The Mathematics of Catastrophe
A mathematical description of catastrophe in complex systems modeled as a network is presented with emphasis on network topology and its relationship to risk and resilience. We present mathematical formulas for computing risk, resilience, and likelihood of faults in nodes/links of network models of...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-09-01
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| Series: | AppliedMath |
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| Online Access: | https://www.mdpi.com/2673-9909/2/3/28 |
| _version_ | 1797466023094386688 |
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| author | Ted Gyle Lewis |
| author_facet | Ted Gyle Lewis |
| author_sort | Ted Gyle Lewis |
| collection | DOAJ |
| description | A mathematical description of catastrophe in complex systems modeled as a network is presented with emphasis on network topology and its relationship to risk and resilience. We present mathematical formulas for computing risk, resilience, and likelihood of faults in nodes/links of network models of complex systems and illustrate the application of the formulas to simulation of catastrophic failure. This model is not related to nonlinear “Catastrophe theory” by René Thom, E.C. Zeeman and others. Instead, we present a strictly probabilistic network model for estimating risk and resilience—two useful metrics used in practice. We propose a mathematical model of exceedance probability, risk, and resilience and show that these properties depend wholly on vulnerability, consequence, and properties of the network representation of the complex system. We use simulation of the network under simulated stress causing one or more nodes/links to fail, to extract properties of risk and resilience. In this paper two types of stress are considered: viral cascades and flow cascades. One unified definition of risk, MPL, is proposed, and three kinds of resilience illustrated—viral cascading, blocking node/link, and flow resilience. The principal contributions of this work are new equations for risk and resilience and measures of resilience based on vulnerability of individual nodes/links and network topology expressed in terms of spectral radius, bushy, and branchy metrics. We apply the model to a variety of networks—hypothetical and real—and show that network topology needs to be included in any definition of network risk and resilience. In addition, we show how simulations can identify likely future faults due to viral and flow cascades. Simulations of this nature are useful to the practitioner. |
| first_indexed | 2024-03-09T18:29:58Z |
| format | Article |
| id | doaj.art-13e71fc4e6d5414b815fa8c0320f42a6 |
| institution | Directory Open Access Journal |
| issn | 2673-9909 |
| language | English |
| last_indexed | 2024-03-09T18:29:58Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | AppliedMath |
| spelling | doaj.art-13e71fc4e6d5414b815fa8c0320f42a62023-11-24T07:33:10ZengMDPI AGAppliedMath2673-99092022-09-012348050010.3390/appliedmath2030028The Mathematics of CatastropheTed Gyle Lewis0Naval Postgraduate School (ret), Monterey, CA 93943, USAA mathematical description of catastrophe in complex systems modeled as a network is presented with emphasis on network topology and its relationship to risk and resilience. We present mathematical formulas for computing risk, resilience, and likelihood of faults in nodes/links of network models of complex systems and illustrate the application of the formulas to simulation of catastrophic failure. This model is not related to nonlinear “Catastrophe theory” by René Thom, E.C. Zeeman and others. Instead, we present a strictly probabilistic network model for estimating risk and resilience—two useful metrics used in practice. We propose a mathematical model of exceedance probability, risk, and resilience and show that these properties depend wholly on vulnerability, consequence, and properties of the network representation of the complex system. We use simulation of the network under simulated stress causing one or more nodes/links to fail, to extract properties of risk and resilience. In this paper two types of stress are considered: viral cascades and flow cascades. One unified definition of risk, MPL, is proposed, and three kinds of resilience illustrated—viral cascading, blocking node/link, and flow resilience. The principal contributions of this work are new equations for risk and resilience and measures of resilience based on vulnerability of individual nodes/links and network topology expressed in terms of spectral radius, bushy, and branchy metrics. We apply the model to a variety of networks—hypothetical and real—and show that network topology needs to be included in any definition of network risk and resilience. In addition, we show how simulations can identify likely future faults due to viral and flow cascades. Simulations of this nature are useful to the practitioner.https://www.mdpi.com/2673-9909/2/3/28risk and resilience in complex infrastructureconsequenceexceedance probabilityfractal dimensionnetwork simulation modelsblocking nodes/links |
| spellingShingle | Ted Gyle Lewis The Mathematics of Catastrophe AppliedMath risk and resilience in complex infrastructure consequence exceedance probability fractal dimension network simulation models blocking nodes/links |
| title | The Mathematics of Catastrophe |
| title_full | The Mathematics of Catastrophe |
| title_fullStr | The Mathematics of Catastrophe |
| title_full_unstemmed | The Mathematics of Catastrophe |
| title_short | The Mathematics of Catastrophe |
| title_sort | mathematics of catastrophe |
| topic | risk and resilience in complex infrastructure consequence exceedance probability fractal dimension network simulation models blocking nodes/links |
| url | https://www.mdpi.com/2673-9909/2/3/28 |
| work_keys_str_mv | AT tedgylelewis themathematicsofcatastrophe AT tedgylelewis mathematicsofcatastrophe |