Finite-size effect in the Eguíluz and Zimmermann model of herd formation and information transmission

The Eguíluz and Zimmermann model of information transmission and herd formation in a financial market is studied analytically. Starting from a formal description on the rate of change of the system from one partition of agents in the system to another, a mean-field theory is systematically developed. The validity of the mean-field theory is carefully studied against fluctuations. When the number of agents N is sufficiently large and the probability of making a transaction $a\ll 1/N\ln N,$ finite-size effect is found to be significant. In this case, the system has a large probability of becoming a single cluster containing all the agents. For small clusters of agents, the cluster size distribution still obeys a power law but with a much reduced magnitude. The exponent is found to be modified to the value of $-3$ by the fluctuation effects from the value of $-5/2$ in the mean-field theory.