Fig. 5: Branching factor.
From: The random cascading origin of abrupt transitions in interdependent systems
a Experimental (top network of an L = 416 ISN) and (b) theoretical measurements of the average branching factor during the cascading near criticality of the abrupt transition. These are extracted from resistance versus time curves shown in Fig. S2. Above the critical point (ΔI > 0), the average branching factor \(\langle {{\mathscr{R}}}\rangle\) is smaller than one, and each segment impacts on average less than one segment. Below the critical point (ΔI < 0), the average branching factor is larger than one, and each segment impacts on average more than one segment, as hypothesized. Note also that \(\langle {{\mathscr{R}}}\rangle\) approaches 1 as I approaches Ic from both directions. Exactly at the critical point (ΔI = 0) a critical branching factor of \({\langle {{\mathscr{R}}}\rangle }_{c}=1\) is observed, where each segment impacts on average exactly one segment.
