Random Latin square graphs

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power-law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real-world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007

In this talk, we discuss some abstract models of sensor networks and also some basic algorithmic problems. We first go through the basic architecture and communication capabilities of a single, ultra small sized, sensor. We distinguish broadcast capability (at a certain radius r around the sensor) and directional broadcasting at an angle (which can be reduced to almost zero in case of optical communication). Based on such features, we first go through the (well known) model of Random Geometric Graphs and its threshold properties for connectivity, average degree, chromatic number etc. Then, we define the model of Random Sector Graphs, in order to capture unidirectional sending at an angle. We provide thresholds for connectivity, clique and chromatic number there also and point out its differences from the Random Geometric Graphs model. We also discuss other models related to ways of configuring sensor nets, such as “for each point (sensor) connect to its k nearest neighbours” or “for each sensor, connect to k points within its communication range r”. We also refer to the Combinatorial model of Random Intersection Graphs (which abstracts the geometry out). It is interesting to see that some properties (e.g., sensors density necessary for connectivity) are “invariant” in all these models. We then go through some fundamental algorithmic issues and their existing solutions. We first discuss the problem of finding efficient local protocols aiming in propagating a local event E to a sink. We discuss several efficiency measures like hop count, energy spent, short paths. Randomised local protocols are then sketched, useful for robust and energy efficient local event reporting in case of a sensor net with failures at some nodes (or with energy depleted nodes). In this framework we indicate some energy-time tradeoffs for propagation of local information and notice their similarity with tradeoffs between area and time in VLSI circuits. The problem of propagation of local information in areas with obstacles (lakes) is also discussed. We then examine efficient routing policies and especially refer to greedy techniques in the case of existence of virtual coordinates. The topology control problem is the next topic. We present several local protocols for configuring the net and establishing global properties like connectivity or short paths. Finally, we discuss problems related to scenarios where a sensor net is “embedded” into a larger fixed-connectivity network. There, we provide the abstraction of a “sensors cloud” and discuss the possible enhancement of properties like maximum flow or connectivity or diameter of the resulting hybrid network. Our talk aims in showing the emergence of a new algorithmic subfield, useful to the pragmatic considerations in the actual design and control of such networks.

Entropy of the ensemble of acyclic graphs

Random graphs with latent geometric structure are popular models of social and biological networks, with applications ranging from network user profiling to circuit design. These graphs are also of purely theoretical interest within computer science, probability and statistics. A fundamental initial question regarding these models is: when are these random graphs affected by their latent geometry and when are they indistinguishable from simpler models without latent structure, such as the Erdős–Rényi graph $${\mathcal {G}}(n, p)$$ ? We address this question for two of the most well-studied models of random graphs with latent geometry—the random intersection and random geometric graph. Our results are as follows: (a) The random intersection graph is defined by sampling n random sets $$S_1, S_2, \ldots , S_n$$ by including each element of a set of size d in each $$S_i$$ independently with probability $$\delta $$ , and including the edge $$\{i, j\}$$ if $$S_i \cap S_j \ne \emptyset $$ . We prove that the random intersection graph converges in total variation to an Erdős–Rényi graph if $$d = {\tilde{\omega }}(n^3)$$ , and does not if $$d = o(n^3)$$ , for both dense and sparse edge densities p. This resolves an open problem in Fill et al. (Random Struct Algorithms 16(2):156–176, 2000), Rybarczyk (Random Struct Algorithms 38(1–2):205–234, 2011) and Kim et al. (Random Struct Algorithms 52(4):662–679, 2018). The same result was obtained simultaneously and independently by Bubeck et al. (When random intersection graphs lose geometry. Manuscript, 2019). (b) We strengthen the preceding argument to show that the matrix of random intersection sizes $$|S_i \cap S_j|$$ converges in total variation to a symmetric matrix with independent Poisson entries. This yields the first total variation convergence result for $$\tau $$ -random intersection graphs, where the edge $$\{i, j\}$$ is included if $$|S_i \cap S_j| \ge \tau $$ . More precisely, our results imply that, if p is bounded away from 1, then the $$\tau $$ -random intersection graph with edge density p converges to $${\mathcal {G}}(n, p)$$ if $$d = \omega (\tau ^3 n^3)$$ . (c) The random geometric graph on $${\mathbb {S}}^{d - 1}$$ is defined by sampling $$X_1, X_2, \dots , X_n$$ uniformly at random from $${\mathbb {S}}^{d - 1}$$ and including the edge $$\{i, j\}$$ if $$\Vert X_i - X_j \Vert _2 \le \tau $$ . A result of Bubeck et al. (Random Struct Algorithms 49:503–532, 2016) showed that this model converges to $${\mathcal {G}}(n, p)$$ in total variation, where p is chosen so that the models have matching edge densities, as long as $$d = \omega (n^3)$$ . It was conjectured in Bubeck et al. (2016) that this threshold decreases drastically for p small. We make the first progress towards this conjecture by showing convergence if $$d = {\tilde{\omega }}\left( \min \{ pn^3, p^2 n^{7/2} \} \right) $$ . Our proofs are a hybrid of combinatorial arguments, direct couplings and applications of information inequalities. Previous upper bounds on the total variation distance between random graphs with latent geometry and $${\mathcal {G}}(n, p)$$ have typically not been both combinatorial and information-theoretic, while this interplay is essential to the sharpness of our bounds.

