Network extraction by routing optimization

Abstract

Routing optimization is a relevant problem in many contexts. Solving directly this type of optimization problem is often computationally intractable. Recent studies suggest that one can instead turn this problem into one of solving a dynamical system of equations, which can instead be solved efficiently using numerical methods. This results in enabling the acquisition of optimal network topologies from a variety of routing problems. However, the actual extraction of the solution in terms of a final network topology relies on numerical details which can prevent an accurate investigation of their topological properties. In fact, in this context, theoretical results are fully accessible only to an expert audience and ready-to-use implementations for non-experts are rarely available or insufficiently documented. In particular, in this framework, final graph acquisition is a challenging problem in-and-of-itself. Here we introduce a method to extract network topologies from dynamical equations related to routing optimization under various parameters’ settings. Our method is made of three steps: first, it extracts an optimal trajectory by solving a dynamical system, then it pre-extracts a network, and finally, it filters out potential redundancies. Remarkably, we propose a principled model to address the filtering in the last step, and give a quantitative interpretation in terms of a transport-related cost function. This principled filtering can be applied to more general problems such as network extraction from images, thus going beyond the scenarios envisioned in the first step. Overall, this novel algorithm allows practitioners to easily extract optimal network topologies by combining basic tools from numerical methods, optimization and network theory. Thus, we provide an alternative to manual graph extraction which allows a grounded extraction from a large variety of optimal topologies. The analysis of these may open up the possibility to gain new insights into the structure and function of optimal networks. We provide an open source implementation of the code online.

Optimal Transport in Multilayer Networks for Traffic Flow Optimization

“…We consider the formalism of optimal transport theory, and in particular, recent works that map the setting of solving a standard optimization problem into that of solving a dynamical system of equations [24][25][26][27][28][29][30]40]. Specifically, we model two main quantities defined on network edges: (i) fluxes F e of passengers traveling through an edge e; and (ii) conductivities µ e , which are quantities determining the flux passing through an edge e. Intuitively, the conductivity µ e of an edge can be seen as proportional to the size of the edge e. To keep track of the different routes that passengers have, we consider multicommodity formalism as in [40], i.e., we distinguish passengers based on their entry station a ∈ S, where S ⊆ V is the set of stations where passengers enter, and we denote with M = |S| the number of passenger types.…”

“…The regularization is obtained via a parameter β that allows to flexibly tune the cost between settings where traffic is penalized or consolidated. Optimal transport is a proven, powerful tool to model traffic in networks and optimal network design [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Recent works [30,40] extended this formalism to a multi-commodity case that properly accounts for passengers with different origins and destinations.…”

Principled network extraction from images

“…Here, we present a method that addresses these issues by considering the framework of optimal transport. Specifically, inspired by a recently developed model to extract network topologies from solutions of routing optimization problems [13], we adapt this formalism to our specific and different setting. We start from a raw image as input and propose a model that outputs a network representing the topological structure contained in the image.…”

“…These are not immediately associated with a network meant as a set of nodes and a set of edges connecting them. However, Baptista et al [13] propose principled rules to automatically extract network topologies from these solutions in continuous space. While the main focus of that work was to extract network topologies from this particular type of input (functions defined in a continuous domain, e.g.…”

“…Here, we expand on this insight, and adapt this principled network extraction to inputs that are images. In particular, we propose an algorithm to effectively tackle two problems that are relevant for images and that were only briefly discussed for general applications in [13]: how to select source and sinks and how to obtain loopy structures.…”