Random spanning trees for expanders, sparsifiers, and virtual network security

This work describes probabilistic methods for utilizing random spanning trees generated via a random walk process. We generalize a 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. Particularly, we show that our union of random spanning trees is a spectral sparsifier of the graph, and we show that for 1/root n < epsilon <= 1, O(log n/e(2)) random spanning trees are required in order to spectrally approximate a bounded degree expander graph. In another part of our research work, we show that our random spanning trees based construction provides security features for virtual networks, in context of Software-Defined Networking. Namely, we demonstrate that our construction allows on-demand efficient monitoring or anonymity services.