Enyi Graphs Research Articles

Fully localized and partially delocalized states in the tails of Erdös-Rényi graphs in the critical regime

In this work, we study the spectral properties of the adjacency matrix of critical Erd\"os-R\'enyi (ER) graphs, i.e., when the average degree is of order $lnN$. In a series of recent inspiring papers, Alt, Ducatez, and Knowles have rigorously shown that these systems exhibit a ``semilocalized'' phase in the tails of the spectrum where the eigenvectors are exponentially localized on a subextensive set of nodes with anomalously large degree. We propose two approximate analytical strategies to analyze this regime based respectively on the simple ``rules of thumb'' for localization and ergodicity and on an approximate treatment of the self-consistent cavity equation for the resolvent. Both approaches suggest the existence of two different regimes: a fully Anderson localized phase at the spectral edges, in which the eigenvectors are localized around a unique center, and an intermediate partially delocalized but nonergodic phase, where the eigenvectors spread over many resonant localization centers. In this phase, the exponential decay of the effective tunneling amplitudes between the localization centers is counterbalanced by the large number of nodes towards which tunneling can occur, and the system exhibits minibands in the local spectrum over which the Wigner-Dyson statistics establishes. We complement these results by a detailed numerical study of the finite-size scaling behavior of several observables that supports the theoretical predictions and allows us to determine the critical properties of the two transitions. Critical ER graphs provide a pictorial representation of the Hilbert space of a generic many-body Hamiltonian with short-range interactions. In particular, we argue that their phase diagram can be mapped onto the out-of-equilibrium phase diagram of the quantum random energy model.

Banach space actions and L2-spectral gap

\.{Z}uk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap $> \frac{1}{2}$, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example an $L^p$-space with $1 < p < \infty$ or an interpolation space between a Hilbert space and an arbitrary Banach space) as soon as the Laplacian on the links has a two-sided spectral gap $>1-\varepsilon$. This criterion applies to random groups in the triangular density model for densities $> \frac{1}{3}$. In this way, we are able to generalize recent results of Dru\c{t}u and Mackay to affine isometric actions of random groups on uniformly curved Banach spaces. Also, in the setting of actions on $L^p$-spaces, our results are quantitatively stronger, even in the case $p=2$. This naturally leads to new estimates on the conformal dimension of the boundary of random groups in the triangular model. Additionally, we obtain results on the eigenvalues of the $p$-Laplacian on graphs, and on the spectrum and degree distribution of Erd\H{o}s-R\'enyi graphs.

Random geometric graphs in high dimension.

Many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbors to each point of a data set to perform their tasks. These proximity relations define a so-called geometric graph, where two nodes are linked if they are sufficiently close to each other. Random geometric graphs, where the positions of nodes are randomly generated in a subset of R^{d}, offer a null model to study typical properties of data sets and of machine learning algorithms. Up to now, most of the literature focused on the characterization of low-dimensional random geometric graphs whereas typical data sets of interest in machine learning live in high-dimensional spaces (d≫10^{2}). In this work, we consider the infinite dimensions limit of hard and soft random geometric graphs and we show how to compute the average number of subgraphs of given finite size k, e.g., the average number of k cliques. This analysis highlights that local observables display different behaviors depending on the chosen ensemble: soft random geometric graphs with continuous activation functions converge to the naive infinite-dimensional limit provided by Erdös-Rényi graphs, whereas hard random geometric graphs can show systematic deviations from it. We present numerical evidence that our analytical results, exact in infinite dimensions, provide a good approximation also for dimension d≳10.

