Related papers: Shortest paths on systems with power-law distribut…
In this paper we address two different related problems. We first study the problem of finding a simple shortest path in a $d$-dimensional real space subdivided in several polyhedra endowed with different $\ell_p$-norms. This problem is a…
The length function $\ell_q(r,R)$ is the smallest length of a $q$-ary linear code of codimension (redundancy) $r$ and covering radius $R$. The $d$-length function $\ell_q(r,R,d)$ is the smallest length of a $q$-ary linear code with…
We consider the optimal paths in a $d$-dimensional lattice, where the bonds have isotropically correlated random weights. These paths can be interpreted as the ground state configuration of a simplified polymer model in a random potential.…
Among all characteristics exhibited by natural and man-made networks the small-world phenomenon is surely the most relevant and popular. But despite its significance, a reliable and comparable quantification of the question `how small is a…
A network topology with low average shortest path length (ASPL) provides efficient data transmission while the number of nodes and the number of links incident to each node are often limited due to physical constraints. In this paper, we…
Properties of networks are often characterized in terms of features such as node degree distributions, average path lengths, diameters, or clustering coefficients. Here, we study shortest path length distributions. On the one hand, average…
In this paper we revisit random linear under-determined systems with sparse solutions. We consider $\ell_1$ optimization heuristic known to work very well when used to solve these systems. A collection of fundamental results that relate to…
Among the several topological properties of complex networks, the shortest path represents a particularly important characteristic because of its potential impact not only on other topological properties, but mainly for its influence on…
Given a network infrastructure (e.g., data-center or on-chip-network) and a distribution on the source-destination requests, the expected path (route) length is an important measure for the performance, efficiency and power consumption of…
We derive exact expressions for the shortest path length to a point of a Poisson line Cox process (PLCP) from the typical point of the PLCP and from the typical intersection of the underlying Poisson line process (PLP), restricted to a…
Random subspaces $X$ of $\mathbb{R}^n$ of dimension proportional to $n$ are, with high probability, well-spread with respect to the $\ell_2$-norm. Namely, every nonzero $x \in X$ is "robustly non-sparse" in the following sense: $x$ is…
We study the behavior of the optimal path between two sites separated by a distance $r$ on a $d$-dimensional lattice of linear size $L$ with weight assigned to each site. We focus on the strong disorder limit, i.e., when the weight of a…
Reachability questions are one of the most fundamental algorithmic primitives in temporal graphs -- graphs whose edge set changes over discrete time steps. A core problem here is the NP-hard Short Restless Temporal Path: given a temporal…
We describe a distributed randomized algorithm computing approximate distances and routes that approximate shortest paths. Let n denote the number of nodes in the graph, and let HD denote the hop diameter of the graph, i.e., the diameter of…
We study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path $\ell_{\rm opt}$ in a disordered Erd\H{o}s-R\'enyi (ER) random network and scale-free (SF) network. Each link $i$ is…
We study network loss tomography based on observing average loss rates over a set of paths forming a tree -- a severely underdetermined linear problem for the unknown link loss probabilities. We examine in detail the role of sparsity as a…
We consider the mean distribution functions Phi(r|l), Phi(B)(r|l), and Phi(S)(r|l), giving the probability that two sites on the incipient percolation cluster, on its backbone and on its skeleton, respectively, connected by a shortest path…
Supplementing a lattice with long-range connections effectively models small-world networks characterized by a high local and global interconnectedness observed in systems ranging from society to the brain. If the links have a wiring cost…
We investigate the localization of electrons hopping on quasi-1D strips in the presence of random magnetic field. In the weak-disorder region, by perturbative analytical techniques, we derive scaling laws for the localization length,…
Recently, the worse-case analysis, probabilistic analysis and empirical justification have been employed to address the fundamental question: When does $\ell_1$-minimization find the sparsest solution to an underdetermined linear system? In…