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Jarek 31

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- 31

- TL;DR Summary
- local vs global entropy maximization?

To choose random walk on a graph, it seems natural to to assume that the walker jumps using each possible edge with the same probability (1/degree) - such

Discretizing continuous space and taking infinitesimal limit we get various used diffusion models this way.

However, looking at

It brings a crucial question

GRW

- uses "local" approximation of (Jaynes) https://en.wikipedia.org/wiki/Principle_of_maximum_entropy

- has no localization property (nearly uniform stationary probability distribution),

- has characteristic length of one step - this way e.g. depends on chosen discretization of a continuous system.

MERW

- is the one maximizing mean entropy, "most random among random walks",

- has strong localization property ( https://en.wikipedia.org/wiki/Anderson_localization ) - stationary probability distribution exactly as quantum ground state,

- is limit of characteristic step to infinity - is discretization independent.

Simulator of both for electron conductance: https://demonstrations.wolfram.com/ElectronConductanceModelsUsingMaximalEntropyRandomWalks/

Diagram with example of evolution and stationary denstity, also some formulas (MERW uses dominant eigenvalue):

**GRW (generic random walk) maximizes entropy locally (for each step)**.Discretizing continuous space and taking infinitesimal limit we get various used diffusion models this way.

However, looking at

**mean entropy production**: averaged over stationary probability distribution of nodes, its maximization leads to usually a bit different**MERW**: https://en.wikipedia.org/wiki/Maximal_entropy_random_walkIt brings a crucial question

**which philosophy should we choose**for various applications - I would like to discuss.GRW

- uses "local" approximation of (Jaynes) https://en.wikipedia.org/wiki/Principle_of_maximum_entropy

- has no localization property (nearly uniform stationary probability distribution),

- has characteristic length of one step - this way e.g. depends on chosen discretization of a continuous system.

MERW

- is the one maximizing mean entropy, "most random among random walks",

- has strong localization property ( https://en.wikipedia.org/wiki/Anderson_localization ) - stationary probability distribution exactly as quantum ground state,

- is limit of characteristic step to infinity - is discretization independent.

Simulator of both for electron conductance: https://demonstrations.wolfram.com/ElectronConductanceModelsUsingMaximalEntropyRandomWalks/

Diagram with example of evolution and stationary denstity, also some formulas (MERW uses dominant eigenvalue):

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