Related papers: Expected Crossing Numbers
We study the expected value of the length $L_n$ of the minimum spanning tree of the complete graph $K_n$ when each edge $e$ is given an independent uniform $[0,1]$ edge weight. We sharpen the result of Frieze \cite{F1} that…
Consider a random geometric graph $G$ with a vertex set defined by a Poisson point process with intensity $t>0$ in a convex body. We can generate a drawing of the graph by projecting the construction onto some plane $L$. Choosing different…
We consider the P\'olya random walk in $\mathbb{Z}^2$. The paper establishes a number of results for the distributions and expectations of the number of usual (undirected) and specifically defined in the paper up- and down-directed…
The number of triangles in a graph is useful to deduce a plethora of important features of the network that the graph is modeling. However, finding the exact value of this number is computationally expensive. Hence, a number of…
Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone…
We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include…
Let $V$ be a set of $n$ vertices, ${\cal M}$ a set of $m$ labels, and let $\mathbf{R}$ be an $m \times n$ matrix of independent Bernoulli random variables with success probability $p$. A random instance $G(V,E,\mathbf{R}^T\mathbf{R})$ of…
Exchangeable random graphs serve as an important probabilistic framework for the statistical analysis of network data. In this work we develop an alternative parameterization for a large class of exchangeable random graphs, where the nodes…
We derive two conditional expectation bounds, which we use to simplify cryptographic security proofs. The first bound relates the expectation of a bounded random variable and the average of its conditional expectations with respect to a set…
Reinforced random walks are random walks on graphs whose transition probabilities along edges from a vertex are proportional to the weights of those edges, but where the weight of an edge evolves in a way that depends on the past traversals…
We consider the length of {\em ordered loose paths} in the random $r$-uniform hypergraph $H=H^{(r)}(n, p)$. A ordered loose path is a sequence of edges $E_1,E_2,\ldots,E_\ell$ where $\max\{j\in E_i\}=\min\{j\in E_{i+1}\}$ for $1\leq…
The aim of this paper is to establish Hoeffding and Bernstein type concentration inequalities for weighted sums of exchangeable random variables. A special case is the i.i.d. setting, where random variables are sampled independently from…
Nowadays, the emergence of online services provides various multi-relation information to support the comprehensive understanding of the epidemic spreading process. In this Letter, we consider the edge weights to represent such multi-role…
Karonski, Luczak, and Thomason (2004) conjectured that, for any connected graph G on at least three vertices, there exists an edge weighting from {1,2,3} such that adjacent vertices receive different sums of incident edge weights.…
We consider a variation on Maker-Breaker games on graphs or digraphs where the edges have random costs. We assume that Maker wishes to choose the edges of a spanning tree, but wishes to minimise his cost. Meanwhile Breaker wants to make…
Let $Q_n(x)=\sum_{i=0}^{n} A_{i}x^{i}$ be a random polynomial where the coefficients $A_0,A_1,... $ form a sequence of centered Gaussian random variables. Moreover, assume that the increments $\Delta_j=A_j-A_{j-1}$, $j=0,1,2,...$ are…
We find exact and asymptotic formulas for the average values of several statistics on set partitions: of Carlitz's $q$-Stirling distributions, of the numbers of crossings in linear and circular representations of set partitions, of the…
We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of…
We study random graphs with latent geometric structure, where the probability of each edge depends on the underlying random positions corresponding to the two endpoints. We focus on the setting where this conditional probability is a…
We analyze the properties of degree-preserving Markov chains based on elementary edge switchings in undirected and directed graphs. We give exact yet simple formulas for the mobility of a graph (the number of possible moves) in terms of its…