Correction to: Bull Math Biol https://doi.org/10.1007/s11538-022-01029-z
The original version of the article unfortunately contained mistakes. It has been corrected in this correction.
1 Corrections
There was an incorrect citation of a theorem from Morse Theory (Forman 1998, Corollary 3.6, page 107). Some assumptions were missing. This mistake is corrected as follows.
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1.
Add the necessary assumptions in the theorem.
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2.
Complete the proof that uses this theorem (by showing that the assumptions hold in the context of our proof).
2 Morse Function Assumption
Theorem 2, Section 4, page 3 of 12, is a reference to a result from Morse Theory given by Forman (1998, Corollary 3.6, page 107). The result was wrongly cited since there is a missing assumption about the properties of the functions that the theorem applies to. This does not invalidate the conclusions of Theorem 2 since the function we construct satisfies these missing properties.
The necessary changes are as follows.
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1.
Theorem 2, Section 4, page 4 of 12, adds the assumption that the function \(f : V \cup E \rightarrow \mathbb {R}\) needs to be a Morse function.
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2.
Section 4.1, “Necessary definitions”, page 5 of 12, introduces the definition of Morse function for graphs. Formally, the following definition.
Definition 1
(Morse function) Let \(G = (V, E)\) be a graph. The function \(f:V \cup E \rightarrow \mathbb {R}\) is a Morse function if the following conditions hold.
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(a)
All vertices have at most one edge with lower or equal value. Formally, for all \(v \in V\),
$$\begin{aligned} \vert \{ u \in V : e = \{ u, v \} \in E, \quad f(e) \le f(v) \} \vert \le 1 \,. \end{aligned}$$ -
(b)
All edges have at most one vertex with lower or equal value. Formally, for all \(e = \{ u, v \} \in E\),
$$\begin{aligned} \vert \{ u \in V : \exists v \in V \quad e = \{u, v\}, \quad f(e) \ge f(v) \} \vert \le 1 \,. \end{aligned}$$
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3.
Section 4.2, “Proof”, page 5 of 12, adds a step in the proof consisting of proving that the function defined is a Morse function.
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4.
Section 4.2, “Proof”, page 5 of 12, proves that the function defined is a Morse function as follows.
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By the definition of Morse functions, we have to show a property for each vertex and each edge. For each vertex v, we need to show that there is at most one edge connected to v whose value is lower or equal to f(v). Indeed this is the case since edges in \(E_2\) have the highest value possible and, by construction, there is at most one edge in \(E_1\) connecting v with its fittest mutation. All other edges connected to v come from a vertex with strictly lower fitness. Therefore, if v is not a peak, there is exactly one edge connected to v with a lower value. If v is a peak, all edges connected to v have strictly higher values than f(v).
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For each edge e, we need to show that at most one of its vertices has a value larger or equal to f(e), but not both. Indeed this is the case for \(e \in E_1\), since the value of f(e) is the average of the value of its vertices. Edges in \(E_2\) have the highest possible value, so this property also holds. In conclusion, the function f is indeed a Morse function.
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Reference
Forman R (1998) Morse theory for cell complexes. Advances in mathematics, 134(1), pp.90 – 145. https://doi.org/10.1006/aima.1997.1650
Acknowledgements
We would like to thank Dr. Suman G. Das for spotting the incomplete citation.
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Saona, R., Kondrashov, F.A. & Khudiakova, K.A. Correction to: Relation Between the Number of Peaks and the Number of Reciprocal Sign Epistatic Interactions. Bull Math Biol 85, 17 (2023). https://doi.org/10.1007/s11538-022-01118-z
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DOI: https://doi.org/10.1007/s11538-022-01118-z