Impact of directionality and correlation on contagion

The threshold model has been widely adopted for modelling contagion processes on social networks, where individuals are assumed to be in one of two states: inactive or active. This paper studies the model on directed networks where nodal inand out-degrees may be correlated. To understand how directionality and correlation affect the breakdown of the system, a theoretical framework based on generating function technology is developed. First, the effects of degree and threshold heterogeneities are identified. It is found that both heterogeneities always decrease systematic robustness. Then, the impact of the correlation between nodal in- and out-degrees is investigated. It turns out that the positive correlation increases the systematic robustness in a wide range of the average in-degree, while the negative correlation has an opposite effect. Finally, a comparison between undirected and directed networks shows that the presence of directionality and correlation always make the system more vulnerable.