complex Networks”. May 21, 2018 DRAFT 2 In this technical note, we study this relationship, focusing on the role played by structural features that can be extracted from localized samples of the network structure. Our objectiv e is then to efficiently aggre gate these local samples of the network structure to infer global properties of the Laplacian spectrum. W e propose a graph-theoretical approach to relate structural features of a network with algebraic properties of its Laplacian matrix. Our analysis rev eals that there are certain spectral properties, such as the so-called spectral moments, that can be efficiently computed from these structural features. Furthermore, applying a recent result by Lasserre [4], we propose a series of semidefinite programs to compute bounds on the Laplacian spectral radius and spectral gap from a truncated sequence of spectral moments. The paper is organized as follows. In the next subsection, we define terminology needed in our deri v ations. In Section II, we introduce a graph-theoretical methodology to deriv e closed- form expressions for the so-called Laplacian spectral moments in terms of structural features of the network. In Section III, we use semidefinite programming to deriv e optimal bounds on the Laplacian spectral radius and spectral gap from a truncated sequence of spectral moments. W e v alidate our results numerically in Section IV. A. Notations & Pr eliminaries Let G = ( V , E ) be an undirected graph, where V = { v 1 , . . . , v n } denotes a set of n nodes and E ⊆ V × V denotes a set of e undirected edges. If { v i , v j } ∈ E , we call nodes v i and v j adjacent (or first-neighbors), which we denote by v i ∼ v j . W e define the set of first-neighbors of a node v i as N i = { w ∈ V : { v i , w } ∈ E } . The de gr ee d i of a vertex v i is the number of nodes adjacent to it, i.e., d i = |N i | . W e consider three types of undirected graphs: ( i ) A graph is called simple if its edges are unweighted and it has no self-loops 1 , ( ii ) a graph is loopy if it has self-loops, and ( iii ) a graph is weighted if there is a real number associated with e very edge in the graph. More formally , a weighted graph H can be defined as the triad H = ( V , E , W ) , where V and E are the sets of nodes and edges in H , and W = { w ij ∈ R , for all { v i , v j } ∈ E } is the set of (possibly negati ve) weights. The adjacency matrix of a simple graph G , denoted by A G = [ a ij ] , is an n × n symmetric matrix defined entry-wise as a ij = 1 if nodes v i and v j are adjacent, and a ij = 0 otherwise. In 1 A self-loop is an edge of the type { v i , v i } . May 21, 2018 DRAFT 3 the case of weighted graphs (and possibly non-simple), the weighted adjacency matrix is defined by W G = [ w ij ] , where w ij = 0 if v i is not adjacent to v j . W e define the de gr ee matrix of a simple graph G as the diagonal matrix D G = diag ( d i ) . W e define the Laplacian matrix L G (also kno wn as combinatorial Laplacian, or Kirchhoff matrix) of a simple graph as L G = D G − A G . For simple graphs, L G is a symmetric, positi ve semidefinite matrix, which we denote by L G  0 [5]. Thus, L G has a full set of n real and orthogonal eigen vectors with real nonnegati ve eigen values 0 = λ 1 ≤ λ 2 ≤ ... ≤ λ n . The second smallest and largest eigen v alues of L G , λ 2 and λ n , are called the spectral gap and spectral radius of L G , respecti vely . Giv en a n × n real and symmetric matrix B with (real) eigen values σ 1 , ..., σ n , we define the k -th spectral moment of B as m k ( B ) , 1 n P n i =1 σ k i . As we shall sho w in Section II, there is an interesting connection between the spectral moments of the Laplacian matrix, m k ( L G ) , and structural features of the network. W e now define a collection of structural properties that are important in our deriv ations. The de gr ee sequence of a simple graph G is the ordered list of its degrees, ( d 