Exact formulas for fixation probabilities on a complete oriented star
Introduction
The evolutionary graph theory is a new branch of population dynamics, which studies the evolution of structured populations [3], [5]. In this framework, the structure of a population is modeled by a weighted digraph on vertices 1, 2, … , N, where wij stands for the weight on the edge {i, j}, if any. It is assumed that every individual of the population occupies a unique vertex of the digraph. In each iteration, an individual i is chosen for reproduction with a probability proportional to its fitness, and the resulting offspring will occupy vertex j with probability wij. The intrinsic weights of a digraph are defined this way: For a vertex i with k outgoing edges, let wij = 1/k if edge {i, j} exists. Fig. 1 gives a digraph with intrinsic weights. In this paper, we focus our attention on the structures with intrinsic weights.
Consider a homogeneous population on a weighted digraph whose individuals all have fitness 1. Suppose m new mutants with fitness r > 1 are introduced by placing them on m randomly chosen vertices of the digraph. These mutants have a certain chance of fixation, i.e., to generate a lineage that takes over the population. It is a major issue in population dynamics to find the mth-order fixation probability, i.e., the fixation probability of m mutants, on a population.
An unstructured population can be modeled by such a looped complete digraph that all edges have the same weight. The evolution of an unstructured population is often modeled by the Moran process, whose mth-order fixation probability iswhich defines a balance between natural selection and random drift. It is known that a structured population has the same first fixation probability as the corresponding Moran process if and only if the structure is an isothermal digraph [5]. A structure is referred to as an amplifier of selection (respectively, suppressor of selection) if the first-order fixation probability of one advantageous mutant on this structure is greater than (respectively, less than) that for the corresponding Moran process. An important issue in evolutionary graph theory is to answer whether a given structure is an amplifier of selection or a suppressor of selection [3], [4], [8], [11].
A complete oriented star (COS) is a digraph with a single central vertex so that (a) there is an edge from the center to each peripheral vertex, (b) there is an edge from each peripheral vertex to the center, and (c) there is no other edge. The digraph given in Fig. 1 is a COS. COSs are basic digraphs [1], [10] and are popular network topologies.
This paper addresses the evolutionary dynamics on COSs. First, we give the exact formulas for the fixation probabilities on a COS. We then apply these formulas to study some properties of COSs. The obtained results partially reveal how the fixation probabilities are affected by COSs.
The subsequent materials are organized this way: Section 2 introduces basic notions and notations. In Section 3 we present the formulas for the intrinsic fixation probabilities on a COS. Some interesting properties of these fixation probabilities are given in Section 4. This work is closed with some remarks.
Section snippets
Notions and notations
A complete oriented star (COS) of size N, denoted SN, is a digraph with vertex set V = {1, 2, … , N} and edge set E = {〈1, i〉, 〈i, 1〉∣2 ⩽ i ⩽ N}. Fig. 2 gives two COSs with intrinsic weights. In the sequel, the term “COS” means “COS with intrinsic weights”.
Let ρm(N; r) denote the mth-order fixation probability on SN. For technical reason, we need the following notations:
At time t, the configuration of a population on SN is described by a vector M(t) = (m1(t), m2(t)), where m1(t) = 1 or 0 according as a mutant
Fixation probabilities on a complete oriented star
The main results of this paper are stated as follows. Theorem 1 Let r and N be given. For 1 ⩽ m ⩽ N − 1, there holdsIn particular,
To prove this theorem, let us first introduce two lemmas. Lemma 1 Let r and N be given. LetFor 1 ⩽ m ⩽ N − 1, the following recursions hold: Proof It is
Applications
Theorem 1 allows us to calculate the fixation probabilities for a given COS. For example, we have the following: Theorem 2 S3, S4 and S5 are all amplifiers of selection. Proof When r > 1, it follows from Eq. (3) and by algebraic calculations thatand Another utility of Theorem 1 is to study the asymptotic
Concluding remarks
We have found the exact formulas for fixation probabilities on complete oriented stars, and have applied these formulas to study some dynamic properties of populations on complete oriented stars.
After careful consideration, we find that, except isothermal digraphs, rooted digraphs and multiple-rooted digraphs, complete oriented stars are the only digraphs whose fixation probabilities can be expressed analytically. Therefore, a wise strategy of studying the fixation probabilities on other kinds
Acknowledgements
This work was supported in part by Major Program of NNSFC (Grant No. 90818028), General Program of NNSFC (Grant No. 10771227), Program for New Century Excellent Talent of China (Grant No. NCET-05-0759) and Hong Kong Research Grants Council (Grant No. 210508).
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