Phenomenological models of socioeconomic network dynamics

We study a general set of models of social network evolution and dynamics. The models consist of both a dynamics on the network and evolution of the network. Links are formed preferentially between “similar” nodes, where the similarity is defined by the particular process taking place on the network. The interplay between the two processes produces phase transitions and hysteresis, as seen using numerical simulations for three specific processes. We obtain analytic results using mean-field approximations, and for a particular case we derive an exact solution for the network. In common with real-world social networks, we find coexistence of high and low connectivity phases and history dependence.

Figure 1

(Color online) Mean degree $⟨k⟩$ as a function of $\lambda$ ($\eta$ has been set to 1) for $ϵ=0.2$ when $d_{ij}$ is the distance on the graph and $\overline{d}>n-2$. The results of a mean-field theory for $n=\infty$ (solid line) are compared to numerical simulations (×) starting from both low and high connected states with $n=20000$. The dashed line corresponds to an unstable solution of the mean-field equations that separates the basins of stability of the two solutions. For finite $n$, the low-density state “flips” to the high-density state when a random fluctuation in $⟨k⟩$ brings the system across the stability boundary (i.e., when a sizable giant component forms). These fluctuations become more and more rare as $n$ increases. Inset: Phase diagram in mean-field theory. Coexistence occurs in the shaded region whereas below (above) only the dense (sparse) network phase is stable. Numerical simulations (symbols) agree qualitatively with the mean-field prediction. The high- (low-) density state is stable up (down) to the points marked with × ◇ and is unstable at points marked with 엯 (+). The behavior of $⟨k⟩$ along the dashed line is reported in the main part of the figure.

Figure 2

Mean degree $⟨k⟩$ (top) and growth rate $v$ (bottom) as a function of $\eta$, found from numerical simulations of the model with Eq. (9). Shown are simulations with $n=500$ (plusses) and 1000 (crosses). Arrows denote the approximate point at which the system jumps from one phase to the other (this point can be dependent on $n$). Here $ϵ=0.001$, noise strength $\Delta =0.1$, and similarity threshold $\overline{d}=2$. The system was run up to $t=1000$ for equilibration, then from $t=1000$ to 1100 for data taking.

Figure 3

Mean degree $⟨k⟩$ (top) and probability that two randomly chosen nodes are within $\overline{d}$ of each other, $\pi$ (bottom), as a function of $\eta$. Shown are simulations with $n=200$ (plusses) and $n=500$ (crosses). Also theory for the high connected phase (dashed line). For large $\eta$, the data points converge toward the theory curve as $n$ increases. Arrows denote the approximate point at which the system jumps from one phase to the other. Here $ϵ=0.001$, noise strength $\Delta =1$, and similarity threshold $\overline{d}=2$. The system was run up to $t=1000$ for equilibration, then from $t=1000$ to 1100 for data taking.

Figure 4

Results from the analytic solution. Upper plot: Plot of the degree distribution, $p\left(k\right)$ for $\eta ∕\lambda =4$, showing also the Poisson distribution with the same $⟨k⟩$ for comparison (dashed lines). Note that the degree distribution is not Poissonian. Lower plot: Plot of the average magnetization of a node of degree $k$ for $\eta ∕\lambda =4$, showing that more highly connected nodes are, on average, more magnetized. $q=10$.

Figure 5

Upper plot: Plots of average degree $\left(⟨k⟩\right)$ as a function of “temperature” $T=1∕\beta$. Lower plot: Plots of the probability that two randomly chosen nodes are in the same state $\left(\pi \right)$, as a function $1∕\beta$. All plots are for $\eta ∕\lambda =4$ (lower curves) and $\eta ∕\lambda =10$ (higher curves). Points are results of simulations. $q=10$ for $n=1000$.

Figure 6

Phase diagram in $\eta$ $(\lambda =1)$ and $T$, showing the high and low connectivity phases and the hysteretic region. Crosses are simulations $(n=5000,10000)$, plusses are theory, and the points are in pairs, one on each side of the phase line. The curves are theory; the leftmost curve is ${\eta }_c=(q∕2)[\exp (1∕T)+q-1]∕[\exp (1∕T)-1]$, which is the expected transition line if the system gets stuck in the metastable, unmagnetized state. The right curve is found using the normal method described above. Upper left: higher connectivity, coordinated region. Lower right: lower connectivity, uncoordinated region. The central region is the hysteretic region.

Figure 7

Plots of the mean connectivity $⟨k⟩$ as a function of $\eta ∕\lambda$. Lines are theory, crosses are simulations, $n=10000$, run to $t=100$ for equilibration, then to $t=200$ for data taking. For $n=10000$ the $q=10$ low state is unstable below the predicted value of 5 due to fluctuations being significant for finite $n$. The crosses that do not lie on the theory curves are systems that “jumped” during the data taking.

Figure 8

Plots of mean connectivity $k$ against time for networks starting in an initially unconnected state with $n=5000$ and $q=10$ and $\lambda =1$. The curve on the top right is $\eta =20$ and the other is $\eta =10$.