Giacomo Como

Robust Distributed Routing in Dynamical Networks–Part II: Strong Resilience, Equilibrium Selection and Cascaded Failures

Strong resilience properties of dynamical networks are analyzed for distributed routing policies. The latter are characterized by the property that the way the outflow at a non-destination node gets split among its outgoing links is allowed to depend only on local information about the current particle densities on the outgoing links. The strong resilience of the network is defined as the infimum sum of link-wise flow capacity reductions making the asymptotic total inflow to the destination node strictly less than the total outflow at the origin. A class of distributed routing policies that are responsive to local information is shown to yield the maximum possible strong resilience under such local information constraints for an acyclic dynamical network with a single origin-destination pair. The maximal achievable strong resilience is shown to be equal to the minimum node residual capacity of the network. The latter depends on the limit flow of the unperturbed network and is defined as the minimum, among all the non-destination nodes, of the sum, over all the links outgoing from the node, of the differences between the maximum flow capacity and the limit flow of the unperturbed network. We propose a simple convex optimization problem to solve for equilibrium flows of the unperturbed network that minimize average delay subject to strong resilience guarantees, and discuss the use of tolls to induce such an equilibrium flow in traffic networks. Finally, we present illustrative simulations to discuss the connection between cascaded failures and the resilience properties of the network.

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

Robustness of distributed routing policies is studied for dynamical networks, with respect to adversarial disturbances that reduce the link flow capacities. A dynamical 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 total outflow at the origin. Routing policies regulate the way the total outflow at each nondestination 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 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 disturbances that make it not partially transferring. The weak resilience of a dynamical network with arbitrary routing policy is shown to be upper bounded by the networks min-cut capacity and, hence, is independent 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 loss of weak resilience. Fundamental properties of dynamical networks driven by locally responsive distributed routing policies are analyzed in detail, including global convergence to a unique limit flow. The derivation of these properties exploits the cooperative nature of these dynamical systems, together with an additional stability property implied by the assumption of monotonicity of the flow as a function of the density on each link.

Anytime Reliable Transmission of Real-Valued Information through Digital Noisy Channels

Motivated by distributed sensor networks scenarios, we consider a problem of state estimation under communication constraints, in which a real-valued random vector needs to be reliably transmitted through a digital noisy channel. Estimations are sequentially updated by the receiver, as more and more channel outputs are observed. Assuming that no channel feedback is available at the transmitter, we study the rates at which the mean squared error of the estimation can be made to converge to zero with time. First, simple low-complexity schemes are considered, and trade-offs between performance and encoder/decoder complexity are found. Then, information-theoretic bounds on the best achievable error exponent are obtained.

Throughput Optimality and Overload Behavior of Dynamical Flow Networks Under Monotone Distributed Routing

This paper investigates the throughput behavior of single-commodity dynamical flow networks governed by monotone distributed routing policies. The networks are modeled as systems of ordinary differential equations based on mass conversation laws on directed graphs with limited flow capacities on the links and constant external inflows at certain origin nodes. Under monotonicity assumptions on the routing policies, it is proven that, if the external inflow at the origin nodes does not violate any cut capacity constraints, then there exists a globally asymptotically stable equilibrium, and the network achieves maximal throughput. On the contrary, should such a constraint be violated, the network overload behavior is characterized. In particular, it is established that there exists a cut with respect to which the flow densities on every link grow linearly over time (respectively, reach their respective limits simultaneously) in the case where the buffer capacities are infinite (respectively, finite).

The Capacity of Finite Abelian Group Codes Over Symmetric Memoryless Channels

The capacity of finite Abelian group codes over symmetric memoryless channels is determined. For certain important examples, such as m -PSK constellations over additive white Gaussian noise (AWGN) channels, with m a prime power, it is shown that this capacity coincides with the Shannon capacity; i.e., there is no loss in capacity using group codes. (This had previously been known for binary-linear codes used over binary-input output-symmetric memoryless channels.) On the other hand, a counterexample involving a three-dimensional geometrically uniform constellation is presented in which the use of Abelian group codes leads to a loss in capacity. The error exponent of the average group code is determined, and it is shown to be bounded away from the random-coding error exponent, at low rates, for finite Abelian groups not admitting Galois field structure.

Opinion fluctuations and persistent disagreement in social networks

We study a tractable opinion dynamics model that generates long-run disagreements and persistent opinion fluctuations. Our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents: regular agents, who update their beliefs according to information that they receive from their social neighbors; and stubborn agents, who never update their opinions and might represent leaders, political parties or media sources attempting to influence the beliefs in the rest of the society. When the society contains stubborn agents with different opinions, the belief dynamics never lead to a consensus (among the regular agents). Instead, beliefs in the society almost surely fail to converge, the belief profile keeps on oscillating in an ergodic fashion, and it converges in law to a non-degenerate random vector.

Stability of monotone dynamical flow networks

We study stability properties of monotone dynamical flow networks. Demand and supply functions relate states and flows of the network, and the dynamics at junctions are subject to fixed turning rates. Our main result consists in the characterization of a stability region such that: If the inflow vector in the network lies strictly inside the stability region and a certain graph theoretical condition is satisfied, then a globally asymptotically stable equilibrium exists. In contrast, if the inflow vector lies strictly outside the region, then every trajectory grows unbounded in time. As a special case, our framework allows for the stability analysis of the Cell Transmission Model on networks with arbitrary topologies. These results extend and unify previous work by Gomes et al. on stability of the Cell Transmission Model on a line topology as well as that by the authors on throughput optimality in monotone dynamical flow networks.

On the Capacity of Memoryless Finite-State Multiple-Access Channels With Asymmetric State Information at the Encoders

A single-letter characterization is provided for the capacity region of finite-state multiple-access channels, when the channel state process is an independent and identically distributed sequence, the transmitters have access to partial (quantized) state information, and complete channel state information is available at the receiver. The partial channel state information is assumed to be asymmetric at the encoders. As a main contribution, a tight converse coding theorem is presented. The difficulties associated with the case when the channel state has memory are discussed and connections to decentralized stochastic control theory are presented.

The Error Exponent of Variable-Length Codes Over Markov Channels With Feedback

The error exponent of Markov channels with feedback is studied in the variable-length block-coding setting. Burnashevs classic result is extended to finite-state ergodic Markov channels. For these channels, a single-letter characterization of the reliability function is presented, under the assumption of full causal output feedback, and full causal observation of the channel state both at the transmitter and at the receiver side. Tools from stochastic control theory are used in order to treat channels with intersymbol interference (ISI). Specifically, the convex-analytic approach to Markov decision processes is adopted in order to handle problems with stopping time horizons induced by variable-length coding schemes.

Average Spectra and Minimum Distances of Low-Density Parity-Check Codes over Abelian Groups

Ensembles of regular low-density parity-check codes over any finite Abelian group