How damage diversification can reduce systemic risk

We study the influence of risk diversification on cascading failures in weighted complex networks, where weighted directed links represent exposures between nodes. These weights result from different diversification strategies and their adjustment allows us to reduce systemic risk significantly by topological means. As an example, we contrast a classical exposure diversification (ED) approach with a damage diversification (DD) variant. The latter reduces the loss that the failure of high degree nodes generally inflict to their network neighbors and thus hampers the cascade amplification. To quantify the final cascade size and obtain our results, we develop a branching process approximation taking into account that inflicted losses cannot only depend on properties of the exposed, but also of the failing node. This analytic extension is a natural consequence of the paradigm shift from individual to system safety. To deepen our understanding of the cascade process, we complement this systemic perspective by a mesoscopic one: an analysis of the failure risk of nodes dependent on their degree. Additionally, we ask for the role of these failures in the cascade amplification.

Figure 1

Illustration of the local tree approximation. The green node is the focal node. Its conditional failure probability $\mathbb{P}(s=1|k=5)$ can be computed according to Eq. (5), and depends on the state of its neighbors: Here, the two red ones and the gray one have failed, while the two blue ones are still functional. The neighbors' conditional failure probabilities $\mathbb{P}(s_{\mathrm{nb}}=1|k)$ rely on the failure probabilities of their own neighbors without regarding the green focal point.

Figure 2

(a) The studied degree distributions: Poisson distribution with parameter $\lambda =2.82$ and cutoff degree $c=50$ (blue), scale-free distribution with exponent $\gamma =3$ and maximal degree $c=200$ and average degree $z=3$ (red) in log-log scale. (b) Comparison of the final fraction of failed nodes obtained by simulations (symbols) on networks with $100000$ nodes as average over 500 independent realizations with numerical results from the cHMF (lines) for the DD case. The thresholds $\mathrm{\Theta }$ are normally distributed with mean $\mu$ and standard deviation $\sigma$ ($\mathrm{\Theta }\sim \mathcal{N}(\mu ,{\sigma }^2)$). Nodes with negative thresholds fail initially. Results for scale free networks are depicted in blue for $\sigma =0.2$ and red for $\sigma =0.5$. Results for networks with Poisson degree distribution are shown in black for $\sigma =0.2$ and green for $\sigma =0.5$.

Figure 3

(a) Fraction of failed nodes obtained by (simpHMF) for the DD case on Poisson random networks with $\lambda =2.82, z=3$, and $c=50$. (b) Difference between the correct version (cHMF) and (a). (c) Fraction of failed nodes obtained by (simpHMF) for the DD case on scale free networks with $\gamma =3, z=3$, and $c=200$. (d) Difference between the corresponding correct version (cHMF) version and (c). The thresholds $\mathrm{\Theta }$ are normally distributed with mean $\mu$ and standard deviation $\sigma$ ($\mathrm{\Theta }\sim \mathcal{N}(\mu ,{\sigma }^2)$).

Figure 4

Phase diagram for the fraction of failed nodes $\rho$ with normally distributed thresholds ($\mathrm{\Theta }\sim \mathcal{N}(\mu ,{\sigma }^2)$) for (a) fully connected networks, and (c) regular networks with degree $z=3$. The darker the color the higher is the systemic risk. The middle panel (b) shows their difference ${\rho }^{\left(a\right)}-{\rho }^{\left(b\right)}$.