1 , ..., d n ) . A walk of length k from v i 1 to v i k +1 is an ordered sequence of nodes  v i 1 , v i 2 , ..., v i k +1  such that v i j ∼ v i j +1 for j = 1 , 2 , ..., k . One says that the walk touches each of the nodes that comprises it. If v i 1 = v i k +1 , then the walk is closed. A closed walk with no repeated nodes (with the exception of the first and last nodes) is called a cycle . Gi ven a walk p =  v i 1 , v i 2 , ..., v i k +1  in a weighted graph H , we define the weight of the walk as, ω ( p ) = w i 1 i 2 w i 2 i 3 ...w i k i k +1 . I I . M O M E N T - B A S E D A N A LY S I S O F T H E L A P L A C I A N M A T R I X In this paper , we use algebraic graph theory to study the relationship between structural properties of a netw ork and its Laplacian spectrum based on the spectral moments. A well- kno wn result in algebraic graph theory relates the diagonal entries of the k -th power of the adjacency matrix,  A k G  ii , to the number of closed w alks of length k in G that start and finish at node v i [5]. Using this result, it is possible to relate algebraic properties of the adjacency matrix A G to the presence of certain subgraphs in the network [6]. W e can generalize this result to weighted graphs as follo ws: Pr oposition 1: Let H = ( V , E , W ) be a weighted graph with weighted adjacenc y matrix W H = [ w ij ] . Then,  W k H  ii = P p ∈ P k,i ω ( p ) , where P k,i is the set of closed walks of length k from v i to itself in H . May 21, 2018 DRAFT 4 Pr oof: By recursiv ely applying the multiplication rule for matrices, we have the follo wing expansion  W k H  ii = n X i =1 n X i 2 =1 · · · n X i k =1 w i,i 2 w i 2 , i 3 · · · w i k ,i . (1) Using the graph-theoretic nomenclature introduced in Section I-A, we ha ve that w i,i 2 w i 2 , i 3 ...w i k ,i = ω ( p ) , for p = ( v i , v i 2 , v i 3 , ..., v i k , v i ) . Hence, the summations in (1) can be written as  W k H  ii = P 1 ≤ i,i 2 ,...,i k ≤ n ω ( p ) . Finally , the set of closed walks p = ( v i , v i 2 , v i 3 , ..., v i k , v i ) with indices 1 ≤ i, i 2 , ..., i k ≤ n is equal to the set of closed walks of length k from v i to itself in H (which we hav e denoted by P k,i in the statement of the Proposition). The abov e Proposition allo ws us to write the relate moments of the weighted adjacency matrix of a weighted graph H to closed walks in H , as follows: Lemma 2.1: Let H = ( V , E , W ) be a weighted graph with weighted adjacency matrix W H = [ w ij ] . Then, m k ( W H ) = 1 n X v i ∈V X p ∈ P k,i ω ( p ) , where P k,i is the set of closed walks of length k from v i to itself in H . Pr oof: Let us denote by µ 1 , ..., µ n the set of (real) eigen values of the (symmetric) weighted adjacency matrix W H . W e hav e that the moments can be written as m k ( W H ) , 1 n n X i =1 µ k i = 1 n T race  W k H  , since W H is a symmetric (and diagonalizable) matrix. W e then apply Proposition 1 to rewrite the moments as follo ws, m k ( W H ) = 1 n n X i =1  W k H  ii = 1 n X v i ∈V X p ∈ P k,i ω ( p ) . In Subsections II-B, we shall apply this result to compute spectral moments of the Laplacian matrix in terms of structural features of the network. First, we need to introduce a weighted graph that is useful in our deriv ations: May 21, 2018 DRAFT 5 Definition 2.1: Giv en a simple graph G = ( V , E ) , we define the Laplacian graph of G as the weighted graph L ( G ) , ( V , E ∪ S n , Γ) , where S n = {{ v , v } for all v ∈ V } (the set of all self- loops), and Γ = [ γ ij ] is a set of weights defined as: γ ij ,          − 1 , for { v i , v j } ∈ E d i , for i = j 0 , otherwise. Remark 2.1: Note that the weighted adjacency matrix of the Laplacian graph L ( G ) is equal to the Laplacian matrix of the simple graph G . Hence, we can apply Lemma 2.1 to express the spectral moments of the Laplacian matrix L G in terms of weighted walks in the Laplacian graph L ( G ) . Before we apply Lemma 2.1 