Broom, M.

Hadjichrysanthou, C.

Rychtar, J.

Stadler, B. T.

English

Broom M.

Tucker A. C.

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Two results on evolutionary processes on general non-directed graphs

City Research Online

Broom, M., Hadjichrysanthou, C., Rychtar, J. & Stadler, B. T. (2010). Two results on evolutionary processes on general non-directed graphs. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466(2121), 2795 - 2798. doi: 10.1098/rspa.2010.0067 <http://dx.doi.org/10.1098/rspa.2010.0067>City Research OnlineOriginal citation: Broom, M., Hadjichrysanthou, C., Rychtar, J. & Stadler, B. T. (2010). Two results on evolutionary processes on general non-directed graphs. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466(2121), 2795 - 2798. doi: 10.1098/rspa.2010.0067 <http://dx.doi.org/10.1098/rspa.2010.0067>Permanent City Research Online URL: http://openaccess.city.ac.uk/972/ Copyright & reuseCity  University  London has developed City  Research Online  so that  its  users  may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders. Users may download and/ or print one  copy  of  any  article(s)  in  City  Research  Online  to  facilitate  their  private  study  or  for  non-commercial research. Users may not engage in further distribution of the material or use it for any profit-making activities or any commercial gain. All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of researchThe version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper.EnquiriesIf you have any enquiries about any aspect of City Research Online, or if you wish to make contact  with the author(s) of this paper, please email the team at publications@city.ac.uk.Two results on evolutionary processes ongeneral non-directed graphsBy M. Broom1, C. Hadjichrysanthou1, J. Rychta´rˇ2, B.T. Stadler31 Centre for Mathematical Science, City University, London EC1V 0HB, UK2 Department of Mathematics and Statistics, The University of North Carolina atGreensboro, NC 27402, USA3 Department of Computer Science, The University of North Carolina atGreensboro, NC 27402, USA1. IntroductionThe paper Broom & Rychta´rˇ (2008) analytically investigated the probability formutants to fixate in an otherwise uniform population on two types of heteroge-neous graphs (lines and stars) by evolutionary dynamics. The main motivation forconcentrating on those two types of graphs only was the potentially exponential sizeof the system of linear equations (see (1.1) below) yielding the fixation probabilityon general heterogenous graphs. The size of the system was given by formula (4.1)from Broom & Rychta´rˇ (2008). It turns out that formula (4.1) is in fact only alower bound for the size of the system and in this paper we correct this by derivinga formula for the exact size of the system (1.1). We also solve the system (1.1) forgeneral heterogeneous graphs in the case of random drift.Let G = (V,E) be an undirected graph, where V is the set of vertices and E isthe set of edges. We assume that the graph is finite, connected and simple, i.e. novertex is connected to itself and there are no parallel edges. The graph structure isrepresented by a matrix W = (wij), wherewij ={d−1i , if i and j are connected,0, otherwise,where di is the degree of the vertex i, i.e. the number of edges incident to the vertexi.The evolutionary dynamics on graphs is described e.g. in Lieberman et al. (2005)and is treated as a discrete time Markov chain. At the beginning, a vertex is chosenat random and replaced by a mutant with fitness r, all remaining vertices hav-ing fitness 1. At subsequent steps, a randomly chosen individual replicates with aprobability proportional to its fitness and its offspring replaces an individual at arandomly chosen neighbouring vertex. The process stops when there are no mutantsor no resident individuals in the graph. Each state of the dynamics is described bya set C ⊆ V , a set of vertices inhabited by mutants.Let PC denote the probability of mutant fixation given that mutants currentlyinhabit a set C. The rules of the dynamics yield (see Broom & Rychta´rˇ (2008);Article submitted to Royal Society TEX PaperLieberman et al. (2005)),PC =∑i∈C∑j 6∈C(rwijPC∪{j} + wjiPC\{i})∑i∈C∑j 6∈C(rwij + wji) (1.1)with P∅ = 0 and PV = 1. This system has a unique solution following Broom &Rychta´rˇ (2008).For what classes of graphs can the system (1.1) be solved explicitly? Liebermanet al. (2005) solved it for regular graphs (where wij = wji). Broom & Rychta´rˇ(2008) solved