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The threshold model is a simple but classic model of contagion spreading incomplex social systems. To capture the complex nature of social influencing weinvestigate numerically and analytically the transition in the behavior ofthreshold-limited cascades in the presence of multiple initiators as thedistribution of thresholds is varied between the two extreme cases of identicalthresholds and a uniform distribution. We accomplish this by employing atruncated normal distribution of the nodes' thresholds and observe anon-monotonic change in the cascade size as we vary the standard deviation.Further, for a sufficiently large spread in the threshold distribution, thetipping-point behavior of the social influencing process disappears and isreplaced by a smooth crossover governed by the size of initiator set. Wedemonstrate that for a given size of the initiator set, there is a specificvariance of the threshold distribution for which an opinion spreads optimally.Furthermore, in the case of synthetic graphs we show that the spreadasymptotically becomes independent of the system size, and that global cascadescan arise just by the addition of a single node to the initiator set.

The Impact of Heterogeneous Thresholds on Social Contagion with Multiple Initiators