A moment inequality and positivity for signed graph Laplacians
aa r X i v : . [ m a t h . SP ] M a y A MOMENT INEQUALITY AND POSITIVITY FOR SIGNED GRAPH LAPLACIANS ∗ IKEMEFUNA AGBANUSI † , JARED C. BRONSKI ‡ , AND
DEREK KIELTY ‡ Abstract.
A number of recent papers have considered signed graph Laplacians, a generalization of the classical graphLaplacian, where the edge weights are allowed to take either sign. In the classical case, where the edge weights are all positive,the Laplacian is positive semi-definite with the dimension of the kernel representing the number of connected components ofthe graph. In many applications one is interested in establishing conditions which guarantee the positive semi-definiteness ofthe matrix. In this paper we present an inequality on the eigenvalues of a weighted graph Laplacian (where the weights neednot have any particular sign) in terms of the first two moments of the edge weights. This bound involves the eigenvalues ofthe equally weighted Laplacian on the graph as well as the eigenvalues of the adjacency matrix of the line graph (the edge-to-vertex dual graph). For a regular graph the bound can be expressed entirely in terms of the second eigenvalue of the equallyweighted Laplacian, an object that has been extensively studied in connection with expander graphs and spectral measures ofgraph connectivity. We present several examples including Erd˝os–R´enyi random graphs in the critical and subcritical regimes,random large d -regular graphs, and the complete graph, for which the inequalities here are tight. Key words.
Signed Laplacian, Eigenvalue Inequality
1. Introduction.
There are a number of problems in applied mathematics where one is led to considerthe eigenvalues of a signed (combinatorial) graph Laplacian: given a graph G with N vertices and E edgesthe signed combinatorial Laplacian is an N × N matrix of the form(1.1) L ij ( γ ) = P k = i γ ik i = j − γ ij i = j, i ∼ j i = j, i j. Here i ∼ j denotes the relation that distinct vertices i and j are connected by an edge, γ ij denotes the weightof edge ij and γ ∈ R E is the vector of all edge weights. In this note Laplacian matrices are always symmetric,that is, γ ij = γ ji for each i and j . Note that the vector N = (1 , , , . . . , t is always in the kernel of L ( γ ).In the classical case where the edge weights are positive, γ ij ≥
0, the matrix is positive semi-definite butin this paper the edge weights γ ij are not assumed to have any particular sign. An incomplete list of theapplications of such matrices includes: • Data mining in social networks [1, 2]. • Hypergraph clustering algorithms [3]. • Models for the evolution of multi-agent networks [4, 5]. • Finding the fastest mixing linear consensus model [6, 7]. • The stability of phase-locked solutions to the Kuramoto and related models [8, 9, 10, 11, 12, 13].In many of these applications one is interested in establishing the semi-definiteness of a Laplacianmatrix, which typically implies stability of the associated fixed point or a consensus state. For this reason anumber of papers have considered the problem of establishing semi-definiteness of the Laplacian matrix, asin [14, 15, 16, 17, 18].The purpose of this note is to present an inequality (Theorem 1.2) on the eigenvalues of a Laplacianmatrix in terms of the first two moments — the mean and variance — of the edge weights. Following thiswe give a proof of the inequality and applications to both deterministic and random graphs. We first define:(1.2) Q = 1 E X i>j γ ij and P = 1 E X i>j γ ij . Note that the Cauchy–Schwartz inequality implies that P − Q ≥ i th eigenvalue of a symmetric N × N matrix L by λ i ( L ), numberedin increasing order of absolute values | λ ( L ) | ≤ | λ ( L ) | ≤ · · · ≤ | λ N ( L ) | . ∗ Funding:
This work was supported by the National Science Foundation under grant NSF-DMS 1615418 † Colorado College, Department of Mathematics and Computer Science, Colorado Springs, CO 80903 ([email protected].) ‡ University of Illinois, Department of Mathematics, Urbana, IL 61801 ([email protected], [email protected].)1
