Extreme Thouless effect in a minimal model of dynamic social networks

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 $I\text{−}E$ 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 $N_I=N_E$.

Figure 1

The nodes of the two groups are denoted by circles: blue open $\left(I\right)$ and red closed $\left(E\right)$. The black lines represent the active cross-links and the red dashed lines the frozen $E\text{−}E$ links. For this network, the sets of $k$'s are $k_I=\left\{1,0,1,1,1,2,1\right\}$ and $k_E=\left\{6,5,4,4\right\}$. Thus, this configuration contributes $1,5,1$ to ${\rho }_I$ $\left(k=0,1,2\right)$ and $2,1,1$ to ${\rho }_E$ $\left(k=4,5,6\right)$, respectively.

Figure 2

The distributions, $P\left(X\right)$, compiled with the time traces of $X$ for three cases with $\left(N_I,N_E\right)$ near and at criticality: $\left(101,99\right)$ (green dashed line), $\left(99,101\right)$ (blue dot-dashed line), and $\left(100,100\right)$ (red solid line).

Figure 3

Power spectra, $I\left(\omega \right)$, associated with the time traces of $X\left(t\right)$. The dashed black line is proportional to $1/{\omega }^2$. Note that the spectra associated with the two off-critical cases (farthest from the dashed line, shown as green dots and a blue line) are statistically identical, as expected from particle-hole symmetry. Both scales are ${\log }_{10}$.

Figure 4

(a) $m\left(h\right)$ for various $N$'s and allowed $\Delta$'s with $h\le 0.01$. (b) Log-log plots of indicated subset of points in (a). Solid and dashed lines are linear fits with slope $-0.363$ and $-0.707$, respectively. Both correlation coefficients are greater than $0.999$. (c) Scaled plots of the points in (a), showing tolerable data collapse. Dashed line is linear, as a guide to the eye.

Figure 5

Degree distributions, $\rho$, for several cases with $N_I+N_E=200$. Simulation results for the low or high $k$ components, associated with introverts or extroverts, are denoted by open and solid symbols, respectively. (a) The symbols for $(N_I,N_E)$ are orange triangles $(150,50)$, purple squares $(125,75)$, and green circles $(101,99)$. The solid black lines are predictions from a self-consistent mean-field theory. (b) When two introverts “change sides,” a dramatic jump in $\rho \left(k\right)$ results, with the case of $(99,101)$ shown as blue diamonds.

Figure 6

Simulation results of the degree distribution for the symmetric $\left(100,100\right)$ case. The two components, denoted by open and solid diamonds, can be associated with the separate distributions ${\rho }_I$ and ${\rho }_E$, respectively.