Expansion Properties Of Graphs Research Articles

Robust and efficient membership management in large-scale dynamic networks

Epidemic protocols are a bio-inspired communication and computation paradigm for large-scale networked systems based on randomised communication. These protocols rely on a membership service to build decentralised and random overlay topologies. In large-scale, dynamic network environments, node churn and failures may have a detrimental effect on the structure of the overlay topologies with negative impact on the efficiency and the accuracy of applications. Most importantly, there exists the risk of a permanent loss of global connectivity that would prevent the correct convergence of applications. This work investigates to what extent a dynamic network environment may negatively affect the performance of Epidemic membership protocols. A novel Enhanced Expander Membership Protocol (EMP+) based on the expansion properties of graphs is presented. The proposed protocol is evaluated against other membership protocols and the comparative analysis shows that EMP+ can support faster application convergence and is the first membership protocol to provide robustness against global network connectivity problems.

Optimal Design of Process Flexibility for General Production Systems

Process flexibility has been widely adopted as an effective strategy to improve responsiveness to demand uncertainty. Many algorithms for constructing sparse flexibility designs with good theoretical guarantees have been developed for balanced and symmetrical production systems, which assume that the number of plants equals the number of products, the supplies have the same capacity, and demands are independently and identically distributed.In this paper, we relax these restrictions and consider a very general class of production systems. We provide a simple construction of flexibility design to fulfill (1-\epsilon) fraction of the expected total demand with high probability (w.h.p.) with an average degree of O (ln(1/\epsilon)). To motivate our construction, we first consider a natural weighted probabilistic construction from Chou et. al. (2011), in which the degree of each node is proportional to its expected capacity. However, we show that this strategy is sub-optimal. To obtain an optimal construction, we develop a simple, yet effective thresholding scheme. Our theoretical analysis extends the classical analysis of expander graphs by overcoming several technical difficulties. The developed techniques might be useful for other applications that require certain expansion properties of graphs with non-uniform degree sequence.

Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs

It is shown that the edges of any n-point vertex expander can be replaced by new edges so that the resulting graph is an edge expander, and such that any two vertices that are joined by a new edge are at distance O ( log n ) in the original graph. This result is optimal, and is shown to have various geometric consequences. In particular, it is used to obtain an alternative perspective on the recent algorithm of Arora et al. [Proceedings of the 36th Annual ACM Symposium on the Theory of Computing, 2004, pp. 222–231.] for approximating the edge expansion of a graph, and to give a nearly optimal lower bound on the ratio between the observable diameter and the diameter of doubling metric measure spaces which are quasisymmetrically embeddable in Hilbert space.