Related papers: Closed Graph Theorem For Linear Random Operators

Let $G$ be any graph of minimum degree at least $k$, and let $G_p$ be the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. Recently, Krivelevich, Lee and Sudakov showed that if $pk\to\infty$ then with…

Graph-theoretic methods have seen wide use throughout the literature on multi-agent control and optimization. When communications are intermittent and unpredictable, such networks have been modeled using random communication graphs. When…

The Rado Graph, sometimes also known as the (countable) Random Graph, can be generated almost surely by putting an edge between any pair of vertices with some fixed probability $p \in (0, 1)$, independently of other pairs. In this article,…

A class of doubly stochastic graph shift operators (GSO) is proposed, which is shown to exhibit: (i) lower and upper $L_{2}$-boundedness for locally stationary random graph signals; (ii) $L_{2}$-isometry for \textit{i.i.d.} random graph…

In many branches of engineering, Banach contraction mapping theorem is employed to establish the convergence of certain deterministic algorithms. Randomized versions of these algorithms have been developed that have proved useful in…

Starting from Sinclair's 1976 work {\it Automatic Continuity of Linear Operators}, Cambridge University Press, (1976), on automatic continuity of linear operators on Banach spaces, we prove that sequences of intertwining continuous linear…

One of the main questions that arise when studying random and quasi-random structures is which properties P are such that any object that satisfies P "behaves" like a truly random one. In the context of graphs, Chung, Graham, and Wilson…

There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable structured class, such as the class of all planar graphs. Here we consider a general 'bridge-addable' class of graphs - if a graph…

This is the first in a series of six articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. Many of…

A closure operator on a set $X$ is a function $\operatorname{cl}: \wp(X) \to \wp(X)$ satisfying, for all $A, B \subseteq X$, the following properties: extensivity, $A \subseteq \operatorname{cl}(A)$; monotonicity, which states that if $A…

We define a growing model of random graphs. Given a sequence of nonnegative integers $\{d_n\}_{n=0}^\infty$ with the property that $d_i\leq i$, we construct a random graph on countably infinitely many vertices $v_0,v_1\ldots$ by the…

We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables.…

Let $E,F$ be two topological spaces and $u:E\rightarrow F$ be a map. \ If $F$ is Haudorff and $u$ is continuous, then its graph is closed. \ \ The Closed Graph Theorem establishes the converse when $E$ and $F$ are suitable objects of…

I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key…

This paper presents an alternative proof of the celebrated friendship theorem, originally established by Erd\H{o}s, R\'{e}nyi, and S\'{o}s (1966). The proof relies on a closed-form expression for the Lov\'{a}sz $\vartheta$-function of…

The concept of graph powers has been extensively studied in graph theory. Analogous to graph powers, Chandran et al. [3] introduced the notion of bipartite powers for bipartite graphs. In this paper, we show that the class of interval…

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"…

We introduce a class of generative network models that insert edges by connecting the starting and terminal vertices of a random walk on the network graph. Within the taxonomy of statistical network models, this class is distinguished by…

The stochastic Kronecker Graph model can generate large random graph that closely resembles many real world networks. For example, the output graph has a heavy-tailed degree distribution, has a (low) diameter that effectively remains…

The theory of random graphs goes back to the late 1950s when Paul Erd\H{o}s and Alfr\'ed R\'enyi introduced the Erd\H{o}s-R\'enyi random graph. Since then many models have been developed, and the study of random graph models has become…