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The emergence of collective dynamics in neural networks is a mechanism of the animal and human brain for information processing. In this paper, we develop a computational technique using distributed processing elements in a complex network,…

The study of time-varying (dynamic) networks (graphs) is of fundamental importance for computer network analytics. Several methods have been proposed to detect the effect of significant structural changes in a time series of graphs. The…

A graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge-length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius…

Graph Signal Processing generalizes classical signal processing to signal or data indexed by the vertices of a weighted graph. So far, the research efforts have been focused on static graph signals. However numerous applications involve…

A $\delta$ once-reinforced random walk ($\delta$-ORRW) on connected graph is a self-interacting random walk which moves to its neighbors at each step according to the weights of the edges at that time, where the weights are $1$ on edges…

Consider a stationary sequence $X=(X_n)$ of integer-valued random variables with mean $m \in [-\infty, \infty]$. Let $S=(S_n)$ be the stochastic process with increments $X$ and such that $S_0=0$. For each time $i$, draw an edge from…

The past few years have seen intensive research efforts carried out in some apparently unrelated areas of dynamic systems -- delay-tolerant networks, opportunistic-mobility networks, social networks -- obtaining closely related insights.…

We define an operation of jets on graphs inspired by the corresponding notion in commutative algebra and algebraic geometry. We examine a few graph theoretic properties and invariants of this construction, including chromatic numbers,…

Coherent evolution governs the behaviour of all quantum systems, but in nature it is often subjected to influence of a classical environment. For analysing quantum transport phenomena quantum walks emerge as suitable model systems. In…

We introduce a cellular automaton model coupled with a transport equation for flows on graphs. The direction of the flow is described by a switching process where the switching probability dynamically changes according to the value of the…

The integration and transmission of information in the brain are dependent on the interplay between structural and dynamical properties. Implicit in any pursuit aimed at understanding neural dynamics from appropriate sets of mathematically…

We give the detale description from various points of view of Pascal automorphism,--- a natural transformation of the space of paths in the Pascal graph (= infinite Pascal triangle), and describetha plan of the proof of continuiuty of its…

Many different classification tasks need to manage structured data, which are usually modeled as graphs. Moreover, these graphs can be dynamic, meaning that the vertices/edges of each graph may change during time. Our goal is to jointly…

Many tools from the field of graph signal processing exploit knowledge of the underlying graph's structure (e.g., as encoded in the Laplacian matrix) to process signals on the graph. Therefore, in the case when no graph is available, graph…

We study an exactly solvable random walk model with long-range memory on arbitrary networks. The walker performs unbiased random steps to nearest-neighbor nodes and intermittently resets to previously visited nodes in a preferential way,…

A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. If the isolated vertex is excluded as trivial, nut graphs have seven or more vertices;…

Dynamic networks consist of interconnected dynamical systems. The subsystems can be viewed as transformations of input signals into output signals, where signals flow from one system into another through interconnections. The signal flows…

The monography examines the problem of constructing a group of automorphisms of a graph. A graph automorphism is a mapping of a set of vertices onto itself that preserves adjacency. The set of such automorphisms forms a vertex group of a…

The central question of this paper is: how do algebraic invariants of edge ideals change under natural graph operations? We study this question through the lens of suspensions. The (full) suspension of a graph is obtained by adjoining a new…

The structural properties of graphs are usually characterized in terms of invariants, which are functions of graphs that do not depend on the labeling of the nodes. In this paper we study convex graph invariants, which are graph invariants…