Peer-to-peer networks based on random transformations of connected regular undirected graphs

• Authors: Peter Mahlmann;Christian Schindelhauer
• Venue: Proceedings of the seventeenth annual ACM symposium on Parallelism in algorithms and architectures
• Year: 2005

We present k-Flipper, a graph transformation algorithmthat transforms regular undirected graphs. Given a path ofk+2 edges it interchanges the end vertices of the path. Bydefinition this operation preserves regularity and connectivity. Weshow that every regular connected graph can be reached by a seriesof these operations for all k ¡Ý 1. We use arandomized version, called Random k-Flipper, in order tocreate random regular connected undirected graphs that may serve asa backbone for peer-to-peer networks. We prove for degreed¡Ê ¦¸(log n) that a seriesof O(dn) Random k-Flipper operations withk ∈¦¨(d2n2 log1/¦Å) transforms any graph into an expander graph withhigh probability, i.e. 1-n-¦¨(1).The Random 1-Flipper is symmetric, i.e. the transformationprobability from any labeled d-regular graphG to G' is equal to thosefrom G' to G. From this andthe reachability property we conclude that in the limit a series ofRandom 1-Flipper operations converges against an uniformprobability distribution over all connected labeledd-regular graphs. For degreed ∈ ω(1) growing with the graph sizethis implies that iteratively applying Random 1-Flipper transformsany given graph into an expander asymptotically almost surely.We use these operations as a maintenance operation for apeer-to-peer network based on random regular connected graphs thatprovides high robustness and recovers from degenerate networkstructures by continuously applying these random graphtransformations. For this, we describe how network operations forjoining and leaving the network can be designed and how theconcurrency of the graph transformations can be handled.