Local Algorithms, Regular Graphs of Large Girth, and Random Regular Graphs

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.

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.

A k-dominating set of a graph, G, is a set of vertices, D⊆ V(G), such that for every vertex v ∈ V(G), either v ∈ D or there exists a vertex u ∈ D that is at distance at most k from v in G. We are interested in finding k-dominating sets of small cardinality. In this paper we consider a simple, yet efficient, randomised greedy algorithm for finding a small k-dominating set of regular graphs. We analyse the average-case performance of this heuristic by analysing its performance on random regular graphs using differential equations. This, in turn, proves an upper bound on the size of a minimum k-dominating set of random regular graphs.

Let G be a given graph (modelling a communication network) which we assume suffers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1 - p). In particular we are interested in robustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree. Our result is: If the degree d is fixed then p = 1/(d - 1) is a threshold probability for the existence of a linear-sized component in a faulty version of almost all random regular graphs. We show: If each edge of an arbitrary graph G with maximum degree bounded above by d is present with, probability p = Λ/(d - 1) where Λ 1 then for almost all G the faulted version of G has a linear-sized component with high probability. Note that these results imply some kind of optimality of random regular graphs among the class of graphs with the same degree bound. The theme is: Use the known expansion properties of almost all random regular graphs to obtain strong robustness results. This has not been done systematically before.

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random d-regular graph on n vertices, and let \lambda be the largest absolute value of a non-trivial eigenvalue of its adjacency matrix. It was conjectured by Alon [86'] that a random d-regular graph is almost Ramanujan, in the following sense: for every e>0, \lambda<2\sqrt{d-1} + e asymptotically almost surely. Friedman famously presented a proof of this conjecture in [08']. Here we suggest a new, substantially simpler proof of a nearly-optimal result: we show that a random d-regular graph satisfies \lambda < 2\sqrt{d-1} + 1 a.a.s. A main advantage of our approach is that it is applicable to a generalized conjecture: For d even, a d-regular graph on n vertices is an n-covering space of a bouquet of d/2 loops. More generally, fixing an arbitrary base graph H, we study the spectrum of G, a random n-covering of H. Let \lambda be the largest absolute value of a non-trivial eigenvalue of G. Extending Alon's conjecture to this more general model, Friedman [03'] conjectured that for every e>0, a.a.s. \lambda < \rho+e, where \rho is the spectral radius of the universal cover of H. When H is regular we get a bound of \rho+0.84, and for an arbitrary H, we prove a nearly optimal upper bound of \sqrt{3}\rho. This is a substantial improvement upon all known results (by Friedman, Linial-Puder, Lubetzky-Sudakov-Vu and Addario-Berry-Griffiths).

Connected domination of regular graphs

We determine the asymptotics of the independence number of the random $d$-regular graph for all $d \ge d_0$. It is highly concentrated, with constant-order fluctuations around $n\alpha_* - c_*\log n$ for explicit constants $\alpha_*(d)$ and $c_*(d)$. Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs.

OF THE DISSERTATION Random matrices and random boxes by Linh V. Tran Dissertation Director: Prof. Van Vu This thesis concerns two questions on random structures: the semi-circular law for adjacency matrix of regular random graph and the piercing number for random boxes. • Random matrices We proved in full generality the semi-circular law for random d-regular graph model in the case d tends to infinity as n does. Our result complements the McKay law [19], which applied for the case d is an absolute constant. • Random boxes. Take n random boxes with axis-parallel edges inside the unit cube [0, 1]d, the piercing number τd(n) is the minimum number of points needed to pierce all boxes. Using hypergraph setting, we was able to prove a near sharp estimation for τd(n): for any d ≥ 2 Ωd( √ n(log n)d/2−1) = τd(n) = Od( √ n(log n)d/2−1 log log n). This thesis is based on two papers by the author [31] and [30] (joint work with Van Vu and Ke Wang).

Large independent sets in random regular graphs

We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core.

