Graph diameter in long‐range percolation

We propose a general model for testing graph properties, which extends and simplifies the bounded degree model of Goldreich and Ron [Property Testing in Bounded Degree Graphs, Proc. 31st Annual ACM Symposium on the Theory of Computing, 1997, pp. 406–415.] In this model, we present a family of algorithms that test whether the diameter of a graph is bounded by a given parameterD, or is ϵ‐far from any graph with diameter at most β(D). The function β(D) ranges betweenD+4 and 4D+2, depending on the algorithm. All our algorithms run in time polynomial in 1/ϵ. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 165–183, 2002

We study the cover time of random geometric graphs. Let $I(d)=[0,1]^{d}$ denote the unit torus in d dimensions. Let $D(x,r)$ denote the ball (disc) of radius r. Let $\Upsilon_d$ be the volume of the unit ball $D(0,1)$ in d dimensions. A random geometric graph $G=G(d,r,n)$ in d dimensions is defined as follows: Sample n points V independently and uniformly at random from $I(d)$. For each point x draw a ball $D(x,r)$ of radius r about x. The vertex set $V(G)=V$ and the edge set $E(G)=\{\{v,w\}: w\ne v,\,w\in D(v,r)\}$. Let $G(d,r,n),\,d\geq 3$ be a random geometric graph. Let $C_G$ denote the cover time of a simple random walk on G. Let $c>1$ be constant, and let $r=(c\log n/(\Upsilon_dn))^{1/d}$. Then whp the cover time satisfies © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 324–349, 2011

We destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. Informally, the associated cut‐tree describes the genealogy of the connected components created by this destruction process. We provide a general criterion for the convergence of the rescaled cut‐tree in the Gromov‐Prohorov topology to an interval endowed with the Euclidean distance and a certain probability measure, when the underlying tree has branching points close to the root and height of order . In particular, we consider uniform random recursive trees, binary search trees, scale‐free random trees and a mixture of regular trees. This yields extensions of a result in Bertoin (Probab Stat 5 (2015), 478–488) for the cut‐tree of uniform random recursive trees and also allows us to generalize some results of Kuba and Panholzer (Online J Anal Combin (2014), 26) on the multiple isolation of vertices. The approach relies in the close relationship between the destruction process and Bernoulli bond percolation, which may be useful for studying the cut‐tree of other classes of trees. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 404–427, 2017

In this chapter, we discuss two classical problems lying on the edge of graph theory and combinatorial geometry. The first problem is due to E. Nelson. It consists of coloring metric spaces in such a way that pairs of points at some prescribed distances receive different colors. The second problem is attributed to K. Borsuk and involves finding the minimum number of parts of smaller diameter into which an arbitrary bounded nonsingleton point set in a metric space can be partitioned. Both problems are easily translated into the language of graph theory, provided we consider, instead of the whole space, any (finite) distance graph or any (finite) graph of diameters. During the last decades, a huge number of ideas have been proposed for solving both problems, and many results in both directions have been obtained. In the survey below, we try to give an entire picture of this beautiful area of geometric combinatorics.

We study Maker‐Breaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker's win in a game in which Maker's goal is to build a graph which admits some monotone increasing property \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{P}\end{align*} \end{document}. We focus on three natural target properties for Maker's graph, namely being k ‐vertex‐connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k ‐vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojaković and Szabó. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011

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

We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieltjes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdös‐Rényi graphs and graphs with a given degree sequence. We give examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

In this paper, we present a method for the numerical minimization of the Mumford–Shah functional that is based on the idea of topological asymptotic expansions. The basic idea is to cover the expected edge set with balls of radius \epsilon > 0 and use the number of balls, multiplied with 2 \epsilon , as an estimate for the length of the edge set. We introduce a functional based on this idea and prove that it converges in the sense of \Gamma -limits to the Mumford–Shah functional. Moreover, we show that ideas from topological asymptotic analysis can be used for determining where to position the balls covering the edge set. The results of the proposed method are presented by means of two numerical examples and compared with the results of the classical approximation due to Ambrosio and Tortorelli.

Cognitive Radio (CR) networks have become an important component of the modern communication infrastructure due to their capability of improving spectrum usage efficiency by exploiting channels opportunistically. In CR networks, the network topology changes very frequently because of the temporarily available channels and dynamic transmitting parameters (e.g. transmission power and transmitting frequency), which may even result in network disconnectivity from time to time. Hence an interesting and open question is that: are there bounds on end-to-end delay between a source-destination pair with Euclidean distance d apart in such networks? These bounds are required for time-critical applications. This paper first investigates the nature of topology-incurred end-to-end delay in large multi-channel CR networks and then identifies the conditions under which the asymptotic topology-incurred delay scales linearly with the Euclidean distance (d); that is, the conditions under which the end-to-end delay is bounded. The results in this paper are validated through extensive simulations and can advance our understanding of CR network performance.

