Information Technology Reference
In-Depth Information
1
d
= 10
d
= 1
d
= 0
Ideal
0.9
0.8
0.7
0.6
0.5
1
0.8
0.4
d
= 0
d
= 1
d
= 6
d
= 10
0.6
0.3
0.4
0.2
0.2
0.1
0
0
50
100
150
t (s)
0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
h
Fig. 17.3. Order parameter v
a
vs. when long-range connections are introduced. The various
curves refer to dierent values of . Other parameters are N = 100, L = 5, = 4, r = 1, = 0.03,
t = 1. Simulation time is 200 steps beyond the time necessary to reach convergence in the case
= 0. Inset: time evolution of the average velocity for the case N = 100, = 4, = 1:0, with
dierent .
Eq. (17.11). With the aim of keeping the communication scheme mostly local,
while approaching the performance of the ideal all-to-all scheme, we evaluate in
the following the introduction of long-range connections: at each time step, some
randomly selected agents are allowed to have long-range connections with other
randomly chosen agents, which can lie out of the interaction radius, by including
the heading of these last agents in the computation of the average headingh
i
i
r
.
The idea underlying this new model is that of obtaining a group which is much
more connected, by exploiting almost the same radius of communication, plus a few
long-range links. The eect expected is that of a massive connection without the
communication overhead imposed by this conguration.
At rst, the eects of long-range connections have been evaluated by taking
into account the same set of parameters as in [Vicsek et al. (1995)]. In particular
N = 100 agents are allowed to move on a plane of dimension L = 5, that is the
density = 4. The number of long-range connections at each time step has been
indicated as and ranges from 0 to 10, in order to keep the number of long-range
connections small. The case = 0 corresponds to the original Vicsek's model, where
interactions are only local. The performance of the system has been evaluated by
taking into account the order parameter v
a
with respect to increasing values of .
Results are shown in Fig. 17.3, where it is clear that the performance is not
strongly improved by increasing the value of . Another important eect of long-
range connections is shown in the inset of Fig. 17.3, illustrating that the time
required to reach the steady value of v
a
decreases when long-range connections are
introduced.
