Conductance and canonical paths for directed non-lazy walks
نویسنده
چکیده
We show two Conductance-like theorems for mixing time of finite non-reversible non-lazy Markov Chains, i.e. when the Markov kernel is neither self-adjoint nor positive definite. The first holds for walks with small holding probability, while the second theorem holds even for walks with no holding probability. These are used to derive two canonical path theorems for such non-reversible non-lazy walks. As an application we show that a known bound for mixing time of walks on undirected Cayley graphs applies to all finite directed Cayley graphs.
منابع مشابه
Intersection Conductance and Canonical Alternating Paths: Methods for General Finite Markov Chains
We extend the conductance and canonical paths methods to the setting of general finite Markov chains, including non-reversible non-lazy walks. The new path method is used to show that a known bound for mixing time of a lazy walk on a Cayley graph with symmetric generating set also applies to the non-lazy non-symmetric case, often even when there is no holding probability.
متن کاملTwo conductance theorems, two canonical path theorems, and two walks on directed Cayley graphs
We show two Conductance-like theorems for mixing time of non-reversible non-lazy walks. These bounds involve a measure of expansion which expresses how well ergodic flow is distributed among vertices, which while conceptually similar to Blocking Conductance apply to non-lazy non-reversible Markov chains as well. As an application we derive two canonical path theorems for mixing time of non-reve...
متن کاملDuality and evolving set bounds on mixing times
We sharpen the Evolving set methodology of Morris and Peres and extend it to study convergence in total variation, relative entropy, L2 and other distances. Bounds in terms of a modified form of conductance are given which apply even for walks with no holding probability. These bounds are found to be strictly better than earlier Evolving set bounds, may be substantially better than conductance ...
متن کاملThe simple random walk and max-degree walk on a directed graph
We show bounds on total variation and L∞ mixing times, spectral gap and magnitudes of the complex valued eigenvalues of a general (non-reversible non-lazy) Markov chain with a minor expansion property. This leads to the first known bounds for the non-lazy simple and max-degree walks on a (directed) graph, and even in the lazy case they are the first bounds of the optimal order. In particular, i...
متن کاملRandom walks on directed Cayley graphs
Previous authors have shown bounds on mixing time of random walks on finite undirected Cayley graphs, both with and without self-loops. We extend this to the most general case by showing that a similar bound holds for walks on all finite directed Cayley graphs. These are shown by use of two new canonical path theorems for mixing time of non-reversible Markov chains. The first result is related ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008