Gossip algorithms have recently received significant attention, mainly because they constitute simple and robust message-passing schemes for distributed information processing over networks. However for many topologies that are realistic for wireless ad-hoc and sensor networks (like grids and random geometric graphs), the standard nearest-neighbor gossip converges as slowly as flooding (O(n 2) messages). A recently proposed algorithm called geographic gossip improves gossip efficiency by a √ n factor, by exploiting geographic information to enable multi-hop long distance communications. In this paper we prove that a variation of geographic gossip that averages along routed paths, improves efficiency by an additional √ n factor and is order optimal (O(n) messages) for grids and random geometric graphs. We develop a general technique (travel agency method) based on Markov chain mixing time inequalities, which can give bounds on the performance of randomized message-passing algorithms operating over various graph topologies.

We consider the problem of cooperatively minimizing the sum of convex functions, where the functions represent local objective functions of the agents. We assume that each agent has information about his local function, and communicate with the other agents over a time-varying network topology. For this problem, we propose a distributed subgradient method that uses averaging algorithms for locally sharing information among the agents. In contrast to previous works on multi-agent optimization that make worst-case assumptions about the connectivity of the agents (such as bounded communication intervals between nodes), we assume that links fail according to a given stochastic process. Under the assumption that the link failures are independent and identically distributed over time (possibly correlated across links), we provide almost sure convergence results for our subgradient algorithm.

We present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time-varying connectivity. Our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem, where the global objective function is a combination of local agent objective functions. Our main focus is on constrained problems where the estimate of each agent is restricted to lie in a different constraint set. To highlight the effects of constraints, we first consider a constrained consensus problem and present a distributed “projected consensus algorithm ” in which agents combine their local averaging operation with projection on their individual constraint sets. This algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents. We establish convergence and convergence rate results for the projected consensus algorithm. We next study a constrained optimization problem for optimizing the

We study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents. We assume that each agent knows only his own local objective function and constraint set, and exchanges information with the other agents over a randomly varying network topology to update his information state. We assume a statedependent communication model over this topology: communication is Markovian with respect to the states of the agents and the probability with which the links are available depends on the states of the agents. In this paper, we study a projected multi-agent subgradient algorithm under state-dependent communication. The algorithm involves each agent performing a local averaging to combine his estimate with the other agents’ estimates, taking a subgradient step along his local objective function, and projecting the estimates

Abstract — In this paper we show how interference can be exploited to perform gossip computations over a larger local neighborhood, rather than only pairs of nodes. We use a recently introduced technique called computation coding to perform reliable computation over noisy multiple access channels. Since many nodes can simultaneously average in a single round, neighborhood gossip clearly converges faster than nearest neighbor gossip. We characterize how many gossip rounds are required for a given neighborhood size. Also, we show that if the size of the collaboration neighborhood is larger than a critical value that depends on the path loss exponent and the network size, interference can yield exponential benefits in the energy required to compute the average. I.

Abstract — We consider a convex unconstrained optimization problem that arises in a network of agents whose goal is to cooperatively optimize the sum of the individual agent objective functions through local computations and communications. For this problem, we use averaging algorithms to develop distributed subgradient methods that can operate over a timevarying topology. Our focus is on the convergence rate of these methods and the degradation in performance when only quantized information is available. Based on our recent results on the convergence time of distributed averaging algorithms, we derive improved upper bounds on the convergence rate of the unquantized subgradient method. We then propose a distributed subgradient method under the additional constraint that agents can only store and communicate quantized information, and we provide bounds on its convergence rate that highlight the dependence on the number of quantization levels. I.

Abstract — The central goal in multiagent systems is to design local control laws for the individual agents to ensure that the emergent global behavior is desirable with respect to a given system level objective. Ideally, a system designer seeks to satisfy this goal while conditioning each agent’s control law on the least amount of information possible. Unfortunately, there are no existing methodologies for addressing this design challenge. The goal of this paper is to address this challenge using the field of game theory. Utilizing game theory for the design and control of multiagent systems requires two steps: (i) defining a local objective function for each decision maker and (ii) specifying a distributed learning algorithm to reach a desirable operating point. One of the core advantages of this game theoretic approach is that this two step process can be decoupled by utilizing specific classes of games. For example, if the designed objective functions result in a potential game then the system designer can utilize distributed learning algorithms for potential games to complete step (ii) of the design process. Unfortunately, designing agent objective functions to meet objectives such as locality of information and efficiency of resulting equilibria within the framework of potential games is fundamentally challenging and in many case impossible. In this paper we develop a systematic methodology for meeting these objectives using a broader framework of games termed state based potential games. State based potential games is an extension of potential games where an additional state variable is introduced into the game environment hence permitting more flexibility in our design space. Furthermore, state based potential games possess an underlying structure that can be exploited by distributed learning algorithms in a similar fashion to potential games hence providing a new baseline for our decomposition. I.

Abstract—In many application scenarios sensors need to calculate the average of some local values, e.g. of local measurements. A possible solution is to rely on consensus algorithms. In this case each sensor maintains a local estimate of the global average, and keeps improving it by performing a weighted sum of the estimates of all its neighbors. The number of iterations neededtoreach anaccurate estimate dependson theweightsused ateach sensor.Speedinguptheconvergence rate isimportantalso to reduce the number of messages exchanged among neighbors and then the energetic cost of these algorithms. While it is possible in principle to calculate the optimal weights, the known algorithm requires a single sensor to discover the topology of the whole network and perform the calculations. This may be unfeasible for large and dynamic sensor networks, because of sensor computational constraints and of the communication overhead due to the need to acquire the new topology after each change. In this paper we propose a new average consensus algorithm, where each sensor selects its own weights on the basis of some local information about its neighborhood. Our algorithm is tailored for networks having cluster structure, like it is common for wireless sensor networks. In realistic sensor network topologies, the algorithm shows faster convergence than other existing consensus protocols. I.

Abstract—Influence systems form a large class of multiagent systems designed to model how influence, broadly defined, spreads across a dynamic network. We build a general analytical framework which we then use to prove that, while Turing-complete, influence dynamics of the diffusive type is almost surely asymptotically periodic. Besides resolving the dynamics of a popular family of multiagent systems, the other contribution of this work is to introduce a new type of renormalization-based bifurcation analysis for multiagent systems. I.

Abstract—This paper presents a general class of gossipbased averaging algorithms, which are inspired from Uniform Gossip [1]. While Uniform Gossip works synchronously on complete graphs, weighted gossip algorithms allow asynchronous rounds and converge on any connected, directed or undirected graph. Unlike most previous gossip algorithms [2]–[6], Weighted Gossip admits stochastic update matrices which need not be doubly stochastic. Double-stochasticity being very restrictive in a distributed setting [7], this novel degree of freedom is essential and it opens the perspective of designing a large number of new gossip-based algorithms. To give an example, we present one of these algorithms, which we call One-Way Averaging. It is based on random geographic routing, just like Path Averaging [5], except that routes are used unidirectionally instead of back and forth. Hence in this example, getting rid of double stochasticity allows us to add robustness to Path Averaging. I.