Gossip Graphs Research Articles

Convergence rate on periodic gossiping

In a sensor network in which each sensor controls a real-valued state, the goal of a distributed averaging problem is to compute the global average in a decentralized way, which is the average of all sensors’ initial state values across the entire network. A T-periodic gossiping protocol can solve such a problem, which stipulates that each agent must gossip with each of its neighbors exactly once every T time unit. The convergence rate of a T-periodic gossiping protocol is determined by the magnitude of the second largest eigenvalue of the stochastic matrix associated with the gossip sequence occurring over one period. An interesting result is that when the allowable gossip graph is a tree, the convergence rate is independent of gossip orders within one period. This paper will prove this result by developing several properties of doubly stochastic matrices. The properties derived also can be used in analyzing convergence rate problems of other periodic gossip protocols.

On the minimum path problem in Knödel graphs

AbstractThe Knödel graphs Wd,n are regular graphs, of even order n and degree d ≤ ⌊log n⌋. They have been introduced by W. Knödel as gossip graphs for d = ⌊log n⌋. A logarithmic algorithm for the minimum path problem in Knödel graphs is an open problem despite the fact that they are bipartite and highly symmetric. In this paper, we describe a logarithmic time two‐approximation algorithm for the shortest path in the Knödel graph on 2d vertices with degree d. We also prove that for a subset of the set of vertices, the algorithm gives a minimum length path. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 86–91 2007

Minimum linear gossip graphs and maximal linear (Δ, k)‐gossip graphs

AbstractGossiping is an information dissemination problem in which each node of a communication network has a unique piece of information that must be transmitted to all other nodes using two‐way communications between pairs of nodes along the communication links of the network. In this paper, we study gossiping using a linear‐cost model of communication which includes a start‐up time and a propagation time which is proportional to the amount of information transmitted. A minimum linear gossip graph is a graph (modeling a network), with the minimum possible number of links, in which gossiping can be completed in minimum time under the linear‐cost model. For networks with an even number of nodes, we prove that the structure of minimum linear gossip graphs is independent of the relative values of the start‐up and unit propagation times. We prove that this is not true when the number of nodes is odd. We present four infinite families of minimum linear gossip graphs. We also present minimum linear gossip graphs for all even numbers of nodes n ≤ 32 except n = 22. A linear (Δ, k)‐gossip graph is a graph with maximum degree Δ in which gossiping can be completed in k rounds with minimum propagation time. We present three infinite families of maximal linear (Δ, k)‐gossip graphs, that is, linear (Δ, k)‐gossip graphs with a maximum number of nodes. We show that not all minimum broadcast graphs are maximal linear (Δ, k)‐gossip graphs. © 2001 John Wiley & Sons, Inc.

A study of minimum gossip graphs

Gossiping and broadcasting are two problems of information dissemination described in a group of individuals connected by a communication network. In broadcasting, one individual has an item of information and needs to communicate it to everyone else. In gossiping, every person in the network knows a unique item of information and needs to communicate it to everyone else. Those two communication patterns find their main applications in the field of interconnection networks for parallel and distributed architecture. After reviewing some of the main results that have been obtained on these problems, we give new properties concerning gossiping, mainly about the structure of the networks with n nodes gossiping in minimum time and their minimum number, G( n), of communication links. These properties lead to new bounds for G( n) in the general case, and in particular when 1⩽ n⩽32. Moreover, for some values of n (namely n=15,24,28), we show the exact value of G( n).

Compounding of gossip graphs

Gossiping refers to the following task: In a group of individuals connected by a communication network, every node has a piece of information and needs to transmit it to all the nodes in the network. The networks are modeled by graphs, where the vertices represent the nodes, and the edges, the communication links. In this paper, we concentrate on minimum gossip graphs of even order, that is, graphs able to achieve gossiping in minimum time and with a minimum number of links. More precisely, we derive upper bounds for their number of edges from a compounding method, the k-way split method, previously introduced for broadcasting by Farley [Networks 9 (1979), 313–332]. We show that this method can be applied to gossiping in some cases and that this generalizes some compounding methods for gossip graphs given in [5]. We also show that, when applicable, this method gives the best-known upper bounds on the size of minimum gossip graphs in most cases, either improving or matching them. Notably, we present for the first time two families of regular gossip graphs of order n and of degree ⌈log2(n)⌉ − 3 and ⌈log2(n)⌉ − 4, respectively. We also give some lower bounds on the number of edges of gossip graphs which improve the ones given by Fertin [5]. Moreover, we show that the above compounding method also applies for minimum linear gossip graphs (or MLGGs) of even order, which corresponds to a variant of gossiping where the time of information transmission between two nodes depends on the amount of information exchanged. We also prove that this gives the best-known upper bounds for Gβ,τ(n)—the size of an MLGG of order n—in most cases. In particular, we derive from this method the exact value of Gβ,τ(72), which was previously unknown. © 2000 John Wiley & Sons, Inc.

Gossiping in Vertex-Disjoint Paths Mode in d-Dimensional Grids and Planar Graphs

This paper continues with the study of the communication modes introduced by J. Hromkovič, R. Klasing, E. A. Stöhr ["Dissemination of Information in Vertex-Disjoint Paths Mode, Part 1: General Bounds and Gossiping in Hypercube-Like Networks," Technical Report No. tr-ri-93-129, University of Paderborn. (Extended abstract presented at WG′93.)] as a generalization of the standard one-way and two-way modes allowing messages to be sent between processors of interconnection networks via vertex-disjoint paths in one communication step. The complexity of communication algorithms is measured by the number of communication steps (rounds). Here, the complexity of gossiping in grids and in planar graphs is investigated. The main results are the following: 1. Effective one-way and two-way gossip algorithms for d-dimensional grids, d ≥ 2, are designed. 2. The lower bound 2 log2n − log2k − log2 log2n − 4 is established on the number of rounds of every two-way gossip algorithm working on any graph of n nodes and vertex bisection k. This proves that the designed two-way gossip algorithms on d-dimensional grids, d ≥ 3, are almost optimal, and it also shows that the 2-dimensional grid belongs to the best gossip graphs among all planar graphs. 3. Another lower bound proof is developed to get some tight lower bounds on one-way "well-structured" gossip algorithms on planar graphs (to the best of the authors′ knowledge, to date all gossip algorithms designed in vertex-disjoint paths mode have been "well-structured").

Some minimum gossip graphs

AbstractWhat is the minimum number of edges that a graph on n vertices must have to allow gossiping by bidirectional telephone calls within ⌈log2 n⌉ rounds? Besides general estimates, this question is finally answered for integers of the form n = 2p − 2 (p ≥ 3) or n = 2p − 4 (p ≥ 6), and for n = 10, 12. Moreover, for n = 14, we prove the uniqueness of such a graph. © 1993 by John Wiley & Sons, Inc.