Edge rigidity and universality of random regular graphs of intermediate degree

For random $d$-regular graphs on $N$ vertices with $1 \ll d \ll N^{2/3}$, we develop a $d^{-1/2}$ expansion of the local eigenvalue distribution about the Kesten-McKay law up to order $d^{-3}$. This result is valid up to the edge of the spectrum. It implies that the eigenvalues of such random regular graphs are more rigid than those of Erd\H{o}s-R\'enyi graphs of the same average degree. As a first application, for $1 \ll d \ll N^{2/3}$, we show that all nontrivial eigenvalues of the adjacency matrix are with very high probability bounded in absolute value by $(2 + o(1)) \sqrt{d - 1}$. As a second application, for $N^{2/9} \ll d \ll N^{1/3}$, we prove that the extremal eigenvalues are concentrated at scale $N^{-2/3}$ and their fluctuations are governed by Tracy-Widom statistics. Thus, in the same regime of $d$, $52\%$ of all $d$-regular graphs have second-largest eigenvalue strictly less than $2 \sqrt{d - 1}$.

Phase transitions in atypical systems induced by a condensation transition on graphs.

Random graphs undergo structural phase transitions that are crucial for dynamical processes and cooperative behavior of models defined on graphs. In this work we investigate the impact of a first-order structural transition on the thermodynamics of the Ising model defined on Erd\"os-R\'enyi random graphs, as well as on the eigenvalue distribution of the adjacency matrix of the same graphical model. The structural transition in question yields graph samples exhibiting condensation, characterized by a large number of nodes having degrees in a narrow interval. We show that this condensation transition induces distinct thermodynamic first-order transitions between the paramagnetic and the ferromagnetic phases of the Ising model. The condensation transition also leads to an abrupt change in the global eigenvalue statistics of the adjacency matrix, which renders the second moment of the eigenvalue distribution discontinuous. As a side result, we derive the critical line determining the percolation transition in Erd\"os-R\'enyi graph samples that feature condensation of degrees.

Graph Fourier Transform: A Stable Approximation

In Graph Signal Processing (GSP), data dependencies are represented by a graph whose nodes label the data and the edges capture dependencies among nodes. The graph is represented by a weighted adjacency matrix $A$ that, in GSP, generalizes the Discrete Signal Processing (DSP) shift operator $z^{-1}$. The (right) eigenvectors of the shift $A$ (graph spectral components) diagonalize $A$ and lead to a graph Fourier basis $F$ that provides a graph spectral representation of the graph signal. The inverse of the (matrix of the) graph Fourier basis $F$ is the Graph Fourier transform (GFT), $F^{-1}$. Often, including in real world examples, this diagonalization is numerically unstable. This paper develops an approach to compute an accurate approximation to $F$ and $F^{-1}$, while insuring their numerical stability, by means of solving a non convex optimization problem. To address the non-convexity, we propose an algorithm, the stable graph Fourier basis algorithm (SGFA) that we prove to exponentially increase the accuracy of the approximating $F$ per iteration. Likewise, we can apply SGFA to $A^H$ and, hence, approximate the stable left eigenvectors for the graph shift $A$ and directly compute the GFT. We evaluate empirically the quality of SGFA by applying it to graph shifts $A$ drawn from two real world problems, the 2004 US political blogs graph and the Manhattan road map, carrying out a comprehensive study on tradeoffs between different SGFA parameters. We also confirm our conclusions by applying SGFA on very sparse and very dense directed Erd\H os-R\'enyi graphs.

Critical Stretching of Mean-Field Regimes in Spatial Networks.

We study a spatial network model with exponentially distributed link lengths on an underlying grid of points, undergoing a structural crossover from a random, Erdős-Rényi graph, to a -dimensional lattice at the characteristic interaction range . We find that, whilst far from the percolation threshold the random part of the giant component scales linearly with , close to criticality it extends in space until the universal length scale , for , before crossing over to the spatial one. We demonstrate the universal behavior of the spatiotemporal scales characterizing this critical stretching phenomenon of mean-field regimes in percolation and in dynamical processes on networks, and we discuss its general implications to real-world phenomena, such as neural activation, traffic flows or epidemic spreading.

