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We investigate by random-walk simulations and a mean-field theory how growth by biased addition of nodes affects flow of the current through the emergent conducting graph, representing a digital circuit. In the interior of a large network…
A simple random walk on a graph is a sequence of movements from one vertex to another where at each step an edge is chosen uniformly at random from the set of edges incident on the current vertex, and then transitioned to next vertex.…
We develop a path-based approach to continuous-time random walks on networks with arbitrarily weighted edges. We describe an efficient numerical algorithm for calculating statistical properties of the stochastic path ensemble. After…
In recent years, graph neural networks (GNNs) have gained increasing popularity and have shown very promising results for data that are represented by graphs. The majority of GNN architectures are designed based on developing new…
Representations based on random walks can exploit discrete data distributions for clustering and classification. We extend such representations from discrete to continuous distributions. Transition probabilities are now calculated using a…
A large and sparse random graph with independent exponentially distributed link weights can be used to model the propagation of messages or diseases in a network with an unknown connectivity structure. In this article we study an extended…
It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results…
A particular case of a causal set is considered that is a directed dyadic acyclic graph. This is a model of a discrete pregeometry on a microscopic scale. The dynamics is a stochastic sequential growth of the graph. New vertexes of the…
Currently there are three major paradigms of quantum computation, the gate model, annealing, and walks on graphs. The gate model and quantum walks on graphs are universal computation models, while annealing plays within a specific subset of…
Many polynomial invariants are defined on graphs for encoding the combinatorial information and researching them algebraically. In this paper, we introduce the cycle polynomial and the path polynomial of directed graphs for counting cycles…
We consider the distribution of cycle counts in a random regular graph, which is closely linked to the graph's spectral properties. We broaden the asymptotic regime in which the cycle counts are known to be approximately Poisson, and we…
We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool…
We study the distribution of the number of (non-backtracking) periodic walks on large regular graphs. We propose a formula for the ratio between the variance of the number of $t$-periodic walks and its mean, when the cardinality of the…
We present a general model for the growth of weighted networks in which the structural growth is coupled with the edges' weight dynamical evolution. The model is based on a simple weight-driven dynamics and a weights' reinforcement…
We investigate network exploration by random walks defined via stationary and adaptive transition probabilities on large graphs. We derive an exact formula valid for arbitrary graphs and arbitrary walks with stationary transition…
Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph and, therefore, we can count the number of linear…
There was recent interest in Motzkin paths without peaks (peak: up-step followed immediately by down-step); additional results about this interesting family is worked out. The new results are the enumeration of such paths that live in a…
The aim of this paper is to present an algorithm which gives all the possible paths that start from a specific node to another of a weighted multi-graph. This algorithm is intended to be applied for the direct topological method.
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
We set the ground for a theory of quantum walks on graphs- the generalization of random walks on finite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible.…