Measuring the hierarchical influence in social contagions and the emergence of crossover phase transitions

In social networks, a minority of individuals are more likely to influence others and called opinion leaders. Distinguishing the influence of opinion leaders and the masses is critical to the understanding of how information spreads and to the identification of the opinion leaders. Thus we introduce a social contagion model with heterogeneous influence strength, and analyze the spreading process by developing an edge-based compartmental theory and utilize the permutation and combination theory to examine how heterogeneous individuals exert their influence. We find a hierarchy characteristic of behavior influence: the opinion leaders initially affect the susceptible individuals, and then stimulate the masses who influence other susceptible individuals with opinion leaders and finally do it independently. Moreover, crossover phase transitions exist when varying the proportion of opinion leaders: it changes from being first-order to second-order when the influence of the masses is high; interestingly, when the influence of the masses is relatively low, the changes from first-order to hybrid, and then to second-order have been observed. Similar crossover phenomenon is seen by varying the influence ratio between the masses and opinion leaders. Finally, we consider the situation that both the influence strength and adoption thresholds are heterogeneous, and find that it is not always beneficial for spreading if heterogeneity increases. Our theoretical predictions agree well with the simulation results.