Figure 3: Density histogram of the prediction k^* for the onset of excitation propagation (point A in

Fig. 1) as a function of the observed transition value 1/ κ_c. The basis of this prediction is that the node with the highest degree on the easiest path (path along which the maximal degree is minimal) from the input node to the output node is limiting, with degree k^*. Scanning different values of κ_c is obtained by running the dynamics on networks having N = 80 nodes and M = 100...1900 edges (in steps of 100), considering 10 realizations for each values of M and for each network all the possibilities for the input node, while observing the topological quantity k^*. Data are aggregated into a heat map (color bar on the right, in %) representing the probability density (normalized histogram) of values of k^* for each value of 1/κ_c.