Large-deviation properties of the largest biconnected component for random graphs

We study the size of the largest biconnected components in sparse Erd\H{o}s-R\'enyi graphs with finite connectivity and Barab\'asi-Albert graphs with non-integer mean degree. Using a statistical-mechanics inspired Monte Carlo approach we obtain numerically the distributions for different sets of parameters over almost their whole support, especially down to the rare-event tails with probabilities far less than $10^{-100}$. This enables us to observe a qualitative difference in the behavior of the size of the largest biconnected component and the largest $2$-core in the region of very small components, which is unreachable using simple sampling methods. Also, we observe a convergence to a rate function even for small sizes, which is a hint that the large deviation principle holds for these distributions.

Sensitivity of directed networks to the addition and pruning of edges and vertices.

Directed networks have various topologically different extensive components, in contrast to a single giant component in undirected networks. We study the sensitivity (response) of the sizes of these extensive components in directed complex networks to the addition and pruning of edges and vertices. We introduce the susceptibility, which quantifies this sensitivity. We show that topologically different parts of a directed network have different sensitivity to the addition and pruning of edges and vertices and, therefore, they are characterized by different susceptibilities. These susceptibilities diverge at the critical point of the directed percolation transition, signaling the appearance (or disappearance) of the giant strongly connected component in the infinite size limit. We demonstrate this behavior in randomly damaged real and synthetic directed complex networks, such as the World Wide Web, Twitter, the Caenorhabditis elegans neural network, directed Erdős-Rényi graphs, and others. We reveal a nonmonotonic dependence of the sensitivity to random pruning of edges or vertices in the case of C. elegans and Twitter that manifests specific structural peculiarities of these networks. We propose the measurements of the susceptibilities during the addition or pruning of edges and vertices as a new method for studying structural peculiarities of directed networks.

Information Limits for Recovering a Hidden Community

We study the problem of recovering a hidden community of cardinality $K$ from an $n \times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} \sim P$ if $i, j$ both belong to the community and $A_{ij} \sim Q$ otherwise, for two known probability distributions $P$ and $Q$ depending on $n$. If $P={\rm Bern}(p)$ and $Q={\rm Bern}(q)$ with $p>q$, it reduces to the problem of finding a densely-connected $K$-subgraph planted in a large Erd\"os-R\'enyi graph; if $P=\mathcal{N}(\mu,1)$ and $Q=\mathcal{N}(0,1)$ with $\mu>0$, it corresponds to the problem of locating a $K \times K$ principal submatrix of elevated means in a large Gaussian random matrix. We focus on two types of asymptotic recovery guarantees as $n \to \infty$: (1) weak recovery: expected number of classification errors is $o(K)$; (2) exact recovery: probability of classifying all indices correctly converges to one. Under mild assumptions on $P$ and $Q$, and allowing the community size to scale sublinearly with $n$, we derive a set of sufficient conditions and a set of necessary conditions for recovery, which are asymptotically tight with sharp constants. The results hold in particular for the Gaussian case, and for the case of bounded log likelihood ratio, including the Bernoulli case whenever $\frac{p}{q}$ and $\frac{1-p}{1-q}$ are bounded away from zero and infinity. An important algorithmic implication is that, whenever exact recovery is information theoretically possible, any algorithm that provides weak recovery when the community size is concentrated near $K$ can be upgraded to achieve exact recovery in linear additional time by a simple voting procedure.

Exotic phase transitions of k-cores in clustered networks.

The giant k-core-maximal connected subgraph of a network where each node has at least k neighbors-is important in the study of phase transitions and in applications of network theory. Unlike Erdős-Rényi graphs and other random networks where k-cores emerge discontinuously for k≥3, we show that transitive linking (or triadic closure) leads to 3-cores emerging through single or double phase transitions of both discontinuous and continuous nature. We also develop a k-core calculation that includes clustering and provides insights into how high-level connectivity emerges.

