Degree fluctuations and the convergence time of consensus algorithms

We consider a consensus algorithm in which every node in a time-varying undirected connected graph assigns equal weight to each of its neighbors. Under the assumption that the degree of any given node is constant in time, we show that the algorithm achieves consensus within a given accuracy ∈ on n nodes in time O(n[superscript 3]ln(n=∈)). Because there is a direct relation between consensus algorithms in time-varying environments and inhomogeneous random walks, our result also translates into a general statement on such random walks. Moreover, we give simple proofs that the convergence time becomes exponentially large in the number of nodes n under slight relaxations of the above assumptions. We prove that exponential convergence time is possible for consensus algorithms on fixed directed graphs, and we use an example of Cao, Spielman, and Morse to give a simple argument that the same is possible if the constant degrees assumption is even slightly relaxed

Approximate Consensus in Highly Dynamic Networks: The Role of Averaging Algorithms

In this paper, we investigate the approximate consensus problem in highly dynamic networks in which topology may change continually and unpredictably. We prove that in both synchronous and partially synchronous systems, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree, i.e., there is a coordinator at each time. The striking point in this result is that the coordinator is not required to be unique and can change arbitrarily from round to round. Interestingly, the class of averaging algorithms, which are memoryless and require no process identifiers, entirely captures the solvability issue of approximate consensus in that the problem is solvable if and only if it can be solved using any averaging algorithm. Concerning the time complexity of averaging algorithms, we show that approximate consensus can be achieved with precision of $\varepsilon$ in a coordinated network model in $O(n^{n+1} \log\frac{1}{\varepsilon})$ synchronous rounds, and in $O(\Delta n^{n\Delta+1} \log\frac{1}{\varepsilon})$ rounds when the maximum round delay for a message to be delivered is $\Delta$. While in general, an upper bound on the time complexity of averaging algorithms has to be exponential, we investigate various network models in which this exponential bound in the number of nodes reduces to a polynomial bound. We apply our results to networked systems with a fixed topology and classical benign fault models, and deduce both known and new results for approximate consensus in these systems. In particular, we show that for solving approximate consensus, a complete network can tolerate up to 2n-3 arbitrarily located link faults at every round, in contrast with the impossibility result established by Santoro and Widmayer (STACS '89) showing that exact consensus is not solvable with n-1 link faults per round originating from the same node

Lyapunov Approach to Consensus Problems

This paper investigates the weighted-averaging dynamic for unconstrained and constrained consensus problems. Through the use of a suitably defined adjoint dynamic, quadratic Lyapunov comparison functions are constructed to analyze the behavior of weighted-averaging dynamic. As a result, new convergence rate results are obtained that capture the graph structure in a novel way. In particular, the exponential convergence rate is established for unconstrained consensus with the exponent of the order of $1-O(1/(m\log_2m))$. Also, the exponential convergence rate is established for constrained consensus, which extends the existing results limited to the use of doubly stochastic weight matrices

Persistent Graphs and Consensus Convergence

Abstract-This paper investigates the role persistent arcs play for averaging algorithms to reach a global consensus under discrete-time or continuous-time dynamics. Each (directed) arc in the underlying communication graph is assumed to be associated with a time-dependent weight function. An arc is said to be persistent if its weight function has infinite L1 or ℓ1 norm for continuous-time or discrete-time models, respectively. The graph that consists of all persistent arcs is called the persistent graph of the underlying network. Three necessary and sufficient conditions on agreement or ϵ-agreement are established, by which we prove that the persistent graph fully determines the convergence to a consensus. It is also shown how the convergence rates explicitly depend on the diameter of the persistent graph

Degree Fluctuations and the Convergence Time of Consensus Algorithms

We consider a consensus algorithm in which every nodein a sequence of undirected, B-connected graphs assigns equal weight to each of its neighbors. Under the assumption that the degree of each node is fixed (except for times when the node has no connections to other nodes), we show that consensus is achieved within a given accuracy on nodes in time B+4n3In(2n/ε). Because there is a direct relation between consensus algorithms in time-varying environments and in homogeneous random walks, our result also translates into a general statement on such random walks.Moreover, we give a simple proof of a result of Cao, Spielman, and Morse that the worst case convergence time becomes exponentially large inthe number of nodes under slight relaxation of the degree constancy assumption. ©2013 IEEE