A Non-asymptotic Approach to Analyzing Kidney Exchange Graphs

We propose a non-asymptotic approach to analyze kidney exchange that builds on the random graph model of kidney exchange introduced in Ashlagi, Garmarnik, Rees and Roth's "The need for (long) chains in kidney exchange" (2012). We analyze a two phase procedure where random walks are used to allocate chains, followed by allocation via matching in cycles. Random walks preserve the probabilistic structure of residual graphs, greatly facilitating analysis without sending the number of nodes to infinity. We derive useful analytical bounds that illustrate the performance of our procedure and more general kidney allocation procedures. Our results complement previous asymptotic results for large (limit) graphs on the benefits of using chains in kidney exchange and empirical results based on data from fielded kidney exchanges.