Large Deviations for Marked Sparse Random Graphs with Applications to Interacting Diffusions

The genus of the binomial random graph $G(n,p)$ is well understood for a wide range of $p=p(n)$. Recently, the study of the genus of the random bipartite graph $G(n_1,n_2,p)$, with partition classes of size $n_1$ and $n_2$, was initiated by Mohar and Ying, who showed that when $n_1$ and $n_2$ are comparable in size and $p=p(n_1,n_2)$ is significantly larger than $(n_1n_2)^{-\frac{1}{2}}$ the genus of the random bipartite graph has a similar behaviour to that of the binomial random graph. In this paper we show that there is a threshold for planarity of the random bipartite graph at $p=(n_1n_2)^{-\frac{1}{2}}$ and investigate the genus close to this threshold, extending the results of Mohar and Ying. It turns out that there is qualitatively different behaviour in the case where $n_1$ and $n_2$ are comparable, when whp the genus is linear in the number of edges, than in the case where $n_1$ is asymptotically smaller than $n_2$, when whp the genus behaves like the genus of a sparse random graph $G(n_1,q)$ for an appropriately chosen $q=q(p,n_1,n_2)$.

In this article for a finite typed random geometric graph we define the\nempirical locality distribution, which records the number of nodes of a given\ntype linked to a given number of nodes of each type. We find large deviation\nprinciple (LDP) for the empirical locality measure given the empirical pair\nmeasure and the empirical type measure of the typed random geometric graphs.\nFrom this LDP, we derive large deviation principles for the degree measure and\nthe proportion of detached nodes in the classical Erdos-Renyi graph defined on\n[0, 1]^d. This graphs have been suggested by (Canning and Penman, 2003) as a\npossible extension to the randomly typed random graphs.\n

The authors study here low-energy linear excitations in XY and Heisenberg spin glasses at zero temperature, considering systems defined on sparse random regular graphs. The localization properties of the soft modes are linked to the spin glass transition in a field. The authors discover a new kind of low-energy excitation, called a ``Concentrated and Delocalized Low Energy Mode'' (CDLEM), given that this excitation is restricted to a sub-extensive number of vertices spread over the largest possible distances.

Two ensembles are frequently used to model random graphs subject to constraints: the microcanonical ensemble (= hard constraint) and the canonical ensemble (= soft constraint). It is said that breaking of ensemble equivalence (BEE) occurs when the specific relative entropy of the two ensembles does not vanish as the size of the graph tends to infinity. Various examples have been analysed in the literature. It was found that BEE is the rule rather than the exception for two classes of constraints: sparse random graphs when the number of constraints is of the order of the number of vertices, and dense random graphs when there are two or more constraints that are frustrated. We establish BEE for a third class: dense random graphs with a single constraint on the density of a given simple graph. We show that BEE occurs in a certain range of choices for the density and the number of edges of the simple graph, which we refer to as the BEE-phase. We also show that, in part of the BEE-phase, there is a gap between the scaling limits of the averages of the maximal eigenvalue of the adjacency matrix of the random graph under the two ensembles, a property that is referred to as the spectral signature of BEE. We further show that in the replica symmetric region of the BEE-phase, BEE is due to the coexistence of two densities in the canonical ensemble.

ABSTRACTAn independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let . In 1987, Alavi, Malde, Schwenk, and Erdős asked if the independent set sequence of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk, and Erdős is “four‐fifths true,” with high probability. We also show unimodality of the independent set sequence of Erdős–Rényi random graphs, when the expected degree of a single vertex is large (with [exponentially] high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs. The structure of independent sets of size as varies is of interest in probability, statistical physics, combinatorics, and computer science.

Consider an interacting particle system indexed by the vertices of a (possibly random) locally finite graph whose vertices and edges are equipped with weights or marks that represent parameters of the model, such as the environment and initial conditions. Each particle takes values in a countable state space and evolves according to a pure jump process whose jump rates depend only on its own state (or history) and marks, and states (or histories) and marks of particles and edges in its neighborhood. Under mild conditions on the jump rates, it is shown that if the sequence of (marked) interaction graphs converges in probability in the local weak sense to a limit (marked) graph that satisfies a certain finite dissociability property, then the corresponding sequence of empirical measures of the particle trajectories converges weakly to the law of the marginal dynamics at the root vertex of the limit graph. The proof of this hydrodynamic limit relies on several auxiliary results of potentially independent interest. First, such interacting particle systems are shown to be well-posed on (almost surely) finitely dissociable graphs, which include graphs with uniformly bounded maximum degrees and any Galton-Watson tree whose offspring distribution has a finite first moment. A counterexample is also provided to show that well-posedness can fail for dynamics on graphs outside this class. Next, given any sequence of graphs that converges in the local weak sense to a finitely dissociable graph, it is shown that the corresponding sequence of jump processes also converges in the same sense to a jump process on the limit graph. Finally, the dynamics are also shown to exhibit an (annealed) asymptotic correlation decay property. These results complement recent work on hydrodynamic limits of locally interacting probabilistic cellular automata and diffusions on sparse random graphs. However, the analysis of jump processes requires very different techniques, including percolation arguments and notions such as consistent spatial localization and causal chains.

