  In this paper, we study how the influence of initiators (seeds) affects the
tipping point of information cascades in networks. We consider an extended
version of the Watts model, in which each node is either active (i.e., having
adopted an innovation) or inactive. In this extended model, the adoption
threshold, defined as the fraction of active neighbors required for an inactive
node to become active, depends on whether the node is a seed neighbor (i.e.,
connected to one or more initiators) or an ordinary node (i.e., not connected
to any initiators). Using the tree approximation on random graphs, we determine
the tipping point, at which the fraction of active nodes in the final state
increases discontinuously with an increasing seed fraction. The occurrence of a
tipping point and the scale of cascades depend on two factors: whether a giant
component of seed neighbors is formed when the seed fraction is large enough to
trigger cascades among seed neighbors, and whether the giant component of
ordinary nodes is maintained when newly activated nodes trigger further
activations among ordinary nodes. The coexistence of two giant components
suggests that a tipping point can appear twice. We present an example
demonstrating the existence of two tipping points when there is a gap between
the adoption thresholds of seed neighbors and ordinary nodes. Monte Carlo
simulations clearly show that the first cascade, occurring at a small tipping
point, occurs in the giant component of seed neighbors, while the second
cascade, occurring at a larger tipping point, extends into the giant component
of ordinary nodes.
