# Model.p.Fitness.Servedio: Multiplicative Fitness Model for Power Law In systemicrisk: A Toolbox for Systemic Risk

## 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

 `1` ```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 alph=-1a then

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

Otherwise,

g(x)=(g0^(α+1)+(1-g0^(α+1))*x)^(1/(α+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.

## 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

 ```1 2 3 4 5 6 7 8``` ```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) ```

systemicrisk documentation built on Nov. 17, 2017, 6:13 a.m.