to study the Laplacian spectral moments, we must introduce the concept of subgraph cov ered by a walk. Definition 2.2: Consider a walk p =  v i 1 , v i 2 , ..., v i k +1  of length k in a (possibly loopy) graph. W e define the subgraph cov ered by p as the simple graph C ( p ) = ( V c ( p ) , E c ( p )) , with node-set V c ( p ) = S k +1 r =1 v i r , and edge-set E c ( p ) = S v i r 6 = v i r +1  v i r , v i r +1  , for 1 ≤ r ≤ k + 1 . Based on the above, we define triangles , quadrangles and pentagons as the subgraphs covered by cycles of length three, four , and fiv e, respectiv ely . Notice that self-loops are excluded from E c ( p ) in Definition 2.2. For example, consider a walk p = ( v 1 , v 2 , v 2 , v 3 , v 3 , v 1 , v 3 , v 1 ) in a graph with self-loops. Then, C ( p ) has node-set V c ( p ) = { v 1 , v 2 , v 3 } and edge-set E c ( p ) = {{ v 1 , v 2 } , { v 2 , v 3 } , { v 3 , v 1 }} . In other words, C ( p ) is a simple triangle. In what follows, we build on the abov e results to deriv e closed-form expressions for the first fi ve spectral moments of the Laplacian matrix in terms of rele vant structural features of the network. A. Low-Or der Laplacian Spectral Moments The following theorem, prov ed in [7] via algebraic techniques, allows us to compute the first three Laplacian spectral moments in terms of the degree sequence and the number of triangles in the graph. Theor em 2.2: Let G be a simple graph with Laplacian matrix L G . Then, the first three spectral May 21, 2018 DRAFT 6 moments of the Laplacian matrix are m 1 ( L G ) = 1 n S 1 , (2) m 2 ( L G ) = 1 n ( S 1 + S 2 ) , m 3 ( L G ) = 1 n (3 S 2 + S 3 − 6∆) , where S p , P v i ∈V d p i , and ∆ is the total number of triangles in G . In [7], Theorem 2.2 was proved using a purely algebraic approach. This algebraic approach presents the limitation of not being applicable to compute moments of order greater than three. In what follo ws, we propose an alternati ve graph-theoretical approach that allo ws to compute higher-order spectral moments of the Laplacian matrix, beyond the third order . In particular , according to Lemma 2.1, we can compute the k -th spectral moment of the Laplacian of G by analyzing the set of closed walks of length k in the Laplacian graph L ( G ) . B. Higher-Or der Laplacian Spectral Moments In this Subsection, we apply the set of graph-theoretical tools introduced abov e to compute the fourth- and fifth-order spectral moments of the Laplacian matrix. W e first define the collection of structural measurements that are in volv ed in our expressions. Let us denote by t i , q i , and p i the number of triangles, quadrangles, and pentagons touching node v i in G , respectiv ely . The total number of quadrangles and pentagons in G are denoted by Q and P , respectiv ely . The follo wing terms define structural correlations that are relev ant in our analysis: C dd , 1 n X v i ∼ v j d i d j , C d 2 d , 1 n X v i ∼ v j d 2 i d j , (3) C dt , 1 n X v i ∈V d i t i , C d 2 t , 1 n X v i ∈V d 2 i t i , C dq , 1 n X v i ∈V d i q i , D dd , 1 n X v i ∼ v j d i d j |N i ∩ N j | , where |N i ∩ N j | is the number of common neighbors shared by v i and v j . The main result in this section relates the fourth and fifth Laplacian spectral moments to local structural measurements and correlation terms, as follo ws: May 21, 2018 DRAFT 7 Theor em 2.3: Let G be a simple graph with Laplacian matrix L G . Then, the fourth and fifth Laplacian moments can be written as m 4 ( L G ) = 1 n ( − S 1 + 2 S 2 + 4 S 3 + S 4 + 8 Q ) (4) +4 C dd − 8 C dt , m 5 ( L G ) = 1 n ( − 5 S 2 + 5 S 3 + 5 S 4 + S 5 + 30∆ − 10 P ) +10 ( C dd + C d 2 d − C dt − C d 2 t + C dq − D dd ) where S p , P v i ∈V d p i , and the correlation terms C dd , C dt , C dq , C d 2 d , C d 2 t , and D dd are defined in (3). Pr oof: In the Appendix. Remark 2.2: Theorem 2.2 relates purely algebraic properties – the spectral moments – to