Model.p.Fitness.Servedio {systemicrisk}

Multiplicative Fitness Model for Power Law

Description

This model has a power law of the degree distribution with a parameter \alpha and is tuned to a desired link existence probability. It is based on a fitness model.

Usage

Model.p.Fitness.Servedio(n, alpha, meandegree, sdprop = 0.1)

Arguments

n

dimension of matrix.

alpha

exponent for power law. Must be <=-1.

meandegree

overall mean degree (expected degree divided by number of nodes). Must be in (0,1).

sdprop

standard deviation of updated steps.

Details

Every node i has a fitness \theta_i being an independent realisation of a U[0,1] distribution. The probability of a link between a node with fitness x and a node with fitness y is g(x)g(y) where g is as follows. If \alpha=-1 then

g(x)=g0*\exp(-\log(g0)*x)

Otherwise,

g(x)=(g0^(\alpha+1)+(1-g0^(\alpha+1))*x)^(1/(\alpha+1))

where g0 is tuned numerically to achieve the desired overall mean degree.

Updating of the model parameters in the MCMC setup is done via a Metropolis-Hastings step, adding independent centered normal random variables to each node fitness in \theta.

Value

the resulting model.

References

Servedio V. D. P. and Caldarelli G. and Butta P. (2004) Vertex intrinsic fitness: How to produce arbitrary scale-free networks. Physical Review E 70, 056126.

Examples

n <- 5
mf <- Model.p.Fitness.Servedio(n=n,alpha=-2.5,meandegree=0.5)
m <- Model.Indep.p.lambda(model.p=mf,
                          model.lambda=Model.lambda.GammaPrior(n,scale=1e-1))
x <- genL(m)
l <- rowSums(x$L)
a <- colSums(x$L)
res <- sample_HierarchicalModel(l,a,model=m,nsamples=10,thin=10)