I. AGBANUSI, J.C. BRONSKI, AND D. KIELTY
Definition
For a connected graph let L ( E ) denote the equally weighted Laplacian—the (combina-torial) Laplacian on the graph G with all edge weights taken to be unity: γ ij = 1 . L ij ( E ) = deg( v i ) i = j − i = j, i ∼ j i = j, i ∼ j, where deg( v i ) is the degree of vertex i . In particular, let λ Gi = λ i ( L ( E )) denote the i th eigenvalue of L ( E ) ,numbered in increasing order (1.3) 0 = λ G < λ G ≤ λ G ≤ · · · ≤ λ GN . To state our main result let A LG ( G ) denote the adjacency matrix of the line graph of G and define thequantity µ = max γ ∈ R E : γ ⊥ E h γ , (4 I + A LG ( G ) ) γ ik γ k . One has the inequalities(1.4) 4 + λ E − ( A LG ( G ) ) ≤ µ ≤ λ E ( A LG ( G ) ) ≤ d max + 2 , where d max is the maximum degree of the vertices in the graph G . In the case of a d -regular graph we havethe equality µ = 2 d + 2 − λ G = 4 + λ E − ( A LG ( G ) ) (see the proof of Theorem 1.2). Theorem
Consider a weighted Laplacian matrix L ( γ ) defined as in (1.1) on a connected graph G with λ G and λ GN as in (1.3) and N ≥ . If P and Q are defined as in (1.2) then the N − eigenvalues of L ( γ ) corresponding to eigenvectors orthogonal to N satisfy the inequality Qλ G − r E ( P − Q ) µ N − N − ≤ λ i ( L ( γ )) ≤ Qλ GN + r E ( P − Q ) µ N − N − . Further this inequality is tight: for the complete graph there are choices of edge weights realizing the upperand lower bounds.In particular, if G is a d -regular graph then Qλ G − r E ( P − Q )(2 d + 2 − λ G ) N − N − ≤ λ i ( L ( γ )) ≤ Qλ GN + r E ( P − Q )(2 d + 2 − λ G ) N − N − . It is notable that, at least in the case of a regular graph, the lower bound depends only on λ G , the secondlargest eigenvalue of the graph Laplacian. The second eigenvalue is, of course, a well-studied object thatencodes important geometric information on the connectivity of the graph, dating back at least to the workof Feidler [19], and is closely connected with the theory of expander graphs. See, for instance, the reviewarticle of Hoory, Linial, and Wigderson [20] for an overview of this area.The lower bound in Theorem 1.2 is most important when considering the question of the positivity of L ( γ ). It shows that L ( γ ) is positive definite if the variance P − Q ≥ L ( γ ) is positive definite if(1.5) ( λ G ) µ E − Q > P − Q . Of course if the variance is small enough then each edge weight is necessarily positive, and hence theLaplacian is necessarily positive semi-definite. A computation shows that all the edge weights are positive if(1.6) E − Q > P − Q . Thus, when ( λ G ) /µ > λ G ) /µ > OSITIVITY FOR SIGNED GRAPH LAPLACIANS λ G ) /µ ≤
1. A simple exampleis the extreme case where G is disconnected so that λ G = 0.The inequalities in Theorem 1.2 are also reminiscent of the Samuelson inequality for a finite set of realnumbers. The original Samuelson inequality states that a finite set of real numbers is contained in a ballwith center equal to the mean and radius proportional to the standard deviation of its elements (see [21] forexample). Proof of Theorem 1.2.
Recall that E ∈ R E represents the vector E = (1 , , , . . . , t so we have theorthogonal decomposition γ = Q E + ˜ γ , where ˜ γ satisfies(1.7) h ˜ γ , E i = 0 and k ˜ γ k = E ( P − Q ) , and k · k is the Euclidean norm.This gives a decomposition of the graph Laplacian into a “mean” and “fluctuation” as follows L ( γ ) = L ( Q E + ˜ γ ) = QL ( E ) + L (˜ γ ) . Recall that for symmetric matrices A and B we have the inequalities λ min ( A + B ) ≥ λ min ( A ) + λ min ( B ) and λ max ( A + B ) ≤ λ max ( A ) + λ max ( B ) , where λ min ( A ) = min i λ i ( A ) and λ max ( A ) = max i λ i ( A ) are the smallest and largest eigenvalues of A . Thus,to prove the theorem it is enough to bound the spectral radius of the fluctuation L (˜ γ ) and apply the aboveinequalities with A = QL ( E ) and B = L (˜ γ ).To bound the fluctuation recall that the square of the Hilbert–Schmidt norm k · k HS of a matrix is thesum of the squares of its eigenvalues, that