If $G$ is a connected graph with vertex set $V$, then the eccentric connectivity index of $G$, $xi^c(G)$, is defined as $sum_{vin V(G)}deg(v)ecc(v)$ where $deg(v)$ is the degree of a vertex $v$ and $ecc(v)$ is its eccentricity. In this paper we show some convergence in probability and an asymptotic normality based on this index in random bucket recursive trees.

Abstract Kernelized Gram matrix $W$ constructed from data points $\{x_i\}_{i=1}^N$ as $W_{ij}= k_0( \frac{ \| x_i - x_j \|^2} {\sigma ^2} ) $ is widely used in graph-based geometric analysis and unsupervised learning. An important question how to choose the kernel bandwidth $\sigma $, a common practice called self-tuned adaptively sets _i$ at each point $x_i$ by $k$-nearest neighbor (kNN) dista...

The distributions of traffics are defined and are applied for families of larges random matrices, random groups and infinite random rooted graphs with uniformly bounded degree. There are constructed by adding axioms in Voiculescu’s definition of ∗-distribution of non commutative random variables. The convergence in distribution of traffics generalizes Benjamini, Schramm, Aldous, Lyons’ weak loc...

In this paper we introduce a new notion of convergence of sparse graphs which we call Large Deviations or LD-convergence and which is based on the theory of large deviations. The notion is introduced by ”decorating” the nodes of the graph with random uniform i.i.d. weights and constructing random measures on [0, 1] and [0, 1]2 based on the decoration of nodes and edges. A graph sequence is defi...

This paper presents estimates of the convergence rate and complexity of an algebraic multilevel preconditioner based on piecewise constant coarse vector spaces applied to the graph Laplacian. A bound is derived on the energy norm of the projection operator onto any piecewise constant vector space, which results in an estimate of the two-level convergence rate where the coarse level graph is obt...

We consider the convergence problem of consensus seeking agents and the effect of probabilistic switching on the convergence rate. Using a probabilistic setting, we develop a framework to study the convergence rate for different classes of graph topologies, including Small World networks. We show that by making the effective diameter of the underlying graph small, probabilistic switching can be...

We present a notion of convergence for sequences of finite graphs {Gn} that can be seen as a generalization of the Benjamini-Schramm convergence notion for bounded degree graphs, regarding the distribution of r-neighbourhoods of the vertices, and the left-convergence notion for dense graphs, regarding, given any finite graph F , the limit of the probabilities that a random map from V (F ) to V ...