Study of the correlation between the clustering coefficient (defined a l=
a Watts-Strogatz) and the degree of the nodes of a network. The correlation=
is expressed through the function *c(k)*, which represents the aver=
age clustering coefficient of all nodes with degree *k*.

The network to analyze must be undirected, otherwise there are no specia= l constraints.

=20The correlation function *c(k)* can help to identify hierarchical=
organization in networks.

The algorithm requires two inputs, the file where the edges of the netwo=
rk are listed and the number of points for the binned correlation function =
described below. A first read-in of the inputfile will set the values of th=
e number of nodes and edges of the network. In the second read-in the degre=
e of all nodes is calculated and the edges are stored in an array. Then the=
clustering coefficients for all nodes are calculated. The program generate=
s one output file, corresponding to the *binned* correlation functio=
n, i.e. the interval spanned by the values of the degree is divided into bi=
ns whose size grows while going to higher values of the variable. The size =
of each bin is obtained by multiplying by a fixed number the size of the pr=
evious bin. The program calculates the average clustering coefficient of no=
des whose degree falls within each bin. Because of the different sizes of t=
he bins, these averages must be divided by the respective bin size, to have=
consistent results.

This technique is particularly suitable to study s=
kewed correlation functions: the fact that the size of the bins grows large=
for large values of the degree compensates for the fact that not many node=
s have high degrees, so it suppresses the fluctuations that one would obser=
ve by using bins of equal size. On a double logarithmic scale, which is ver=
y useful to determine the possible power law behavior of the correlation fu=
nction, the points of the latter will appear equally spaced on the x-axis.<=
br> The algorithm runs in a time , where is the number of nodes of the netw=
ork and is the average degree squared.

A simple application of this algorithm could be to calculate *c(k) for networks created by the modeling algorithms of the NWB. For instance=
, the inputfile can be created through the Barabasi-Albert model.*

- =20
- Source Code =20

The algorithm was implemented and documented by S. Fortunato, integrated= by S. Fortunato and W. Huang.

=20Vazquez, A., Pastor-Satorras, R., Vespignani, A. (2002) Large-scale topological and dynamical properties o=
f Internet

Physical Review E 65:066130.

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