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Assume for simplicity that the considered graph is indirected, connected and aperiodic, what allows to conclude from the [[Perron-Frobenius theorem]] that the dominant eigenvector is unique. Hence <math>A^l</math> can be asymptotically <math>(l\to\infty)</math> approximated by <math>\lambda^l \psi \psi^T</math> (or <math>\lambda^l |\psi\rangle \langle \psi|</math> in [[bra-ket notation]]).
MERW is uniform distribution among paths. The number of length <math>2l</math> paths with vertex <math>i</math> in the center is <math>\sum_{jk} (A^l)_{ji} (A^l)_{ik} </math> what asymptotically grows like <math>\lambda^{2l} \psi_i^2</math>, getting the <math>\rho_i\propto \psi_i^2</math> behavior.
Analogously calculating probability distribution for two succeeding vertices <math>ij</math>, we get <math>\textrm{Pr}(ij)\propto \psi_i A_{ij} \psi_j</math>. Dividing by <math>\textrm{Pr}(i)=\rho_i</math> and normalizing to <math>\sum_j S_{ij}=1</math>, we get <math>S_{ij} = \frac{A_{ij}}{\lambda} \frac{\psi_j}{\psi_i}</math>.
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== External links ==
* Gábor Simonyi, [http://www.nature.com/articles/srep05365 Y. Lin, Z. Zhang, "Mean first-passage time for maximal-entropy random walks in complex networks"].
* [http://demonstrations.wolfram.com/ElectronConductanceModelsUsingMaximalEntropyRandomWalks/ Electron Conductance Models Using Maximal Entropy Random Walks] Wolfram Demonstration Project
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