structural features of the network, namely the degree sequence, the number of cycles of length 3 and 5, and all the correlation terms defined in (3). The ke y steps behind the proof are: ( i ) Relate the spectral moments m 4 ( L G ) and m 5 ( L G ) with closed walks of length four and fi ve in the Laplacian graph L ( G ) , and ( ii ) classify the set of closed walks in L ( G ) into subsets according to the subgraph cov ered by each walk. In the ne xt section, we present a series of semidefinite programs (SDP’ s) whose solutions provide optimal bounds on the Laplacian spectral radius and spectral gap in terms of Laplacian spectral moments. I I I . O P T I M A L L A P L AC I A N B O U N D S F R O M S P E C T R A L M O M E N T S In this section, we introduce a nov el approach to compute bounds on the spectral gap and the spectral radius of the Laplacian matrix from a truncated sequence of Laplacian spectral moments. More explicitly , the problem solved in this section can be stated as follows: Pr oblem 1 (Moment-based bounds): Giv en a truncated sequence of Laplacian spectral mo- ments ( m k ( L G )) K k =1 , find bounds on the spectral gap and the spectral radius of the Laplacian matrix L G . In this section, we propose a solution to the abov e problem based on a recent result in [4]. In [4], Lasserre developed an approach to find bounds on the support of an unkno wn density function when only a sequence of its moments is av ailable. In order to adapt Problem 1 to May 21, 2018 DRAFT 8 this framework, we need to introduce some definitions. Giv en a simple connected graph G with Laplacian eigen v alues { λ i } n i =1 , we define the spectral density of the nontrivial eigen v alue spectrum as ρ G ( λ ) , 1 n − 1 X i ≥ 2 δ ( λ − λ i ) , (5) where δ ( · ) is the Dirac delta function. Notice how we hav e excluded the trivial eigen v alue, λ 1 = 0 , from the spectral density; hence, the support 2 of ρ G ( λ ) is equal to supp ( ρ G ) = { λ i } n i =2 . The moments of the spectral density in (5), denoted by m k ( L G ) , can be written in terms of the spectral moments of L G , as follows m k ( L G ) , Z R λ k 1 n − 1 n X i =2 δ ( λ − λ i ) dλ = 1 n − 1 n X i =2 λ k i = n n − 1 m k ( L G ) , (6) for all k ≥ 1 (where we hav e used the fact that λ 1 = 0 , in our deri vations). In what follows, we propose a solution to Problem 1 using a technique proposed by Lasserre in [4]. In that paper , the following problem was addressed: Pr oblem 2: Consider a truncated sequence of moments ( M k ) 1 ≤ r ≤ K corresponding to an un- kno wn density function µ ( λ ) , i.e., M k , R λ k dµ ( λ ) . Denote by [ a, b ] the smallest interval containing the support of µ . Compute an upper bound α ≥ a and a lower bound β ≤ b when only the truncated sequence of moments is av ailable. In the context of Problem 1, we hav e access to a truncated sequence of fiv e spectral moments, ( m k ( L G )) 1 ≤ k ≤ 5 , corresponding to the unkno wn spectral density function ρ G and giv en by the expressions (2), (4), and (6). In this conte xt, the smallest interv al [ a, b ] containing supp ( ρ G ) is equal to [ λ 2 , λ n ] . Therefore, a solution to Problem 2 would directly provide an upper bound on the spectral gap, α ≥ λ 2 , and a lo wer bound on the spectral radius, β ≤ λ n . W e now describe a numerical scheme proposed in [4] to solve Problem 2. This solution is based on a series of semidefinite programs in one v ariable. In order to formulate this series of SDP’ s, we need to introduce some definitions. For any s ∈ N , let us consider a truncated sequence of moments 2 Recall that the support of a density function µ on R , denoted by supp ( µ ) , is the smallest closed set B such that µ ( R \ B ) = 0 . May 21, 2018 DRAFT 9 M = ( M k ) 2 s +1 k =1 , associated with an unknown density function µ . W e define the following Hankel matrices of moments: R 2 s ( M ) ,        1 M 1 · · · M s M 1 M 2 · · · M s +1 . . . . . . . . . . . . M s M s +1 · · · M 2 s        , R 2 s +1 ( M ) ,        M 1 M 2 · · · M s +1 M 2 M 3 · · · M s +2 . . . . . . . . . . . . M s +1 M s +2 · · · M 2 s +1        . (7) W e also define the localizing matrix 3 H s ( x, M ) as, H s ( x, M ) , R 2 s +1 ( M ) − x R 2 s ( M ) . (8) Using the abo ve matrices, Lasserre proposed in [4] the following series of SDP’ s to find a solution for Problem 2: Solution to Pr oblem 2 : Let M = ( M k ) 2 s +1 k =1 be a truncated sequence of moments associated with an unknown density function µ . Then a ≤ α s ( M ) , max x { x : H s ( x, M )  0 } , (9) b ≥ β s ( M ) , min x { x : − H s ( x, M )  0 } , (10) where [ a, b ] is the smallest interval containing the support of µ . Therefore, we can directly apply the above result to solve Problem 1 by considering the sequence of moments m , ( m r ( L G )) 2 s +1 r =1 = ( n n − 1 m r ( L G )) 2 s +1 r =1 in the statement of the solution to Problem 2. Since this sequence of moments corresponds to the spectral density ρ G , with support { λ i } n i =2 , the solutions in (9) and (10) directly provide the follo wing bounds on the spectral radius and spectral gap: Solution to Pr oblem 1 : Let m , ( n n − 1 m r ( L G )) 2 s +1 r =1 be a truncated sequence of (scaled) Laplacian spectral moments associated with a graph G . Then the Laplacian spectral gap and spectral radius of G satisfy the following bounds: λ 2 ≤ α s ( m ) , max x { x : H s ( x, m )  0 } , (11) λ n ≥ β s ( m ) , min x { x : − H s ( x, m )  0 } . (12) 3 A more general definition of localizing matrix can be found in [ ? ]. For simplicity , we restrict our definition to the particular form used in our problem. May 21, 2018 DRAFT 10 In Section II, we deri ved expressions for the first fiv e Laplacian spectral moments, ( m r ( L G )) 5 r =1 , in terms of structural features of the network, namely , the degree sequence, the number of triangles, quadrangles and pentagons, and the correlation terms in (3). Therefore, we can apply the Solution to Pr oblem 1 to find bounds on λ 2 and λ n . In this section, we hav e presented an optimization-based approach to compute optimal bounds on the Laplacian spectral gap and spectral radius from a truncated sequence of Laplacian spectral moments. The truncated sequence of spectral moments ( m k ( L G )) 5 k =1 can be written in terms of local structural measurements using (2) and (4). Hence, the above methodology allows to compute bounds on the spectral radius and spectral gap of the Laplacian matrix giv en a collection of local structural features of the network. In the follo wing section, we illustrate the usage of this approach with numerical examples. I V . S T R U C T U R A L A N A L Y S I S A N D S I M U L A T I O N S In this section, we apply the moment-based approach herein proposed to study the relationship between structural and spectral properties of an unweighted, undirected graph representing the structure of the high-voltage transmission network of Spain (the adjacency of this network is av ailable, in MA TLAB format, in [8]). The number of nodes (buses) and edges (transmission lines) in this network are n = 98 and e = 175 , respectiv ely . From this dataset, we compute the set of structural properties in volv ed in (2) and (4), namely , the power -sums of the de grees ( S r ) 5 r =1 = (350 , 1692 , 9836 , 64056 , 44942) , the number of cycles ∆ = 79 , Q = 134 , P = 232 , and the correlation terms C dd = 42 . 58 , C d 2 d = 249 . 41 , C dt = 13 . 98 , C d 2 t = 88 . 69 , C dq = 33 . 11 , and D dd = 80 . 77 . Using this collection of structural measurements, we use (2) and (4) to compute the first fi ve Laplacian spectral moments of the Spanish transmission network: ( m k ( L G )) 5 k =1 = (3 . 571 , 20 . 83 , 147 . 33 , 1155 . 5 , 9686 . 