Random Regular Graphs Research Articles

Asymptotic bounds on total domination in regular graphs

We obtain new upper bounds on the size of a minimum total dominating set for random regular graphs and for regular graphs with large girth. In particular, they imply that an upper bound conjectured by Thomassé and Yeo (2007) holds asymptotically almost surely for 5-regular graphs and holds for all 5-regular graphs with sufficiently large girth. Our bounds are obtained through the analysis of a local algorithm using a method due to Hoppen and Wormald (2018).

Minimal model of many-body localization

We present a fully analytical description of a many body localization (MBL) transition in a microscopically defined model. Its Hamiltonian is the sum of one- and two-body operators, where both contributions obey a maximum-entropy principle and have no symmetries except hermiticity (not even particle number conservation). These two criteria paraphrase that our system is a variant of the Sachdev-Ye-Kitaev (SYK) model. We will demonstrate how this simple `zero-dimensional' system displays numerous features seen in more complex realizations of MBL. Specifically, it shows a transition between an ergodic and a localized phase, and non-trivial wave function statistics indicating the presence of `non-ergodic extended states'. We check our analytical description of these phenomena by parameter free comparison to high performance numerics for systems of up to $N=15$ fermions. In this way, our study becomes a testbed for concepts of high-dimensional quantum localization, previously applied to synthetic systems such as Cayley trees or random regular graphs. We believe that this is the first many body system for which an effective theory is derived and solved from first principles. The hope is that the novel analytical concepts developed in this study may become a stepping stone for the description of MBL in more complex systems.

Network community structure and resilience to localized damage: Application to brain microcirculation

In cerebrovascular networks, some vertices are more connected to each other than with the rest of the vasculature, defining a community structure. Here, we introduce a class of model networks built by rewiring Random Regular Graphs, which enables reproduction of this community structure and other topological properties of cerebrovascular networks. We use these model networks to study the global flow reduction induced by the removal of a single edge. We analytically show that this global flow reduction can be expressed as a function of the initial flow rate in the removed edge and of a topological quantity, both of which display probability distributions following Cauchy laws, i.e. with large tails. As a result, we show that the distribution of blood flow reductions is strongly influenced by the community structure. In particular, the probability of large flow reductions increases substantially when the community structure is stronger, weakening the network resilience to single capillary occlusions. We discuss the implications of these findings in the context of Alzheimers Disease, in which the importance of vascular mechanisms, including capillary occlusions, is beginning to be uncovered. Statement of significance“Occlusions of capillary vessels, the smallest blood vessels in the brain, are involved in major diseases, including Alzheimers Disease and ischemic stroke. To better understand their impact on cerebral blood flow, we theoretically study the vessel network response to a single occlusion. We show that the reduction of blood flow at network scale is a function of the initial blood flow in the occluded vessel and of a topological quantity, both of which have broad distributions, that is, with significant probabilities of extreme values. Using model networks built from Random Regular Graphs, we show that the presence of communities in the network (subparts more connected to each other than with the rest of the vasculature) yield a broader distribution of the topological quantity. This weakens the resilience of brain vessel networks to single capillary occlusions, which may contribute to the pathogenicity of capillary occlusions in the brain”.

Zero-forcing in random regular graphs

Zero-forcing in random regular graphs

Precise asymptotics of some meeting times arising from the voter model on large random regular graphs

We consider two independent stationary random walks on large random regular graphs of degree k≥3 with N vertices. On these graphs, the exponential approximations of the meeting times are known to follow from existing methods and form a basis for the voter model’s diffusion approximations. The main result of this note improves the exponential approximations to an explicit form such that the first moments are asymptotically equivalent to N(k−1)∕[2(k−2)].

Localization on Certain Graphs with Strongly Correlated Disorder.