it for stars and significantly reduced the size of the system for lines.Below we shall solve the system for general graphs and r = 1, but first we considerthe size of the system.2. The number of mutant-resident formationsAt every vertex of a graph G there can be either a resident or a mutant; and thusup to 2|V | potential mutant-resident formations or patterns. Let a mutant-residentformation be represented by a function m : V 7→ {0, 1} (0 for resident, 1 for mutant).The (finite) automorphism group Aut(G) of the graph G acts on a set of for-mations M = {m : V 7→ {0, 1}} by(f ◦m)(v) = m(f−1(v)) (2.1)for every vertex v whenever f ∈ Aut(G), m ∈ M . We say that two formations mand m′ are equivalent, if there is an automorphism f such that m′ = f ◦m (and thusm = f−1 ◦m′). The number of unknowns in the system (1.1) is equal to the numberof equivalence classes of mutant-resident formations |M/Aut(G)|. Burnside’s orbitcounting theorem (Tucker, 1994) yields|M/Aut(G)| = 1|Aut(G)| ∑f∈Aut(G)|Mf | (2.2)where |Mf | denotes the number of the elements of M fixed by f .It is easy to see that if f is any permutation of vertices (this includes anyautomorphism of the graph), then f ◦m = m exactly for those m that are constanton the cycles of permutation f . Hence, if C(f) denotes the number of cycles of apermutation f (fixed points - as elements of V - of the permutation f count as onecycle), then every automorphism f fixes exactly 2C(f) formations m because one canhave only all 0’s or all 1’s on every cycle of f . Thus, the total count of equivalenceclasses of m and thus the number of mutant-resident formations, MRF (G), is givenbyMRF (G) =1|Aut(G)| ∑f∈Aut(G)2C(f) . (2.3)Consequently, since the number given by formula (2.3) is at least as big as thenumber given by formula (4.1) from Broom & Rychta´rˇ (2008), the main point of2that paper is still valid as formula (4.1) was shown to emphasize the large size ofthe system (1.1) in a general heterogeneous graph.3. Random drift, the case when r = 1For the case of random drift, r = 1, the solution of (1.1) is given byPC =∑i∈Cdi−1∑k∈Vdk−1 . (3.1)which can be checked by direct substitution. To derive the formula (3.1), assumethat, for disjoint sets C,D ⊂ V ,PC∪D = PC + PD. (3.2)By (3.2), the system (1.1) is equivalent to∑i∈C∑j 6∈CwjiP{i} =∑i∈C∑j 6∈CwijP{j}which is satisfied if, for all i, j,wjiP{i} = wijP{j}.Consequently, whenever vertices i and j are connected,P{i}P{j}=djdi(3.3)Since the graph is connected, the repeated application of (3.3) along a path betweenany two vertices i, j shows that (3.3) holds even when i, j are not connected by anedge. Since, by (3.2) and (3.3),1 = PV =∑j∈VP{j} = P{i}∑j∈Vdidj, (3.4)we getP{i} =(∑k∈Vdidk)−1. (3.5)The formula (3.1) then follows from (3.5) and (3.2). The assumptions (3.2) and (3.3)can now be justified by the uniqueness of the solution of the system (1.1). Note that(3.2) cannot hold for general r. For example, if r is very large, PC∪D ≈ PC ≈ PD ≈ 1which violates (3.2). Yet, as shown above, (3.2) holds when r = 1.34. DiscussionThe two results that we have presented apply for general graphs. Previous analyticalwork, including Broom & Rychta´rˇ (2008), only consider very special graphs. Thelarge size of the system of equations (2.3) makes it very difficult to find analyticalresults in the general case, of which (3.1) is a very special example. In fact (2.3) isvery useful in considering whether an analytical approach should be made, as thelarger the value of MRF (G) the more difficult the system of equations is to dealwith in general. Two of the simplest graphs that have received the most attentionso far, the complete graph and the star, have values of MRF (G) of |V |+1 and 2|V |respectively. Interestingly the circle and the line, which have also been investigated,have much larger values of MRF (G), but many states cannot be accessed from theinitial state of a single mutant (it was shown in Broom & Rychta´rˇ (2008) that thesewere the only graphs for which this was true).ReferencesBroom, M. & Rychta´rˇ, J. 2008 An analysis of the fixation probability of a mutanton special classes of non-directed graphs. Proc. R. Soc. A 464 2609–2627. (doi:10.1098/rspa.2008.0058)Lieberman, E., Hauert, C., Nowak, M.A. 2005 Evolutionary Dynamics on Graphs.Nature 433, 312–316. (doi: 10.1038/nature03204)Tucker, A.C. 1994 Applied combinatorics, 3rd ed. New York: Wiley.4

Two results on evolutionary processes on general non-directed graphs

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