Robust Transport over Networks

We consider transportation over a strongly connected, directed graph. The scheduling amounts to selecting transition probabilities for a discrete-time Markov evolution which is designed to be consistent with initial and final marginal constraints on mass transport. We address the situation where initially the mass is concentrated on certain nodes and needs to be transported over a certain time period to another set of nodes, possibly disjoint from the first. The evolution is selected to be closest to a {\em prior} measure on paths in the relative entropy sense--such a construction is known as a Schroedinger bridge between the two given marginals. It may be viewed as an atypical stochastic control problem where the control consists in suitably modifying the prior transition mechanism. The prior can be chosen to incorporate constraints and costs for traversing specific edges of the graph, but it can also be selected to allocate equal probability to all paths of equal length connecting any two nodes (i.e., a uniform distribution on paths). This latter choice for prior transitions relies on the so-called Ruelle-Bowen random walker and gives rise to scheduling that tends to utilize all paths as uniformly as the topology allows. Thus, this Ruelle-Bowen law (${\mathfrak M}_{\rm RB}$) taken as prior, leads to a transportation plan that tends to lessen congestion and ensures a level of robustness. We also show that the distribution ${\mathfrak M}_{\rm RB}$ on paths, which attains the maximum entropy rate for the random walker given by the topological entropy, can itself be obtained as the time-homogeneous solution of a maximum entropy problem for measures on paths (also a Schr\"{o}dinger bridge problem, albeit with prior that is not a probability measure). Finally we show that the paradigm of Schr\"odinger bridges as a mechanism for scheduling transport on networks can be adapted to graphs that are not strongly connected, as well as to weighted graphs. In the latter case, our approach may be used to design a transportation plan which effectively compromises between robustness and other criteria such as cost. Indeed, we explicitly provide a robust transportation plan which assigns {\em maximum probability} to {\em minimum cost paths} and therefore compares favorably with Optimal Mass Transportation strategies

Efficient robust routing for single commodity network flows

We study single commodity network flows with suitable robustness and efficiency specs. An original use of a maximum entropy problem for distributions on the paths of the graph turns this problem into a steering problem for Markov chains with prescribed initial and final marginals. From a computational standpoint, viewing scheduling this way is especially attractive in light of the existence of an iterative algorithm to compute the solution. The present paper builds on [13] by introducing an index of efficiency of a transportation plan and points, accordingly, to efficient-robust transport policies. In developing the theory, we establish two new invariance properties of the solution (called bridge) \u2013 an iterated bridge invariance property and the invariance of the most probable paths. These properties, which were tangentially mentioned in our previous work, are fully developed here. We also show that the distribution on paths of the optimal transport policy, which depends on a \u201ctemperature\u201d parameter, tends to the solution of the \u201cmost economical\u201d but possibly less robust optimal mass transport problem as the temperature goes to zero. The relevance of all of these properties for transport over networks is illustrated in an example

Matricial Wasserstein-1 Distance

In this note, we propose an extension of the Wasserstein 1-metric ($W_1$) for matrix probability densities, matrix-valued density measures, and an unbalanced interpretation of mass transport. The key is using duality theory, in particular, a "dual of the dual" formulation of $W_1$. This matrix analogue of the Earth Mover's Distance has several attractive features including ease of computation.Comment: 8 page

Estimating ensemble flows on a hidden Markov chain

We propose a new framework to estimate the evolution of an ensemble of indistinguishable agents on a hidden Markov chain using only aggregate output data. This work can be viewed as an extension of the recent developments in optimal mass transport and Schr\"odinger bridges to the finite state space hidden Markov chain setting. The flow of the ensemble is estimated by solving a maximum likelihood problem, which has a convex formulation at the infinite-particle limit, and we develop a fast numerical algorithm for it. We illustrate in two numerical examples how this framework can be used to track the flow of identical and indistinguishable dynamical systems.Comment: 8 pages, 4 figure

Control and estimation of multi-commodity network flow under aggregation

A paradigm put forth by E. Schr\"odinger in 1931/32, known as Schr\"odinger bridges, represents a formalism to pose and solve control and estimation problems seeking a perturbation from an initial control schedule (in the case of control), or from a prior probability law (in the case of estimation), sufficient to reconcile data in the form of marginal distributions and minimal in the sense of relative entropy to the prior. In the same spirit, we consider traffic-flow and apply a Schr\"odinger-type dictum, to perturb minimally with respect to a suitable relative entropy functional a prior schedule/law so as to reconcile the traffic flow with scarce aggregate distributions on families of indistinguishable individuals. Specifically, we consider the problem to regulate/estimate multi-commodity network flow rates based only on empirical distributions of commodities being transported (e.g., types of vehicles through a network, in motion) at two given times. Thus, building on Schr\"odinger's large deviation rationale, we develop a method to identify {\em the most likely flow rates (traffic flow)}, given prior information and aggregate observations. Our method further extends the Schr\"odinger bridge formalism to the multi-commodity setting, allowing commodities to exit or enter the flow field as well (e.g., vehicles to enter and stop and park) at any time. The behavior of entering or exiting the flow field, by commodities or vehicles, is modeled by a Markov chains with killing and creation states. Our method is illustrated with a numerical experiment.Comment: 12 pages, 5 figure

Exploring Robustness of Neural Networks through Graph Measures

Motivated by graph theory, artificial neural networks (ANNs) are traditionally structured as layers of neurons (nodes), which learn useful information by the passage of data through interconnections (edges). In the machine learning realm, graph structures (i.e., neurons and connections) of ANNs have recently been explored using various graph-theoretic measures linked to their predictive performance. On the other hand, in network science (NetSci), certain graph measures including entropy and curvature are known to provide insight into the robustness and fragility of real-world networks. In this work, we use these graph measures to explore the robustness of various ANNs to adversarial attacks. To this end, we (1) explore the design space of inter-layer and intra-layers connectivity regimes of ANNs in the graph domain and record their predictive performance after training under different types of adversarial attacks, (2) use graph representations for both inter-layer and intra-layers connectivity regimes to calculate various graph-theoretic measures, including curvature and entropy, and (3) analyze the relationship between these graph measures and the adversarial performance of ANNs. We show that curvature and entropy, while operating in the graph domain, can quantify the robustness of ANNs without having to train these ANNs. Our results suggest that the real-world networks, including brain networks, financial networks, and social networks may provide important clues to the neural architecture search for robust ANNs. We propose a search strategy that efficiently finds robust ANNs amongst a set of well-performing ANNs without having a need to train all of these ANNs.Comment: 18 pages, 15 figure