Local Large deviation: A McMillian Theorem for Coloured Random Graph Processes
K Doku-Amponsah - arXiv preprint arXiv:1707.01978, 2017 - arxiv.org
arXiv preprint arXiv:1707.01978, 2017•arxiv.org
For a finite typed graph on $ n $ nodes and with type law $\mu, $ we define the so-called
spectral potential $\rho_ {\lambda}(\,\cdot,\,\mu), $ of the graph. From the $\rho_
{\lambda}(\,\cdot,\,\mu) $ we obtain Kullback action or the deviation function, $\mathcal {H} _
{\lambda}(\pi\,\|\,\nu), $ with respect to an empirical pair measure, $\pi, $ as the Legendre
dual. For the finite typed random graph conditioned to have an empirical link measure $\pi $
and empirical type measure $\mu $, we prove a Local large deviation principle (LLDP), with …
spectral potential $\rho_ {\lambda}(\,\cdot,\,\mu), $ of the graph. From the $\rho_
{\lambda}(\,\cdot,\,\mu) $ we obtain Kullback action or the deviation function, $\mathcal {H} _
{\lambda}(\pi\,\|\,\nu), $ with respect to an empirical pair measure, $\pi, $ as the Legendre
dual. For the finite typed random graph conditioned to have an empirical link measure $\pi $
and empirical type measure $\mu $, we prove a Local large deviation principle (LLDP), with …
For a finite typed graph on nodes and with type law we define the so-called spectral potential of the graph.From the we obtain Kullback action or the deviation function, with respect to an empirical pair measure, as the Legendre dual. For the finite typed random graph conditioned to have an empirical link measure and empirical type measure , we prove a Local large deviation principle (LLDP), with rate function and speed We deduce from this LLDP, a full conditional large deviation principle and a weak variant of the classical McMillian Theorem for the typed random graphs. Given the typical empirical link measure, the number of typed random graphs is approximately equal Note that we do not require any topological restrictions on the space of finite graphs for these LLDPs.
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