Smoothed analysis on connected graphs

The main paradigm of smoothed analysis on graphs suggests that for any large graph G in a certain class of graphs, perturbing slightly the edge set of G at random (usually adding few random edges to G) typically results in a graph having much "nicer" properties. In this work, we study smoothed analysis on trees or, equivalently, on connected graphs. Given an n-vertex connected graph G, form a random supergraph Gâ of G by turning every pair of vertices of G into an edge with probability Îμ n, where Îμ is a small positive constant. This perturbation model has been studied previously in several contexts, including smoothed analysis, small world networks, and combinatorics. Connected graphs can be bad expanders, can have a very large diameter, and can possibly contain no long paths. In contrast, we show that if G is an n-vertex connected graph, then typically Ghas edge expansion ω( 1 log n ), diameter O(log n), and vertex expansion ω( 1 log n ) and contains a path of length ω(n), where for the last two properties we additionally assume that G has bounded maximum degree. Moreover, we show that if G has bounded degeneracy, then typically the mixing time of the lazy random walk on Gis O(log2 n). All these results are asymptotically tight.