In a paper that initiated the modern study of the stochastic block model,\nDecelle et al., backed by Mossel et al., made the following conjecture: Denote\nby $k$ the number of balanced communities, $a/n$ the probability of connecting\ninside communities and $b/n$ across, and set\n$\\mathrm{SNR}=(a-b)^2/(k(a+(k-1)b)$; for any $k \\geq 2$, it is possible to\ndetect communities efficiently whenever $\\mathrm{SNR}&gt;1$ (the KS threshold),\nwhereas for $k\\geq 4$, it is possible to detect communities\ninformation-theoretically for some $\\mathrm{SNR}&lt;1$. Massouli\\'e, Mossel et\nal.\\ and Bordenave et al.\\ succeeded in proving that the KS threshold is\nefficiently achievable for $k=2$, while Mossel et al.\\ proved that it cannot be\ncrossed information-theoretically for $k=2$. The above conjecture remained open\nfor $k \\geq 3$.\n This paper proves this conjecture, further extending the efficient detection\nto non-symmetrical SBMs with a generalized notion of detection and KS\nthreshold. For the efficient part, a linearized acyclic belief propagation\n(ABP) algorithm is developed and proved to detect communities for any $k$ down\nto the KS threshold in time $O(n \\log n)$. Achieving this requires showing\noptimality of ABP in the presence of cycles, a challenge for message passing\nalgorithms. The paper further connects ABP to a power iteration method with a\nnonbacktracking operator of generalized order, formalizing the interplay\nbetween message passing and spectral methods. For the information-theoretic\n(IT) part, a non-efficient algorithm sampling a typical clustering is shown to\nbreak down the KS threshold at $k=4$. The emerging gap is shown to be large in\nsome cases; if $a=0$, the KS threshold reads $b \\gtrsim k^2$ whereas the IT\nbound reads $b \\gtrsim k \\ln(k)$, making the SBM a good study-case for\ninformation-computation gaps.\n

We study the return probability for the Anderson model on the random regular graph and give evidence of the existence of two distinct phases: a fully ergodic and nonergodic one. In the ergodic phase, the return probability decays polynomially with time with oscillations, being the attribute of the Wigner-Dyson-like behavior, while in the nonergodic phase the decay follows a stretched exponential decay.We give a phenomenological interpretation of the stretched exponential decay in terms of a classical random walker. Furthermore, comparing typical and mean values of the return probability, we show how to differentiate an ergodic phase from a nonergodic one. We benchmark this method first in two random matrix models, the power-law random banded matrices, and the Rosenzweig-Porter matrices, which host both phases. Second, we apply this method to the Anderson model on the random regular graph to give further evidence of the existence of the two phases.

We study the coordination game with an aspiration-driven update rule in regular graphs and scale-free networks. We prove that the model coincides exactly with the Ising model and shows a phase transition at the critical selection noise when the aspiration level is zero. It is found that the critical selection noise decreases with clustering in random regular graphs. With a non-zero aspiration level, the model also exhibits a phase transition as long as the aspiration level is smaller than the degree of graphs. We also show that the critical exponents are independent of clustering and aspiration level to confirm that the coordination game belongs to the Ising universality class. As for scale-free networks, the effect of aspiration level on the order parameter at a low selection noise is examined. In model networks (the Barabási-Albert network and the Holme-Kim network), the order parameter abruptly decreases when the aspiration level is the same as the average degree of the network. In contrast, in real-world networks, the order parameter decreases gradually. We explain this difference by proposing the concepts of hub centrality and local hub. The histogram of hub centrality of real-world networks separates into two parts unlike model networks, and local hubs exist only in real-world networks. We conclude that the difference of network structures in model and real-world networks induces qualitatively different behavior in the coordination game.

We consider the contact process with infection rate $\lambda $ on a random $(d+1)$-regular graph with $n$ vertices, $G_n$. We study the extinction time $\tau _{G_n}$ (that is, the random amount of time until the infection disappears) as $n$ is taken to infinity. We establish a phase transition depending on whether $\lambda $ is smaller or larger than $\lambda _1(\mathbb{T} ^d)$, the lower critical value for the contact process on the infinite, $(d+1)$-regular tree: if $\lambda < \lambda _1(\mathbb{T} ^d)$, $\tau _{G_n}$ grows logarithmically with $n$, while if $\lambda > \lambda _1(\mathbb{T} ^d)$, it grows exponentially with $n$. This result differs from the situation where, instead of $G_n$, the contact process is considered on the $d$-ary tree of finite height, since in this case, the transition is known to happen instead at the upper critical value for the contact process on $\mathbb{T} ^d$.

We show that the contact process on a random $d$-regular graph initiated by a single infected vertex obeys the “cutoff phenomenon” in its supercritical phase. In particular, we prove that, when the infection rate is larger than the lower critical value of the contact process on the infinite $d$-regular tree, there are positive constants $C$, $p$ depending on the infection rate such that for any $\varepsilon>0$, when the number $n$ of vertices is large then (a) at times $t<(C-\varepsilon)\log n$ the fraction of infected vertices is vanishingly small, but (b) at time $(C+\varepsilon)\log n$ the fraction of infected vertices is within $\varepsilon$ of $p$, with probability $p$.