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arise in physical, social and biological phenomena, ecosystems and telecommunica-
tions [Galstyan and Lerman (2002); Wang and Wilde (2004); Zimmermann et al.
(2004); Frasca et al. (2006, 2008)].
In network theory, the eect of long-range connections (also called weak ties) in
static networks is well known. An example is that of small-world networks [Watts
and Strogatz (1998)], which are built by starting from a regular ring lattice network
and rewiring links with a given probability. If the probability is low, this results
in a network structure which is very similar to the original one, except for a few
long-distance links. The presence of such links makes the characteristic path length
shorter while keeping the clustering coecient high. In other words, small-world
topologies conjugate the advantages of fast information transfer typical of random
networks with the robustness features typical of regular networks. Moreover, in gen-
eral long-range connections enhance signicantly the system performance in terms
of synchronization and cooperation. One may expect that a similar result can be
achieved also in time-variant networks.
In this section we report on the eects of the introduction of long-range connec-
tions in the interaction schemes of the two models introduced in Section 17.2. This
is motivated by possible applications in decentralized control of autonomous mobile
robots. In fact, the addition of a few long-range connections is expected to provide
better performance with a moderate, if not negligible, change in the communication
scheme, which is kept mostly local.
When applied to robotics, the aforementioned models can be used to control the
cooperative behavior of robots through decentralized strategies. These strategies
exploit mostly local connections realized through sensors able to collect information
from neighboring robots, without needing a central coordination and supervision
unit. Recently, technology involved in control techniques for single robots has been
pushed to a high level of robustness as long as computational and communication ca-
pabilities are growing. These enhancements make possible to equip each robot with
sensors, wireless transmission systems and a processor, dedicated to the processing
of information collected by sensors through the communication system [Petriu et al.
(2004)]. In this perspective, each robot is able to control its movement by exploiting
information coming from other robots acting in the same environment. Thus, the
development of distributed algorithms for controlling the robots is becoming very
important to achieve a global coordinated and cooperative behavior without the
use of central coordination [Klavins and Murray (2004)].
17.3.1. Eect of long-range connections in the Vicsek's model
The behavior of the Vicsek's model (17.1) in absence of noise has been investigated
in several papers. In particular, in [Jadbabaie et al. (2003)], under a simplifying
hypothesis for the heading update rule, it is proved that all particles shall move in
the same heading, if the graphs describing their interactions are periodically linked
together. A weaker condition is proved in [Li and Wang (2004)], where the sucient
