Effective resistance and other graph measures for network robustness

Robustness is the ability of a network to continue performing well when it is subject to failures or attacks. In this thesis we survey robustness measures on simple, undirected and unweighted graphs, network failures being interpreted as vertex or edge deletions. We study graph measures based on connectivity, distance, betweenness and clustering. Besides these, reliability polynomials and measures based on the Laplacian eigenvalues are considered. In addition to surveying existing measures, we propose a new robustness measure, the normalized effective resistance, which is derived from the total effective resistance. Total effective resistance is — within the field of electric circuit analysis — defined as the sum of the pairwise effective resistances over all pairs of vertices. The strength of this measure lies in the fact that all (not necessarily disjoint) paths are considered, in other words, the more backup possibilities, the larger the normalized effective resistance and the larger the robustness. A chapter is dedicated to optimizing the normalized effective resistance, first for graphs with a fixed number of vertices and diameter, and second for the addition of an edge to a given graph. For all of the measures described above we evaluate the effectiveness as a measure of network robustness. The discussion and comparison of robustness measures is illustrated by a number of examples. Where possible we make extensions to weighted graphs and for all statements we provide either an elaboration of the original proof, or — when a rigorous proof is not available — we provide one ourselves.