Maximal entropy random walk
Maximal entropy random walk (MERW) is a popular type of biased random walk on a graph, in which transition probabilities are chosen accordingly to the principle of maximum entropy, which says that the probability distribution which best represents the current state of knowledge is the one with largest entropy. While standard random walk chooses for every vertex uniform probability distribution among its outgoing edges, locally maximizing entropy rate, MERW maximizes it globally (average entropy production) by assuming uniform probability distribution among all paths in a given graph. MERW is used in various fields of science. A  direct application is choosing probabilities to maximize transmission rate through a constrained channel, analogously to Fibonacci coding. Its properties also made it useful for example in analysis of complex networks, like link prediction, community detection, robust transport over networks and centrality measures. Also in image analysis, for example for detecting visual saliency regions, object localization, tampering detection or tractography problem. Additionally, it recreates some properties of quantum mechanics, suggesting a way to repair the discrepancy between diffusion models and quantum predictions, like Anderson localization. Basic model. Consider a graph with formula_1 vertices, defined by an adjacency matrix formula_2: formula_3 if  there is an edge from vertex formula_4 to formula_5, 0 otherwise. For simplicity assume it is an undirected graph, which corresponds to a symmetric formula_6; however, MERW can also be generalized for directed and weighted graphs (for example Boltzmann distribution among paths instead of uniform). We would like to choose a random walk as a Markov process on this graph: for every vertex formula_4 and its outgoing edge to formula_5, choose probability formula_9 of the walker randomly using this edge after visiting formula_4. Formally, find a stochastic matrix formula_11 (containing the transition probabilities of a Markov chain) such that Assuming  this graph is connected and not periodic, ergodic theory says that evolution of this stochastic process leads to some stationary probability distribution formula_16 such that formula_17. Using Shannon entropy for every vertex and averaging over probability of visiting this vertex (to be able to use its entropy), we get the following formula for average entropy production (entropy rate) of the stochastic process: This definition turns out to be equivalent to the asymptotic average entropy (per length) of the probability distribution in the space of paths for this stochastic process. In the standard random walk, referred to here as generic random  walk (GRW), we naturally choose that each outgoing edge is equally probable: For a symmetric formula_6 it leads to a stationary probability distribution formula_16 with It locally maximizes entropy production (uncertainty) for every vertex, but usually leads to a suboptimal averaged global entropy rate formula_23. MERW chooses the stochastic matrix which maximizes formula_23, or equivalently assumes uniform probability distribution among all paths in a given graph. Its formula is obtained by first calculating the dominant eigenvalue formula_25 and corresponding eigenvector formula_26 of the adjacency matrix, i.e. the largest formula_27 with corresponding formula_28 such that formula_29. Then stochastic matrix and stationary  probability distribution are given by for which every possible path of length formula_31 from the formula_4-th to formula_5-th vertex has probability Its entropy rate is formula_35 and the stationary probability distribution formula_16 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 formula_38 can be asymptotically (formula_39) approximated by formula_40 (or formula_41 in bra–ket notation). MERW requires uniform distribution along paths. The number formula_42 of paths with length formula_43 and vertex formula_4 in the center  is hence for all formula_4, Analogously calculating probability distribution for two succeeding vertices, one obtains that the probability of being at the formula_4-th vertex and next at the formula_5-th vertex is Dividing by the probability of being at the formula_4-th vertex, i.e. formula_52, gives for the conditional probability formula_9 of the formula_5-th vertex being next after the formula_4-th vertex Weighted MERW: Boltzmann path ensemble. We have assumed that formula_57 for MERW corresponding to uniform ensemble among paths. However, the above derivation works for real nonnegative formula_6. Parametrizing formula_59 and asking for probability of length formula_60  path formula_61, we get: As in Boltzmann distribution of paths for energy defined as sum of formula_63 over given path. For example it allows to calculate probability distribution of patterns in Ising model. Examples. Let us first look at a simple nontrivial situation: Fibonacci coding, where we want to transmit a message as a sequence of 0s and 1s, but not using two successive 1s: after a 1 there has to be a 0. To maximize the amount of information transmitted in such sequence, we should assume uniform probability distribution in the space of all possible sequences fulfilling this constraint . To practically use such long sequences, after 1 we have to use 0, but there remains a freedom of choosing the probability of 0 after 0. Let us denote this probability by formula_64, then entropy coding would allow encoding a message using this chosen probability distribution. The stationary probability distribution of symbols for a given formula_64 turns out to be formula_66. Hence, entropy production is formula_67, which is maximized for formula_68, known as the golden ratio. In contrast, standard random walk would choose suboptimal formula_69. While choosing larger formula_64 reduces the amount of information produced after 0, it also reduces  frequency of 1, after which we cannot write any information. A more complex example is the defected one-dimensional cyclic lattice: let say 1000 nodes connected in a ring, for which all nodes but the defects have a self-loop (edge to itself). In standard random walk (GRW) the stationary probability distribution would have defect probability being 2/3 of probability of the non-defect vertices – there is nearly no localization, also analogously for standard diffusion, which is infinitesimal limit of GRW. For MERW we have to first find the dominant eigenvector of the adjacency matrix – maximizing formula_25 in: formula_72  for all positions formula_73, where formula_74 for defects, 0 otherwise. Substituting formula_75 and multiplying the equation by −1 we get: formula_76 where formula_77 is minimized now, becoming the analog of energy. The formula inside the bracket is discrete Laplace operator, making this equation a discrete analogue of stationary Schrodinger equation. As in quantum mechanics, MERW predicts that the probability distribution should lead exactly to the one of quantum ground state: formula_78 with its strongly localized density (in contrast to standard diffusion). Taking the infinitesimal limit, we can get standard continuous stationary (time-independent) Schrodinger equation (formula_79 for formula_80) here. Maximal