TY  - JOUR
AU  - Doku-Amponsah, Kwabena 
PY  - 2017
TI  - Local Large Deviations: A McMillian Theorem for Typed Random Graph Processes
JF  - Journal of Mathematics and Statistics
VL  - 13
IS  - 4
DO  - 10.3844/jmssp.2017.347.352
UR  - https://thescipub.com/abstract/jmssp.2017.347.352
AB  - For a finite typed graph on n nodes and with type law µ on finite alphabet Γ, we define the spectral potential ρλ(., µ), of the graph. From the ρλ(., µ) we define the Kullback action or the divergence function, Hλ(π || v), with respect to an empirical link measure, π, as the Legendre dual. For the finite typed random graph conditioned to have an empirical link measure π and empirical type measure µ, we prove a Local large Deviation Principle (LLDP), with rate function Hλ(π || v) and speed n. From this LLDP, we obtain a full conditional large deviation principle and a weak variant of the classical McMillian Theorem for the typed random graph models. Given the typical empirical link measure, λµ⊗µ, we show that, the number of typed random graphs is equal to en||λµ⊗µ||H(λµ⊗µ⁄||λµ⊗µ||) approximately. Note that we do not require any topological restrictions on the space of finite graphs for these LLDPs.