Correction to: Relation Between the Number of Peaks and the Number of Reciprocal Sign Epistatic Interactions

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.

• 1. Add the necessary assumptions in the theorem.

• 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.

• 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.

• 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.

• (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 $e = \{ u, v \} \in E$,

  $$\begin{aligned} \vert \{ u \in V : \exists v \in V \quad e = \{u, v\}, \quad f(e) \ge f(v) \} \vert \le 1 \,. \end{aligned}$$

• 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.

• 4. Section 4.2, “Proof”, page 5 of 12, proves that the function defined is a Morse function as follows.

  • 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).

  • 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.