Figure 2: Analytical and numerical evidence for super-stable nodes.

(a, b) Pagerank gap Δ(p_m) and the fluctuation σ(p_m) plotted as function of the rank m for two scale-free networks with the same exponent, γ=3.5, and different system sizes, N=10² and 10⁴¸ showing the emergence of super-stability for larger N. (c) The (N, γ) phase diagram showing the stable and unstable regimes that emerge for γ>γ_c=3. (d) The critical rank m_c as a function of γ where the curves correspond to the calculated m_c and the points represent the measured m_c in numerical simulations on multiple networks. The peak near γ_c=3 is more pronounced as N increases, with different scaling behaviors above and below γ_c. (e, f) Pagerank distributions of the top-ranked nodes in the www (scale-free) and the food web (exponential) shown in Table 1. The separated curves correspond to m_c (measured) values shown in the table. Note that in the www the top-ranked nodes are clearly identifiable, whereas in the food web there are no super-stable nodes as predicted by the calculation.