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Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of low-degree dependencies such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants $kge 3$ and $lambda > 0$, an ErdH os--Renyi random graph $Gsimmathbb{G}(n,lambda/n)$ with $n$ vertices and edge probability $lambda/n$ typically has the property that its $k$-core (its largest subgraph with minimum degree at least $k$) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for extremely sparse random matrices with density $O(1/n)$. A key aspect of our proof is a technique to extract high-degree vertices and use them to boost the rank, starting from approximate rank bounds obtainable from (non-quantitative) spectral convergence machinery due to Bordenave, Lelarge and Salez.
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Let $xi$ be a non-constant real-valued random variable with finite support, and let $M_{n}(xi)$ denote an $ntimes n$ random matrix with entries that are independent copies of $xi$. For $xi$ which is not uniform on its support, we show that begin{align*} mathbb{P}[M_{n}(xi)text{ is singular}] &= mathbb{P}[text{zero row or column}] + (1+o_n(1))mathbb{P}[text{two equal (up to sign) rows or columns}], end{align*} thereby confirming a folklore conjecture. As special cases, we obtain: (1) For $xi = text{Bernoulli}(p)$ with fixed $p in (0,1/2)$, [mathbb{P}[M_{n}(xi)text{ is singular}] = 2n(1-p)^{n} + (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n},] which determines the singularity probability to two asymptotic terms. Previously, no result of such precision was available in the study of the singularity of random matrices. (2) For $xi = text{Bernoulli}(p)$ with fixed $p in (1/2,1)$, [mathbb{P}[M_{n}(xi)text{ is singular}] = (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n}.] Previously, only the much weaker upper bound of $(sqrt{p} + o_n(1))^{n}$ was known due to the work of Bourgain-Vu-Wood. For $xi$ which is uniform on its support: (1) We show that begin{align*} mathbb{P}[M_{n}(xi)text{ is singular}] &= (1+o_n(1))^{n}mathbb{P}[text{two rows or columns are equal}]. end{align*} (2) Perhaps more importantly, we provide a sharp analysis of the contribution of the `compressible part of the unit sphere to the lower tail of the smallest singular value of $M_{n}(xi)$.
We determine the size of $k$-core in a large class of dense graph sequences. Let $G_n$ be a sequence of undirected, $n$-vertex graphs with edge weights ${a^n_{i,j}}_{i,j in [n]}$ that converges to a kernel $W:[0,1]^2to [0,+infty)$ in the cut metric. Keeping an edge $(i,j)$ of $G_n$ with probability $min { {a^n_{i,j}}/{n},1 }$ independently, we obtain a sequence of random graphs $G_n(frac{1}{n})$. Denote by $C_k(G)$ the size of $k$-core in graph $G$, by $X^W$ the branching process associated with the kernel $W$, by $mathcal{A}$ the property of a branching process that the initial particle has at least $k$ children, each of which has at least $k-1$ children, each of which has at least $k-1$ children, and so on. Using branching process and theory of dense graph limits, under mild assumptions we obtain the size of $k$-core of random graphs $G_n(frac{1}{n})$, begin{align*} C_kleft(G_nleft(frac{1}{n}right)right) =n mathbb{P}_{X^W}left(mathcal{A}right) +o_p(n). end{align*} Our result can also be used to obtain the threshold of appearance of a $k$-core of order $n$. In addition, we obtain a probabilistic result concerning cut-norm and branching process which might be of independent interest.
We study random walks on the giant component of the ErdH{o}s-Renyi random graph ${cal G}(n,p)$ where $p=lambda/n$ for $lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $( u {bf d})^{-1}log n pm (log n)^{1/2+o(1)}$, where $ u$ and ${bf d}$ are the speed of random walk and dimension of harmonic measure on a ${rm Poisson}(lambda)$-Galton-Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the non-backtracking random walk.
A wide array of random graph models have been postulated to understand properties of observed networks. Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges. It is conjectured that for a large class of models, the nature of this emergence is similar to that of the ErdH{o}s-Renyi random graph, in the sense that (a) the sizes of the maximal components in the critical regime scale like $n^{2/3}$, and (b) the structure of the maximal components at criticality (rescaled by $n^{-1/3}$) converges to random fractals. To date, (a) has been proven for a number of models using different techniques. This paper develops a general program for proving (b) that requires three ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent, (ii) scaling exponents of susceptibility functions are the same as that of the ErdH{o}s-Renyi random graph, and (iii) macroscopic averaging of distances between vertices in the barely subcritical regime. We show that these apply to two fundamental random graph models: the configuration model and inhomogeneous random graphs with a finite ground space. For these models, we also obtain new results for component sizes at criticality and structural properties in the barely subcritical regime.
We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As applications, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph.
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