Jonah Blasiak, Rick Durrett
Inspired by a concept in comparative genomics, we investigate properties of randomly chosen members of G_1(m,n,t), the set of bipartite graphs with $m$ left vertices, n right vertices, t edges, and each vertex of degree at least one. We give asymptotic results for the number of such graphs and the number of $(i,j)$ trees they contain. We compute the thresholds for the emergence of a giant component and for the graph to be connected.
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Given independent random points $\mathcal{X}_n=\{X_1,...,X_n\}$ in $\mathbb{R}^2$, drawn according to some probability density function $f$ on $\mathbb{R}^2$, and a cutoff $r_n>0$ we construct a random geometric digraph $G(\mathcal{X}_n,\mathcal{Y}_n,r_n)$ with vertex set $\mathcal{X}_n$. Each vertex $X_i$ is assigned uniformly at random a sector $S_i$, of central angle $α$ with inclination $Y_i$, in a circle of radius $r_n$ (with vertex $X_i$ as the origin). An arc is present from $X_i$ to $X_j$, if $X_j$ falls in $S_i$. We also introduce another random geometric digraph $G(\mathcal{X}_n,\mat…
Collisions of random walks in unimodular random graphs: applications to the random geometric graph and long-range percolation
研究单模随机根图中简单随机游走的碰撞性质，在常返及根的可积条件下两独立随机游走几乎必然无限次碰撞，瞬态及相关可积假设下几乎必然仅有限次碰撞，还应用于多种无界度随机图及长程渗流模型。
Random bipartite graphs with i.i.d. weights and applications to inhomogeneous random intersection graphs
本文构建带独立同分布权重的随机二部图并导出非齐次随机交图，刻画了中等聚类临界区中大连通分量的三类标度极限。
We examine the mixing time for random walks on graphs. In particular we are interested on investigating graphs with bottlenecks. Furthermore, the cutoff phenomenon is examined.
We study a random graph model which combines properties of the edge percolation model on Z^d and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so-called "rank 1 case" of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe completely the phase diagram in the case d=1. The phase transition is similar to the classical random graph, it is of the second order. We also find the scaled size of the largest connected component above the phase transition.
A stationary random graph is a random rooted graph whose distribution is invariant under re-rooting along the simple random walk. We adapt the entropy technique developed for Cayley graphs and show in particular that stationary random graphs of subexponential growth are almost surely Liouville, that is, admit no non constant bounded harmonic function. Applications include the uniform infinite planar quadrangulation and long-range percolation clusters.
Temporal connectivity of Random Geometric Graphs
该研究针对时间随机几何图，确定了全顶点对间存在单调递增时间戳路径的连通性阈值，发现其所需边密度远高于底层图的简单连通阈值，且该结论适用于标准及一类软随机几何图。
In [17], the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in [11], we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem that the magnetization scaled by n 3/4 converges to a specific random variable, with n the number of vertices of random regular graphs.
The theory of random graphs goes back to the late 1950s when Paul Erdős and Alfréd Rényi introduced the Erdős-Rényi random graph. Since then many models have been developed, and the study of random graph models has become popular for real-life network modelling such as social networks and financial networks. The aim of this overview is to review relevant random graph models for real-life network modelling. Therefore, we analyse their properties in terms of stylised facts of real-life networks.
Dyadic lattice graphs and their duals are commonly used as discrete approximations to the hyperbolic plane. We use them to give examples of random rooted graphs that are stationary for simple random walk, but whose duals have only a singular stationary measure. This answers a question of Curien and shows behaviour different from the unimodular case. The consequence is that planar duality does not combine well with stationary random graphs. We also study harmonic measure on dyadic lattice graphs and show its singularity.
We prove that any stationary random graph satisfying a growth condition and having positive entropy almost surely admits an infinite dimensional space of bounded harmonic functions. Applications to random infinite planar triangulations and Delaunay graphs are given.
We study logical limit laws for uniform attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time $n+1$, the vertex $n+1$ is introduced together with $m$ edges joining the new vertex with $m$ different vertices chosen uniformly at random from $1,\ldots,n$. We prove that this random graph obeys convergence law for first-order sentences with at most $m-2$ variables.
We prove that in any recurrent reversible random rooted graph, two independent simple random walks started at the same vertex collide infinitely often almost surely. This applies to the Uniform Infinite Planar Triangulation and Quadrangulation and to the Incipient Infinite Cluster in $\mathbb{Z}^2$.
