An Extensive Study of Two-Node McCulloch-Pitts Networks
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915858001952768 |
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| author | Li, Wentian Provata, Astero MacCarthy, Thomas |
| author_facet | Li, Wentian Provata, Astero MacCarthy, Thomas |
| contents | Networks with two nodes are previously grouped into either two classes (mutually interactive, master-slave) or five classes (mutualism, competition, predator-prey, commensalism, amensalism). By allowing self-loops, the number of signed regulatory graphs increases to 39. We provide a complete summary of dynamical behaviors of the 39 two-node McCulloch-Pitts models when the link weights are constrained to three values [$-1$,0,$+1$] and Boolean node variables. Depending on whether the Boolean values are [$-1,1$] (bipolar) or [0,1] (binary), we show that the dynamics could also be different with the same signed regulatory graphs. We demonstrate that slight variations in the McCulloch-Pitts model (called variants) may lead to fundamentally different dynamics. We study the full model space and three kinds of robustness or stability: a) of a rule against parameter change on its overall dynamics, b) for a given state against parameter change on its final state, and c) against an initial state change on its final state. All these stability properties are loosely related to a model's limiting dynamics, with the fixed-point rules to be more stable in the first two types of robustness, but less stable in the third robustness type. These analyses pave the way towards a better understanding of a minimum complex system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_00254 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Extensive Study of Two-Node McCulloch-Pitts Networks Li, Wentian Provata, Astero MacCarthy, Thomas Social and Information Networks Physics and Society Networks with two nodes are previously grouped into either two classes (mutually interactive, master-slave) or five classes (mutualism, competition, predator-prey, commensalism, amensalism). By allowing self-loops, the number of signed regulatory graphs increases to 39. We provide a complete summary of dynamical behaviors of the 39 two-node McCulloch-Pitts models when the link weights are constrained to three values [$-1$,0,$+1$] and Boolean node variables. Depending on whether the Boolean values are [$-1,1$] (bipolar) or [0,1] (binary), we show that the dynamics could also be different with the same signed regulatory graphs. We demonstrate that slight variations in the McCulloch-Pitts model (called variants) may lead to fundamentally different dynamics. We study the full model space and three kinds of robustness or stability: a) of a rule against parameter change on its overall dynamics, b) for a given state against parameter change on its final state, and c) against an initial state change on its final state. All these stability properties are loosely related to a model's limiting dynamics, with the fixed-point rules to be more stable in the first two types of robustness, but less stable in the third robustness type. These analyses pave the way towards a better understanding of a minimum complex system. |
| title | An Extensive Study of Two-Node McCulloch-Pitts Networks |
| topic | Social and Information Networks Physics and Society |
| url | https://arxiv.org/abs/2407.00254 |