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1
0.9
0.8
0.7
0.6
0.5
0.4
slope = 0.36
slope = 0.43
0.3
d
= 0
d
= 0
d
= 10
d
= 10
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7 0.8 0.9 1
ln[(
h
c
(L) −
h
)/
h
c
(L)]
Fig. 17.5. Power-law approximation of v
a
vs. (
c
(L)-)/
c
in logarithmic scale. It can be seen
that the slope of the tting curve decreases from 0.43 to 0.36 introducing 10 long-range connections.
c
=L
0:5
as in [Czirok et al. (1997)].
17.3.2. Eect of long-range connections in the Couzin's model
The behavior of the Couzin's model is here investigated by carrying out numerical
simulations for dierent values of the model parameters, and by evaluating the order
parameters v
a
and v
d
.
At rst, model behavior is characterized with respect to dierent values of the
density . When long-range connections are not considered, v
a
increases as a func-
tion of , as shown in Fig. 17.6(a). In this case, in fact, the coordination between
individuals is favored by an increased density, because agents may exchange more
information.
The comparison between the cases with = 0 and 6= 0 shows that the pres-
ence of long-range connections improves system performance in terms of v
a
, i.e. the
degree of group coordination. It should be noticed that v
a
for 6= 0 decreases as
increases. This is due to the fact that at high densities avoidance tasks occur
more frequently. Moreover, Fig. 17.6(a) shows that the improvements due to the
introduction of long-range connections are more important for low density values,
as observed also in the Vicsek's model. It should be noticed that a few long-range
connections improve system performance, but a further increase in the number of
long-range connections leads to poor advantages. This behavior also occurs in small-
world networks, when the changes in the characteristic path length are considered
with respect to the number of long-range connections. In fact, in small-world net-
works, the introduction of a few long-range links decreases the characteristic path
length, but further increases of the number of long-range links have small eects on
this parameter.