We construct an example of a bounded degree, nonamenable, unimodular random rooted graph with $p_c=p_u$ for Bernoulli bond percolation, as well as an example of a bounded degree, unimodular random rooted graph with $p_c<1$ but with an infinite cluster at criticality. These examples show that two well-known conjectures of Benjamini and Schramm are false when generalised from transitive graphs to unimodular random rooted graphs.
We get central limit type theorems for the total number of edges in the generalized random graphs with random vertex weights under different moment conditions on distributions of the weights.
We show that a random graph studied by Ioffe and Levit is an example of an inhomogeneous random graph of the type studied by Bollobas, Janson and Riordan, which enables us to give a new, simple, proof of their result on a phase transition.
In this paper, we investigate the induced subgraphs of percolated random geometric graphs, and get some asymptotic results for the expected number of the subgraphs. Moreover, we get the Poisson approximation for the counting by Stein's method. We also present some similar results for the expectation of Betti number of the associated percolated random geometric complexes.
In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to supercritical percolation clusters, graph trees converging to the continuum random tree and the homogenisation problem for nested fractals. A subsequential local limit theorem for the simple random walks on generalised Sierpinski carpet graphs is also presented.
Wildberger gave a method to construct a finite hermitian discrete hypergroup from a random walk on a certain kind of graphs. In this article, we reveal that his method is applicable to a random walk on a certain kind of infinite graphs. Moreover, we make some observations of finite or infinite graphs on which a random walk produces a hermitian discrete hypergroup.
In this paper we study the one dimensional random geometric graph when the location of the nodes are independent and exponentially distributed. We derive exact results and the limit theorems for the connectivity and other properties associated with this random graph. We show that the asymptotic properties of a graph with a truncated exponential distribution can be obtained using the exponential random geometric graph.
We consider a class of biased random walks on infinite graphs and present several general results on the spectral radius of biased random walk.
Due to the popularity of randomly evolving graph processes, there exists a randomized version of many recursively defined graph models. This is also the case with the cherry tree, which was introduced by Bukszár and Prékopa to improve Bonferroni type upper bounds on the probability of the union of random events. Here we consider a substantially extended random analogue of that model, embedding it into a general time dependent branching process.
An infinite graph G has the property that a random walk in random environment on G defined by i.i.d. resistances with any common distribution is almost surely transient, if and only if for some p<1, simple random walk is transient on a percolation cluster of G under bond percolation with parameter p.
We give asymptotics for the cumulative distribution function (CDF) for degrees of large dense random graphs sampled from a graphon. The proof is based on precise asymptotics for binomial random variables. Replacing the indicator function in the empirical CDF by a smoother function, we get general asymptotic results for functionals of homomorphism densities for partially labeled graphs with smoother functions. This general setting allows to recover recent results on asymptotics for homomorphism densities of sampled graphon.
We establish universality of cutoff for simple random walk on a class of random graphs defined as follows. Given a finite graph $G=(V,E)$ with $|V|$ even we define a random graph $ G^*=(V,E \cup E')$ obtained by picking $E'$ to be the (unordered) pairs of a random perfect matching of $V$. We show that for a sequence of such graphs $G_n$ of diverging sizes and of uniformly bounded degree, if the minimal size of a connected component of $G_n$ is at least 3 for all $n$, then the random walk on $G_n^*$ exhibits cutoff w.h.p. This provides a simple generic operation of adding some randomness to a g…
We prove that the trace of a transient branching random walk on a planar hyperbolic Cayley graph has a.s. continuum many ends and no isolated end.
Recently, Kigami's resistance form framework has been applied to provide a general approach for deriving the scaling limits of random walks on graphs with a fractal scaling limit. As an illustrative example, this article describes an application to the random conductance model with heavy tails on nested fractal graphs.
Coalescing random walk on a unimodular random rooted graph for which the root has finite expected degree visits each site infinitely often almost surely. A corollary is that an opinion in the voter model on such graphs has infinite expected lifetime. Additionally, we deduce an adaptation of our main theorem that holds uniformly for coalescing random walk on finite random unimodular graphs with degree distribution stochastically dominated by a probability measure with finite mean.
