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On the Second Eigenvalue of Random Bipartite Biregular Graphs

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On the Second Eigenvalue of Random Bipartite Biregular Graphs ON THE SECOND EIGENVALUE OF RANDOM BIPARTITE BIREGULAR GRAPHS YIZHE ZHU Abstract. We consider the spectral gap of a uniformly chosen random (d1; d2)-biregular bipartite graph G with jV j = n; jV j = m, where d ; d could possibly grow with n and m. Let A be the adjacency matrix 1 2 1 2 p 2=3 of G. Under the assumption that d1 ≥ d2 and d2 = O(n ); we show that λ2(A) = O( d1) with high probability. As a corollary, combining the results from [47],p we confirm a conjecture in [14] that the second singular value of a uniform random d-regular digraph is O( d) for 1 ≤ d ≤ n=2 with highp probability. This also implies that the second eigenvalue of a uniform random d-regular digraph is O( d) for 1 ≤ d ≤ n=2 with high probability. 2 Assuming d2 = O(1) and d1 = O(n ), we further prove that for a random (d1; d2)-biregular bipartite 2 p graph, jλi (A) − d1j = O( d1(d2 − 1)) for all 2 ≤ i ≤ n + m − 1 with high probability. The proofs of the two results are based on the size biased coupling method introduced in [13] for random d-regular graphs and several new switching operations we defined for random bipartite biregular graphs. 1. Introduction An expander graph is a sparse graph that has strong connectivity properties and exhibits rapid mixing. Expander graphs play an important role in computer science including sampling, complexity theory, and the design of error-correcting codes (see [27]). When a graph is d-regular, i.e., each vertex has degree d, quantification of expansion is possible based on the eigenvalues of the adjacency matrix. Let A be the adjacency matrix of a d-regular graph. The first eigenvalue λ1(A) is always d. The second eigenvalue in absolute value λ(A) = maxfλ2(A); −λn(A)g is of particular interest, since the difference between d and λ, also known as the spectral gap, provides an estimate on the expansion property of the graph. The study of the spectral gap in d-regular graphs with fixed d had the first breakthroughp in the Alon- Boppana bound. Alon and Boppanap in [1, 44] showed that for d-regular graph λ(A) ≥ 2 d − 1 − o(1): Regular graphs with λ(A) ≤ 2 d − 1 are calledpRamanujan. In [22], Friedman proved Alon's conjecture in [1] that for any fixed d ≥ 3 and " > 0, λ(A) ≤ 2 d − 1 + " asymptotically almost surely. The result implies almost allp random regular graphs are nearly Ramanujan. Recently, Bordenave [7] gave a simpler proof that λ(A) ≤ 2 d1 + "n for a sequence "n ! 0 asymptotically almost surely. A generalization of Alon's question is to consider the spectral gap of random d-regular graphs when d 1=2 grows with n. In [9]p the authors showed that for d = o(n ), a uniformly distributed random d-regular graph satisfies λ(A) = O( d) with high probability. The authors worked with random regular multigraphs drawn from the configuration model and translated the result for the uniform modelp by the contiguity argument, 1=2 2=3 which hit a barrier at d = o(n ). The range of d for the bound λ(A) = O( d) was extended to d = O(np ) in [13] by proving concentration results directly for the uniform model. In [47] it was proved that the O( d) bound holds for n" ≤ d ≤ n=2 with " 2 (0; 1). Vu in [50, 51] conjectured that λ(A) = (2 + o(1))pd(1 − d=n) arXiv:2005.08103v3 [math.PR] 24 Oct 2020 with high probability when d ≤ n=2 and d tends to infinity with n. The combination of the results in [13] and [47] confirms Vu's conjecture up to a multiplicative constant.p Very recently, the authors in [4] showed that for n" ≤ d ≤ n2=3−" for any " > 0, λ(A) = (2 + o(1)) d − 1 with high probability, which proves Vu's conjecture in this regime with the exact multiplicative constant. 1.1. Random bipartite biregular graphs. In many applications, one would like to construct bipartite expander graphs with two unbalanced disjoint vertex sets, among which bipartite biregular graphs are of particular interest. An (n; m; d1; d2)-bipartite biregular graph is a bipartite graph G = (V1;V2;E) where jV1j = n; jV2j = m and every vertex in V1 has degree d1 and every vertex in V2 has degree d2. Note that we Date: October 27, 2020. 