Related papers: The Distribution of Path Lengths On Directed Weigh…
The $k$-th power of the adjacency matrix of a simple undirected graph represents the number of walks with length $k$ between pairs of nodes. As a walk where no node repeats, a path is a walk where each node is only visited once. The set of…
We show that weighted path orders are special instances of a variant of semantic path orders. Exploiting this fact, we introduce a generalization of weighted path orders that goes beyond the realm of simple termination. Experimental data…
We investigate the paramater of the average range of $M$-Lipschitz mapping of a given graph. We focus on well-known classes such as paths, complete graphs, complete bipartite graphs and cycles and show closed formulas for computing this…
The shortest-path, commute time, and diffusion distances on undirected graphs have been widely employed in applications such as dimensionality reduction, link prediction, and trip planning. Increasingly, there is interest in using…
Directed acyclic graphs provide a fundamental tool for representing directed dependence structures in multivariate network data, and are widely used to model financial and economic networks. However, accurate and interpretable estimation…
We point out that the total number of trails and the total number of paths of given length, between two vertices of a simple undirected graph, are obtained as expectation values of specifically engineered quantum mechanical observables.…
Large unweighted directed graphs are commonly used to capture relations between entities. A fundamental problem in the analysis of such networks is to properly define the similarity or dissimilarity between any two vertices. Despite the…
We find rational expressions for all minors of the weighted path matrix of a directed graph, generalizing the classical Lindstrom/Gessel-Viennot result for acyclic directed graphs. The formulas are given in terms of certain flows in the…
A weighted coloured-edge graph is a graph for which each edge is assigned both a positive weight and a discrete colour, and can be used to model transportation and computer networks in which there are multiple transportation modes. In such…
We define a random walk problem which admits analytic results, on a class of infinite periodic lattices which are directed and colored. Our approach is motivated from the fact that such lattices arise in string theoretic constructs of…
In this article, we consider limit theorems for some weighted type random sums (or discrete rough integrals). We introduce a general transfer principle from limit theorems for unweighted sums to limit theorems for weighted sums via rough…
Let P_{n,m} denote the graph taken uniformly at random from the set of all planar graphs on {1,2,..., n} with exactly m(n) edges. We use counting arguments to investigate the probability that P_{n,m} will contain given components and…
This paper gives the quantum walks determined by graph zeta functions. The result enables us to obtain the characteristic polynomial of the transition matrix of the quantum walk, and it determines the behavior of the quantum walk. We treat…
We recently introduced a formalism for the modeling of temporal networks, that we call stream graphs. It emphasizes the streaming nature of data and allows rigorous definitions of many important concepts generalizing classical graphs. This…
Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple…
We study directed random graphs (random graphs whose edges are directed), and present new results on the so-called strong components of those graphs. We provide analytic and simulation results on two special classes of strong component,…
We study the directed polymer model for general graphs (beyond $\mathbb Z^d$) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an $L^2$ region, and of very strong disorder,…
We analyze the statistics of the shortest and fastest paths on the road network between randomly sampled end points. To a good approximation, these optimal paths are found to be directed in that their lengths (at large scales) are linearly…
Random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology. In recent years, relevant mathematical results have been obtained in this field, and…
The paper considers the behaviour of the number of paths of length $N$ on graphs with two heavy roots. Such vertices can be entropic traps. Numerical analysis is carried out for graphs with different degrees of root vertices. In the…