Many-body localization in interacting quantum systems can be cast as a disordered hopping problem on the underlying Fock-space graph. A crucial feature of the effective Fock-space disorder is that the Fock-space site energies are strongly correlated-maximally so for sites separated by a finite distance on the graph. Motivated by this, and to understand the effect of such correlations more fundamentally, we study Anderson localization on Cayley trees and random regular graphs, with maximally correlated disorder. Since such correlations suppress short distance fluctuations in the disorder potential, one might naively suppose they disfavor localization. We find however that there exists an Anderson transition, and indeed that localization is more robust in the sense that the critical disorder scales with graph connectivity K as sqrt[K], in marked contrast to KlnK in the uncorrelated case. This scaling is argued to be intimately connected to the stability of many-body localization. Our analysis centers on an exact recursive formulation for the local propagators as well as a self-consistent mean-field theory; with results corroborated using exact diagonalization.

On the unbalanced cut problem and the generalized Sherrington–Kirkpatrick model

We establish a strict asymptotic inequality between a class of graph partition problems on the sparse Erdős–Rényi and random regular graph ensembles with the same average degree. Along the way, we establish a variational representation for the ground state energy for generalized mixed p -spin glasses and derive strict comparison inequalities for such models as the alphabet changes.

Finding maximum matchings in random regular graphs in linear expected time

AbstractIn a seminal paper on finding large matchings in sparse random graphs, Karp and Sipser proposed two algorithms for this task. The second algorithm has been intensely studied, but due to technical difficulties, the first algorithm has received less attention. Empirical results by Karp and Sipser suggest that the first algorithm is superior. In this paper we show that this is indeed the case, at least for random k‐regular graphs. We show that w.h.p. the first algorithm will find a matching of size in a random k‐regular graph, k = O(1). We also show that the algorithm can be adapted to find a maximum matching in O(n) time w.h.p. This is to be compared with O(n3/2) time for the worst‐case.

Scaling up the Anderson transition in random-regular graphs

We study the Anderson transition in lattices with the connectivity of a random-regular graph. Our results indicate that fractal dimensions are continuous across the transition, but a discontinuity occurs in their derivatives, implying the non-ergodicity of the metal near the Anderson transition. A critical exponent $\nu = 1.00 \pm0.02$ and critical disorder $W= 18.2\pm 0.1$ are found via a scaling approach. Our data support that the predictions of the relevant Gaussian Ensemble are only recovered at zero disorder.

Publisher's Note: Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase [Phys. Rev. B 96 , 064202 (2017)

Received 30 October 2020DOI:https://doi.org/10.1103/PhysRevB.102.179901©2020 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAnderson localizationPhysical SystemsSpin glassesTechniquesRandom matrix theoryStatistical PhysicsCondensed Matter, Materials & Applied Physics

SEAIR Epidemic spreading model of COVID-19.

We study Susceptible-Exposed-Asymptomatic-Infectious-Recovered (SEAIR) epidemic spreading model of COVID-19. It captures two important characteristics of the infectiousness of COVID-19: delayed start and its appearance before onset of symptoms, or even with total absence of them. The model is theoretically analyzed in continuous-time compartmental version and discrete-time version on random regular graphs and complex networks. We show analytically that there are relationships between the epidemic thresholds and the equations for the susceptible populations at the endemic equilibrium in all three versions, which hold when the epidemic is weak. We provide theoretical arguments that eigenvector centrality of a node approximately determines its risk to become infected.

Measuring pressure in equilibrium and nonequilibrium lattice-gas models.

We develop an algorithm based on the method proposed by Dickman for directly measuring pressure in lattice-gas models. The algorithm gives the possibility to access the equation of state with a single run by adding multiple ghost sites to the original system. This feature considerably improves calculations and makes the algorithm particularly efficient for systems with inhomogeneous density profiles, both in equilibrium and nonequilibrium steady states. We illustrate its broad applicability by considering some paradigmatic systems of statistical mechanics such as the lattice gas under gravity, nearest-neighbor exclusion models in finite dimension and on regular random graphs, and the boundary-driven simple symmetric exclusion process.