Bipodal Structure in Oversaturated Random Graphs

We study the asymptotics of large simple graphs constrained by the limiting density of edges and the limiting subgraph density of an arbitrary xed graph H. We prove that, for all but nitely many values of the edge density, if the density of H is constrained to be slightly higher than that for the corresponding Erd} os-R enyi graph, the typical large graph is bipodal with parameters varying analytically with the densities. Asymptotically, the parameters depend only on the degree sequence of H.

A Note on Dynamical Models on Random Graphs and Fokker–Planck Equations

We address the issue of the proximity of interacting diffusion models on large graphs with a uniform degree property and a corresponding mean field model, i.e. a model on the complete graph with a suitably renormalized interaction parameter. Examples include Erd\H{o}s-R\'enyi graphs with edge probability $p_n$, $n$ is the number of vertices, such that $\lim_{n \to \infty}p_n n= \infty$. The purpose of this note it twofold: (1) to establish this proximity on finite time horizon, by exploiting the fact that both systems are accurately described by a Fokker-Planck PDE (or, equivalently, by a nonlinear diffusion process) in the $n=\infty$ limit; (2) to remark that in reality this result is unsatisfactory when it comes to applying it to systems with $N$ large but finite, for example the values of $N$ that can be reached in simulations or that correspond to the typical number of interacting units in a biological system.

Large Deviation Function for the Number of Eigenvalues of Sparse Random Graphs Inside an Interval.

We present a general method to obtain the exact rate function Ψ_{[a,b]}(k) controlling the large deviation probability Prob[I_{N}[a,b]=kN]≍e^{-NΨ_{[a,b]}(k)} that an N×N sparse random matrix has I_{N}[a,b]=kN eigenvalues inside the interval [a,b]. The method is applied to study the eigenvalue statistics in two distinct examples: (i)the shifted index number of eigenvalues for an ensemble of Erdös-Rényi graphs and (ii)the number of eigenvalues within a bounded region of the spectrum for the Anderson model on regular random graphs. A salient feature of the rate function in both cases is that, unlike rotationally invariant random matrices, it is asymmetric with respect to its minimum. The asymmetric character depends on the disorder in a way that is compatible with the distinct eigenvalue statistics corresponding to localized and delocalized eigenstates. The results also show that the level compressibility κ_{2}/κ_{1} for the Anderson model on a regular graph satisfies 0<κ_{2}/κ_{1}<1 in the bulk regime, in contrast with the behavior found in Gaussian random matrices. Our theoretical findings are thoroughly compared to numerical diagonalization in both cases, showing a reasonable good agreement.

On the Correlation Functions of the Characteristic Polynomials of the Sparse Hermitian Random Matrices

We consider asymptotics of the correlation functions of characteristic polynomials corresponding to random weighted $G(n, \frac{p}{n})$ Erd{\H o}s -- R\'enyi graphs with Gaussian weights in the case of finite $p$ and also when $p \to \infty$. It is shown that for finite $p$ the second correlation function demonstrates a kind of transition: when $p < 2$ it factorizes in the limit $n \to \infty$, while for $p > 2$ there appears an interval $(-\lambda_*(p), \lambda_*(p))$ such that for $\lambda_0 \in (-\lambda_*(p), \lambda_*(p))$ the second correlation function behaves like that for GUE, while for $\lambda_0$ outside the interval the second correlation function is still factorized. For $p \to \infty$ there is also a threshold in the behavior of the second correlation function near $\lambda_0 = \pm 2$: for $p \ll n^{2/3}$ the second correlation function factorizes, whereas for $p \gg n^{2/3}$ it behaves like that for GUE. For any rate of $p \to \infty$ the asymptotics of correlation functions of any even order for $\lambda_0 \in (-2, 2)$ coincide with that for GUE.

