# Álvaro Lozano RojoSuppressors of selection

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 # # The l-family checker # # The graph l_{2n+2} is a complete graph K_{2n} with two more nodes # each of them connected with half the nodes of the complete core. These new # nodes are also joint by an edge. # # The nodes are writen as # # (e,k,k',e') # # where e,e'\in\{0,1\} represents if the external nodes are mutants or not, # and k,k'\in{0,1,...,n} the number of mutants of each halves of the complete # core. Using the symmetries of the graph, it is possible to reduce the nodes # to those with (lexicographically) # # k >= k' e >= e' # # The parametre n is: nn = 2 # # Reduces the state s to a cannonical form # def reduce_state(s): if s[1]s[0]): return (s[3],s[2],s[1],s[0]) return s # # Compute the possible (reduced) states for a given n, it returns the list of # states and the position of the states in the list # def ComputeStates(n): # Compute the states states = [] for k in [0..n]: for kp in [0..k]: if k != kp: states.extend([ (0,k,kp,0), (1,k,kp,0), (0,k,kp,1), (1,k,kp,1) ]) else: states.extend([ (1,k,kp,1), (1,k,kp,0), (0,k,kp,0) ]) istates = { v:k for k,v in enumerate(states)} s0 = states[0] sn = states[-1] states[0] = (0,0,0,0) states[-1] = (1,n,n,1) states[istates[(0,0,0,0)]] = s0 states[istates[(1,n,n,1)]] = sn istates[s0] = istates[(0,0,0,0)] istates[sn] = istates[(1,n,n,1)] istates[(0,0,0,0)] = 0 istates[(1,n,n,1)] = len(states)-1 return states, istates # # Compute the matrix Q and the vector b for the list # def ComputeQb(r,n,states,istates): # Compute matrix Q = matrix(QQ, [[0]*len(states)]*len(states) ) for s in states: m = sum(s) Q[ istates[s], istates[s] ] = r * m + 2 * n + 2 - m for s in states: # Leftmost if s[0] == 1: Q[istates[s], istates[(1,s[1],s[2],1)]] -= r/(n+1) if s[1] == n: Q[istates[s],istates[s]] -= r*n/(n+1) else: Q[istates[s],istates[(s[0],s[1]+1,s[2],s[3])]] -= r*(n-s[1])/(n+1) Q[istates[s],istates[s]] -= r*s[1]/(n+1) else: Q[istates[s], istates[(0,s[1],s[2],0)]] -= 1/(n+1) if s[1] == 0: Q[istates[s],istates[s]] -= n/(n+1) else: Q[istates[s],istates[reduce_state((s[0],s[1]-1,s[2],s[3]))]] -= s[1]/(n+1) Q[istates[s],istates[s]] -= (n-s[1])/(n+1) # Rightmost if s[3] == 1: Q[istates[s], istates[(1,s[1],s[2],1)]] -= r/(n+1) if s[2] == n: Q[istates[s],istates[s]] -= r*n/(n+1) else: Q[istates[s],istates[reduce_state((s[0],s[1],s[2]+1,s[3]))]] -= r*(n-s[2])/(n+1) Q[istates[s],istates[s]] -= r*s[2]/(n+1) else: Q[istates[s], istates[(0,s[1],s[2],0)]] -= 1/(n+1) if s[2] == 0: Q[istates[s],istates[s]] -= n/(n+1) else: Q[istates[s],istates[(s[0],s[1],s[2]-1,s[3])]] -= s[2]/(n+1) Q[istates[s],istates[s]] -= (n-s[2])/(n+1) # Core left if s[1] == n: Q[istates[s],istates[(1,n,s[2],s[3])]] -= r/2 #n*r*1/(2*n) Q[istates[s],istates[s]] -= r*(n-1)/2 #n*r * (n-1)/(2*n) if s[2] == n: Q[istates[s],istates[s]] -= r*n/2 #n*r * n/(2*n) else: Q[istates[s],istates[reduce_state((s[0],n,s[2]+1,s[3]))]] -= r*(n-s[2])/2 #n*r*(n-s[2])/(2*n) Q[istates[s],istates[s]] -= r*s[2]/2 # n*r*s[2]/(2*n) elif s[1] == 0: Q[istates[s],istates[(s[3],0,0,0)]] -= 1/2 #n/(2*n) Q[istates[s],istates[s]] -= (n-1)/2 + n/2 #n*(n-1)/(2*n) + n*n/2*n else: Q[istates[s],istates[(1,s[1],s[2],s[3])]] -= s[1]*r/(2*n) Q[istates[s],istates[reduce_state((0,s[1],s[2],s[3]))]] -= (n-s[1])/(2*n) Q[istates[s],istates[s]] -= s[1]*r*(s[1]-1)/(2*n) + (n-s[1])*(n-s[1]-1)/(2*n) Q[istates[s],istates[(s[0],s[1]+1,s[2],s[3])]] -= s[1]*r*(n-s[1])/(2*n) Q[istates[s],istates[reduce_state((s[0],s[1]-1,s[2],s[3]))]] -= (n-s[1])*s[1]/(2*n) Q[istates[s],istates[s]] -= r*s[1]*s[2]/(2*n) + (n-s[1])*(n-s[2])/(2*n) Q[istates[s],istates[reduce_state((s[0],s[1],s[2]+1,s[3]))]] -= r*s[1]*(n-s[2])/(2*n) if s[2] != 0: Q[istates[s],istates[(s[0],s[1],s[2]-1,s[3])]] -= (n-s[1])*s[2]/(2*n) # Core right if s[2] == n: Q[istates[s],istates[(1,n,n,s[0])]] -= r/2 #n*r*1/(2*n) Q[istates[s],istates[s]] -= r*(2*n-1)/2 # n*r*(2*n-1)/(2*n) elif s[2] == 0: Q[istates[s],istates[(s[0],s[1],0,0)]] -= 1/2 #n/(2*n) if s[1] == 0: Q[istates[s],istates[s]] -= (2*n-1)/2 #n*(2*n-1)/(2*n) else: Q[istates[s],istates[s]] -= (n-1)/2 + (n-s[1])/2 #n*(n-1)/2*n + n*(n-s[1])/(2*n) Q[istates[s],istates[reduce_state((s[0],s[1]-1,0,s[3]))]] -= s[1]/2 #n*s[1]/(2*n) else: Q[istates[s],istates[reduce_state((s[0],s[1],s[2],1))]] -= s[2]*r/(2*n) Q[istates[s],istates[(s[0],s[1],s[2],0)]] -= (n-s[2])/(2*n) Q[istates[s],istates[s]] -= s[2]*r*(s[2]-1)/(2*n) + (n-s[2])*(n-s[2]-1)/(2*n) Q[istates[s],istates[s]] -= r*s[2]*s[1]/(2*n) + (n-s[2])*(n-s[1])/(2*n) Q[istates[s],istates[reduce_state((s[0],s[1],s[2]+1,s[3]))]] -= s[2]*r*(n-s[2])/(2*n) Q[istates[s],istates[(s[0],s[1],s[2]-1,s[3])]] -= (n-s[2])*s[2]/(2*n) if s[1] != n: Q[istates[s],istates[(s[0],s[1]+1,s[2],s[3])]] -= r*s[2]*(n-s[1])/(2*n) Q[istates[s],istates[reduce_state((s[0],s[1]-1,s[2],s[3]))]] -= (n-s[2])*s[1]/(2*n) b=-Q[1:len(states)-1,len(states)-1] Q=Q[1:len(states)-1,1:len(states)-1] return Q,b # # Computes the average fixation probability function # def Phi(Q,b,n,istates): sol = Q.solve_right(b) return sol[istates[(1,0,0,0)]-1,0] / (n + 1) + sol[istates[(0,1,0,0)]-1,0] * n / (n + 1) # # "Entry point" # # Compute the states for the given size states, istates = ComputeStates(nn) # A first estimate of the degree of the rational function Phi num_fac = len(states) - 2 fits=[1..num_fac+1] fits.extend([ 1/f for f in fits[1:-1] ]) # Compute the fixation probabilities for the selected fitnesses fps = { f : Phi(*ComputeQb(f, nn, states, istates), n=nn, istates=istates) for f in fits } # Now we are ready to compute the coefficients of the function. # Construct the matrix of the linear system describing the function \Phi X = matrix(QQ, [[1]*len(fits)]*len(fits) ) X[:,num_fac-2] = [ [r] for r in fits ] X[:,-1] = [ [ -fps[r] ] for r in fits ] X[:,-2] = [ [ -r*fps[r] ] for r in fits ] y = vector(QQ, [(fps[r]-1)*r**num_fac for r in fits ] ) for i in xrange(num_fac-3,-1,-1): X[:,i] = X[:,i+1].elementwise_product( X[:,num_fac-2] ) X[:,num_fac + i] = X[:,num_fac+i+1].elementwise_product( X[:,num_fac-2] ) # Solve it sol = X.solve_right(y) # Create the field of fuctions. Then x is regarded as the variable of the # field Q(x). You can bypass this step to simplify you later the expressions # (that is regarding x as a symbolic variable with no special meaning) F. = Frac(QQ['x']) # Construct the function FB = (sum( sol[num_fac-1-i]* x**i for i in [0..num_fac-1] ) + x**num_fac) / (sum( sol[-(i+1)]* x**i for i in [0..num_fac-1] ) + x**num_fac) pretty_print(FB) # and finally the delta delta = FB - x**(2*nn+1)/sum( x**i for i in range(2*nn+2) ) pretty_print(delta) # take a look to the numerator and denominator pretty_print(delta.denominator()) pretty_print(delta.numerator()/(x-1)) # up to (x-1)