In this paper we derive an analytical expression for the mean load at each node of an arbitrary undirected graph for the uniform multicommodity flow problem under random walk routing. We show the mean load is linearly dependent on the nodal degree with a common multiplier equal to the sum of the inverses of the non-zero eigenvalue of the graph Laplacian. Even though some aspects of the mean load value, such as linear dependence on the nodal degree, are intuitive and may be derived from the equilibrium distribution of the random walk on the undirected graph, the exact expression for the mean load in terms of the full spectrum of the graph has not been known before. Using the explicit expression for the mean load, we give asymptotic estimates for the load on a variety of graphs whose spectral density are well known. We conclude with numerical computation of the mean load for other well-known graphs without known spectral densities.
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