Entropy and expansion

Shearer's inequality bounds the sum of joint entropies of random variables in terms of the total joint entropy. We give another lower bound for the same sum in terms of the individual entropies when the variables are functions of independent random seeds. The inequality involves a constant characterizing the expansion properties of the system. Our results generalize to entropy inequalities used in recent work in invariant settings, including the edge-vertex inequality for factor-of-IID processes, Bowen's entropy inequalities, and Bollob\'as's entropy bounds in random regular graphs. The proof method yields inequalities for other measures of randomness, including covariance. As an application, we give upper bounds for independent sets in both finite and infinite graphs.

Frozen (Δ + 1)-colourings of bounded degree graphs

AbstractLet G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for $k\geq \Delta+2$ . Feghali, Johnson and Paulusma (J. Graph Theory83 (2016) 340–358) showed that the (Δ + 1)-recolouring graph is composed by a unique connected component of size at least 2 and (possibly many) isolated vertices.In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the (Δ + 1)-recolouring graph. Our first contribution is to show that if G is connected, the proportion of frozen colourings of G is exponentially smaller in n than the total number of colourings. This motivates the use of the Glauber dynamics to approximate the number of (Δ + 1)-colourings of a graph. In contrast to the conjectured mixing time of O(nlog n) for $k\geq \Delta+2$ colours, we show that the mixing time of the Glauber dynamics for (Δ + 1)-colourings restricted to non-frozen colourings can be Ω(n2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.

The Cover Time of a Biased Random Walk on a Random Regular Graph of Odd Degree

We consider a random walk process on graphs introduced by Orenshtein and Shinkar (2014). At any time, the random walk moves from its current position along a previously unvisited edge chosen uniformly at random, if such an edge exists. Otherwise, it walks along a previously visited edge chosen uniformly at random. For the random $r$-regular graph, with $r$ a constant odd integer, we show that this random walk process has asymptotic vertex and edge cover times $\frac{1}{r-2}n\log n$ and $\frac{r}{2(r-2)}n\log n$, respectively, generalizing a result of Cooper, Frieze and the author (2018) from $r = 3$ to any odd $r\geqslant 3$. The leading term of the asymptotic vertex cover time is now known for all fixed $r\geqslant 3$, with Berenbrink, Cooper and Friedetzky (2015) having shown that $G_r$ has vertex cover time asymptotic to $\frac{rn}{2}$ when $r\geqslant 4$ is even.

The random transposition dynamics on random regular graphs and the Gaussian free field

Une permutation donnée, vue comme réunion de cycles disjoints, représente un graphe régulier de degré $2$. Considérons $d$ permutations aléatoires indépendantes, en superposons leurs structures de graphes. Ceci est le modèle de permutation bien connu donnant un (multi-)graphe régulier aléatoire de degré $2d$. Nous considérons un champ $2$-dimensionnel de $d$ permutations indexé par la taille et le temps. La taille de chaque permutation croît selon des processus couplés de restaurants chinois, tandis que chaque permutation évolue dans le temps selon une chaîne de transpositions aléatoires. À travers le modèle de permutation, ceci se projette en une famille à deux paramètres de graphes qui croissent en taille (« dimension ») et qui évoluent en temps. Dans cet ensemble de graphes aléatoires, on observe asymptotiquement une évolution remarquable des petits cycles et des statistiques linéaires des valeurs propres en dimension et en temps. En dimension, il avait été montré par Johnson et Pal (Ann. Probab. 42 (2014) 1396–1437) que les nombres de cycles sont décrits par un champ poissonnien de processus de Yule. Ici, nous donnons une description en dimension et en temps du processus des cycles en termes en termes d’un surface aléatoire poissonnienne, pour tout $d$. Lorsque $d$ tend vers l’infini, les fluctuations des nombres de cycles convergent vers un processus gaussien indexé par la dimension et le temps. Les marginales en dimension se trouvent être le champ libre gaussien, et le processus est stationnaire en temps. Une structure similaire de covariance apparaît dans les fluctuations des valeurs propres du processus des mineurs d’une matrice de Wigner réelle symétrique dont les coordonnées évoluent selon des processus stochastiques stationnaires i.i.d.. Ainsi, cet article décrit un analogue poissonnien d’une dynamique markovienne naturelle sur le champ libre gaussien, et étudie ses propriétés trajectorielles.

