Speeding up random walk mixing by starting from a uniform vertex

by   Alberto Espuny Díaz, et al.

The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of bottlenecks in a graph hampers mixing and, in particular, starting inside a small bottleneck significantly slows down the diffusion of the walk in the first steps of the process. To circumvent this problem, the average mixing time is defined to be the mixing time starting at a uniformly random vertex. In this paper we provide a general framework to show logarithmic average mixing time for random walks on graphs with small bottlenecks. The framework is especially effective on certain families of random graphs with heterogeneous properties. We demonstrate its applicability on two random models for which the mixing time was known to be of order log^2n, speeding up the mixing to order log n. First, in the context of smoothed analysis on connected graphs, we show logarithmic average mixing time for randomly perturbed graphs of bounded degeneracy. A particular instance is the Newman-Watts small-world model. Second, we show logarithmic average mixing time for supercritically percolated expander graphs. When the host graph is complete, this application gives an alternative proof that the average mixing time of the giant component in the supercritical Erdős-Rényi graph is logarithmic.


page 1

page 2

page 3

page 4


Estimating graph parameters via random walks with restarts

In this paper we discuss the problem of estimating graph parameters from...

Fast MCMC sampling algorithms on polytopes

We propose and analyze two new MCMC sampling algorithms, the Vaidya walk...

Rapid mixing in unimodal landscapes and efficient simulatedannealing for multimodal distributions

We consider nearest neighbor weighted random walks on the d-dimensional ...

Local Mixing Time: Distributed Computation and Applications

The mixing time of a graph is an important metric, which is not only use...

Complexity and Geometry of Sampling Connected Graph Partitions

In this paper, we prove intractability results about sampling from the s...

Analyzing Ta-Shma's Code via the Expander Mixing Lemma

Random walks in expander graphs and their various derandomizations (e.g....

Towards a Theory of Mixing Graphs: A Characterization of Perfect Mixability

Some microfluidic lab-on-chip devices contain modules whose function is ...

Please sign up or login with your details

Forgot password? Click here to reset