Random graph models are used to describe the complex structure of real-world networks in diverse fields of knowledge. Studying their behaviour and fitting properties are still critical challenges that, in general, require model-specific techniques. An important line of research is to develop generic methods able to fit and select the best model among a collection. Approaches based on spectral density (i.e. distribution of the graph adjacency matrix eigenvalues) appeal to that purpose: they apply to different random graph models. Also, they can benefit from the theoretical background of random matrix theory. This work investigates the convergence properties of model fitting procedures based on the graph spectral density and the corresponding cumulative distribution function. We also review the convergence of the spectral density for the most widely used random graph models. Moreover, we explore through simulations the limits of these graph spectral density convergence results, particularly in the case of the block model, where only partial results have been established. random graphs, spectral density, model fitting, model selection, convergence.

Given a sequence of nonnegative real numbers λ0, λ1… which sum to 1, we consider random graphs having approximately λi n vertices of degree i. Essentially, we show that if Σ i(i ‐ 2)λi > 0, then such graphs almost surely have a giant component, while if Σ i(i ‐2)λ. < 0, then almost surely all components in such graphs are small. We can apply these results to Gn,p,Gn.M, and other well‐known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

Ad-regular graph has largest or rst (adjacency matrix) eigenvalue 1 = d. Consider for an even d 4, a random d-regular graph model formed from d=2 uniform, independent permutations on f1;:::;ng. We shall show that for any >0 we have all eigenvalues aside from 1 = d are bounded by 2 p d 1+ with probability 1 O(n ), where =d p d 1+ 1 =2e 1. We also show that this probability is at most 1 c=n 0 , for a constant c and a 0 that is either or +1 (\more often than + 1). We prove related theorems for other models of random graphs, including models with d odd. These theorems resolve the conjecture of Alon, that says that for any > 0a ndd, the second largest eigenvalue of \most random dregular graphs are at most 2 p d 1+ (Alon did not specify precisely what \most should mean or what model of random graph one should take).

We consider a formal discretisation of Euclidean quantum gravity defined by a statistical model of random 3-regular graphs and making using of the Ollivier curvature, a coarse analogue of the Ricci curvature. Numerical analysis shows that the Hausdorff and spectral dimensions of the model approach 1 in the joint classical-thermodynamic limit and we argue that the scaling limit of the model is the circle of radius r, . Given mild kinematic constraints, these claims can be proven with full mathematical rigour: speaking precisely, it may be shown that for 3-regular graphs of girth at least 4, any sequence of action minimising configurations converges in the sense of Gromov–Hausdorff to . We also present strong evidence for the existence of a second-order phase transition through an analysis of finite size effects. This—essentially solvable—toy model of emergent one-dimensional geometry is meant as a controllable paradigm for the nonperturbative definition of random flat surfaces.

Edge-independent random graphs are a model of random graphs in which each potential edge appears independently with an individual probability. Based on the relative entropy method, we determine the upper and lower bounds for the extremal vertex degrees using the edge probability matrix and its largest eigenvalue. Moreover, an application to random graphs with given expected degree sequences is presented.

