Comparison between the eccentricity of planets in the N-body simulation (blue line) and the analytical value due to secular interaction, calculated according to Murray & Dermott (1999). The left panel corresponds to the eccentricity of planet c and the right panel corresponds to planet e. The blue line shows the average and standard deviation of planet eccentricities in the simulation with different masses of d and the yellow line shows the mean and standard deviation of the Laplace-Lagrante theory averaged over random initial conditions sampled at different time.