Asymptotic Network Robustness
This paper examines the dependence of network performance measures on network size and considers scaling results for large networks. We connect two performance measures that are well studied, but appear to be unrelated. The first measure is concerned with energy metrics, namely the <formula>&l...
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| Format: | Article |
| Language: | English |
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Institute of Electrical and Electronics Engineers (IEEE)
2020
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| Online Access: | https://hdl.handle.net/1721.1/124407 |
| _version_ | 1811088558834843648 |
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| author | Sarkar, Tuhin Roozbehani, Mardavij Dahleh, Munther A |
| author2 | Massachusetts Institute of Technology. Institute for Data, Systems, and Society |
| author_facet | Massachusetts Institute of Technology. Institute for Data, Systems, and Society Sarkar, Tuhin Roozbehani, Mardavij Dahleh, Munther A |
| author_sort | Sarkar, Tuhin |
| collection | MIT |
| description | This paper examines the dependence of network performance measures on network size and considers scaling results for large networks. We connect two performance measures that are well studied, but appear to be unrelated. The first measure is concerned with energy metrics, namely the <formula><tex>$H2$</tex></formula>-norm of a network, which arises in control theory applications. The second measure is concerned with the notion of “tail risk” which arises in economic and financial networks. We study the question of why such performance measures may deteriorate at a faster rate than the growth rate of the network. We first focus on the energy metric and its well known connection to controllability Gramian of the underlying dynamical system. We show that undirected networks exhibit the most graceful energy growth rates as network size grows. This rate is quantified completely by the proximity of spectral radius to unity or distance to instability. In contrast, we show that the simple characterization of energy in terms of network spectrum does not exist for directed networks. We demonstrate that, for any fixed distance to instability, energy of a directed network can grow at an exponentially faster rate. We provide general methods for manipulating networks to reduce energy. In particular, we prove that certain operations that increase the symmetry in a network cannot increase energy (in an order sense). Additionally, we demonstrate that such operations can effectively reduce energy for many network topologies. Secondly, we focus on tail risk in economic and financial networks. In contrast to H2-norm which arises from computing the expectation of energy in the network, tail risk focuses on tail probability behavior of network variables. Although the two measures differ substantially we show that they are precisely connected through the system Gramian. This surprising result explains why topology considerations rather than specific performance measures dictate the large scale behavior of networks. Finally, we demonstrate the consistency of our theory with simulations on synthetic and real life networks. Keywords: Network topology; Topology; Control systems; Robustness; Economics; Symmetric matrices; Electric shock |
| first_indexed | 2024-09-23T14:04:04Z |
| format | Article |
| id | mit-1721.1/124407 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:04:04Z |
| publishDate | 2020 |
| publisher | Institute of Electrical and Electronics Engineers (IEEE) |
| record_format | dspace |
| spelling | mit-1721.1/1244072022-10-01T18:58:11Z Asymptotic Network Robustness Sarkar, Tuhin Roozbehani, Mardavij Dahleh, Munther A Massachusetts Institute of Technology. Institute for Data, Systems, and Society Massachusetts Institute of Technology. Laboratory for Information and Decision Systems This paper examines the dependence of network performance measures on network size and considers scaling results for large networks. We connect two performance measures that are well studied, but appear to be unrelated. The first measure is concerned with energy metrics, namely the <formula><tex>$H2$</tex></formula>-norm of a network, which arises in control theory applications. The second measure is concerned with the notion of “tail risk” which arises in economic and financial networks. We study the question of why such performance measures may deteriorate at a faster rate than the growth rate of the network. We first focus on the energy metric and its well known connection to controllability Gramian of the underlying dynamical system. We show that undirected networks exhibit the most graceful energy growth rates as network size grows. This rate is quantified completely by the proximity of spectral radius to unity or distance to instability. In contrast, we show that the simple characterization of energy in terms of network spectrum does not exist for directed networks. We demonstrate that, for any fixed distance to instability, energy of a directed network can grow at an exponentially faster rate. We provide general methods for manipulating networks to reduce energy. In particular, we prove that certain operations that increase the symmetry in a network cannot increase energy (in an order sense). Additionally, we demonstrate that such operations can effectively reduce energy for many network topologies. Secondly, we focus on tail risk in economic and financial networks. In contrast to H2-norm which arises from computing the expectation of energy in the network, tail risk focuses on tail probability behavior of network variables. Although the two measures differ substantially we show that they are precisely connected through the system Gramian. This surprising result explains why topology considerations rather than specific performance measures dictate the large scale behavior of networks. Finally, we demonstrate the consistency of our theory with simulations on synthetic and real life networks. Keywords: Network topology; Topology; Control systems; Robustness; Economics; Symmetric matrices; Electric shock 2020-03-30T14:30:51Z 2020-03-30T14:30:51Z 2018-10 2019-05-14T17:08:17Z Article http://purl.org/eprint/type/JournalArticle 2325-5870 2372-2533 https://hdl.handle.net/1721.1/124407 Sarkar, Tuhin, Roozbehani, Mardavij and Dahleh, Munther A. "Asymptotic Network Robustness." IEEE Transactions on Control of Network Systems 6, 2 (June 2019): 812 - 821 ©2018 IEEE. en http://dx.doi.org/10.1109/tcns.2018.2878504 IEEE Transactions on Control of Network Systems Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Institute of Electrical and Electronics Engineers (IEEE) arXiv |
| spellingShingle | Sarkar, Tuhin Roozbehani, Mardavij Dahleh, Munther A Asymptotic Network Robustness |
| title | Asymptotic Network Robustness |
| title_full | Asymptotic Network Robustness |
| title_fullStr | Asymptotic Network Robustness |
| title_full_unstemmed | Asymptotic Network Robustness |
| title_short | Asymptotic Network Robustness |
| title_sort | asymptotic network robustness |
| url | https://hdl.handle.net/1721.1/124407 |
| work_keys_str_mv | AT sarkartuhin asymptoticnetworkrobustness AT roozbehanimardavij asymptoticnetworkrobustness AT dahlehmunthera asymptoticnetworkrobustness |