Motivated by applications to sensor, peer-to-peer, and ad hoc networks, we study distributed algorithms, also known as gossip algorithms, for exchanging information and for computing in an arbitrarily connected network of nodes. The topology of such networks changes continuously as new nodes join and old nodes leave the network. Algorithms for such networks need to be robust against changes in topology. Additionally, nodes in sensor networks operate under limited computational, communication, and energy resources. These constraints have motivated the design of “gossip ” algorithms: schemes which distribute the computational burden and in which a node communicates with a randomly chosen neighbor. We analyze the averaging problem under the gossip constraint for an arbitrary network graph, and find that the averaging time of a gossip algorithm depends on the second largest eigenvalue of a doubly stochastic matrix characterizing the algorithm. Designing the fastest gossip algorithm corresponds to minimizing this eigenvalue, which is a semidefinite program (SDP). In general, SDPs cannot be solved in a distributed fashion; however, exploiting problem structure, we propose a distributed subgradient method that solves the optimization problem over the network. The relation of averaging time to the second largest eigenvalue naturally relates it to the mixing time of a random walk with transition probabilities derived from the gossip algorithm. We use this connection to study the performance and scaling of gossip algorithms on two popular networks: Wireless Sensor Networks, which are modeled as Geometric Random Graphs, and the Internet graph under the so-called Preferential Connectivity (PC) model.

Abstract — We consider a network of distributed sensors, where each sensor takes a linear measurement of some unknown parameters, corrupted by independent Gaussian noises. We propose a simple distributed iterative scheme, based on distributed average consensus in the network, to compute the maximum-likelihood estimate of the parameters. This scheme doesn’t involve explicit point-to-point message passing or routing; instead, it diffuses information across the network by updating each node’s data with a weighted average of its neighbors ’ data (they maintain the same data structure). At each step, every node can compute a local weighted least-squares estimate, which converges to the global maximum-likelihood solution. This scheme is robust to unreliable communication links. We show that it works in a network with dynamically changing topology, provided that the infinitely occurring communication graphs are jointly connected. I.

Asynchronous iterations arise naturally parallel computers wants minimize times. This paper reviews certain models asynchronous iterations, using a common theoretical framework. The corresponding convergence theory and various domains applications presented. These include nonsingular linear systems, nonlinear systems, initial value problems.

We study stability properties of a finite set Sigma of n×n-matrices such as paracontractivity, BV- and LCP-stability, and their relations to each other. The conjecture on equivalence of paracontractivity and LCP-stability is proved. Moreover, we prove the equivalence of the uniform BV-stability and the property of vanishing length of steps of any trajectory of Sigma.

Introduction. In the investigation of chaotic iteration procedures for linear consistent systems matrices which are paracontracting with respect to some vector norm play an important role. It was shown in [EKN], that if A 1 ; : : : ; Am are finitely many k--by--k complex matrices which are paracontracting with respect to the same norm, then for any sequence d i ; 1 d i m; i = 1; 2; : : : and any x 0 the sequence x i+1 = A d i x i i = 1; 2; : : : is convergent. In particular A (d) = lim i!1 A d i : : : A d1 exists for all sequences fd i g

We develop conditions under which a product Q 1 i=0 T i of matrices chosen from a possibly infinite set of matrices S = fT j jj 2 Jg converges. We obtain the following conditions which are sufficient for the convergence of the product: There exists a vector norm such that all matrices in S are nonexpansive with respect to this norm and there exists a subsequence fi k g 1 k=0 of the sequence of the nonnegative integers such that the corresponding sequence of operators \Phi T i k \Psi 1 k=0 converges to an operator which is paracontracting with respect to this norm. We deduce the continuity of the limit of the product of matrices as a function of the sequences fi k g 1 k=0 . But more importantly, we apply our results to the question of the convergence of inner--outer iteration schemes for solving singular consistent linear systems of equations, where the outer splitting is regular and the inner splitting is weak regular.

In [6] the authors introduced the concept of paracontracting operators for fixed point problems and their solution with asynchronous iteration methods. We will extend their results essentially with respect to the criterion of contraction for the pool of operators, from which a common fixed point is searched and a more general kind of asynchronous iterations, which we will call confluent. As an application of this theory, we will consider asynchronous iteration methods for consistent linear systems Ax = b ; where A is a singular M-matrix. In [7] Lubachevski and Mitra investigated asynchronous iteration methods for the Perron-vector problem, e.g. determining a positive solution of Tx = ae(T ) x ; where T is an irreducible nonnegative matrix and its spectral radius is known. We will show that we can extend their results by using the developed theory of paracontractions and confluence to the reducible-affine case which arises, if we have those kind of linear systems. Key words: Asynchron...

this paper; see [12], [16], and the references given therein. We point out that, since the number of inner iterations may vary from block to block, the convergence results in Baudet [1] or Chazan and Miranker [5] cannot be applied to our situation. In Sect. 2, we derive two new convergence results for block two-stage iterative methods. We then investigate asynchronous variations of our two-stage methods. These asynchronous methods arise naturally in parallel computations if one tries to reduce idle times of the processors. In Sects. 3 and 4, we introduce two different asynchronous models for block two-stage methods and investigate their convergence. As our major result, we establish convergence for both asynchronous models under the same conditions as those for the synchronous method. In the rest of this section we present some notation, definitions and preliminary results which we refer to later. We say that a vector x is nonnegative (positive), denoted x 0 (x ? 0), if all its entries are nonnegative (positive). Similarly, a matrix B is said to be nonnegative, denoted B O, if all its entries are nonnegative or, equivalently, if it leaves invariant the set of all nonnegative vectors. We compare two matrices

Several consensus protocols have been proposed in the literature and their convergence properties studied via a variety of methods. In all these methods, the communication topologies play a key role in the convergence of consensus processes. The goal of this paper is two fold. First, we explore communication topologies, as implied by the communication assumptions, that lead to consensus among agents. For this, several important results in the literature are examined and the focus is on different classes of communication assumptions being made, such as synchronism, connectivity, and direction of communication. In the latter part of this paper, we show that the confluent iteration graph unifies various communication assumptions and proves to be fundamental in understanding the convergence of consensus processes. In particular, based on asynchronous iteration methods for nonlinear paracontractions, we establish a new result which shows that consensus is reachable under directional, time-varying and asynchronous topologies with nonlinear protocols. This result extends the existing ones in the literature and have many potential applications. 1

In this paper tools are developed to analyse a recently proposed random matrix model of communication networks that employ additive-increase multiplicative-decrease (AIMD) congestion control algorithms. We investigate properties of the Markov process describing the evolution of the window sizes of network users. Using paracontractivity properties of the matrices involved in the model, it is shown that the process has a unique invariant probability, and the support of this probability is characterized. Based on these results we obtain a weak law of large numbers for the average distribution of resources between the users of a network. This shows that under reasonable assumptions such networks have a well-defined stochastic equilibrium. ns2 simulation results are discussed to validate the obtained formulae. (The simulation program ns2, or network simulator, is an industry standard for the simulation of Internet dynamics.)