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Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be useful. To be precise, I am referring to polyhedra and their higher-dimensional analogues. Might mathematicians have developed such a concept?

One idea that occurred to me was to use the number of Hamiltonian cycles. If the polytope is embedded in $\mathbb{R}^3$ this allows us to ask questions such as how many polytopes with distinct topology with $n$ vertices have less than $\lambda n!$ Hamiltonian cycles where $\lambda \in [0,1]$.

Another way of approaching this might be to think about compression algorithms and Kolmogorov Complexity. However, I haven't found much work applying notions of algorithmic complexity to problems in topology.

## Addendum:

Ginestra Bianconi and Kartik Anand have formulated a more general problem: the challenge of defining an entropy measure for complex networks [3]. This problem is motivated in [3] as follows:

Following ten years of active research in the field of complex networks, the state of the art includes, a deep understanding of their evolution , an unveiling of the rich interplay between network topology and dynamics and a description of networks through structural characteristics. Nevertheless, we still lack the means to quantify, how complex is a complex network. In order to answer this question we need a new theory of information of complex networks.

References:

1. Elizabeth Drellich, Andrew Gainer-Dewar, Heather A. Harrington, Qijun He, Christine Heitsch, AND Svetlana Poznanovic. Geometric Combinatorics AND Computational Molecular biology: branching polytopes for RNA sequences. 2016.
2. Paul M. Dodda, Pablo F. Damascenob and Sharon C. Glotzera. Universal folding pathways of polyhedron nets. PNAS. 2018.
3. Kartik Anand & Ginestra Bianconi. Entropy measures for networks: Toward an information theory of complex topologies. Arxiv. 2009.
4. Ginestra Bianconi. The entropy of network ensembles. Arxiv. 2008.

Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be useful. To be precise, I am referring to polyhedra and their higher-dimensional analogues. Might mathematicians have developed such a concept?

One idea that occurred to me was to use the number of Hamiltonian cycles. If the polytope is embedded in $\mathbb{R}^3$ this allows us to ask questions such as how many polytopes with distinct topology with $n$ vertices have less than $\lambda n!$ Hamiltonian cycles where $\lambda \in [0,1]$.

Another way of approaching this might be to think about compression algorithms and Kolmogorov Complexity. However, I haven't found much work applying notions of algorithmic complexity to problems in topology.

Addendum:

Ginestra Bianconi and Kartik Anand have formulated a more general problem: the challenge of defining an entropy measure for complex networks [3]. This problem is motivated in [3] as follows:

Following ten years of active research in the field of complex networks, the state of the art includes, a deep understanding of their evolution , an unveiling of the rich interplay between network topology and dynamics and a description of networks through structural characteristics. Nevertheless, we still lack the means to quantify, how complex is a complex network. In order to answer this question we need a new theory of information of complex networks.

References:

1. Elizabeth Drellich, Andrew Gainer-Dewar, Heather A. Harrington, Qijun He, Christine Heitsch, AND Svetlana Poznanovic. Geometric Combinatorics AND Computational Molecular biology: branching polytopes for RNA sequences. 2016.
2. Paul M. Dodda, Pablo F. Damascenob and Sharon C. Glotzera. Universal folding pathways of polyhedron nets. PNAS. 2018.
3. Kartik Anand & Ginestra Bianconi. Entropy measures for networks: Toward an information theory of complex topologies. Arxiv. 2009.
4. Ginestra Bianconi. The entropy of network ensembles. Arxiv. 2008.

Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be useful. To be precise, I am referring to polyhedra and their higher-dimensional analogues. Might mathematicians have developed such a concept?

One idea that occurred to me was to use the number of Hamiltonian cycles. If the polytope is embedded in $\mathbb{R}^3$ this allows us to ask questions such as how many polytopes with distinct topology with $n$ vertices have less than $\lambda n!$ Hamiltonian cycles where $\lambda \in [0,1]$.

Another way of approaching this might be to think about compression algorithms and Kolmogorov Complexity. However, I haven't found much work applying notions of algorithmic complexity to problems in topology.

