Network risk and forecasting power in phase-flipping dynamical networks

To model volatile real-world network behavior, we analyze a phase-flipping dynamical scale-free network in which nodes and links fail and recover. We investigate how stochasticity in a parameter governing the recovery process affects phase-flipping dynamics, and we find the probability that no more than $q\%$ of nodes and links fail. We derive higher moments of the fractions of active nodes and active links, $f_n\left(t\right)$ and $f_{\ell }\left(t\right)$, and we define two estimators to quantify the level of risk in a network. We find hysteresis in the correlations of $f_n\left(t\right)$ due to failures at the node level, and we derive conditional probabilities for phase-flipping in networks. We apply our model to economic and traffic networks.

Figure 1

Network statistics used in analyzing (in)stability in the (i)–(iii) networks. In (a)–(c) we start with a BA network with $N=10000$ and $\langle k\rangle =3$, and then we introduce the (i)–(iii) network, where $T_h=50\%$ and $\tau =50$. Fixing $p_1$ and increasing $p_2$, for each ($p_1,p_2$) we calculate the fraction of active nodes, $f_n\left(t\right)$. We show hystereses for two network statistics: (a) the average $f_n\left(t\right)$, $\langle f_n\rangle$, and (b) the standard deviation of $f_n\left(t\right)$, ${\sigma }_n$. We use $p_1=0.2(p=0.004)$ in two directions: from $p_2=0$ to 1, and then from $p_2=1$ to 0. As $p_2\to p_2^{\prime }$, $\langle f_n\rangle$ and ${\sigma }_n$ exhibit first-order transitions. (c) Emergence of hysteresis in ($p_1,p_2$) space. The hysteresis point $X_A$ is characterized by $\tau =50$, $p=0.0065$ ($p_1=0.277$), and $p_2=0.65$. The curves in (a)–(c) are the averages over different networks of the same fixed $k$. Different networks are characterized by different stochastic realizations and different choices for the initial nodes and links in BA networks.

Figure 2

Effect of stochasticity in $\tau$ on network phase-flipping. If $\tau$ is a random variable from a homogeneous PDF, $H({\tau }_0-\Delta \tau ,{\tau }_0+\Delta \tau )$, with ${\tau }_0=50$, the phase-flipping phenomenon gradually disappears with increasing $\Delta \tau$.

Figure 3

Dependence of hysteresis on parameters $k$ and $T_h\%$. Fixing $p_1$ and increasing $p_2$ for each ($p_1,p_2$), we calculate the fraction of active nodes, $f_n\left(t\right)$. (a) Average $f_n\left(t\right)$, $\langle f_n\rangle$ vs $p_2$ for $\langle k\rangle =3$ and 6. With increasing $\langle k\rangle$, the hysteresis region is getting larger. We choose $T_h=50\%$, $p=0.0001$, and $\tau =50$. (b) Average $f_n\left(t\right)$ vs $p_2$ for $T_h=35\%$ and $75\%$. We set $p=0.0001$, $\langle k\rangle =6$, and $\tau =50$. With increasing $T_h$, the hysteresis region is getting broader.

Figure 4

Network estimator and forecasting in the (i)–(iii) networks. (a) Network estimator, the ratio $\langle f_n\rangle /{\sigma }_n$, exhibits a strong hysteresis—the larger $\langle f_n\rangle /{\sigma }_n$, the more stable the network. The parameters used are as those in Fig. 1. The curve is the average over different networks. (b) The two fractions, $f_n$ and $f_{\ell }$, of the network with time-dependent $p_1$ that moves from $I_0$ [Fig. 1] at time $t=0$ to $X_A$ during the first $\delta =250$ steps. From $\delta$ to $2\delta$, the network stays in $X_A$. Upon reaching $X_A$, the network phase-flips between mainly “active,” I, and mainly “inactive” phases, II. (c) Moving from $I_0$ to $X_A$ after $t=2\delta$, we calculate two CDFs, $C\left(f_n\right)$ and $C\left(f_{\ell }\right)$, both exhibiting a highly asymmetric bimodal shape. From $C\left(f_n\right)$ we can estimate the percentage of (in)active nodes at future $t$. Shown also is a combination of two Poissonian distributions, $w_{1}P\left({\lambda }_1\right)+w_{1}P\left({\lambda }_2\right)$, where $w_1=0.1$, $w_2=0.9$, ${\lambda }_1=0.59$, and ${\lambda }_2=0.36$, where ${\lambda }_1$ and ${\lambda }_2$ are $\langle f_n\rangle$ values in I and II. (d) For the number of active links, $L\left(t\right)$, we show $P\left[\Delta L\left(t\right)/L\right(t\left)\right]$ and its exponential fit.

Figure 5

Emergence of correlations at the network level due to failure at the node level. (a) For each time series (constant $p_2$) used in Fig. 1, we show the DFA plot of $f_n\left(t\right)$ vs scale $\ell$, $F\left(\ell \right)\propto {\left(\ell \right)}^{\alpha }$, for the I-II transition. With increasing $p_2$ up to 0.94, $F\left(\ell \right)$ moves upward, accompanied by a small (b) increase in the DFA exponent $\alpha$. $\alpha$ exhibits a crossover at a scale $l$ that varies with the recovery times $\tau$ and ${\tau }^{\prime }$. Suddenly, at $p_2\approx 0.94$, the DFA exponent drops in a first-order phase transition. We also show correlations in the fraction of externally failed nodes (dotted lines), responsible for correlations in $f_n$. (b) Exponent $\alpha$, calculated for both I-II and II-I transitions for scales $\ell \le 100$, exhibits a clear hysteresis. The curve is the average over different networks.

Figure 6

Application of dynamical networks. (a) Economics—how far is an economy from a volatile regime? Fraction of developed countries, $f_n\left(t\right)$, not in recession ($\Delta \mathit{gdp}>0$). (b) For a (i)–(iii) network we show the model's $f_n\left(t\right)$ obtained by fitting the first and the second moments in (a), where $\tau =1.3$, $T_h=56\%$, $p_1=0.103$, and $p_2=0.77$. (c) Hysteresis with two spinodals for the (i)–(iii) network with $N=100$, $\tau =2$, and $T_h=50\%$. The parameter set is close to the critical line in the supercritical region.

Figure 7

Air traffic network. Fraction of active northeastern US airports, $f_n$, with more than 40% of canceled flights. We also show $f_{\ell }$.