Multiple random walks on paths and grids

We derive several new results on multiple random walks on "low dimensional" graphs. First, inspired by an example of a weighted random walk on a path of three vertices given by Efremenko and Reingold, we prove the following dichotomy: as the path length n tends to infinity, we have a super-linear speed-up w.r.t. the cover time if and only if the number of walks k is equal to 2. An important ingredient of our proofs is the use of a continuous-time analogue of multiple random walks, which might be of independent interest. Finally, we also present the first tight bounds on the speed-up of the cover time for any d-dimensional grid with d >= 2 being an arbitrary constant, and reveal a sharp transition between linear and logarithmic speed-up

Dispersion processes

We study a synchronous dispersion process in which $M$ particles are initially placed at a distinguished origin vertex of a graph $G$. At each time step, at each vertex $v$ occupied by more than one particle at the beginning of this step, each of these particles moves to a neighbour of $v$ chosen independently and uniformly at random. The dispersion process ends once the particles have all stopped moving, i.e. at the first step at which each vertex is occupied by at most one particle. For the complete graph $K_n$ and star graph $S_n$, we show that for any constant $\delta>1$, with high probability, if $M \le n/2(1-\delta)$, then the process finishes in $O(\log n)$ steps, whereas if $M \ge n/2(1+\delta)$, then the process needs $e^{\Omega(n)}$ steps to complete (if ever). We also show that an analogous lazy variant of the process exhibits the same behaviour but for higher thresholds, allowing faster dispersion of more particles. For paths, trees, grids, hypercubes and Cayley graphs of large enough sizes (in terms of $M$) we give bounds on the time to finish and the maximum distance traveled from the origin as a function of the number of particles $M$

Controllability and observability of grid graphs via reduction and symmetries

In this paper we investigate the controllability and observability properties of a family of linear dynamical systems, whose structure is induced by the Laplacian of a grid graph. This analysis is motivated by several applications in network control and estimation, quantum computation and discretization of partial differential equations. Specifically, we characterize the structure of the grid eigenvectors by means of suitable decompositions of the graph. For each eigenvalue, based on its multiplicity and on suitable symmetries of the corresponding eigenvectors, we provide necessary and sufficient conditions to characterize all and only the nodes from which the induced dynamical system is controllable (observable). We discuss the proposed criteria and show, through suitable examples, how such criteria reduce the complexity of the controllability (respectively observability) analysis of the grid

Intelligibility and First Passage Times In Complex Urban Networks

Topology of urban environments can be represented by means of graphs. We explore the graph representations of several compact urban patterns by random walks. The expected time of recurrence and the expected first passage time to a node scales apparently linearly in all urban patterns we have studied In space syntax theory, a positive relation between the local property of a node (qualified by connectivity or by the recurrence time) and the global property of the node (estimated in our approach by the first passage time to it) is known as intelligibility. Our approach based on random walks allows to extend the notion of intelligibility onto the entire domain of complex networks and graph theory.Comment: 19 pages, 4 figures, English UK, the Harvard style reference