We apply here methods of inhomogeneous random graphs to a class of random distance graphs. This provides an example outside of the rank 1 models which is still solvable as long as the largest connected component is concerned. In particular, we show that some random distance graphs behave exactly as the classical Erdős-Rényi model not only in the supercritical phase (as was already known) but in the subcritical case as well.
A scaling limit for the simple random walk on the largest connected component of the Erdos-Renyi random graph in the critical window is deduced. The limiting diffusion is constructed using resistance form techniques, and is shown to satisfy the same quenched short-time heat kernel asymptotics as the Brownian motion on the continuum random tree.
We study a random graph model which is a superposition of the bond percolation model on $Z^d$ with probability $p$ of an edge, and a classical random graph $G(n, c/n)$. We show that this model, being a {\it homogeneous} random graph, has a natural relation to the so-called "rank 1 case" of {\it inhomogeneous} random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters $c\geq 0$ and $0 \leq p<p_c$, where $p_c=p_c(d)$ is the critical probability for the bond percolation on $Z^d$. The phase transition is sim…
We consider the random walk attachment graph introduced by Saramäki and Kaski and proposed as a mechanism to explain how behaviour similar to preferential attachment may appear requiring only local knowledge. We show that if the length of the random walk is fixed then the resulting graphs can have properties significantly different from those of preferential attachment graphs, and in particular that in the case where the random walks are of length 1 and each new vertex attaches to a single existing vertex the proportion of vertices which have degree 1 tends to 1, in contrast to preferential at…
We derive normal approximation bounds in the Wasserstein distance for sums of weighted U-statistics, based on a general distance bound for functionals of independent random variables of arbitrary distributions. Those bounds are applied to normal approximation for the combined weights of subgraphs in the Erdős-Rényi random graph, extending the graph counting results of [1] to the setting of graph weighting. Our approach relies on a general stochastic analytic framework for functionals of independent random sequences.
On the plane, every random compact set with almost surely uncountable first projection intersects with a high probability the graph of some continuous function. Implication: every black noise over the plane fails to factorize when the plane is split by such graph.
We consider three classes of random graphs: edge random graphs, vertex random graphs, and vertex-edge random graphs. Edge random graphs are Erdos-Renyi random graphs, vertex random graphs are generalizations of geometric random graphs, and vertex-edge random graphs generalize both. The names of these three types of random graphs describe where the randomness in the models lies: in the edges, in the vertices, or in both. We show that vertex-edge random graphs, ostensibly the most general of the three models, can be approximated arbitrarily closely by vertex random graphs, but that the two categ…
We investigate the asymptotic number of induced subgraphs in power-law uniform random graphs. We show that these induced subgraphs appear typically on vertices with specific degrees, which are found by solving an optimization problem. Furthermore, we show that this optimization problem allows to design a linear-time, randomized algorithm that distinguishes between uniform random graphs and random graph models that create graphs with approximately a desired degree sequence: the power-law rank-1 inhomogeneous random graph. This algorithm uses the fact that some specific induced subgraphs appear …
We prove local convergence results for the uniformly random, labelled or unlabelled, graphs from subcritical families. As an example special case, we prove Benjamini-Schramm convergence for the uniform random unlabelled tree. We introduce a compactification of the space of countable (connected) rooted graphs, and use it to generalise the notion of Benjamini-Schramm convergence in order to allow for vertices of infinite degree in the limit object.
We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (i) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (ii) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third m…
In this note we study inhomogeneous random bipartite graphs in random environment. These graphs can be thought of as an extension of the classical Erdös-Rényi random graphs in a random environment. We show that the expected number of perfect matchings obeys a precise quenched asymptotic and that it can be approximated using an iterative process that converges exponentially fast.
We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits.
We use the order complex corresponding to a symmetric matrix (defined by Giusti et al in 2015). In this note, we use it to define a class of models of random graphs, and show some surprising experimental results, showing sharp phase transitions.
In this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models of random graphs when considered at criticality. Specifically, we apply our method to a model of random intersection graph, a random graph obtained through $p$-bond percolation on a general $d$-regular graph, and a model of inhomogeneous random graph.
We introduce a construction that gives rise to a variety of "geometric" finite random graphs, and describe connections to the Poisson boundary, Naim's kernel, and Sznitman's random interlacements.
We obtained a $ψ_1$ estimate for the sum of Rademacher random variables under condition that they are dependent.