2000 Mathematics Subject Classification. Primary 60C05, 60B20; Secondary 05C50. Key words and phrases. random bipartite biregular graph, spectral gap, switching, size biased coupling. 1 must have nd1 = md2 = jEj. When the number of vertices is clear, we call it a (d1; d2)-biregular bipartite graph for simplicity. n×m Let X 2 f0; 1g be a matrix indexed by V1 × V2 such that Xij = 1 if and only if (i; j) 2 E. The adjacency matrix of a (d1; d2)-biregular bipartite graph with V1 = [n];V2 = [m] can be written as 0 X (1.1) A = : X> 0 All eigenvalues of A come in pair as {−λ, λg, where jλj pis a singular value of X along withp at least jn − mj zero eigenvalues. It's easy to see λ1(A) = −λn+m(A) = d1d2. The difference between d1d2 and λ2(A) is called the spectral gap for the bipartite biregular graph. The spectral gap of bipartite biregular graphs has found applications in error correcting codes, matrix completion and community detection, see for example [46, 45, 25,8, 10]. Previous works of [21, 34] showed an analog of Alon-Boppana bound for bipartite biregular graph: for any sequence of (d1; d2)-biregular bipartite graphs with fixed d1 and d2, as the number of vertices tends to infinity, p p lim inf λ2 ≥ d1 − 1 + d2 − 1 − " n!1 p pfor any " > 0. In [21], a (d1; d2)-biregular bipartite graph is defined to be Ramanujan if λ2 ≤ d1 − 1 + d2 − 1. Marcus, Spielman and Srivastava in [40] obtained a breakthrough showing that there exist infinite families of (d1; d2)-biregular bipartite Ramanujan graphs for every d1; d2 ≥ 3. Very recently, for fixed d1 and d2,[8] showed that almost all (d1; d2)-biregular bipartite graphs are almost Ramanujan in the sense that p p (1.2) λ2 ≤ d1 − 1 + d2 − 1 + "n for a sequence "n ! 0 asymptotically almost surely. 1.2. Main results. In this paper, we consider the spectral gap of a uniformly chosen random (d1; d2)- biregular bipartite graph with V1 = [n];V2 = [m], where d1; d2 can possibly grow with n and m. Without n×m loss of generality, we assume d1 ≥ d2. Let X 2 R be the biadjacency matrix of a bipartite biregular graph > m with jV1j = n; jV2j = m. Namely, Xuv = 1 if (u; v) 2 E; u 2 V1 andpv 2 V2. Let 1m = (1;:::; 1) 2 R . Since > X X1m = d1d21m, the largest singular value satisfies σ1(X) = d1d2: Moreover, the second eigenvalue of A in (1.1) is equal to σ2(X). Define ( m ) m−1 m m−1 m−1 X S : = fv 2 R ; kvk2 ≤ 1g;S0 := v 2 S ; vi = 0 : i=1 Then the second singular value of X can be written as (1.3) σ2(X) = sup kXyk2 = sup hx; Xyi: m−1 n−1 m−1 y2S0 x2S ;y2S0 To show the spectral gap of the adjacency matrix for a random bipartite biregular graph, we will work with the biadjacency matrix X and control its singular values using the formula (1.3). Our first result is about the second eigenvalue bound for the random bipartite biregular graph. This is an analog of the results in [13] for the spectral gap of random bipartite biregular graphs. Theorem 1.1. Let A be the adjacency matrix of a uniform random (n; m; d1; d2)−bipartite biregular graph 2=3 n with d1 ≥ d2. For any constants C0; K > 0, if d2 ≤ min C0n ; 2 , then there exists a constant α > 0 depending only on C0;K such that p −K −m (1.4) P λ2(A) ≤ α d1 ≥ 1 − m − e : p p Remark 1.2. The result (1.4) also implies λ2(A) ≤ α( d1 + d2) with high probability, which is an extension of (1.2) to the case where d1; d2 can possibly grow with n; m. A d-regular digraph on n vertices is a digraph with each vertex having d in-neighbors and d out-neighbors. We can interpret the biadjacency matrix X of a random (d; d; n; n)-bipartite biregular graph as the adjacency matrix of a random d-regular digraph. Therefore the following corollary holds. 2 Corollary 1.3. Let A be the adjacency matrix of a uniform random d-regular digraph on n vertices. For 2=3 any constants C0; K > 0, if d ≤ minfC0n ; n=2g, then there exists a constant α > 0 depending only on C0;K such that p −K −n (1.5) P σ2(A) ≤ α d ≥ 1 − n − e : p In [47] the authors showed σ (A) = O( d) with high probability when n" ≤ d ≤ n . Combining Corollary 2 2 p 1.3 and their result, we confirm a conjecture in [14] that a uniform d-regular digraph has σ2(A) = O( d) for 1 ≤ d ≤ n=2 with high probability. p It is also shown in [16] that for random d-regular digraph, the eigenvalue of A satisfies jλ2(A)j ≤ d + " with high probability for any " > 0.
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