Discussion on "Sparse graphs using exchangeable random measures" by F. Caron and E. B. Fox

Discussion on "Sparse graphs using exchangeable random measures" by F. Caron and E. B. Fox. In this discussion we contribute to the analysis of the GGP model as compared to the Erdos-Renyi (ER) and the preferential attachment (AB) models, using different measures such as number of connected components, global clustering coefficient, assortativity coefficient and share of nodes in the core.Comment: 2 pages, 1 figur

Forbidden Subgraphs in Connected Graphs

Given a set $\xi=\{H_1,H_2,...\}$ of connected non acyclic graphs, a $\xi$-free graph is one which does not contain any member of $$ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let {\gr{W}}_{k,\xi} be theexponential generating function (EGF for brief) of connected $\xi$-free graphs of excess equal to $k$ ($k \geq 1$). For each fixed $\xi$, a fundamental differential recurrence satisfied by the EGFs {\gr{W}}_{k,\xi} is derived. We give methods on how to solve this nonlinear recurrence for the first few values of $k$ by means of graph surgery. We also show that for any finite collection $\xi$ of non-acyclic graphs, the EGFs {\gr{W}}_{k,\xi} are always rational functions of the generating function, $T$, of Cayley's rooted (non-planar) labelled trees. From this, we prove that almost all connected graphs with $n$ nodes and $n+k$ edges are $\xi$-free, whenever $k=o(n^{1/3})$ and $|\xi| < \infty$ by means of Wright's inequalities and saddle point method. Limiting distributions are derived for sparse connected $\xi$-free components that are present when a random graph on $n$ nodes has approximately $\frac{n}{2}$ edges. In particular, the probability distribution that it consists of trees, unicyclic components, $...$, $(q+1)$-cyclic components all $\xi$-free is derived. Similar results are also obtained for multigraphs, which are graphs where self-loops and multiple-edges are allowed

Random Graphs with Hidden Color

We propose and investigate a unifying class of sparse random graph models, based on a hidden coloring of edge-vertex incidences, extending an existing approach, Random graphs with a given degree distribution, in a way that admits a nontrivial correlation structure in the resulting graphs. The approach unifies a number of existing random graph ensembles within a common general formalism, and allows for the analytic calculation of observable graph characteristics. In particular, generating function techniques are used to derive the size distribution of connected components (clusters) as well as the location of the percolation threshold where a giant component appears.Comment: 4 pages, no figures, RevTe

Distributed distance-r covering problems on sparse high-girth graphs

We prove that the distance-r dominating set, distance-r connected dominating set, distance-r vertex cover, and distance-r connected vertex cover problems admit constant factor approximations in the CONGEST model of distributed computing in a constant number of rounds on classes of sparse high-girth graphs. In this paper, sparse means bounded expansion, and high-girth means girth at least 4r + 2. Our algorithm is quite simple; however, the proof of its approximation guarantee is non-trivial. To complement the algorithmic results, we show tightness of our approximation by providing a loosely matching lower bound on rings. Our result is the first to show the existence of constant-factor approximations in a constant number of rounds in non-trivial classes of graphs for distance-r covering problems

Robust Geometric Spanners

Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in addition, geometric spanners. We define a property of spanners called robustness. Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices. We show that robust spanners must have a superlinear number of edges, even in one dimension. On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges.Comment: 18 pages, 8 figure

Dynamic Graph Stream Algorithms in o(n)o(n) Space

In this paper we study graph problems in dynamic streaming model, where the input is defined by a sequence of edge insertions and deletions. As many natural problems require $\Omega(n)$ space, where $n$ is the number of vertices, existing works mainly focused on designing $\tilde{O}(n)$ space algorithms. Although sublinear in the number of edges for dense graphs, it could still be too large for many applications (e.g. $n$ is huge or the graph is sparse). In this work, we give single-pass algorithms beating this space barrier for two classes of problems. We present $o(n)$ space algorithms for estimating the number of connected components with additive error $\varepsilon n$ and $(1+\varepsilon)$-approximating the weight of minimum spanning tree, for any small constant $\varepsilon>0$. The latter improves previous $\tilde{O}(n)$ space algorithm given by Ahn et al. (SODA 2012) for connected graphs with bounded edge weights. We initiate the study of approximate graph property testing in the dynamic streaming model, where we want to distinguish graphs satisfying the property from graphs that are $\varepsilon$-far from having the property. We consider the problem of testing $k$-edge connectivity, $k$-vertex connectivity, cycle-freeness and bipartiteness (of planar graphs), for which, we provide algorithms using roughly $\tilde{O}(n^{1-\varepsilon})$ space, which is $o(n)$ for any constant $\varepsilon$. To complement our algorithms, we present $\Omega(n^{1-O(\varepsilon)})$ space lower bounds for these problems, which show that such a dependence on $\varepsilon$ is necessary.Comment: ICALP 201