aa r X i v : . [ m a t h . C O ] F e b On the skew-spectral distribution ofrandomly oriented graphs
Yilun ShangSchool of Mathematical SciencesTongji University, Shanghai 200092, Chinae-mail: [email protected]
Abstract
The randomly oriented graph G σn,p is an Erd˝os-R´enyi randomgraph G n,p with a random orientation σ , which assigns to each edgea direction so that G σn,p becomes a directed graph. Denote by S n the skew-adjacency matrix of G σn,p . Under some mild assumptions,it is proved in this paper that, the spectral distribution of S n (undersome normalization) converges to the standard semicircular law al-most surely as n → ∞ . It is worth mentioning that our result doesnot require finite moments of the entries of the underlying randommatrix. MSC 2010:
Keywords:
Oriented graph, random matrix, semicircular law
Let G be a simple graph with vertex set V = { v , v , · · · , v n } and G σ be anoriented graph of G with the orientation σ , which assigns to each edge of G adirection so that G σ becomes a directed graph. The skew-adjacency matrix S ( G σ ) = ( s ij ) ∈ R n × n is a real skew-symmetric matrix, where s ij = 1 and s ji = − v i , v j ) is an arc of G σ , otherwise s ij = s ji = 0. The well-knownErd˝os-R´enyi random graph model G n,p is a probability space [6], whichconsists of all simple graphs with vertex set V where each of the possible (cid:0) n (cid:1) = n ( n − / p = p ( n ). Fora random graph G n,p ∈ G n,p , the randomly oriented graph G σn,p is obtainedby orienting every edge { v i , v j } ( i < j ) in G n,p as ( v i , v j ) with probability q = q ( n ) and the other way with probability 1 − q independently of eachother. Here, the superscript σ = σ ( q ) indicates the orientation.The above randomly oriented graph model was first studied in [17] and asimilar model based on the lattice structure (instead of G n,p ) was discussedin [13]. The question of whether the existences of directed paths between1arious pairs of vertices are positively or negatively correlated has attractedsome research attention recently; see e.g. [1, 2, 15]. Diclique structure hasbeen studied in [20]. In this paper, we shall explore this model from a spec-tral perspective. Basically, we determine the limit spectral distribution ofthe random matrix underlying the randomly oriented graph. A semicir-cular law reminiscent of Wigner’s famous semicircular law [23] is obtainedby the moment approach (see Theorem 1 below). We mention that thereis recent increased interest in the spectral properties of oriented graphs inclassical graph theory, see e.g. [8, 10, 14, 19].As is customary, we say that a graph property P holds almost surely(a.s., for short) for G n,p if the probability that G n,p ∈ G n,p has the prop-erty P tends to one as n → ∞ . We will also use the standard Landau’sasymptotic notations such as o, O, ∼ etc. Let E be the indicator of theevent E and i = √− In this section, we characterize the spectral properties for the skew-adjacencymatrices of randomly oriented graphs.Recall that a square matrix M = ( m ij ) is said to be skew-symmetric if m ij = − m ji for all i and j . It is evident that the skew-adjacency matrix S n := S ( G σn,p ) = ( s ij ) ∈ R n × n of the randomly oriented graph G σn,p isa skew-symmetric random matrix such that the upper-triangular elements s ij ( i < j ) are i.i.d. random variables satisfyingP( s ij = 1) = pq, P( s ij = −
1) = p (1 − q ) and P( s ij = 0) = 1 − p. Hence, the eigenvalues of S n are all purely imaginary numbers. We assumethe eigenvalues are i λ , i λ , · · · , i λ n , where all λ i ∈ R .Let Y n ∈ R n × n be a skew-symmetric matrix whose elements above thediagonal are 1 and those below the diagonal are −
1. Define a quantity r = r ( p, q ) = p (1 + p (1 − q )) pq + (1 − p (1 − q )) p (1 − q )and a normalized matrix X n = − i S n − i p (1 − q ) Y n r . (1)It is straightforward to check that X n = ( x ij ) ∈ C n × n is a Hermitian matrixwith the diagonal elements x ii = 0 and the upper-triangular elements x ij ( i < j ) being i.i.d. random variables satisfying mean E( x ij ) = 0 andvariance Var( x ij ) = E( x ij x ij ) = 1. 2n general, for a Hermitian matrix M ∈ C n × n with eigenvalues µ ( M ), µ ( M ), · · · , µ n ( M ), the empirical spectral distribution of M is defined by F M ( x ) = 1 n · { µ i ( M ) | µ i ( M ) ≤ x, i = 1 , , · · · , n } , where {· · · } means the cardinality of a set. Theorem 1.
Suppose that nr → ∞ and p (1 − q ) → as n → ∞ .Then lim n →∞ F n − / X n ( x ) = F ( x ) a.s. i.e., with probability 1, the empirical spectral distribution of the matrix n − / X n converges weakly to a distribution F ( x ) as n tends to infinity,where F ( x ) has the density f ( x ) = 12 π p − x | x |≤ . Before presenting the proof of Theorem 1, we first give a couple ofremarks.