is(1.8) k L (˜ γ ) k HS = N X i =1 λ j . Also note that(1.9) λ = 0 and Tr( L (˜ γ )) = 2 X i>j ˜ γ ij = N X i =1 λ i = 0 . Maximizing | λ i | subject to the constraints (1.8) and (1.9) we havemax i | λ i | ≤ r N − N − k L (˜ γ ) k HS . Next we express the Hilbert–Schmidt norm as a quadratic form in the edge-weights ˜ γ ij :(1.10) k L (˜ γ ) k HS = X i λ i ( L (˜ γ )) = 2 X i 2. Examples. In this section we present examples of Theorem 1.2 applied to the complete graph on N vertices, the Erd˝os–R´enyi random graph in the critical and supercritical scaling regime, the cyclic graph,and random d -regular graphs. Recall that L ( γ ) is positive semi-definite whenever P and Q satisfy (1.5)or (1.6). In each example except the cyclic graph, the quantity ( λ G ) /µ > Note that the complete graph is “universal”; since any graph on N vertices isa subgraph of K N the complete graph inequality applies to any graph, although one can expect to do betterwith an inequality that includes information about the topology of the graph in question. The example ofthe complete graph is also interesting in that the upper and lower bounds are actually attained — while itis clear that each inequality in the derivation of Theorem 1.2 is tight it is not immediately clear that thereis a single example for which all of the inequalities are extremized.For the complete graph the mean, QL ( E ) is a constant multiple of the orthogonal projection onto the N − , , , . . . , ⊥ . It is easy to see that the eigenvalues of QL ( E ) are given by0, with multiplicity 1, and N Q , with multiplicity N − 1. It is also noteworthy that in the case where theunderlying topology is the complete graph the mean QL ( E ) commutes with every combinatorial Laplacian,and thus with the fluctuation L (˜ γ ).The line graph of the complete graph K N is the Johnson J N, . Using the fact that K N is regular ofdegree N − QN − r E ( P − Q )(2( N − 1) + 2 − N ) N − N − ≤ λ i ( L ( γ )) ≤ QN + r E ( P − Q )(2( N − 1) + 2 − N ) N − N − , or equivalently, N (cid:18) Q − r N − p P − Q (cid:19) ≤ λ i ( L ( γ )) ≤ N (cid:18) Q + r N − p P − Q (cid:19) . OSITIVITY FOR SIGNED GRAPH LAPLACIANS L ( γ ) is a graph Laplacian with weights given by γ ∈ R E , then(2.1) N (cid:18) Q − √ N − p P − Q (cid:19) ≤ λ i ( L ( γ )) ≤ N (cid:18) Q + √ N − p P − Q (cid:19) so the current inequality improves on the elementary estimate by roughly a factor of √ N .Furthermore an example in the paper of Agbanusi and Bronski [22] shows that the current inequality issharp: there exist explicit Laplace matrices for which the upper and lower limits are achieved. Consider an Erd˝os–R´enyi random graph in the critical regime,where the edge probability is p = p log NN with p > λ ∼ a ( p ) p log N + O ( p log N ) , as N → ∞ , where a ( p ) ∈ (0 , 1) is defined to be the solution to p − ap (1 − log( a )) . The inequality holds in thesense that (cid:12)(cid:12)(cid:12)(cid:12) λ N p − a ( p ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C N p is true with probability at least 1 − C exp {− C √ N p } for some constants C , C , C .We are not aware of any precise results for µ , but it is fairly easy to get (probabilistic) upper boundssince (1.4) says that µ ≤ d max + 2. It follows a union bound argument (see Appendix A) that there is aconstant C > P (cid:16) max i deg( v i ) ≤ Cp log N (cid:17) ≥ − C N β ( C ) where β ( C ) = 2 − p − Cp log C + Cp . We can choose any C such that β ( C ) < 0. For simplicity if we take C = 4 we have that β ≈ − . p . Since this is a random graph we also need an estimate of E , the total number of edges. Since the edgesare independent this essentially follows from the central limit theorem, and we have that E = p ( N − log N + o ( N / ǫ ) for each ǫ > nonzero eigenvaluessatisfy the lower bound λ i ≥ p log( N ) (cid:18) a ( p ) Q (cid:0) o (1) (cid:1) − q N − (cid:0) o (1) (cid:1)p P − Q (cid:19) with probability tending to 1 as N → ∞ .The upper bound follows similarly — we are not aware of any result on the precise distribution of thelargest eigenvalue of the