Figure 5

Phase diagrams for the final fraction of failed nodes $\rho$ calculated numerically for different degree distributions with average degree $z=3$, diversification variants ED and DD, and their differences. The thresholds $\mathrm{\Theta }$ are normally distributed with mean $\mu$ and standard deviation $\sigma$ ($\mathrm{\Theta }\sim \mathcal{N}(\mu ,{\sigma }^2)$). The fraction of initially failed nodes is given by $F_{\mathrm{\Theta }}\left(0\right)$. The darker the color the higher is the systemic risk. First row: Poisson distribution with parameter $\lambda =2.82$, and cutoff degree $c=50$ for the ED (left) and DD (right). The middle panel shows their difference ${\rho }^{\left(\mathrm{ED}\right)}-{\rho }^{\left(\mathrm{DD}\right)}$. Second row: The difference between the diagrams with Poisson and scale-free degree distributions for the ED variant (left). Similarly for the DD variant (right). In the middle panel the initial fraction of failed nodes $\rho \left(0\right)$ is illustrated. ${\rho }_o:=\rho \left(0\right)$ is constant along the lines $\sigma =\mu /{\mathrm{\Phi }}^{-1}\left({\rho }_0\right)$. Third row: Scale-free distribution with exponent $\gamma =3$, and maximal degree $c=200$ for the ED (left) and DD (right). The middle panel again shows their difference ${\rho }^{\left(\mathrm{ED}\right)}-{\rho }^{\left(\mathrm{DD}\right)}$.

Figure 6

(a) Conditional failure probability for ED and normally distributed thresholds with parameters $(\mu ,\sigma )=(0.3,0.2)$ with $\pi =0.2$ (black triangles), $(\mu ,\sigma )=(0.15,0.01)$ with $\pi =0.1$ (red circles), and $(\mu ,\sigma )=(0.7,0.2)$ with $\pi =0.3$ (blue diamonds). Here, $\pi$ is chosen independent of a cascade process. (b) Conditional failure probability for ED and normally distributed thresholds with parameters $(\mu ,\sigma )=(0.27,0.09)$ (red), $(\mu ,\sigma )=(0.3,0.3)$ (blue), $(\mu ,\sigma )=(0.34,0.19)$ (black), $(\mu ,\sigma )=(0.7,0.6)$ (green), and $\pi$ defined by cascade equilibrium. Lines indicate a scale free distribution with $\gamma =3, c=200$, and $z=3$, while symbols indicate Poisson random graphs with $\lambda =2.82, z=3$, and $c=50$. (c) As in (b), but for DD while in this case green color refers to parameters $(\mu ,\sigma )=(0.5,0.5)$. Nodes with degree $k>10$ have a similar failure probability as a node with degree $k=10$.

Figure 7

Phase diagrams for the final fraction of failed nodes $\rho$ calculated numerically for different degree distributions, diversification variants, and their differences. The thresholds $\mathrm{\Theta }$ are normally distributed with mean $\mu$ and standard deviation $\sigma$ ($\mathrm{\Theta }\sim \mathcal{N}(\mu ,{\sigma }^2)$). The fraction of initially failed nodes is given by $F_{\mathrm{\Theta }}\left(0\right)$. The darker the color the higher is the systemic risk. First row: Poisson distribution with parameter $\lambda =8$ and cutoff degree $c=50$ for the ED (left) and DD (right). The middle shows their difference ${\rho }^{\left(\mathrm{ED}\right)}-{\rho }^{\left(\mathrm{DD}\right)}$. Second row: Degree distribution of the Italian interbank lending network on October 1, 2002, for the ED (left) and DD (right). The middle shows their difference ${\rho }^{\left(\mathrm{ED}\right)}-{\rho }^{\left(\mathrm{DD}\right)}$. Third row: Scale-free distribution with exponent $\gamma =3$ and maximal degree $c=200$ for the ED (left) and DD (right). The middle shows their difference ${\rho }^{\left(\mathrm{ED}\right)}-{\rho }^{\left(\mathrm{DD}\right)}$.

Figure 8

The studied degree distributions: Poisson distribution with parameter $\lambda =8$ and cutoff degree $c=50$ (blue), scale-free distribution with exponent $\gamma =3$ and maximal degree $c=200$ (red), and degree distribution measured from a snapshot of the Italian interbank market on October 1, 2002 (black), in log-log scale.

Figure 9

Final fraction of failed nodes $\rho$ in case of the DD variant for random networks with Poisson degree distribution parametrized by $\lambda =8$ and cutoff $c=50$. The thresholds are normally distributed with varying mean $\mu$ and standard deviation $\sigma =0.31$ (red), $\sigma =0.51$ (blue), and $\sigma =0.71$ (black). Circles belong to weights defined as $w_{ji}^{\mathrm{DD}}$, while solid lines belong to net exposures $w_{ji}^{\mathrm{eff}}$.