Watts and Strogatz Degree Distribution

$P(k) = sum_"n=0"^f( "k" , "K" ) C_( "K" /2)^n * (1-beta)^n * beta^( "K" /(2-n) )* ( ( "beta" * "K" /2)^(( "k" - "K" )/(2-n)) * e^(-( "beta" * "K" )/2) ) / ( (( "k" - "K" )/(2-n)!) ), where f(k,K) = min(k-K/2, K/2)$

$"Number Edges for i-th Node "$

$"Mean Edges Incident on Vertex"$

$"Characterizing Constant"$

Inputs

k - number edges, $k_i$ for i-th node

K - mean number of edges incident on a vertex

$beta$ - characteristic constant of the network

Description

This model describes networks that exhibit the "small world" phenomenon, which is the theory that any person on the planet is a small number of connections away from any other person in the world.

Small world phenomena is not limited to people-related networks. Other networks exhibit the same nodal relationships described by this model.