Understanding the extreme Thouless effect in a simple, dynamic social network - the XIE model

A system undergoing a phase transition which exhibits the characteristics of both first and second order transitions is said to display the Thouless effect, e.g., the order parameter suffering a discontinuity and fluctuates through all values within the jump. In a simple of model extreme introverts and extroverts, the former/latter cuts/adds a random link when chosen to act (the XIE model, EPL 100, 66007 and PRE91, 042102). The steady state consists of a networks of crosslinks between the i's and e's.. The fraction of these, f, serves as an order parameter jumping from ~0 to ~1 as the ratio of i's to e's drops through unity. At unity, f wanders between "soft walls" at f₀ and 1- f₀. With f₀ →0, the system is said to exhibit an “extreme Thouless effect.” We present a novel approach based on a self-consistent mean field theory. The predictions agree spectacularly well with all simulation data. Further, we obtain the analytic form of the asymptotic behavior of f₀ : It vanishes as [(lnL² )/L]^1/2, where L is the size of each subgroup. Though this form sets in as late as L~10²⁰⁰, very good bounds (e.g., ~1%) for more accessible L’s (e.g., 2000) can be found by solving a transcendental equation: x+lnx = ln(L²/2π).