Physical expander in Virtual Tree Overlay

TL;DR

This paper introduces a new method for constructing constant-degree expanders in virtual tree overlays, enhancing robustness in P2P networks through a local, self-organizing pairing algorithm validated by simulations.

Contribution

Proposes a novel construction of constant-degree expanders for virtual tree overlays using a random pairing approach and a local algorithm that is resilient to churn.

Findings

Achieves constant node expansion with high probability.

Provides a local, self-organizing pairing algorithm with O(log^2 n) complexity.

Demonstrates robustness and efficiency through extensive simulations.

Abstract

In this paper, we propose a new construction of constantdegree expanders motivated by their application in P2P overlay networks and in particular in the design of robust trees overlay. Our key result can be stated as follows. Consider a complete binary tree T and construct a random pairing {\Pi} between leaf nodes and internal nodes. We prove that the graph G\Pi obtained from T by contracting all pairs (leaf-internal nodes) achieves a constant node expansion with high probability. The use of our result in improving the robustness of tree overlays is straightforward. That is, if each physical node participating to the overlay manages a random pair that couples one virtual internal node and one virtual leaf node then the physical-node layer exhibits a constant expansion with high probability. We encompass the difficulty of obtaining this random tree virtualization by proposing a local, selforganizing and churn resilient uniformly-random pairing algorithm with O(log2 n) running time. Our algorithm has the merit to not modify the original tree virtual overlay (we just control the mapping between physical nodes and virtual nodes). Therefore, our scheme is general and can be applied to a large number of tree overlay implementations. We validate its performances in dynamic environments via extensive simulations.