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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.05200 |
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| _version_ | 1866929530426359808 |
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| author | Aracena, Julio Astete-Elguin, Raúl |
| author_facet | Aracena, Julio Astete-Elguin, Raúl |
| contents | This paper proposes a new parameter for studying Boolean networks: the independence number. We establish that a Boolean network is $k$-independent if, for any set of $k$ variables and any combination of binary values assigned to them, there exists at least one fixed point in the network that takes those values at the given set of $k$ indices. In this context, we define the independence number of a network as the maximum value of $k$ such that the network is $k$-independent. This definition is closely related to widely studied combinatorial designs, such as "$k$-strength covering arrays", also known as Boolean sets with all $k$-projections surjective. Our motivation arises from understanding the relationship between a network's interaction graph and its fixed points, which deepens the classical paradigm of research in this direction by incorporating a particular structure on the set of fixed points, beyond merely observing their quantity. Specifically, among the results of this paper, we highlight a condition on the in-degree of the interaction graph for a network to be $k$-independent, we show that all regulatory networks are at most $n/2$-independent, and we construct $k$-independent networks for all possible $k$ in the case of monotone networks with a complete interaction graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05200 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | K-Independent Boolean Networks Aracena, Julio Astete-Elguin, Raúl Combinatorics Dynamical Systems This paper proposes a new parameter for studying Boolean networks: the independence number. We establish that a Boolean network is $k$-independent if, for any set of $k$ variables and any combination of binary values assigned to them, there exists at least one fixed point in the network that takes those values at the given set of $k$ indices. In this context, we define the independence number of a network as the maximum value of $k$ such that the network is $k$-independent. This definition is closely related to widely studied combinatorial designs, such as "$k$-strength covering arrays", also known as Boolean sets with all $k$-projections surjective. Our motivation arises from understanding the relationship between a network's interaction graph and its fixed points, which deepens the classical paradigm of research in this direction by incorporating a particular structure on the set of fixed points, beyond merely observing their quantity. Specifically, among the results of this paper, we highlight a condition on the in-degree of the interaction graph for a network to be $k$-independent, we show that all regulatory networks are at most $n/2$-independent, and we construct $k$-independent networks for all possible $k$ in the case of monotone networks with a complete interaction graph. |
| title | K-Independent Boolean Networks |
| topic | Combinatorics Dynamical Systems |
| url | https://arxiv.org/abs/2410.05200 |