Ivancevic_Applied-Diff-Geom

1038 Applied Differential Geometry: A Modern Introductionρ(k, ω; t, θ) obeys the equation∂ t ρ(k, ω; t, θ) = −∂ θ [ρ(k, ω; t, θ) (ω ++ Kk ∫ dω ′ ∫ dk ′ ∫ dθ ′ N(ω ′ )P (k ′ )k ′ ρ(k ′ , ω ′ ; t, θ ′ ) sin(θ − θ ′ )∫dk′ P (k ′ )k ′ )].The mean–field solution of this equation was studied by [Ichinomiya (2004)].6.3.7.2 Path–Integral Approach to Complex NetsRecently, [Ichinomiya (2005)] introduced the path–integral (see subsection4.4.6 above) approach in studying the dynamics of complex networks. Heconsidered the stochastic generalization of the Kuramoto network (6.61),given byẋ i = f i (x i ) +N∑a ij g(x i , x j ) + ξ i (t), (6.62)j=1where f i = f i (x i ) and g ij = g(x i , x j ) are functions of network activations x i ,ξ i (t) is a random force that satisfies 〈ξ i (t) = 0〉, 〈ξ i (t)ξ j (t ′ )〉 = δ ij δ(t−t ′ )σ 2 .He assumed x i = x i,0 at t = 0. In order to discuss the dynamics of thissystem, he introduced the so–called Matrin–Siggia–Rose (MSR) generatingfunctional Z given by [de Dominicis (1978)]( ) 〈 NNt ∫ 1 ∏ NZ[{l ik }, {¯l ik }] =πwhere the action S is given by∏N ti=1 k=0dx ik d¯x ik e −S exp(l ik x ik + ¯l ik ¯x ik )J〉,S = ∑ ik[ σ2 ∆t¯x 22ik+i¯x ik {x ik −x i,k−1 −∆t(f i (x i,k−1 )+ ∑ ja ij g(x i,k−1 , x j,k−1 ))}],and 〈· · · 〉 represents the average over the ensemble of networks. J is thefunctional Jacobian term,⎛⎞J = exp ⎝− ∆t ∑ ∂(f i (x ik ) + a ij g(x ik , x jk ))⎠ .2∂x ikijkIchinomiya considered such a form of the network model in which{ 1 with probability pij ,a ij =0 with probability 1 − p ij .