Large Deviations, Basic Information Theorem for Fitness Preferential Attachment Random Networks

Kwabena Doku-Amponsah ,T Ansah-Narh,F.o Mettle

doi:10.5539/ijsp.v3n2p101

Kwabena Doku-Amponsah , T Ansah-Narh + Show 1 more

Open Access

https://doi.org/10.5539/ijsp.v3n2p101

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Abstract

For fitness preferential attachment random networks, we define the empirical degree and pair measure, which counts the number of vertices of a given degree and the number of edges with given fits, and the sample path empirical degree distribution. For the empirical degree and pair distribution for the fitness preferential attachment random networks, we find a large deviation upper bound. From this result we obtain a weak law of large numbers for the empirical degree and pair distribution, and the basic information theorem or an asymptotic equipartition property for fitness preferential attachment random networks.

Highlights

This paper establishes an asymptotic equipartition property (AEP) for fitness preferential attachment
For fitness preferential attachment random networks, we define the empirical degree and pair measure, which counts the number of vertices of a given degree and the number of edges with given fits, and the sample path empirical degree distribution
For the empirical degree and pair distribution for the fitness preferential attachment random networks, we find a large deviation upper bound

Summary

Introduction

This paper establishes an asymptotic equipartition property (AEP) for fitness preferential attachment Dereich and Morters (2009) studied a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a sublinear function of their degree. We prove a large deviation upper bound for the empirical degree and pair distribution, and use it to find an AEP for for P. Empirical degree and pair distribution we prove from the large deviation upper bound a weak law of large numbers, see Theorem 3. From this weak law of large numbers we find the AEP for a networked structure datasets model, see Theorem 7, as a fitness P. Note that the measure L[Xnt]/n, for t ∈ [0, 1) is deterministic and its distribution is degenerate at some ν[nt]/n, for t ∈ [0, 1) converging to νt, t ∈ [0, 1)

Exponential Change-of-Measure

Proof of Theorem 1

Proof of Theorem 2