Figure 6

Montecarlo trajectories connecting a global minimum to its surrounding local minima. The spin configurations of a global and a local minimum are randomly chosen among those provided by our minimization procedure. The first is mapped in the second by a number of moves (single spin flips) equal to the Hamming distance between the two minima. For visualization purposes, the trajectories are depicted as emanating from a unique point and radially distributed according to a polar coordinate. The vertical axis (and color code) represents the energy, the two remaining axes a relative Hamming distance between spin configurations. The three plots essentially confirm the landscape described in Fig. 5. For E.coli-transcr, global and local minima seem to be always separated by a high and steep barrier. In EGFR-signal and Yeast-metab, the landscape is scattered with different local minima, many of which have energies similar to the global ones, see Fig. S11. This results in some trajectories never emerging from the ordered phase while moving from a minimum to the optimal frustration. The lower row shows the average gradient over 1000 Montecarlo trajectories originating in a global minimum. For E.coli-transcr the barriers of the well of global optima is precisely observable in correspondence of the peak of the gradient. For Yeast-metab such kinetic traps are less steep. For EGFR-signal no clear boundary at all is observable. This, together with Fig. S11, suggests that in the last two networks also spin configurations that are distant from the optimum have cheap routes to converge to the optimal frustration through intermediate low-energy local minima.