Random key graphs are random graphs induced by the random key predistribution scheme of Eschenauer and Gligor under the assumption of full visibility. For this class of random graphs we show the existence of a zero-one law for the appearance of triangles, and identify the corresponding critical scaling. This is done by applying the method of first and second moments to the number of triangles in the graph.
We study the random planar graph process introduced by Gerke, Schlatter, Steger, and Taraz [The random planar graph process, Random Structures Algorithms 32 (2008), no. 2, 236--261; MR2387559]: Begin with an empty graph on $n$ vertices, consider the edges of the complete graph $K_n$ one by one in a random ordering, and at each step add an edge to a current graph only if the graph remains planar. They studied the number of edges added up to step $t$ for 'large' $t=ω(n)$. In this paper we extend their results by determining the asymptotic number of edges added up to step $t$ in the early evoluti…
We study a recent model for edge exchangeable random graphs introduced by Crane and Dempsey; in particular we study asymptotic properties of the random simple graph obtained by merging multiple edges. We study a number of examples, and show that the model can produce dense, sparse and extremely sparse random graphs. One example yields a power-law degree distribution. We give some examples where the random graph is dense and converges a.s. in the sense of graph limit theory, but also an example where a.s. every graph limit is the limit of some subsequence. Another example is sparse and yields c…
A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution in time that is at most polynomial in the size of the graph. This fundamental property, however, only holds if the graph does not change over time, while on the other hand many distributed networks are inherently dynamic, and their topology is subjected to potentially drastic changes. In this work we study the mixing (i.e., converging) properties of random …
We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to the core of a random planar graph is of order c log(n) for an explicit constant c. These results provide new information on the structure of random planar graphs.
This paper studies thresholds in random generalized Johnson graphs for containing large cycles, i.e. cycles of variable length growing with the size of the graph. Thresholds are obtained for different growth rates.
Random walks on edge colored random graphs
研究边着色随机图上的随机游走，其中颜色表示使用成本，蓝色免费，红色每次经过产生单位成本。
We develop a clear connection between deFinetti's theorem for exchangeable arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph limits (work of Lovasz and many coauthors). Along the way, we translate the graph theory into more classical probability.
We study the behaviour of random labelled and unlabelled cographs with n vertices as n tends to infinity. Our main result is a novel probabilistic limit in the space of graphons.
We show the asymptotic degree distribution of the typical vertex of a sparse inhomogeneous random intersection graph.
When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman and Singer-Cohen [Combin Probab Comput 8 (1999), 131--159] introduced a random intersection graph by taking randomly assigned sets. The random intersection graph $G(n,m;p)$ has $n$ …
We study the asymptotic shape of random unlabelled graphs subject to certain subcriticality conditions. The graphs are sampled with probability proportional to a product of Boltzmann weights assigned to their $2$-connected components. As their number of vertices tends to infinity, we show that they admit the Brownian tree as Gromov--Hausdorff--Prokhorov scaling limit, and converge in a strengthened Benjamini--Schramm sense toward an infinite random graph. We also consider a family of random graphs that are allowed to be disconnected. Here a giant connected component emerges and the small fragm…
In previous papers, threshold probabilities for the properties of a random distance graph to contain strictly balanced graphs were found. We extend this result to arbitrary graphs and prove that the number of copies of a strictly balanced graph has asymptotically Poisson distribution at the threshold.
We present new results concerning threshold functions for a wide family of random intersection graphs. To this end we apply the coupling method used for establishing threshold functions for homogeneous random intersection graphs introduced by Karoński, Scheinerman, and Singer--Cohen. In the case of inhomogeneous random intersection graphs the method has to be considerably modified and extended. By means of the altered method we are able to establish threshold functions for a general random intersection graph for such properties as $k$-connectivity, matching containment or hamiltonicity. Moreov…
If a vertex $v$ in a graph $G$ has degree larger than the average of the degrees of its neighbors, we call it a groupie in $G$. In the current work, we study the behavior of groupie in random multipartite graphs with the link probability between sets of nodes fixed. Our results extend the previous ones on random (bipartite) graphs.
We show that asymptotic equivalence, in a strong form, holds between two random graph models with slightly differing edge probabilities under substantially weaker conditions than what might naively be expected. One application is a simple proof of a recent result by van den Esker, van der Hofstad and Hooghiemstra on the equivalence between graph distances for some random graph models.
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