6) . Using this sequence of spectral moments and the methodology described in Section III, we compute bounds on the spectral gap and spectral radius, α 2 and β 2 , solving the SDP’ s in (11) and (12). The numerical v alues for these bounds, as well as the exact values for the spectral gap and spectral radius are: β 2 = 9 . 18 ≤ λ n = 10 . 66 and λ 2 = 0 . 077 ≤ α 2 = 0 . 86 . Our numerical analysis re veals that the Laplacian spectral radius and spectral moments of the electrical transmission network are strongly constrained by local structural features of the network. On the other hand, the spectral gap cannot be efficiently bounded using local structural May 21, 2018 DRAFT 11 features only , since the spectral gap strongly depends on the global connectivity of the network. This limitation is inherent to all spectral bounds based on local structural properties (see [9] for a wide collection of spectral bounds). In the following example, we illustrate this limitation with a simple example. Example 4.1: Consider a ring graph with n = 12 nodes, which we denote by R 12 . The eigen v alues of the Laplacian matrix of a ring graph of length l , R l , are equal to λ i = 2 − 2 cos(2 π i/l ) , for i = 0 , ..., l − 1 [5]. Therefore, the Laplacian spectral gap and spectral radius of R 12 are λ 2 = 2 − 2 cos π / 6 ≈ 0 . 2679 and λ n = 4 , respecti vely . W e can also compute the moment-based bounds α 2 and β 2 using local structural measurements, as follows. The degrees of all the nodes in R 12 are d i = 2 ; thus, the power sums of the degrees are equal to S k = 2 k 12 . The number of triangles, quadrangles and pentagons are ∆ = Q = P = 0 . The correlation terms are C dd = 4 , C d 2 d = 8 , and the rest of correlation terms in (3) are equal to zero. Based on these structural measurements, we hav e from (2) and (4) that the first fiv e Laplacian spectral moments are ( m r ( L G )) 5 r =1 = (2 , 6 , 20 , 70 , 252) , and the resulting moment-based bounds from (11) and (12) are β 2 = 3 . 732 ≤ 4 and α 2 = 0 . 2679 ≈ λ 2 . Therefore, both the bounds on the spectral radius and the spectral gap are very tight for R 12 . In particular , α 2 is remarkably close to λ 2 . On the other hand, we can construct graphs with the same local structural properties (and, therefore, the same first fiv e spectral moments, and bounds α 2 and β 2 ), but very different spectral gap, as follows. Consider a graph of 12 nodes consisting in two disconnected rings of length 6. It is easy to verify that this (disconnected) graph presents the same local structural features as a connected ring of length 12, namely , the degrees of all the nodes are d i = 0 , the number of cycles ∆ = Q = P = 0 , and the correlation terms are the same as the ones computed abov e. In contrast to R 12 , the spectral gap of this disconnected graph is equal to zero, λ 2 = 0 , which is very different than the moment-based bound α 2 . In general, the Laplacian spectral gap is a global property that quantifies how ‘well-connected’ a network is [10]. Since the structural measurements used in our bounds (degree sequence, correlation terms, etc.) have a local nature, they do not contain enough information to determine ho w well connected the network is globally . In other words, it is often possible to find two dif ferent graphs with identical local structural features b ut radically different global structure, as we hav e illustrated in the above example. May 21, 2018 DRAFT 12 V . C O N C L U S I O N S This paper studies the relationship between local structural features of large complex networks and global spectral properties of their Laplacian matrices. In Section II, we ha ve proposed a graph-theoreical approach to compute the first fiv e Laplacian spectral moments of a network from a collection of local structural measurements. In Section III, we hav e proposed an optimization- based approach, based on a recent result by Lasserre [4], to compute bounds on the Laplacian spectral radius and spectral gap of a network from a truncated sequence of spectral