Robust Distributed Routing in Dynamical Networks—Part I: Locally Responsive Policies and Weak Resilience

Abstract

Robustness of distributed routing policies is studied for dynamical flow networks, with respect to adversarial disturbances that reduce the link flow capacities. A dynamical flow network is modeled as a system of ordinary differential equations derived from mass conservation laws on a directed acyclic graph with a single origin-destination pair and a constant inflow at the origin. Routing policies regulate the way the inflow at a non-destination node gets split among its outgoing links as a function of the current particle density, while the outflow of a link is modeled to depend on the current particle density on that link through a flow function. The dynamical flow network is called partially transferring if the total inflow at the destination node is asymptotically bounded away from zero, and its weak resilience is measured as the minimum sum of the link-wise magnitude of all disturbances that make it not partially transferring. The weak resilience of a dynamical flow network with arbitrary routing policy is shown to be upper-bounded by the network's min-cut capacity, independently of the initial flow conditions. Moreover, a class of distributed routing policies that rely exclusively on local information on the particle densities, and are locally responsive to that, is shown to yield such maximal weak resilience. These results imply that locality constraints on the information available to the routing policies do not cause

On resilience of multicommodity dynamical flow networks

“…The definition of distributed routing policy follows the definition in [6], [7]. The key novelty is that in the present setting different commodities are allowed to have different routing policies, namely, to have different routing preferences and to respond in a different way to congestion.…”

“…Dynamical models based on ODEs have been proposed in the literature, but heterogeneity is usually embedded in a single-commodity scenario with fixed turning rates, i.e., in which at each junction the fraction of vehicles turning into each subsequent link is fixed [4], [5]. Differently from the latter approach, we extend the framework proposed in [6], [7] and consider a dynamic responsive scenario, in which agents have preferred paths that they would follow when completely isolated in the network, but are also willing to adapt their behavior according to the local state of the network, and avoid preferred, but highly congested, paths.…”

“…is unstable, i.e., the density vector ⇢(t) blows up in the limit of large t. For a single-commodity network, it was shown that the margin of resilience equals the minimun node residual capacity [6], [7]. This implies that the network in Fig.…”

Game Theoretic Decentralized Feedback Controls in Markov Jump Processes

“…The following assumption establishes that for a given feasible target manifold, there always exists a decentralized routing policy u(t) which drives the density ρ toward the relative manifoldM i (see (6)). …”

“…Differently from [4], we consider a decentralized control (as in [5][6][7]), in which the density of each node is controlled locally. We highlight next three distinct approaches relating to routing/jump problems.…”