Local algorithms on graphs are algorithms that run in parallel on the nodes of a graph to compute some global structural feature of the graph. Such algorithms use only local information available at nodes to determine local aspects of the global structure, while also potentially using some randomness. Research over the years has shown that such algorithms can be surprisingly powerful in terms of computing structures like large independent sets in graphs locally. These algorithms have also been implicitly considered in the work on graph limits, where a conjecture due to Hatami, Lovász and Szegedy [17] implied that local algorithms may be able to compute near-maximum independent sets in (sparse) random d-regular graphs. In this paper we refute this conjecture and show that every independent set produced by local algorithms is smaller that the largest one by a multiplicative factor of at least 1/2+1/(2√2) ≈ .853, asymptotically as d → ∞.

We consider random graphs with edge probability βn−α, where n is the number of vertices of the graph, β > 0 is fixed, and α = 1 or α = (l + 1) /l for some fixed positive integer l. We prove that for every first‐order sentence, the probability that the sentence is true for the random graph has an asymptotic limit.

The author considers random graphs with edge probability beta n/sup - alpha /, where n is the number of vertices of the graph, beta >0 is fixed, and alpha =1 or alpha =(l+1)/l for some fixed positive integer l. It is proved that, for every first-order sentence, the probability that the sentence is true for the random graph has an asymptotic limit. Also, there is an effective procedure for generating the value of the limit in closed form. >

In the binomial random graph 𝐺(𝑛, 𝑝) edges between every pair of vertices appear independently with probability 𝑝. The random graph obeys 0-1 𝑘-law, if, for every first order sentence, the probability that 𝐺(𝑛, 𝑝) satisfies it is either 0 or 1 in the limit (as 𝑛 /∞). This paper addresses the following question: given a small 𝑘 and any ℓ, does 𝐺(𝑛, 𝑛−𝑙−1/ℓ) satisfy 0-1 𝑘-law? We answer this question for 𝑘 = 3 and all ℓ. We also prove that for 𝑘 = 4 and ℓ ∈ [1, 40] ∪ {72} 0-1 𝑘-law does not hold.

We prove that the number of Hamilton cycles in the random graph $G(n,p)$ is $n!p^n(1+o(1))^n$ asymptotically almost surely (a.a.s.), provided that $p\geq \frac{\ln n+\ln\ln n+\omega(1)}{n}$. Furthermore, we prove the hitting time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree $2$ creates $(\frac{\ln n}{e})^n(1+o(1))^n$ Hamilton cycles a.a.s.

The large deviation principle for the Erdős-Rényi random graph

This paper deals with a random walk $S_n:=\xi_1+\cdots+\xi_n$, $n=0,1,\ldots,$ in the $d$-dimensional Euclidean space ${\mathbb R}^d$, where $S_0=0$ and $\xi_k$ are independent identically distributed random vectors satisfying Cramér's moment conditions. For random polygons with nodes at the points $(\frac{k}{n},\frac{1}{x}S_k)$, $k=0,1,\ldots,n,$ we obtain the logarithmic asymptotics of the large deviation probabilities in different trajectory spaces when $x\sim \alpha_0 n$, $\alpha_0>0$, as $n\to\infty.$ The results include the so-called local and extended large deviation principles (l.d.p.'s) (see i̧te15) that hold in those cases where the “usual” l.d.p. does not apply. The paper consists of three parts. Part I has two sections. Section 1 presents the key concepts and some facts concerning the l.d.p. in arbitrary metric spaces. In section 2 we formulate the “strong” versions of the “usual” l.d.p. in the large deviation zones that were obtained earlier in [A. A. Borovkov, Theory Probab. Appl., 12 (1967), pp. 575--595], [A. A. Mogul'skii, Theory Probab. Appl., 21 (1976), pp. 300--315] for the space of continuous functions. Besides that, section 2 also contains the l.d.p. for probabilities for the random walk trajectories to hit a convex set. That result was obtained using inequalities from [A. A. Borovkov and A. A. Mogul'skii, Theory Probab. Appl., 56 (2012), pp. 21--43] and does not involve any moment conditions. Part II begins with section 3 presenting an example elucidating the need to extend both the problem formulation and the very concept of the “large deviation principle.” We introduce a new extended functional space, a metric therein, and the deviation functional (integral) of a more general kind that will be used when constructing an “extended” l.d.p. In section 4 we present and prove the key results of the paper, the local and the extended large deviation principles, for the trajectories of univariate random walks in the space ${\mathbb D}$ of functions without discontinuities of the second kind. Section 5 extends to the multivariate case all the results established in section 4. Section 6 in Part III presents results analogous to those from section 4, but now established in the space of functions of bounded variation with a metric stronger than that in $\mathbb D$. In section 7 we establish the so-called conditional large deviation principles for the trajectories of univariate random walks given the location of the walk at the terminal point. As a consequence, we obtain the Sanov's theorem on large deviations of empirical distributions.

Let D(G) be the smallest quantifier depth of a first-order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the descriptive complexity of G in first-order logic. We show that almost surely $D(G)=\Theta(\frac{\ln n}{\ln\ln n})$ , where G is a random tree of order n or the giant component of a random graph $\C G(n,\frac cn)$ with constant c<1. These results rely on computing the maximum of D(T) for a tree T of order n and maximum degree l, so we study this problem as well.

Ther‐dynamic choosability of a graph G, written , is the least k such that whenever each vertex is assigned a list of at least k colors a proper coloring can be chosen from the lists so that every vertex v has at least neighbors of distinct colors. Let ch(G) denote the choice number of G. In this article, we prove when is bounded. We also show that there exists a constant C such that the random graph with almost surely satisfies . Also if G is a triangle‐free regular graph, then we have .