  In many real-world contagion phenomena, the number of contacts to spreading
entities for adoption varies for different individuals. Therefore, we study a
model of contagion dynamics with heterogeneous adoption thresholds. We derive
mean-field equations for the fraction of adopted nodes and obtain phase
diagrams in terms of the transmission probability and fraction of nodes
requiring multiple contacts for adoption. We find a double phase transition
exhibiting a continuous transition and a subsequent discontinuous jump in the
fraction of adopted nodes because of the heterogeneity in adoption thresholds.
Additionally, we observe hysteresis curves in the fraction of adopted nodes
owing to adopted nodes in the densely connected core in a network.
