Keynote Talk Summary: Algorithmic Aspects of Sensor Networks

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.

In this paper we introduce new models of random graphs, arising from Latin squares which include random Cayley graphs as a special case. We investigate some properties of these graphs including their clique, independence and chromatic numbers, their expansion properties as well as their connectivity and Hamiltonicity. The results obtained are compared with other models of random graphs and several similarities and differences are pointed out. For many properties our results for the general case are as strong as the known results for random Cayley graphs and sometimes improve the previously best results for the Cayley case. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011

Entropy of the ensemble of acyclic graphs

Graph rigidity provides the conditions of unique localizability for cooperative localization of wireless ad hoc and sensor networks. Specifically, redundant rigidity and 3-connectivity are necessary and sufficient conditions for unique localizability of generic configurations. In this article, we introduce a graph invariant for 3-connectivity, called 3-connectivity index. Using this index along with the rigidity and redundancy indices provided in previous work, we explore the rigidity and connectivity properties of two classes of graphs, namely, random geometric graphs and clustered graphs. We have found out that, in random geometric graphs and clustered graphs, it needs significantly less effort to achieve 3-connectivity once we obtain redundant rigidity. In reconsidering the general conditions for unique localizability, the most striking finding in random geometric graphs is that it is unlikely to observe a graph, in which 3-connectivity is satisfied before the graph becomes redundantly rigid. Therefore, in random geometric graphs, it is more likely sufficient to test only 3-connectivity for unique localizability. On the contrary to random geometric graphs, our findings indicate that 3-connectivity may be satisfied before the graph becomes redundantly rigid in clustered graphs, which means that, in clustered graphs, we have to test both redundant rigidity and 3-connectivity for unique localizability.

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 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.

Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung–Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical textsf {NP}-hard optimization problems on PLB networks. It is known that on general graphs with maximum degree Delta, a greedy algorithm, which chooses nodes in the order of their degree, only achieves a Omega (ln Delta )-approximation for Minimum Vertex Cover and Minimum Dominating Set, and a Omega (Delta )-approximation for Maximum Independent Set. We prove that the PLB-U property with beta >2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are APX-hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless P = NP. Furthermore, we prove that all three problems are in MAX SNP if the PLB-U property holds.

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.

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.

Mixed-weight open locating-dominating sets (mixed-weight OLD-sets) model systems that use sensors of multiple strengths to locate and detect problems in a network. Randomly-distributed wireless sensor networks, which often use sensors of different strengths, can be modeled by random geometric graphs. We build and study an Integer Linear Program to solve for mixed-weight OLD-sets and use greedy algorithms to generate mixed-weight OLD-set estimates in random geometric graphs. We also extend our previous results for mixed-weight OLD-sets in random graphs [6] to random geometric graphs by estimating the probabilistic upper bound for the size of the set.

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.

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.

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.

We study the connectivity properties of random Bluetooth graphs that model certain \ad hoc wireless networks. The graphs are obtained as \irrigation subgraphs of the well-known random geometric graph model. There are two parameters that control the model: the radius r that determines the \visible neighbors of each node and the number of edges c that each node is allowed to send to these. The randomness comes from the underlying distribution of data points in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters r;c and completely characterize the connectivity threshold (in c) for values of r close the critical value for connectivity in the underlying random geometric graph.