This paper proposes a locally convexified rigid tube model predictive control for obstacle avoidance for linear, discrete‐time, systems subject to additive, bounded disturbances and state and control constraints. The inherently nonconvex obstacle avoidance constraints are locally convexified and converted into closed polyhedral constraints by deploying recent notions of safe sets and tubes. The proposed method leverages these recent notions of safe sets and tubes to reformulate the original nonconvex optimization problem for obstacle avoidance within the context of disturbances into a computationally highly efficient, strictly convex quadratic programming problem. In order to analyze the separation between a predicted state set and each obstacle avoidance constraint set, we employ the Euclidean distance between these two sets and the radius of the Chebyshev ball of their intersection. In the strict separation case, the separating hyperplanes are constructed by utilizing the separation theorem for convex sets, while in the weak separation case, the construction of separating hyperplanes is achieved through the use of the separation theorem and the properties of convex cones. The proposed method provides guarantees of strong system theoretic properties such as robust recursive feasibility, robust positive invariance and robust stability of the controlled uncertain system. A simulation example is also provided to numerically verify the effectiveness of the proposed method.

The graph spectrum of a graph is a sequence of eigenvalues of its Laplacian matrix. It quantitatively provides some graph information, e.g., structure, topology, and connectivity, and is extensively used in the field of structural comparison. In this paper, by adopting the concept of minimum weight maximal bipartite matching, we improve the distance measuring on protein graph spectra. In the analysis of protein subunit matching problem, corroborating protein databases CATH, SCOP, and RCSB PDB, results suggest that when proposed method is applied, a similar single polypeptide chain can be identified in MHC (the Major Histocompatibility Complex) protein complexes. In addition, the proposed method outperforms than typical geometric metric distance. This graph-theoretic approach offers a practical direction for protein subunit comparison.

The strong chromatic index of a graph , denoted by , is defined as the least number of colors in a coloring of edges of , such that each color class is an induced matching (or: if edges and have the same color, then both vertices of are not adjacent to any vertex of ). A graph is a unit distance graph in if vertices of can be uniquely identified with points in , so that is an edge of if and only if the Euclidean distance between the points identified with and is 1. We would like to find the largest possible value of , where is a unit distance graph (in and ) of maximum degree . We show that , where is a unit distance graph in of maximum degree . We also show that the maximum possible size of a strong clique in unit distance graph in is linear in and give a tighter result for unit distance graphs in the plane.

In this study, we consider the problem of estimating the diameter of a graph, that is, the maximum distance between any two vertices, in linear time. We address a question posed in the literature—whether there exists an interesting graph class with arbitrarily large cycles for which breadth first search (BFS) would always return high‐eccentricity vertices. We answer this question positively, defining a class of graphs that generalizes AT‐free graphs and has no bound on the size of induced cycles, yet BFS always returns a vertex whose eccentricity is within a constant difference from the diameter. In addition, we consider the question—also explicitly stated in the literature—whether some variant of the so‐called multisweep algorithm would always return a high‐eccentricity vertex. We show that the answer is negative by describing a family of graphs for which no variant of multisweep BFS can return a vertex of eccentricity higher than half of the diameter plus a constant.

We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most n^{1/4 + o_n(1)} in n units of time. Together with the complementary lower bound proven by Gwynne and Miller (2017) this shows that the typical graph distance displacement of the walk after n steps is n^{1/4 + o_n(1)}, as conjectured by Benjamini and Curien (Geom Funct Anal 2(2):501–531, 2013. arXiv:1202.5454). More generally, we show that the simple random walks on a certain family of random planar maps in the gamma -Liouville quantum gravity (LQG) universality class for gamma in (0,2)—including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps—typically travels graph distance n^{1/d_gamma + o_n(1)} in n units of time, where d_gamma is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on gamma by Ding and Gwynne (Commun Math Phys 374:1877–1934, 2018. arXiv:1807.01072). Since d_gamma > 2, this shows that the simple random walk on each of these maps is subdiffusive. Our proofs are based on an embedding of the random planar maps under consideration into {mathbb {C}} wherein graph distance balls can be compared to Euclidean balls modulo subpolynomial errors. This embedding arises from a coupling of the given random planar map with a mated-CRT map together with the relationship of the latter map to SLE-decorated LQG.

For<i> j ≤ k</i>, the <i>L(j, k)</i>- labeling problem arose form the code assignment problem of the wireless network. That is, let n,j,k be non-negative real numbers with j ≤ k, an n-<i>L(j, k)</i>-labeling of a graph G is mapping f: V(G)→[0, n] such that |f(u)-f(v)| ≥ j if d(u, v)=1, and |f(u)-f(v)|≥k if t d(u, v)=2. The span of f is the difference between the maximum and minimum labeling numbers assigned by <i>f</i>. The <i>L(j, k)</i>-labeling number of graph G, denoted by λ<sub>(j,k)</sub> (G), is the minimum span of all<i> L(j, k)</i>-labeling of G. The infinite distance graph, denoted by D (d<sub>1</sub>,d<sub>2</sub>,…,d<sub>k</sub>), has the set Z of integers as a vertex set and in which two vertices i,j∈Z are adjacent if and only if |i-j|∈D. The finite distance graph, denoted by D<sub>n</sub> (d<sub>1</sub>,d<sub>2</sub>,…,d<sub>k</sub>), is the subgraph of D(d<sub>1</sub>,d<sub>2</sub>,…,d<sub>k</sub>) induced by vertices {0,1,…,n-1}. This paper determines the <i>L(j, k)</i>-labeling number and the circular <i>L(j, k)</i>-labeling number of distance graph D<sub>n</sub> (1,5) for 2j ≤ k.