Evolutionary dynamics on networks of selectively neutral genotypes: Effects of topology and sequence stability

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 80, Issue 6, 2009, Pages

  Aguirre, Jacobo ^a   Buldú, Javier M ^b   Manrubia, Susanna C ^a  

^a CENTRO DE ASTROBIOLOGÍA CSIC INTA   (Spain)

^b URJC   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

CONNECTED REGION; ENERGETIC STABILITY; EVOLUTIONARY DYNAMICS; GENOME SPACE; LARGER NETWORKS; MUTATION RATES; NEUTRAL NETWORK; NUMERICAL STUDIES; QUALITATIVE BEHAVIOR; REPLICATORS; SELECTIVE PRESSURE; SIMPLE NETWORKS; SMALL NETWORKS; TOPOLOGICAL PROPERTIES; TRANSIENT TIME;

GENES;

TOPOLOGY;

EID: 73649091448     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.80.066112     Document Type: Article

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39. A matrix is irreducible when the corresponding graph is connected; in our case any pair of nodes i and j of the network are connected via mutations by definition. Irreducibility plus the condition Mii >0, i makes matrix M primitive. See for further details on matrix theory and spectral graph theory.
40. In order to calculate the evolution of the population to equilibrium we cannot choose an homogeneous distribution for a completely connected graph, since this initial condition is proportional to the equilibrium distribution. The initial condition n (0) = (1,0,0,...,0) applied to the star graph does not modify the reported dependence with m.
41. Two square matrices A and B are called similar if A= N-1 BN for some invertible matrix N. Similar matrices share many properties, such as the eigenvalues (but not the eigenvectors) and the fact that, if one is diagonalizable, the other is also diagonalizable. Applying this definition, M′ =EM is similar to the symmetric matrix E1/2 M E1/2, because E1/2 is invertible [it is diagonal and (E1/2) ii >0, i]. This means that the eigenvalues λi of M′ are real and the matrix diagonalizable, allowing the decomposition shown in Eq..
42. It is important to mention that the highly symmetric configuration of this network forbids to use an homogeneous initial condition n (0) = 1 3 (1,1,1) if we wish to use the approximation tε1, since it yields α2 =0, μ and β, and thus violates the conditions of applicability of the first-order approximation to the time to equilibrium. The actual time to equilibrium tε is in that particular case dominated by the ratio λ1 / λ3.