  This paper discusses first passage percolation and flooding on large weighted
sparse random graphs with two types of nodes: active and passive nodes. In
mathematical physics passive nodes can be interpreted as closed gates where
fluid flow or water cannot pass through and active nodes can be interpreted as
open gates where water may keep flowing further. The model of this paper has
many applications in real life, for example, information spreading, where
passive nodes are interpreted as passive receivers who may read messages but do
not respond to them. In the epidemic context passive nodes may be interpreted
as individuals who self-isolate themselves after having a disease to stop
spreading the disease any further. When all weights on edges between active
nodes and between active and passive nodes are independent and exponentially
distributed (but not necessary identically distributed), this article provides
an approximation formula for the weighted typical flooding time.

Probability