Laplacian of an Erd˝os–R´enyi graph, but the largest eigenvalue is obviously less thantwice the largest degree of the graph, giving λ i ≤ p log N (cid:18) Q (cid:0) o (1) (cid:1) + q N − (cid:0) o (1) (cid:1)p P − Q (cid:19) . Thus for an Erd˝os–R´enyi graph in the critical scaling regime the Laplacian is (with probability tendingto 1) positive definite if the inequality Q > N − a ( p ) ( P − Q ) . Note that with high probability the number of edges will be p N log N so the above estimate is asymptoticallybetter than the naive estimate (1.6) by a factor of log N . The constant in the above is obviously not sharp,as we have used a crude estimate on the largest eigenvalue of the adjacency matrix, and moreover no use hasbeen made of the constraint that ˜ γ is mean zero. We do, however, expect that the scaling with N is tight. I. AGBANUSI, J.C. BRONSKI, AND D. KIELTY Now we consider the Erd˝os–R´enyi graphs in the supercrit-ical regime with fixed edge probability p ∈ (0 , p ≥ p log( N ) /N for large N we havethat the graph is connected almost surely. Moreover, in this regime the average degree of a vertex is pN . Infact, a similar calculation to the one in Appendix A shows that P (cid:18) max i deg( v i ) ≤ (1 + N − / ǫ ) pN (cid:19) → , as N → ∞ , for ǫ ∈ (0 , / µ ≤ N − / ǫ ) pN + 2 and λ GN ≤ N − / ǫ ) pN, with probability tending to 1 as N → ∞ . By Theorem 2 in [24], for each ǫ > λ G = pN + o ( N + ǫ ) , as N → ∞ . The number of edges is E = pN ( N − / o ( N ǫ ) as N → ∞ . Applying these bounds in thenon-regular case of Theorem 1.2 we have pN (cid:18) Q (1+ o (1)) − q ( N − (cid:0) o (1) (cid:1)p P − Q (cid:19) ≤ λ i ≤ pN (cid:18) Q (cid:0) o (1) (cid:1) + q ( N − (cid:0) o (1) (cid:1)p P − Q (cid:19) with probability tending to 1 as N → ∞ , implying positivity when Q & N ( P − Q ) . Notice that when we take p = 1 we recover the non-sharp bounds for the complete graph topology in (2.1)for large N . This is again due to the fact that we do not employ the constraint that ˜ γ has mean zero. Thesame comments that were made for the critical case apply here as well — the constants can be improvedbut we believe the scaling to be optimal. For the Cyclic graph on N vertices, the graph Laplacian with all edge weightsequal to 1 is twice the identity plus the circulant matrix generated by the vector c = (0 , − , , . . . , , − λ G = 2(1 − cos(2 π/N )) and λ GN = 2, respectively. Since theCyclic graph has degree 2 we have the bounds µ = 6 − λ G . Putting all this together we find that2 (cid:18) Q (1 − cos(2 π/N )) − √ r N N − N − p P − Q (cid:19) ≤ λ i ≤ (cid:18) Q + √ r N N − N − p P − Q (cid:19) . In this example the naive inequality (1.6) on P and Q is actually the stronger one since ( λ G ) /µ ≤ (1 − cos(2 π/N )) / < N . This is in contrast to each of the other examples where( λ G ) /µ > N . d -regular graphs. Consider the probability space consisting of d -regular graphs ( d ≥ N vertices with the uniform probability measure. Work of Freidman [25] implies that in this setting λ G ≥ d − √ d − o (1) , as N → ∞ , with high probability. Applying the above inequality and that λ GN ≤ d to the bound for d -regular graphsin Theorem 1.2 we have that λ i ≥ d (cid:18) Q (cid:0) − d − / + o (1) (cid:1) − r N (cid:0) d − / + o (1) (cid:1)p P − Q (cid:19) , and λ i ≤ d (cid:18) Q + r N (cid:0) d − / + o (1) (cid:1)p P − Q (cid:19) . OSITIVITY FOR SIGNED GRAPH LAPLACIANS µ = 2 d + 2 − λ G for d -regular graphs we have(2.2) ( λ G ) µ ≥ d − d √ d − d − 1) + o (1) d + 2 √ d − o (1) , as N → ∞ , with high probability. Thus, for large d the right side of (2.2) is roughly of size d , and in particular,( λ G ) /µ > N → ∞ . 