Community detection in sparse random networks

We consider the problem of detecting a tight community in a sparse random network. This is formalized as testing for the existence of a dense random subgraph in a random graph. Under the null hypothesis, the graph is a realization of an Erd\os-R\'enyi graph on $N$ vertices and with connection probability $p_0$; under the alternative, there is an unknown subgraph on $n$ vertices where the connection probability is p1 > p0. In Arias-Castro and Verzelen (2012), we focused on the asymptotically dense regime where p0 is large enough that np0>(n/N)^{o(1)}. We consider here the asymptotically sparse regime where p0 is small enough that np0 0. As before, we derive information theoretic lower bounds, and also establish the performance of various tests. Compared to our previous work, the arguments for the lower bounds are based on the same technology, but are substantially more technical in the details; also, the methods we study are different: besides a variant of the scan statistic, we study other statistics such as the size of the largest connected component, the number of triangles, the eigengap of the adjacency matrix, etc. Our detection bounds are sharp, except in the Poisson regime where we were not able to fully characterize the constant arising in the bound.

Toward &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$k$ &lt;/tex-math&gt;&lt;/inline-formula&gt;-Connectivity of the Random Graph Induced by a Pairwise Key Predistribution Scheme With Unreliable Links

We study the secure and reliable connectivity of wireless sensor networks. Security is assumed to be ensured by the random pairwise key predistribution scheme of Chan, Perrig, and Song, and unreliable wireless links are represented by independent on/off channels. Modeling the network by an intersection of a random $K$-out graph and an Erd\H{o}s-R\'enyi graph, we present scaling conditions (on the number of nodes, the scheme parameter $K$, and the probability of a wireless channel being on) such that the resulting graph contains no nodes with degree less than $k$ with high probability, when the number of nodes gets large. Results are given in the form of zero-one laws and are shown to improve the previous results by Ya\u{g}an and Makowski on the absence of isolated nodes (i.e., absence of nodes with degree zero). Via simulations, the established zero-one laws are shown to hold also for the property of $k$-connectivity; i.e., the property that graph remains connected despite the deletion of any $k-1$ nodes or edges.

Discrete Signal Processing on Graphs: Sampling Theory

We propose a sampling theory for signals that are supported on either directed or undirected graphs. The theory follows the same paradigm as classical sampling theory. We show that perfect recovery is possible for graph signals bandlimited under the graph Fourier transform. The sampled signal coefficients form a new graph signal, whose corresponding graph structure preserves the first-order difference of the original graph signal. For general graphs, an optimal sampling operator based on experimentally designed sampling is proposed to guarantee perfect recovery and robustness to noise; for graphs whose graph Fourier transforms are frames with maximal robustness to erasures as well as for Erd\H{o}s-R\'enyi graphs, random sampling leads to perfect recovery with high probability. We further establish the connection to the sampling theory of finite discrete-time signal processing and previous work on signal recovery on graphs. To handle full-band graph signals, we propose a graph filter bank based on sampling theory on graphs. Finally, we apply the proposed sampling theory to semi-supervised classification on online blogs and digit images, where we achieve similar or better performance with fewer labeled samples compared to previous work.

Spectrum of Markov Generators on Sparse Random Graphs

We investigate the spectrum of the infinitesimal generator of the continuoustime random walk on a randomly weighted oriented graph. This is the non‐Hermitian random n × n matrix L defined by Ljk = Xjk if k ≠ j and Ljj = – Σk ≠ j Ljk, where (Xjk)j ≠ k are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n → ∞ of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erdős‐Rényi graph where each edge is present independently with probability p(n) → 0 as long as np(n) ≫ (log(n))6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L.© 2014 Wiley Periodicals, Inc.

Bond percolation in higher dimensions

We collect results for bond percolation on various lattices from two to fourteen dimensions that, in the limit of large dimension d or number of neighbors z, smoothly approach a randomly diluted Erdős-Rényi graph. We include results on bond-diluted hypersphere packs in up to nine dimensions, which show the mean coordination, excess kurtosis, and skewness evolving smoothly with dimension towards the Erdős-Rényi limit.