Emergence, Locating and Measuring It - Systematic Design for Emergence in Cellular Nonlinear Networks - page 63

modules implementing the above concept, together with the CA simulator
ca_sim.m presented in the previous chapter:

function y=dif_filter(x0)
% implements a diffusive filter assuming that
x0 is in the
% range 0 1
[m, n]=size(x0);
i=1:m; j=1:n;
jl=[n, 1:n-1]; jr=[2:n,1];
iu=[m, 1:m-1]; id=[2:m,1];
if m==1
y=1-0.5*abs(x0(jl)+x0(jr)-2*x0);
else
y=1-0.25*abs(x0(i,jl)+x0(i,jr)+x0(iu,j)+x0(id,j)-
4*x0);
end

% compute_clus.m
% compute the clustering coefficient during
% each CA iteration
df=dif_filter(x0);
if dim==1
C=[C mean(df)];
elseif dim==2
C=[C mean(mean(df))];
end

The function dif_filter.m is a diffusive filter implementing (4.3) for either one-

dimensional and two-dimensional CAs. This function is called from com-
pute_clus.m, which has to be included into the main loop of the CA simulator.
Before the loop one should include an initialization of C to a void vector. At the
end of the main loop the entire time sequence of clustering coefficients is avail-
able as a vector C with a size T.
To calculate the transient length Tr the following are applied to the time se-
quence

^ ` T

C t :

t 1 ,..

x The sequence is smoothed, for instance using a FIR filter defined as:

10

1

tzf ( t )

¦

C ( t k )

10

k 1

x The admissible variance ' is computed based on the last 20% of the samples

T

2

of tzf(t) : '

¦

tzf ( t ) mean ( tzf )

t 0 . T

x Tr is decremented from T to 1 and its value is outputted when
' tzf ( Tr ) tzf ( T ) !

x An average clustering coefficient Clus is computed as an average over the
time steps of the

sequence after the transient period:

C t

1

T

Clus C ( f )

¦

C ( t )

T trans

t trans 1

In addition to the defined measures of emergence ( Tr , and Clus ) another meas-
ure, the variance Var is determined. It is inspired from other works [73-75] where