brainwaver (version 1.6)

small.world: Computation of some quantitive parameters given in the framework of the small-world graph theory.

Description

For a given graph, computes the size of the largest connex component, degree, minimum path length and clustering coefficient.

Usage

small.world(wave.adj.mat, dat = "reduced", distance = "norm",  coord = 0, export.data = FALSE)

Arguments

wave.adj.mat

adjacency matrix of the graph to be analysed.

dat

character, if dat = "all", the degree, minimum path length and clustering coefficient are computed for each node of the graph. If dat = "reduced", only the mean on all the nodes is computed.

distance

matrix with a distance associated to each pair of nodes of the graph to take into account in the computation of the small-world parameters. By default, the matrix of the distance has a value of 1 in each entry.

coord

optional vector containing the coordinate of the nodes of the graph.

export.data

logical. If TRUE the correlation matrices with the upper and lower bound are exported to text file.

Value

in.degree.mean

real corresponding to the mean of the degree of all the nodes of the graph.

in.degree

vector containing the degree of each node of the graph.

Lp.mean

real corresponding to the mean of minimum path length for each node belonging to the largest connex component of the graph only.

Lp

vector containing the mean minimum path length for each node of the graph. (the computaion of Lp requires at least one neighbourgh per node, if it is not the case the value of Lp is equal to -1)

Cp.mean

real corresponding to the mean of clustering coefficient for each node belonging to the largest connex component of the graph only.

Cp

vector containing the clustering coefficient for each node of the graph. (the computaion of Cp requires at least two neighbourghs per node, if it is not the case the value of Cp is equal to -1)

size.large.connex

real corresponding to the size of the largest connex component.

References

D. J. Watts and S. H. Strogatz (1998) Collective dynamics of ``small-world'' networks. Nature, Vol. 393, pages 440-442.

S. H. Strogatz (2001) Exploring complex networks. Nature, Vol. 410, pages 268-276.

S. Achard, R. Salvador, B. Whitcher, J. Suckling, Ed Bullmore (2006) A Resilient, Low-Frequency, Small-World Human Brain Functional Network with Highly Connected Association Cortical Hubs. Journal of Neuroscience, Vol. 26, N. 1, pages 63-72.

Examples

# fig 3 of Achard (2006)


data(brain)
brain<-as.matrix(brain)

# WARNING : To process only the first five regions
brain<-brain[,1:5]



#Construction of the correlation matrices for each level of the wavelet decomposition
wave.cor.list<-const.cor.list(brain, method = "modwt" ,wf = "la8", n.levels = 6, 
                               boundary = "periodic", p.corr = 0.975)

sup.seq<-((1:10)/10) #sequence of the correlation threshold 
nmax<-length(sup.seq)
in.degree.mean<-matrix(0,6,nmax)
n.levels<-6

#For each value of the correlation thrashold
for(i in 1:nmax){
n.sup<-sup.seq[i]

#Construction of the adjacency matrices associated to each level of the wavelet decomposition
wave.adj.list<-const.adj.list(wave.cor.list, sup = n.sup, proc.length=dim(brain)[1])


#For each level of the wavelet decomposition
for(j in 1:n.levels){

param.sw.brain<-small.world(wave.adj.list[[j]],dat="reduced")
in.degree.mean[j,i]<-param.sw.brain$in.degree.mean

}}

#Plots of the average in-degree in terms of the scale

n.regions<-dim(brain)[2]

plot(sup.seq,(1:nmax)/2,type='n',xlab='Correlation threshold, R',ylab='Mean degree, k',
     cex.axis=2,cex.lab=2,xlim=c(0,1),ylim=c(0,90))


for(i in 1:n.levels){
lines(sup.seq,in.degree.mean[i,],type='l',col=i,lwd=2)
}
lines(sup.seq,rep(log(n.regions),nmax))
legend(x="topright",legend=c("Level 1","Level 2","Level 3","Level 4",
				"Level 5","Level 6"),col=(1:n.levels),lwd=2)