3. Concluding Remarks. In this paper we have derived bounds on the largest and smallest eigenvaluesof a graph Laplacian in terms of the mean and variance of the edge weights and the second largest eigenvalueof the equally weighted graph Laplacian. These inequalities are tight in the case of the complete graphtopology — there exist edge weightings which attain both the upper and the lower bounds.There are a couple of ways in which it might be interesting to extend these results. Firstly while thebounds are tight for the complete graph it is unlikely that this is the case for most graph topologies. In thecourse of the proof we use the inequality λ min ( A + B ) ≥ λ min ( A ) + λ min ( B ) , where A is the equally weighted Laplacian and B is the fluctuation. In the case of the complete graphtopology the equally weighted Laplacian is the identity on mean zero vectors, A and B commute, andthis inequality is actually an equality. This is not true for other underlying graph topologies. It wouldbe interesting to explore the extent to which this inequality fails to be tight for topologies other than thecomplete graph topology.A second question concerns the quantity µ , which is related to the maximum of a Rayleigh quotient forthe adjacency of the line graph over mean zero vectors in R E . In the regular case we can, via a dualityargument, compute µ in terms of the second largest eigenvalue of the graph Laplacian. For the non-regularcase, we only bound µ in terms of the maximum degree, which does not exploit the mean zero condition atall. It would be interesting to develop a bound on µ in the non-regular case that exploits the mean zerocondition. Acknowledgements: J.C.B. would like to acknowledge support from the National Science Foundationunder grant NSF-DMS 1615418. D.K. would like to acknowledge support from the University of IllinoisCampus Research Board award RB19045 (to Richard Laugesen) and the U.S. Department of Educationthrough the Graduate Assistance in Areas of National Need (GAANN) program. Appendix A. Upper bound on the maximum degree. Proposition A.1. Suppose that G is an Erd˝os–R´enyi random graph where each possible edge is presentwith probability p = p log NN with p > . The probability that max i deg( v i ) ≤ Cp log( N ) tends to algebraically as N → ∞ for C large enough. Choosing C ≥ is sufficient.Proof. Observe that since P (cid:18) max i deg( v i ) ≤ Cp log( N ) (cid:19) = 1 − P (cid:18) deg( v i ) > Cp log( N ) for some i (cid:19) , by a union bound it suffices to show that N P (cid:18) deg( v i ) > Cp log( N ) (cid:19) → , as N → ∞ . Recall that deg( v i ) follows a binomial distribution so that for K ∈ N we have P (deg( v i ) > K ) = N − X j = K (cid:18) N − j (cid:19) p j (1 − p ) N − − j ≤ N X j = K (cid:18) Nj (cid:19) p j (1 − p ) N − j . I. AGBANUSI, J.C. BRONSKI, AND D. KIELTY The inequality follows since the probability that an event occurs in N trials is larger than the probabilitythat it occurs in N − K ≥ K max — the K for which the maximum of K (cid:0) NK (cid:1) p K (1 − p ) N − K is achieved. In this case P (deg( v i ) > K ) ≤ N (cid:18) NK (cid:19) p K (1 − p ) N − K . Now we apply Sterlings approximation to estimate (cid:0) NK (cid:1) to find that P (deg( v i ) > K ) . N N N e − N √ πNK K e − K √ πK ( N − K ) N − K e − ( N − K ) p π ( N − K ) p K (1 − p ) N − K . After regrouping and some elementary estimates we have P (deg( v i ) > K ) . N s NK ( N − K ) (cid:18) N pK (cid:19) K (cid:18) N (1 − p ) N − K (cid:19) N − K . Now we choose K = CN p for C > K ≥ K max for large N since K max ≤ ⌊ ( N + 1) p ⌋ . It followsthat P (deg( v i ) > K ) . N (cid:18) C (cid:19) Cp log( N ) (cid:18) − p log( N ) N − Cp log( N ) N (cid:19) N − K . N (cid:18) C (cid:19) Cp log( N ) (cid:18) − p log( N ) N − Cp log( N ) N (cid:19) N Using that (1 − p log( N ) /N ) N = e N log(1 − p log( N ) /N ) and Taylor expanding the outer logarithm we have P (deg( v i ) > K ) . N (cid:18) C (cid:19) Cp log( N ) e − p log( N ) e − Cp log( N ) = N N − Cp log( C ) N − p N Cp . Combining the exponents shows that N P (cid:18) deg( v i ) > Cp log( N ) (cid:19) . N − C log( C ) p + Cp − p . Since p > C > . C = 4 gives P (cid:18) max i deg( v i ) > p log( N ) (cid:19) . N − . p . REFERENCES[1] J. Kunegis, S. Schmidt, A. Lommatzsch, J. Lerner, E. W. De Luca, and S. Albayrak. Spectral analysis of signed graphs forclustering, prediction and visualization. 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