In cluster analysis, the random graph model G n,p and G n,p-based multigraph models have been used for purposes of statistical modelling of data and testing the randomness of outlined clusters. While being appropriate for non-metric data, such models supposing independence of all edges do not take into account the triangle inequality which is valid for metric data. We will introduce graph models I n,dand I t,n,(d1,…,dt) for random intersection graphs in R 1 and multigraphs in R t under which the triangle inequality holds. We derive limit theorems for the distribution of random variables which describe important properties of these random intersection graphs. While being asymptotically equivalent for some properties like the limit distribution of the number of isolated points, the G n,p-model and the I n,d-model differ in numerous aspects.

Dependent phenomena, such as relational, spatial and temporal phenomena, tend to be characterized by local dependence in the sense that units which are close in a well-defined sense are dependent. In contrast with spatial and temporal phenomena, though, relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent. Owing to the challenge of characterizing local dependence and constructing random graph models with local dependence, many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference. We take first steps to characterize local dependence in random graph models, inspired by the notion of finite neighbourhoods in spatial statistics and M-dependence in time series, and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference. We show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy, but conventional exponential family random graph models do not satisfy. In addition, we establish a central limit theorem for random graph models with local dependence, which suggests that random graph models with local dependence are amenable to statistical inference. We discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure. In the absence of observed neighbourhood structure, we take a Bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures. We present simulation results and applications to two real world networks with 'ground truth'.

How spacetime emerges from featureless nothingness is one of the most intriguing questions in fundamental physics. In this thesis, we take on the random geometry approach to study discretized spacetime and follow the assumption that, in the sim- plest form, geometric structures may arise from random connections between dots under certain rules. We study a family of random graph models called Exponential Random Graph Models (ERGMs). Although this family was extensively investigated in the network science community as a proxy to study real-world social networks, its strength is in its formulation as a Gibbs-Boltzmann distribution in equilibrium sta- tistical mechanics. Thus, one can modify, analyze, and simulate the ensemble using familiar tools from statistical mechanics. We are interested in modifying the basic ERGMs to arrive at a random graph model that possesses emergent geometric properties. This would serve as a proof-of-principle that geometric spacetime may emerge from randomly connected dots. Our study leads to novel classes of random graphs whose edges can self-assemble themselves into both simple geometric primitives (e.g. triangles) and more complex structures (e.g. hexagons). The number of such structures is relatively large compared to the amount of dots and connections available in the graph. Lastly, but interestingly, our model is free from the graph collapse problem that is often observed in the traditional ERGMs.

This chapter presents a random graph model for image representation. The first contribution the authors propose is a Geometric-Edge (G-E) Random Graph Model for image representation. The second contribution is that of casting image matching into G-E Random Graph matching by using the random dot product graph based matching algorithm. Experimental results show that the proposed G-E Random Graph model and matching algorithm are effective and robust to structural variations.

Let L be chosen uniformly at random from among the latin squares of order n ≥ 4 and let r,s be arbitrary distinct rows of L. We study the distribution of σr,s, the permutation of the symbols of L which maps r to s. We show that for any constant c > 0, the following events hold with probability 1 ‐ o(1) as n → ∞: (i) σr,s has more than (log n)1−c cycles, (ii) σr,s has fewer than 9$\sqrt{n}$ cycles, (iii) L has fewer than $ 9\over 2$ n5/2 intercalates (latin subsquares of order 2). We also show that the probability that σr,s is an even permutation lies in an interval $[{1\over 4} - o(1), {3\over 4} + o(1)]$ and the probability that it has a single cycle lies in [2n‐2,2n‐2/3]. Indeed, we show that almost all derangements have similar probability (within a factor of n3/2) of occurring as σr,s as they do if chosen uniformly at random from among all derangements of {1,2,…,n}. We conjecture that σr,s shares the asymptotic distribution of a random derangement. Finally, we give computational data on the cycle structure of latin squares of orders n ≤ 11. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008

In the original (1961) Gilbert model of random geometric graphs, nodes are placed according to a Poisson point process, and links formed between those within a fixed range. Motivated by wireless ad hoc networks "soft" or "probabilistic" connection models have recently been introduced, involving a "connection function" H(r) that gives the probability that two nodes at distance r are linked (directly connect). In many applications (not only wireless networks), it is desirable that the graph is connected; that is, every node is linked to every other node in a multihop fashion. Here the connection probability of a dense network in a convex domain in two or three dimensions is expressed in terms of contributions from boundary components for a very general class of connection functions. It turns out that only a few quantities such as moments of the connection function appear. Good agreement is found with special cases from previous studies and with numerical simulations.