Random Spanning Trees for Expanders, Sparsifiers, and Virtual Network Security

TL;DR

This paper introduces probabilistic methods using random spanning trees to approximate graph expansion, spectral sparsifiers, and enhance security in virtual networks, with theoretical guarantees and practical applications in SDN.

Contribution

It generalizes random spanning tree methods to weighted graphs, proves spectral sparsification with optimal bounds, and applies these concepts to improve network security and monitoring in SDN environments.

Findings

Union of random spanning trees approximates graph expansion.

O(log n / ε^2) trees suffice for spectral sparsification.

Network virtualization enables efficient monitoring and anonymity.

Abstract

This work describes probabilistic methods for utilizing random spanning trees generated via a random walk process. Goyal et al. showed that the union of random spanning trees approximates the expansion of every cut of a graph. First, we generalize the method by Goyal et al. for weighted graphs and show that it is possible to approximate the expansion of every cut in a weighted graph with the union of random spanning trees generated by a random walk on a weighted graph. Second, we show that our union of random spanning trees is a spectral sparsifier of the graph. Moreover, we show that $O(\log n /\epsilon^2)$ random spanning trees are required in order to spectrally approximate a bounded degree graph. This result closes a previously open question on the number of random spanning trees required for saprsification. Third, we show that our random spanning trees based construction provides security features for virtual networks, in the context of Software-Defined Networking. Network virtualization coupled with Software-Defined Networking allows new on-demand management capabilities. We demonstrate such a service, namely, on-demand efficient monitoring or anonymity. The proposed service is based on network virtualization of expanders or sparsifiers over the physical network. The defined virtual (or overlay) communication graphs coupled with a multi-hop extension of Valiant randomization based routing lets us monitor the entire traffic in the network, with a very few monitoring nodes. We propose methods that theoretically improve services provided by existing monitoring or anonymity networks, and optimize the degree of monitoring or anonymity.