One of the most remarkable characteristics of systems like a school of fish or a swarm of locusts, is the emergence of states of collective
order in which all the particles move together in the same direction, in spite of the fact that the interactions between the particles
are (presumably) of short range. Such systems are generally out of, and far from, equilibrium. Therefore, the emergence
of ordered states in these systems cannot be accounted for by the standar theorems in statistical mechanics that explain the existence
of long-range order phase transitions in equilibrium systems (like the Ising model).
Cristian Huepe, Hernan Larralde and I have
put forward a formalism, based on networks and network dynamics, to explain the emergence of collective order in systems of autonomous
particles. This formalism not only does explain the existence of colective order and phase transitions in these systems, but
also allows us to analyze within the same framework the dynamics of flocks and majority voter models. In
the first case each particle tends to move in the direction of motion in which the majority of its neighbors are moving. In the
second case, a given individual in a society tends to be of the same opinion as the majority of his friends. In both cases there
may be some noise in the enviroment that prevents the particles or individuals from blindly following the majority rule. In
other words, while it is true that I tend to be of the same opinion as the majority of my friends, with a finite probability I can
have the opposite opinion.
The results that we have obtained using the network approach to analyze the onset of collective
behavior in flocking-like systems and majority voter models suggest that the main ingredients needed for the emergence of ordered
- Majority rule (each particle tends to follow the majority of its "friends" or "neighbors").
- Noise (with a given probability,
the majority rule can be violated).
- Long-range interactions (particles far away from each other can interact).
Point number 3
is interesting. If we do not explicitely assume that distant particles can interact, then collective order is never achived. This
means that for a school of fish to move collectively, two fish at opposite ends of the school should know what each other
is doing. Our last work on this topic relates the topology of the network with the nature of the phase transition. It can be
The emergence of order in flocks and majorities can be analyzed within the same theoretical framework.