  Collective behaviour is known to be the result of diverse dynamics and is
sometimes likened to a living system. Although many studies have revealed the
dynamics of various collective behaviours, their main focus was on the
information process inside the collective, not on the whole system itself. For
example, the qualitative difference between two elements and three elements as
a system has rarely been investigated. Tononi et al. have proposed Integrated
Information Theory (IIT) to measure the degree of consciousness $\Phi$. IIT
postulates that the amount of information loss caused by certain partitions is
equivalent to the degree of information integration in the system. This measure
is not only useful for estimating the degree of consciousness but can also be
applied to more general network systems. Here we applied IIT (in particular,
IIT 3.0 using PyPhi) to analyse real fish schools ({\it Plecoglossus
altivelis}). Our hypothesis in this study is a very simple one: a living system
evolves to raise its $\Phi$ value. If we accept this hypothesis, IIT reveals
the existence of continuous and discontinuous properties as group size varies.
For example, leadership in the fish school emerged for a school size of four or
above; but not below three. Furthermore, this transition was not observed by
measuring mutual information or in a simple Boids model. This result suggests
that integrated information $\Phi$ can reveal some inherent properties which
cannot be observed using other measures. We also discuss how the fish
recognition of the figure-ground relation, that is, what determines the
relevant ON and OFF states, may reveal various optimal paths for obtaining the
functional evolution of collective behaviour.
