Robustness of networks against fluctuation-induced cascading failures

Dominik Heide, Mirko Schäfer, and Martin Greiner

Phys. Rev. E 77, 056103 – Published 8 May 2008

Abstract

Fluctuating fluxes on a complex network lead to load fluctuations at the vertices, which may cause them to become overloaded and to induce a cascading failure. A characterization of the one-point load fluctuations is presented, revealing their dependence on the nature of the flux fluctuations and on the underlying network structure. Based on these findings, an alternate robustness layout of the network is proposed. Taking load correlations between the vertices into account, an analytical prediction of the probability for the network to remain fully efficient is confirmed by simulations. Compared to previously proposed mean-flux layouts, the alternate layout comes with significantly less investment costs in the high-confidence limit.

Received 9 November 2007

DOI:https://doi.org/10.1103/PhysRevE.77.056103

Authors & Affiliations

Dominik Heide^1,*, Mirko Schäfer^1,†, and Martin Greiner^2,‡

¹Frankfurt Institute for Advanced Studies and Frankfurt International Graduate School for Science, Johann Wolfgang Goethe Universität, Ruth-Moufang-Straße 1, D-60438 Frankfurt am Main, Germany
²Corporate Technology, Information & Communications, Siemens AG, D-81730 München, Germany

^*dheide@fias.uni-frankfurt.de
^†schaefer@fias.uni-frankfurt.de
^‡martin.greiner@siemens.com

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Issue

Vol. 77, Iss. 5 — May 2008

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Images

Figure 1
(Color online) Simulated load distributions $p (L_{v})$ due to sourcelike flux fluctuations for three fluctuation strengths. Two vertices with minimum (a) and maximum (b) load are depicted. One typical realization of a random scale-free network, for which the number of vertices, scale-free exponent, and minimum degree have been set to $N = 1000$ , $γ = 2.5$ and $k_{min} = 2$ , has been used with $10^{4}$ fluctuation realizations. The dashed curves correspond to the three-parameter lognormal distribution (TPLN, 3) with predicted parameters (4, 5).Reuse & Permissions
Figure 2
(Color online) The moment $σ (L_{v}) = \sqrt{{⟨ L_{v}^{2} ⟩}_{c}}$ of Eqs. (4, 5) as a function of the average vertex load $⟨ L_{v} ⟩$ for one realization of pathlike (inset) and sourcelike fluctuations on random scale-free (circles) and Poisson (triangles) networks. Parameters are $N = 1000$ , $γ = 2.5$ , $k_{min} = 2$ for the scale-free networks, and $N = 1000$ , $⟨ k ⟩ = 5$ for the Poisson networks. Fifty out of $10^{3}$ vertices are shown. The dashed and dashed-dotted straight lines represent the scaling exponents $β = 0.5$ and 0.1.Reuse & Permissions
Figure 3
(Color online) Probability distributions for the number of directly failing vertices due to sourcelike flux fluctuations in a scale-free network. Network parameters are as in the previous figures. $10^{4}$ flux realizations have been used for various fluctuation strengths. The distributions only depend on the quantile (6): $q = 0.8$ (right), 0.9 (center), and 0.99 (left). The dotted and dashed curves correspond to binomial and beta-binomial distributions with the same mean $(1 - q) N$ . The correlation parameter $ρ$ of the beta-binomial distribution has been calculated with Eqs. (9, 10).Reuse & Permissions
Figure 4
(Color online) Relation between the parameters $ρ$ and $(1 - q)$ of the beta-binomial distributions (8), which have been directly fitted to the sampled distributions of Fig. 3.Reuse & Permissions
Figure 5
(Color online) Probability for the relative efficiency to remain $\frac{E}{E_{0}} = 1$ given the capacity layout (7) against sourcelike flux fluctuations. The symbols have been determined from a simulation of $10^{3}$ fluctuation realizations with strength $σ = 0.5$ on a representative random scale-free (triangles) and Poisson (circles) network of size $N = 1000$ (other parameters as stated before). The dashed and dotted curves represent the analytical prediction based on the beta-binomial distribution (8) and the binomial simplification, respectively.Reuse & Permissions
Figure 6
(Color online) Relative efficiency of a typical $N = 1000$ scale-free network as a function of investment costs against direct vertex failures as well as subsequent cascading failures induced by pathlike (inset) and sourcelike fluctuations of strength $σ = 0.8$ . Dashed-dotted and dashed curves represent the capacity layouts (2, 7), respectively. All curves come with a confidence level of 0.99, meaning that 10 out of the simulated $10^{3}$ fluctuation realizations result in relative efficiencies below the curve.Reuse & Permissions

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