We seek network routing towards a desired final distribution that can mediate possible random link failures. In other words, we seek a routing plan that utilizes alternative routes so as to be relatively robust to link failures. To this end, we provide a mathematical formulation of a relaxed transport problem where the final distribution only needs to be close to the desired one. The problem is cast as a maximum entropy problem for probability distributions on paths with an added terminal cost. The entropic regularizing penalty aims at distributing the choice of paths amongst possible alternatives. We prove that the unique solution may be obtained by solving a of equations. An iterative algorithm to compute the solution is provided. Each iteration of the algorithm contracts the distance (in the Hilbert metric) to the optimal solution by more than 1/2, leading to extremely fast convergence.

A popular approach to characterizing activity in neuronal networks is the use of statistical models that describe neurons in terms of their firing rates (i.e., the number of spikes produced per unit time). The output realization of a statistical model is, in essence, an -dimensional binary time series, or pattern. While such models are commonly fit to data, they can also be postulated , as a theoretical description of a given spiking network. More generally, they can model any network producing binary events as a function of time. In this paper, we rigorously develop a set of analyses that may be used to assay the controllability of a particular statistical spiking model, the point-process generalized linear model (PPGLM). Our analysis quantifies the ease or difficulty of inducing desired spiking patterns via an extrinsic input signal, thus providing a framework for basic network analysis, as well as for emerging applications such as neurostimulation design.

A large variety of dynamical systems, such as chemical and biomolecular systems, can be seen as networks of nonlinear entities. Prediction, control, and identification of such nonlinear networks require knowledge of the state of the system. However, network states are usually unknown, and only a fraction of the state variables are directly measurable. The observability problem concerns reconstructing the network state from this limited information. Here, we propose a general optimization-based approach for observing the states of nonlinear networks and for optimally selecting the observed variables. Our results reveal several fundamental limitations in network observability, such as the trade-off between the fraction of observed variables and the observation length on one side, and the estimation error on the other side. We also show that, owing to the crucial role played by the dynamics, purely graph-theoretic observability approaches cannot provide conclusions about one's practical ability to estimate the states. We demonstrate the effectiveness of our methods by finding the key components in biological and combustion reaction networks from which we determine the full system state. Our results can lead to the design of novel sensing principles that can greatly advance prediction and control of the dynamics of such networks.