  In common descriptions of phase transitions, first order transitions are
characterized by discontinuous jumps in the order parameter and normal
fluctuations, while second order transitions are associated with no jumps and
anomalous fluctuations. Outside this paradigm are systems exhibiting `mixed
order transitions' displaying a mixture of these characteristics. When the jump
is maximal and the fluctuations range over the entire range of allowed values,
the behavior has been coined an `extreme Thouless effect'. Here, we report
findings of such a phenomenon, in the context of dynamic, social networks.
Defined by minimal rules of evolution, it describes a population of extreme
introverts and extroverts, who prefer to have contacts with, respectively, no
one or everyone. From the dynamics, we derive an exact distribution of
microstates in the stationary state. With only two control parameters,
$N_{I,E}$ (the number of each subgroup), we study collective variables of
interest, e.g., $X$, the total number of $I$-$E $ links and the degree
distributions. Using simulations and mean-field theory, we provide evidence
that this system displays an extreme Thouless effect. Specifically, the
fraction $X/\left( N_{I}N_{E}\right) $ jumps from $0$ to $1$ (in the
thermodynamic limit) when $N_{I}$ crosses $N_{E}$, while all values appear with
equal probability at $N_{I}=N_{E}$.
