We study a fully connected network ͑cluster͒ of interacting two-state units as a model of cooperative decision making. Each unit in isolation generates a Poisson process with rate g. We show that when the number of nodes is finite, the decision-making process becomes intermittent. The decision-time distribution density is characterized by inverse power-law behavior with index = 1.5 and is exponentially truncated. We find that the condition of perfect consensus is recovered by means of a fat tail that becomes more and more extended with increasing number of nodes N. The intermittent dynamics of the global variable are described by the motion of a particle in a double well potential. The particle spends a portion of the total time S at the top of the potential barrier. Using theoretical and numerical arguments it is proved that S ϰ ͑1 / g͒ln͑constϫ N͒. The second portion of its time, K , is spent by the particle at the bottom of the potential well and it is given by K = ͑1 / g͒exp͑constϫ N͒. We show that the time K is responsible for the Kramers fat tail. This generates a stronger ergodicity breakdown than that generated by the inverse power law without truncation. We establish that the condition of partial consensus can be transmitted from one cluster to another provided that both networks are in a cooperative condition. No significant information transmission is possible if one of the two networks is not yet self-organized. We find that partitioning a large network into a set of smaller interacting clusters has the effect of converting the fat Kramers tail into an inverse power law with = 1.5.

We show that the intelligence of a swarm of cooperative units (birds) emerges at criticality, as an effect of the joint action of frequent organizational collapses and of spatial correlation as extended as the flock size. The organizational collapses make the birds become independent of one another, thereby allowing the flock to follow the direction of the lookout birds. Long-range correlation violates the principle of locality, making the lookout birds transmit information on either danger or resources with a time delay determined by the time distance between two consecutive collapses. DOI: 10.1103/PhysRevLett.107.078103 PACS numbers: 87.23.Cc, 64.60.fd, 89.70.Àa, 89.75.Hc In the recent few years, there has been intense activity to explain why a swarm of birds behaves as a single individual [1,2]. How is it possible that when a predator comes, the swarm changes direction to escape from danger? How is it possible that a subset of a swarm becoming aware of the right direction toward a resource [3] convinces the whole flock to pursue that specific course? In which sense, using a metaphor made popular by Couzin [4], can we interpret the swarm as a cognitive mind?The main purpose of this Letter is to prove that this form of intelligence is the effect of the joint action of frequent organizational collapses, allowing the single birds to recover independence of the others, and of a correlation length as extended as the flock size. Although the environment perceiving units are a small fraction of the total number of units, they exert a determinant action on the swarm during the short rearrangement phase after an organizational collapse, which makes each unit free to select a new direction. This freewill condition allows the swarm to select the new flying directions that are transmitted to all the units by the few danger-or resource-perceiving birds, thanks to the criticality-induced long-range correlation. The connection between dynamic instability and information transfer has been recently observed in locust nymphs [3]. The results of this Letter confirm the importance of this observation, establishing at the same time that the information transfer is made possible by the nonlocal nature of the criticality condition, with a time delay depending on the time distance between two consecutive organizational collapses. The correlation length between birds becomes as extended as the finite swarm size, thereby allowing the lookout birds to transmit their flying direction to the whole swarm. Thus, the mean value hi, proportional to the correlation length, remains finite.To afford a convincing proof that the swarm's intelligence is determined by the joint action of organizational collapses and criticality-induced nonlocality, we proceed in two main steps. In the first step we use a model of bird organization, referred to in this Letter as the bird model, to illustrate the concept of temporal complexity. The second step is based on a simpler model, where the relative positions of the birds are fixed and they have only to ch...

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