The Average Distance and the Diameter of Dense Random Regular Graphs

Let $\mathrm{AD}(G_{n,d})$ be the average distance of $G_{n,d}$, a random $n$-vertex $d$-regular graph.
 For $d=(\beta+o(1))n^{\alpha}$ with two arbitrary constants $\alpha\in(0,1)$ and $\beta>0$, we prove that $|\mathrm{AD}(G_{n,d})-\mu|<\epsilon$ holds with high probability for any constant $\epsilon>0$, where $\mu$ is equal to $\alpha^{-1}+\exp(-\beta^{1/\alpha})$ if $\alpha^{-1}\in\mathbb{N}$ and to $\lceil\alpha^{-1}\rceil$ otherwise.
 Consequently, we show that the diameter of the $G_{n,d}$ is equal to $\lfloor\alpha^{-1}\rfloor+1$ with high probability.

From entropic to energetic barriers in glassy dynamics: the Barrat–Mézard trap model on sparse networks

Trap models describe glassy dynamics as a stochastic process on a network of configurations representing local energy minima. We study within this class the paradigmatic Barrat–Mézard model, which has Glauber transition rates. Our focus is on the effects of the network connectivity, where we go beyond the usual mean field (fully connected) approximation and consider sparse networks, specifically random regular graphs (RRG). We obtain the spectral density of relaxation rates of the master operator using the cavity method, revealing very rich behaviour as a function of network connectivity c and temperature T. We trace this back to a crossover from initially entropic barriers, resulting from a paucity of downhill directions, to energy barriers that govern the escape from local minima at long times. The insights gained are used to rationalize the relaxation of the energy after a quench from high T, as well as the corresponding correlation and persistence functions.

Pair approximation for the q-voter model with independence on multiplex networks.

The q-voter model with independence is investigated on multiplex networks with full overlap of nodes in the layers. The layers are various complex networks corresponding to different levels of social influence. Detailed studies are performed for the model on multiplex networks with two layers with identical degree distributions, obeying the LOCAL&AND and GLOBAL&AND spin update rules differing by the way in which the q-lobbies of neighbors within different layers exert their joint influence on the opinion of a given agent. Homogeneous pair approximation is derived for a general case of a two-state spin model on a multiplex network and its predictions are compared with results of mean-field approximation and Monte Carlo simulations of the above-mentioned q-voter model with independence for a broad range of parameters. As the parameter controlling the level of agents' independence is changed ferromagnetic phase transition occurs which can be first- or second-order, depending on the size of the lobby q. Details of this transition, e.g., position of the critical points, critical exponents and the width of the possible hysteresis loop, depend on the topology and other features of the layers, in particular on the mean degree of nodes in the layers which is directly predicted by the homogeneous pair approximation. If the mean degree of nodes is substantially larger than the size of the q-lobby good agreement is obtained between numerical results and theoretical predictions based on the homogeneous pair approximation concerning the order and details of the ferromagnetic transition. In the case of the model on multiplex networks with layers in the form of homogeneous Erdős-Rényi and random regular graphs as well as weakly heterogeneous scale-free networks this agreement is quantitative, while in the case of layers in the form of strongly heterogeneous scale-free networks it is only qualitative. If the mean degree of nodes is small and comparable with q predictions of the homogeneous pair approximation are in general even qualitatively wrong.

Dirac’s theorem for random regular graphs

AbstractWe prove a ‘resilience’ version of Dirac’s theorem in the setting of random regular graphs. More precisely, we show that whenever d is sufficiently large compared to $\epsilon > 0$ , a.a.s. the following holds. Let $G'$ be any subgraph of the random n-vertex d-regular graph $G_{n,d}$ with minimum degree at least $$(1/2 + \epsilon )d$$ . Then $G'$ is Hamiltonian.This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly the condition that d is large cannot be omitted, and secondly the minimum degree bound cannot be improved.