moments. Our bounds take into account the effect of important structural properties that are usually neglected in most of the bounds found in the literature, such as the distribution of cycles and other structural correlations. Our analysis shows that local structural features of the network strongly constrain the Laplacian spectral moments and spectral radius. On the other hand, local structural features are not enough to characterize the Laplacian spectral gap, since this quantity strongly depends on how ‘well-connected’ the network is globally . A P P E N D I X Theor em 2.3 Let G be a simple graph with Laplacian matrix L G . Then, the fourth and fifth Laplacian moments can be written as m 4 ( L G ) = 1 n ( − S 1 + 2 S 2 + 4 S 3 + S 4 + 8 Q ) +4 C dd − 8 C dt , m 5 ( L G ) = 1 n ( − 5 S 2 + 5 S 3 + 5 S 4 + S 5 + 30∆ − 10 P ) +10 ( C dd + C d 2 d − C dt − C d 2 t + C dq − D dd ) where S r = P v i ∈V d r i , and the correlation terms C dd , C dt , C dq , C d 2 d , C d 2 t , and D dd are defined in (3). Pr oof: As in Theorem 2.2, we use Lemma 2.1 to compute the Laplacian spectral moments in terms of weighted sums of closed walks in the weighted Laplacian graph L G . In order to compute the fourth Laplacian spectral moment, we classify the types of possible closed w alks of length 4 into subsets according to the structure of the underlying graph covered by the walk. Specifically , two walks p 1 and p 2 belong to the same type if the subgraphs cov ered by the walks, denoted by C ( p 1 ) and C ( p 2 ) according to Definition 2.2, are isomorphic. W e enumerate May 21, 2018 DRAFT 13 Fig. 1. Collection of possible graphs cov ered by closed walks of length 4. Fig. 2. Collection of possible graphs cov ered by closed walks of length 5. the possible types in Fig. 1 and we denote the corresponding sets of walks as P ( i ) 4 a , P ( i ) 4 b , P ( i ) 4 c , P ( i ) 4 d , and P ( i ) 4 e . These sets P ( i ) 4 a ,..., P ( i ) 4 e partition the set of closed walks P ( i ) 4 ,n . Hence, we hav e m 4 ( L G ) = 1 n P v i ∈V P x ∈{ a,b,c,d,e } P p ∈ P ( i ) 4 x ω ( p ) . W e no w analyze each one of the terms in the above summations. For con venience, we define T 4 x , 1 n P v i ∈V P p ∈ P ( i ) 4 x ω ( p ) , and analyze the term T 4 x for x ∈ { a, b, c, d } : ( a ) For x = a , we hav e that the weights ω ( p ) of the walks in P ( i ) 4 a are all the same, and equal to d 4 i . Hence, T 4 a = 1 n P i d 4 i = S 4 /n. ( b ) For x = b , the weights of the walks in P ( i ) 4 b are equal to 2 + 4  d 2 i + d 2 j + d i d j  . Hence, T 4 b = 1 n P v i ∼ v j 2 + 4  d 2 i + d 2 j + d i d j  = 1 n ( S 1 + 4 S 3 ) + 4 C dd . ( c ) For x = c , the weights of the w alks in P ( i ) 4 c (i.e., walks that cover the tw o-chain graph) are equal to 4 . Hence, T 4 c = 1 n P v j ∼ v i ∼ v k 4 ( i ) = 1 n P n i =1  d i 2  4 = 2 n ( S 2 − S 1 ) , where in equality ( i ) we hav e used the fact that the number of two-chain graphs whose center node is v i is equal to  d i 2  . ( d ) For x = d , the weights of the walks in P ( i ) 4 d are equal to − 8 ( d i + d j + d k ) . Hence, T 4 d = 1 n P v i ∼ v j ∼ v k ∼ v i − 8 ( d i + d j + d k ) = − 8 n P n i =1 P n j =1 P n k =1 3 t ij k d i , where t ij k is an indica- tor function that takes value 1 if v i ∼ v j ∼ v k ∼ v i . Since P n j =1 P n k =1 3 t ij k = t i (the number of triangles touching node v i ), we have that T 4 d = − 8 n P n i =1 t i d i = − 8 C dt . May 21, 2018 DRAFT 14 ( e ) F or x = e , the weights of the walks in P ( i ) 4 e are equal to 8 . Hence, T 4 e = 1 n P v i ∼ v j ∼ v k ∼ v r ∼ v i s.t. 1 ≤ i [9] N.M.M. de Abreu, “Old and New Results on Algebraic Connectivity of Graphs, ” Linear Algebra and its Applications , vol. 423, pp. 53–73, 2007. [10] Y . Kim and M. Mesbahi, “On Maximizing the Second Smallest Eigen value of a State-Dependent Graph Laplacian, ” IEEE T ransactions on Automatic Contr ol , v ol. 51, pp. 116–120, 2006. May 21, 2018 DRAFT