Figure 1: Subcritical paths and collective influence of spreaders.
From: Efficient collective influence maximization in cascading processes with first-order transitions
(a), Three combinations of neighbors P1, P2 and P3 corresponding to νi→j in message passing equation. Node i has a threshold mi = 2. The full activation of at least one combination will lead to νi→j = 1. (b), For link i → j with an active neighbor k1 and inactive ones k2 and k3, 
since i has 0 (<mi − 1) active neighbor excluding k1 and j, while 
because i has 1 (=mi − 1) active neighbor k1 excluding k2 and j. (c), Illustrations of subcritical paths ending with link i → j for L = 0, 1, 2. Red dots stand for seeds, while squares represent m − 1 active neighbors attached to subcritical nodes. Subcritical paths are highlighted by thick links. (d), The contribution of seed i to 
exerted through subcritical paths of length L = 0, 1, 2. (e), Calculation method of CI − TML(i). Subcritical paths starting from i with length 
are displayed by different colors. (f), An example of subcritical cluster. Assuming a uniform threshold m = 3, nodes inside the circle are subcritical since they all have 2 active neighbors, represented by blue nodes. Activation of the red node will trigger a cascade covering all subcritical nodes.
