Related papers: Exact Solution for Optimal Navigation with Total C…
We study the Kleinberg problem of navigation in Small World networks when the underlying lattice is stretched along a preferred direction. Extensive simulations confirm that maximally efficient navigation is attained when the length $r$ of…
We investigate the optimal design of networks for a general transport system. Our network is built from a regular two-dimensional ($d=2$) square lattice to be improved by adding long-range connections (shortcuts) with probability $P_{ij}…
We study the Kleinberg problem of navigation in Small World networks when the underlying lattice is a fractal consisting of N>>1 nodes. Our extensive numerical simulations confirm the prediction that most efficient navigation is attained…
The small-world property is known to have a profound effect on the navigation efficiency of complex networks [J. M. Kleinberg, Nature 406, 845 (2000)]. Accordingly, the proper addition of shortcuts to a regular substrate can lead to the…
We investigate by numerical simulation and finite-size analysis the impact of long-range shortcuts on a spatially embedded transportation network. Our networks are built from two-dimensional ($d=2$) square lattices to be improved by the…
Consider designing a transportation network on $n$ vertices in the plane, with traffic demand uniform over all source-destination pairs. Suppose the cost of a link of length $\ell$ and capacity $c$ scales as $\ell c^\beta$ for fixed…
Navigability of networks, that is the ability to find any given destination vertex starting from any other vertex, is crucial to their usefulness. In 2000 Kleinberg showed that optimal navigability could be achieved in small-world networks…
We study the problem of assigning transmission ranges to radio stations placed arbitrarily in a $d$-dimensional ($d$-D) Euclidean space in order to achieve a strongly connected communication network with minimum total power consumption. The…
In this Rapid Communication we investigate spatially constrained networks that realize optimal synchronization properties. After arguing that spatial constraints can be imposed by limiting the amount of `wire' available to connect nodes…
Navigation process is studied on a variant of the Watts-Strogatz small world network model embedded on a square lattice. With probability $p$, each vertex sends out a long range link, and the probability of the other end of this link…
We prove that for an $L$-layer fully-connected linear neural network, if the width of every hidden layer is $\tilde\Omega (L \cdot r \cdot d_{\mathrm{out}} \cdot \kappa^3 )$, where $r$ and $\kappa$ are the rank and the condition number of…
We initiate the study of the inherent tradeoffs between the size of a neural network and its robustness, as measured by its Lipschitz constant. We make a precise conjecture that, for any Lipschitz activation function and for most datasets,…
Networks are integral parts of modern safety-critical systems and certification demands the provision of guarantees for data transmissions. Deterministic Network Calculus (DNC) can compute a worst-case bound on a data flow's end-to-end…
It is commonly known that there exist short paths between vertices in a network showing the small-world effect. Yet vertices, for example, the individuals living in society, usually are not able to find the shortest paths, due to the very…
We present theoretical and numerical results concerning the problem to find the path that minimizes the time to navigate between two given points in a complex fluid under realistic navigation constraints. We contrast deterministic Optimal…
In this paper we consider spatial networks that realize a balance between an infrastructure cost (the cost of wire needed to connect the network in space) and communication efficiency, measured by average shortest pathlength. A global…
This papers considers the problem of maximizing the load that can be served by a power network. We use the commonly accepted Linear DC power network model and consider wo configuration options: switching lines and using FACTS devices. We…
This paper focuses on designing edge-weighted networks, whose robustness is characterized by maximizing algebraic connectivity, or the second smallest eigenvalue of the Laplacian matrix. This problem is motivated by cooperative vehicle…
The branching geometry of biological transport networks is characterized by a diameter scaling exponent $\alpha$. Two structural attractors compete: impedance matching ($\alpha \sim 2$) for pulsatile flow and viscous-metabolic minimization…
We discuss the optimal matching solution for both the assignment problem and the matching problem in one dimension for a large class of convex cost functions. We consider the problem in a compact set with the topology both of the interval…