Figure 6. Phase diagram of bootstrap percolation on undirected Kleinberg’s spatial networks in parameter spaces (k, α, k_l).

The color of data points in (a–c) marks the value of p_c1, where there is a hybrid phase transition (or a first-order phase transition in the trivial cases where , and the color of data points in (d–f) marks the value of p_c2, where the transition is of second-order. Blank areas stand for the absent of the corresponding phase transitions. Separated by the vertical dash line α = −1, on the right side, the color of data points is nearly unchanged for the same parameter k, meaning that the values of p_c1 and p_c2 are almost invariant. is found to be a parameter-independent critical value, above which the critical points for the double phase transition are almost constant. When , p_c1 decreases and p_c2 increases as α decreases. When α < α_c, p_c2 increases as α decreases. Results are averaged over 1000 realizations with fixed network size L = 400.