that each outgoing edge is equally probable: . For a symmetric it leads to a stationary probability distribution with . It locally maximizes entropy production (uncertainty) for every vertex, but usually leads to a suboptimal averaged global entropy rate . MERW chooses the stochastic matrix which maximizes , or equivalently assumes uniform probability distribution among all paths in a given graph. Its formula is obtained by first calculating the dominant eigenvalue and corresponding eigenvector of the adjacency matrix, i.e. the largest with corresponding such that . Then stochastic matrix and stationary probability distribution are given by for which every possible path of length from the -th to -th vertex has probability . Its entropy rate is and the stationary probability distribution is . In contrast to GRW, the MERW transition probabilities generally depend on the structure of the entire graph (are nonlocal). Hence, they should not be imagined as directly applied by the walker – if random-looking decisions are made based on the local situation, like a person would make, the GRW approach is more appropriate. MERW is based on the principle of maximum entropy, making it the safest assumption when we don't have any additional knowledge about the system. For example, it would be appropriate for modelling our knowledge about an object performing some complex dynamics – not necessarily random, like a particle. Sketch of derivation Assume for simplicity that the considered graph is indirected, connected and aperiodic, allowing to conclude from the Perron–Frobenius theorem that the dominant eigenvector is unique. Hence can be asymptotically () approximated by (or in bra–ket notation). MERW requires uniform distribution along paths. The number of paths with length and vertex in the center is , hence for all , . Analogously calculating probability distribution for two succeeding vertices, one obtains that the probability of being at the -th vertex and next at the -th vertex is . Dividing by the probability of being at the -th vertex, i.e. , gives for the conditional probability of the -th vertex being next after the -th vertex . Weighted MERW: Boltzmann path ensemble We have assumed that for MERW corresponding to uniform ensemble among paths. However, the above derivation works for real nonnegative . Parametrizing and asking for probability of length path , we get: As in Boltzmann distribution of paths for energy defined as sum of over given path. For example it allows to calculate