Constrained Markovian Dynamics of Random Graphs |
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Authors: | A C C Coolen A De Martino and A Annibale |
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Institution: | (2) University of California, San Diego, USA; |
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Abstract: | We introduce a statistical mechanics formalism for the study of constrained graph evolution as a Markovian stochastic process,
in analogy with that available for spin systems, deriving its basic properties and highlighting the role of the ‘mobility’
(the number of allowed moves for any given graph). As an application of the general theory we analyze the properties of degree-preserving
Markov chains based on elementary edge switchings. We give an exact yet simple formula for the mobility in terms of the graph’s
adjacency matrix and its spectrum. This formula allows us to define acceptance probabilities for edge switchings, such that
the Markov chains become controlled Glauber-type detailed balance processes, designed to evolve to any required invariant
measure (representing the asymptotic frequencies with which the allowed graphs are visited during the process). As a corollary
we also derive a condition in terms of simple degree statistics, sufficient to guarantee that, in the limit where the number
of nodes diverges, even for state-independent acceptance probabilities of proposed moves the invariant measure of the process
will be uniform. We test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology, showing
explicitly that, for instances where the simple edge swap dynamics fails to converge to the uniform measure, a suitably modified
Markov chain instead generates the correct phase space sampling. |
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