Exploring the step function distribution of the threshold fraction of adopted neighbors versus minimum fraction of nodes as initial adopters to assess the cascade blocking intra-cluster density of complex real-world networks

We first propose a binary search algorithm to determine the minimum fraction of nodes in a network to be used as initial adopters ( $$f_{IA}^{\min }$$ ) for a particular threshold fraction (q) of adopted neighbors (related to the cascade capacity of the network) leading to a complete information cascade. We observe the q versus $$f_{IA}^{\min }$$ distribution for several complex real-world networks to exhibit a step function pattern wherein there is an abrupt increase in $$f_{IA}^{\min }$$ beyond a certain value of q (qstep); the $$f_{IA}^{\min }$$ values at qstep and the next measurable value of q are represented as $$\underline{{f_{IA}^{\min } }}$$ and $$\overline{{f_{IA}^{\min } }}$$ respectively. The difference $$\overline{{f_{IA}^{\min } }} - \underline{{f_{IA}^{\min } }}$$ is observed to be significantly high (a median of 0.44 for a suite of 40 real-world networks studied in this paper) such that we claim the 1 − qstep value (we propose to refer 1 − qstep as the Cascade Blocking Index, CBI) for a network could be perceived as a measure of the intra-cluster density of the blocking cluster of the network that cannot be penetrated without including an appreciable number of nodes from the cluster to the set of initial adopters (justifying a relatively larger $$\overline{{f_{IA}^{\min } }}$$ value).