Remark 1.
The above function F ( x ) follows the standard semicirculardistribution according to Wigner. However, Theorem 1 extends the clas-sical result of Wigner [23]. To see this, set q = 1 /
2. The assumptionsin Theorem 1 reduce to np → ∞ . It is easy to check that r = √ p and | E( x k +212 ) | = 1 /p k/ if k is even. Hence, if p = o (1), the condition inWigner’s semicircular law that E( | x | k ) < ∞ for any k ∈ N is violated(see e.g. [9, 23]). In the more recent study of spectral convergence resultsfor Hermitian random matrices, it is common to assume finite lower-ordermoments (e.g. fourth-order or eighth-order moments) of the elements ofthe underlying matrices [4, 5, 7, 11, 12, 18]. Therefore, our result does notfit in these frames either. Remark 2.
Apart from Theorem 1, we can also derive an estimate for theeigenvalues i λ , i λ , · · · , i λ n of the matrix S n . Note that the eigenvaluesof Y n are µ i ( Y n ) = i cot( π (2 i − / n ) for i = 1 , , · · · , n . It follows fromTheorem 2.12 in [3] that ρ ( n − / X n ) → ρ ( · ) stands for thespectral radius. By (1), we have − i S n r √ n = X n √ n + i p (1 − q ) Y n r √ n . If we arrange the eigenvalues of a Hermitian matrix M ∈ C n × n as ˆ µ ( M ) ≥ ˆ µ ( M ) ≥ · · · ≥ ˆ µ n ( M ), then the Weyl’s inequality [22] implies that for all3 = 1 , , · · · , n ,ˆ µ n (cid:18) X n √ n (cid:19) + ˆ µ i (cid:18) i p (1 − q ) Y n r √ n (cid:19) ≤ ˆ µ i (cid:18) − i S n r √ n (cid:19) ≤ ˆ µ (cid:18) X n √ n (cid:19) + ˆ µ i (cid:18) i p (1 − q ) Y n r √ n (cid:19) . Putting the above comments together, we obtain r √ n (cid:18) − p (2 q −
1) cot (cid:18) π (2 i − n (cid:19) + o (1) (cid:19) ≤ ˆ µ i ( − i S n ) ≤ r √ n (cid:18) p (2 q −
1) cot (cid:18) π (2 i − n (cid:19) + o (1) (cid:19) a.s. (2)when q ≥ /
2, and r √ n (cid:18) − p (2 q −
1) cot (cid:18) π (2 n − i + 1)2 n (cid:19) + o (1) (cid:19) ≤ ˆ µ i ( − i S n ) ≤ r √ n (cid:18) p (2 q −
1) cot (cid:18) π (2 n − i + 1)2 n (cid:19) + o (1) (cid:19) a.s. (3)when q < /
2. Since ˆ µ i ( − i S n ) is the i -th largest values in the collection { λ , λ , · · · , λ n } by our notation, the estimates for the eigenvalues of S n readily follow from (2) and (3).Now comes the proof of Theorem 1. Proof of Theorem 1.
By the moment approach, it suffices to showthat the moments of the empirical spectral distribution converge to thecorresponding moments of the semicircular law almost surely (see e.g. [3]).That is, lim n →∞ Z x k dF n − / X n ( x ) = Z x k dF ( x ) a.s. (4)for each k ∈ N .First note that under the assumptions of Theorem 1, it can be checkedthat E( x k ) ∼ r k − k ≡
0( mod 4) − i r k − k ≡
1( mod 4) − r k − k ≡
2( mod 4) i r k − k ≡
3( mod 4) (5)for any k ∈ N and k >
1. Recall that x ij (1 ≤ i < j ≤ n ) are independentlyand identically distributed as x , and x ij = − x ji for all i and j . To show(4), we consider the following two scenarios according to whether k is oddor even. 4 A) k is odd. Fix a k = 2 t + 1 with t ∈ N ∪ { } . By symmetry, wehave R − x k f ( x ) dx = 0. On the other hand, the integral on the left-handside of (4) yields Z x k dF n − / X n ( x ) = 1 n E (cid:18) Trace (cid:18) X kn √ n k (cid:19)(cid:19) = 1 n k/ E(Trace( X kn ))= 1 n k/ X ≤ i ,i , ··· ,i k ≤ n E( x i i x i i · · · x i k i ) , (6)where each summand in (6) can be viewed as a closed walk of length k following the arcs ( v i , v i ) , ( v i , v i ) , · · · , ( v i k , v i ) in the complete graph G = K n of order n . Clearly, each such walk contains an edge, say { v i , v j } ,that the total number n ij of times that arcs ( v i , v j ) and ( v j , v i ) are traveledduring this walk is odd. Given a closed walk of length k , denote by Ω theset of edges in it as described above. Thus, we have Ω = ∅ . Now considerthe following two cases: (A1) there exists { v i , v j } ∈ Ω such that n ij = 1;(A2) n i ′ j ′ ≥ { v i ′ , v j ′ } ∈ Ω .For (A1), note that the variables in the summands in (6) are inde-pendent and E( x ij ) = 0. Therefore, such walks contribute zero to theright-hand side of (6).For (A2), let m denote the number of distinct vertices in a closed walk oflength k . Hence, m is less than or equal to the number of distinct vertices ina closed walk of length 2 t , in which each edge (in either direction) appearseven times. Clearly, m ≤ t + 1 (the equality m = t + 1 is attained by a walkin which each arc and the one of opposite direction are traveled exactlyonce, respectively, and all edges in the walk form a tree). Therefore, thesewalks will contribute1 n k/ t +1 X m =1 X { i ,i , ··· ,i k } = m | E( x i i x i i · · · x i k i ) |≤ n / t t +1 X m =1 n m m k (cid:18) r (cid:19) k − m − ≤ n / t ( t + 1) n t +1 ( t + 1) k (cid:18) r (cid:19) t +1 − t = ( t + 1) k +1 n / r , where the first inequality is due to (5), (6) and the fact that the numberof such closed walks is at most n m m k . Consequently, combining (A1) and(A2), it follows from (6) that Z x k dF n − / X n ( x ) = O (cid:18) n / r (cid:19) → n → ∞ , by our assumptions. We complete the proof of (4) for odd k . (B) k is even. Fix a k = 2 t with t ∈ N ∪ { } . We have Z − x k f ( x ) dx = 12 π Z − x k p − x dx = 1 π Z x t p − x dx = 2 t +1 π Z y t − / (1 − y ) / dy = 2 t +1 π · Γ( t + 1 / / t + 2)= 1 t + 1 (cid:18) tt (cid:19) . (7)Given a closed walk of length k in K n , we still set m as the number ofdistinct vertices in it. To analyze the terms in (6), we consider the followingthree cases: (B1) there exists an edge, say { v i , v j } , in the closed walk suchthat the total number of times that arcs ( v i , v j ) and ( v j , v i ) are traveledduring this walk is odd; (B2) no such { v i , v j } exists, and m ≤ t ; (B3)no such { v i , v j } exists, and m = t + 1. Note that if each edge (in eitherdirection) of the closed walk appears even times, we have m ≤ t + 1. Theequality holds if and only if each arc and the one of opposite direction aretraveled exactly once, respectively, and all edges in the walk form a tree.For (B1), we argue similarly as in (A1) and know that the contributionto the right-hand side of (6) is zero.For (B2), an analogous derivation as in (A2) reveals that the contribu-tion to the right-hand side of (6) amounts to1 n k/ t X m =1 X { i ,i , ··· ,i k } = m | E( x i i x i i · · · x i k i ) |≤ n t t X m =1 n m m k (cid:18) r (cid:19) k − m − ≤ n t · t · n t · t k · (cid:18) r (cid:19) t − t − = t k +1 nr . For (B3), noting that E( x x ) = − E( x ) = 1 and the independenceof the variables, we obtain that each term E( x i i x i i · · · x i k i ) in (6) equals1. Recall that a combinatorial result [5, Lemma 2.4] says that the numberof the closed walks of length 2 t on t + 1 vertices, which satisfy that eacheach arc and the one of opposite direction both appear exactly once, and6ll edges in the walk form a tree, equals t +1 (cid:0) tt (cid:1) ( t + 1)!. Since there are (cid:0) nt +1 (cid:1) choices of a set of t + 1 vertices, we conclude that the contribution tothe left-hand side of (6) amounts to1 n k/ · t + 1 (cid:18) tt (cid:19) ( t + 1)! · (cid:18) nt + 1 (cid:19) = n ( n − · · · ( n − t ) n t · t + 1 (cid:18) tt (cid:19) . Finally, combining (B1), (B2) and (B3), it follows from (6) that Z x k dF n − / X n ( x ) = O (cid:18) nr (cid:19) + n ( n − · · · ( n − t ) n t · t + 1 (cid:18) tt (cid:19) → t + 1 (cid:18) tt (cid:19) , as n → ∞ , by our assumptions. In view of (7), the proof of (4) for even k is complete. ✷ To conclude the paper, we simulate the randomly oriented graph modeland computed the eigenvalue distribution for the matrix n − / X n (see Fig-ure 1). The simulation results show a perfect agreement with our theoreticalprediction. For future work, it would be interesting to explore some otherproperties (see e.g. [16, 21]) in the setting of randomly oriented graphs. −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.500.050.10.150.20.250.30.35 F r equen cy Eigenvalues of n −1/2 X n (a) −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.500.050.10.150.20.250.30.35 S pe c t r a l den s i t y Eigenvalues of n −1/2 X n (b) Figure 1: Limiting skew-spectral distribution for G σ ( q ) n,p with n = 1000, p = 0 . q = 0 .
5. (a) Histogram of the spectrum of n − / X n . A solidline shows the semicircular distribution for comparison. (b) The averagespectral density for n − / X n over 500 graphs. Acknowledgements
The author is indebted to the referees for helpful comments. The work issupported by the National Natural Science Foundation of China (11505127)7nd the Shanghai Pujiang Program (15PJ1408300).
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