2 research outputs found
Fiedler Vector Approximation via Interacting Random Walks
The Fiedler vector of a graph, namely the eigenvector corresponding to the
second smallest eigenvalue of a graph Laplacian matrix, plays an important role
in spectral graph theory with applications in problems such as graph
bi-partitioning and envelope reduction. Algorithms designed to estimate this
quantity usually rely on a priori knowledge of the entire graph, and employ
techniques such as graph sparsification and power iterations, which have
obvious shortcomings in cases where the graph is unknown, or changing
dynamically. In this paper, we develop a framework in which we construct a
stochastic process based on a set of interacting random walks on a graph and
show that a suitably scaled version of our stochastic process converges to the
Fiedler vector for a sufficiently large number of walks. Like other techniques
based on exploratory random walks and on-the-fly computations, such as Markov
Chain Monte Carlo (MCMC), our algorithm overcomes challenges typically faced by
power iteration based approaches. But, unlike any existing random walk based
method such as MCMCs where the focus is on the leading eigenvector, our
framework with interacting random walks converges to the Fiedler vector (second
eigenvector). We also provide numerical results to confirm our theoretical
findings on different graphs, and show that our algorithm performs well over a
wide range of parameters and the number of random walks. Simulations results
over time varying dynamic graphs are also provided to show the efficacy of our
random walk based technique in such settings. As an important contribution, we
extend our results and show that our framework is applicable for approximating
not just the Fiedler vector of graph Laplacians, but also the second
eigenvector of any time reversible Markov Chain kernel via interacting random
walks.Comment: in ACM SIGMETRICS, Boston, MA, June 2020, to appear. (Also will be in
Proc. ACM Meas. Anal. Comput. Syst (POMACS), March 2020