References:

1. Elizabeth Drellich, Andrew Gainer-Dewar, Heather A. Harrington, Qijun He, Christine Heitsch, AND Svetlana Poznanovic. Geometric Combinatorics AND Computational Molecular biology: branching polytopes for RNA sequences. 2016.
2. Paul M. Dodda, Pablo F. Damascenob and Sharon C. Glotzera. Universal folding pathways of polyhedron nets. PNAS. 2018.

Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be useful. To be precise, I am referring to polyhedra and their higher-dimensional analogues. Might mathematicians have developed such a concept?

One idea that occurred to me was to use the number of Hamiltonian cycles. If the polytope is embedded in $\mathbb{R}^3$ this allows us to ask questions such as how many polytopes with distinct topology with $n$ vertices have less than $\lambda n!$ Hamiltonian cycles where $\lambda \in [0,1]$.

Another way of approaching this might be to think about compression algorithms and Kolmogorov Complexity. However, I haven't found much work applying notions of algorithmic complexity to problems in topology.

## Addendum:

Ginestra Bianconi and Kartik Anand have formulated a more general problem: the challenge of defining an entropy measure for complex networks [3]. This problem is motivated in [3] as follows:

Following ten years of active research in the field of complex networks, the state of the art includes, a deep understanding of their evolution , an unveiling of the rich interplay between network topology and dynamics and a description of networks through structural characteristics. Nevertheless, we still lack the means to quantify, how complex is a complex network. In order to answer this question we need a new theory of information of complex networks.

References:

1. Elizabeth Drellich, Andrew Gainer-Dewar, Heather A. Harrington, Qijun He, Christine Heitsch, AND Svetlana Poznanovic. Geometric Combinatorics AND Computational Molecular biology: branching polytopes for RNA sequences. 2016.
2. Paul M. Dodda, Pablo F. Damascenob and Sharon C. Glotzera. Universal folding pathways of polyhedron nets. PNAS. 2018.
3. Kartik Anand & Ginestra Bianconi. Entropy measures for networks: Toward an information theory of complex topologies. Arxiv. 2009.
4. Ginestra Bianconi. The entropy of network ensembles. Arxiv. 2008.

Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be useful. Might there be mathematicians that have tried to develop such a concept?

One idea that occurred to me was to use the number of Hamiltonian cycles. If the polytope is embedded in $\mathbb{R}^3$ this allows us to ask questions such as how many polytopes with distinct topology with $n$ vertices have less than $\lambda n!$ Hamiltonian cycles where $\lambda \in [0,1]$.

Another way of approaching this might be to think about compression algorithms and Kolmogorov Complexity. However, I haven't found much work applying notions of algorithmic complexity to problems in topology.

References:

1. ELIZABETH DRELLICH, ANDREW GAINER-DEWAR, HEATHER A. HARRINGTON, QIJUN HE, CHRISTINE HEITSCH, AND SVETLANA POZNANOVIĆ. GEOMETRIC COMBINATORICS AND COMPUTATIONAL. 2016.
2. Paul M. Dodda, Pablo F. Damascenob and Sharon C. Glotzera. Universal folding pathways of polyhedron nets. MOLECULAR BIOLOGY: BRANCHING POLYTOPES FOR RNA SEQUENCES. PNAS. 2018.

Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be useful. To be precise, I am referring to polyhedra and their higher-dimensional analogues. Might mathematicians have developed such a concept?

One idea that occurred to me was to use the number of Hamiltonian cycles. If the polytope is embedded in $\mathbb{R}^3$ this allows us to ask questions such as how many polytopes with distinct topology with $n$ vertices have less than $\lambda n!$ Hamiltonian cycles where $\lambda \in [0,1]$.

Another way of approaching this might be to think about compression algorithms and Kolmogorov Complexity. However, I haven't found much work applying notions of algorithmic complexity to problems in topology.

References:

1. Elizabeth Drellich, Andrew Gainer-Dewar, Heather A. Harrington, Qijun He, Christine Heitsch, AND Svetlana Poznanovic. Geometric Combinatorics AND Computational Molecular biology: branching polytopes for RNA sequences. 2016.
2. Paul M. Dodda, Pablo F. Damascenob and Sharon C. Glotzera. Universal folding pathways of polyhedron nets. PNAS. 2018.