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Note that before using this program, a file called “result” has to be stored in the
running directory. This file contains a list with the
tb parameters for all ID
from 1 to 1,022 (the ID0 and ID1,023 are trivial constant functions with no
relevance for cellular computation). The file is organied around three matrix
variables all having the line as index for a given ID (line 1 corresponds to ID0,
etc.). The matrix tab contains the value of b, the matrix Ntrtab conatins the value
of the number of transients and finally the matrix Ttab contains the effective val-
ues of the transients
,
1 ,...
t
m
1
tt . The information to be processed shall be previously
loaded into a matrix x0 and saved with a name, which is an input parameter of the
above function.
1 ,...
m
1
3.6 Modeling and Simulation of “Small-Worlds” Systems
Note that in the above atlab function
semit_ca.m we introduced a variable
called f , allowing to randomie part of the local connections such that a fraction f
of cells is not connected locally anymore. Instead of connecting the output to one
of the neighbors, the output of such a cell is rerouted randomly to another cell in
the array (Fig. 3.). atts [1] introduced this model as a more natural model of an
interconnection network. Further research revealed that many systems in nature
have this property; most “cells” are locally connected but some large-distance
connections are also present. y taking two arbitrary cells in a network atts de-
fined the average degree of separation d ” as the number of connections connect-
ing a cell to another. A small distance is desirable because it means fast computation.
Indeed for small d , changes are rapidly transmitted to all cells in a network. For
instance it turns out that in a social network this average distance is around 6,
a fact which may appear surprising given the number of several billion of humans
on Earth. ut this is the effect of the “small-world” model where each person
knows a few important or distant fellows besides her very close friends.
In a cellular array, due to the fact that interconnectivity is purely local, the av-
erage separation distance d is large, i.e. the information cannot propagate faster
than 1 unit of discrete space per iteration. At the opposite extreme, a fully con-
nected network has d=1 because each cell is connected to any other possible cell.
But fully connected networks are not natural and they require a large amount
( N ) of interconnecting devices. Much more than mN devices required in the
case of cellular systems with m cells in the neighborhood. The most important dis-
covery of Watts is that relatively small separation lengths d, much smaller than in
the case of the cellular model, can be obtained by altering a small fraction f of
cells from a cellular model such that their neighboring connections are replaced
with randomly chosen connections with distant cells.
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