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Many migratory animals regularly travel thousands of kilometers, exactly finding their destinations. It is assumed that migrants have both a compass sense to hold their course, and a map sense --- a kind of "biological" GPS --- to correct…

An encounter-based network is a frequently disconnected wireless ad-hoc network requiring nearby neighbors to store and forward data utilizing mobility and encounters over time. Using traditional approaches such as gateways or firewalls for…

In this work we study asymptotic properties of a long range memory random walk known as elephant random walk. First we prove recurrence and positive recurrence for the elephant random walk. Then, we establish the transience regime of the…

We show that the recently introduced logarithmic metrics used to predict disease arrival times on complex networks are approximations of more general network-based measures derived from random walks theory. Using the daily air-traffic…

Among the realistic ingredients to be considered in the computational modeling of infectious diseases, human mobility represents a crucial challenge both on the theoretical side and in view of the limited availability of empirical data. In…

Dynamic graphs have emerged as an appropriate model to capture the changing nature of many modern networks, such as peer-to-peer overlays and mobile ad hoc networks. Most of the recent research on dynamic networks has only addressed the…

Understanding the spread of infectious diseases requires integrating movement, physical constraints, and spatial configurations into epidemiological models. In this study, we investigate how particle diffusivity, hardcore interactions, and…

Advances in satellite-based data collection techniques have served as a catalyst for new statistical methodology to analyze these data. In wildlife ecological studies, satellite-based data and methodology have provided a wealth of…

This paper describes the formulation and experimental testing of a novel method for the estimation and approximation of submanifold models of animal motion. It is assumed that the animal motion is supported on a configuration manifold $Q$…

Quantifying and comparing patterns of dynamical ecological systems require averaging over measurable quantities. For example, to infer variation in movement and behavior, metrics such as step length and velocity are averaged over large…

We consider discrete dynamical systems of "ant-like" agents engaged in a sequence of pursuits on a graph environment. The agents emerge one by one at equal time intervals from a source vertex $s$ and pursue each other by greedily attempting…

We study the graph-theoretic properties of the trace of random walks on pseudorandom graphs. We show that for any $\varepsilon>0$, there exists a constant $C$ such that the cover time of an $(n,d,\lambda)$-graph $G$ with $d/\lambda\ge C$ is…

Outbreaks of infectious diseases present a global threat to human health and are considered a major health-care challenge. One major driver for the rapid spatial spread of diseases is human mobility. In particular, the travel patterns of…

Disease propagation between countries strongly depends on their effective distance, a measure derived from the world air transportation network (WAN). It reduces the complex spreading patterns of a pandemic to a wave-like propagation from…

The Abelian Sandpile Model is a discrete diffusion process defined on graphs (Dhar \cite{DD90}, Dhar et al. \cite{DD95}) which serves as the standard model of \textit{self-organized criticality}. The transience class of a sandpile is…

Human mobility forms the backbone of contact patterns through which infectious diseases propagate, fundamentally shaping the spatio-temporal dynamics of epidemics and pandemics. While traditional models are often based on the assumption…

Random walk is one of the basic mechanisms found in many network applications. We study the epidemic spreading dynamics driven by biased random walks on complex networks. In our epidemic model, each time infected nodes constantly spread…

The adjacency matrix of a graph G is the Hamiltonian for a continuous-time quantum walk on the vertices of G. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the…

We formalize the treatment of directed (or chiral) quantum walks using Hermitian adjacency matrices, bridging two developing fields of research in quantum information and spectral graph theory. We display results and simulations which…

For more than a century lattice random walks have been employed ubiquitously, both as a theoretical laboratory to develop intuition about more complex stochastic processes and as a tool to interpret a vast array of empirical observations.…