  We consider a wireless ad hoc network in the presence of eavesdroppers (EDs),
where the nodes are distributed according to independent Poisson point
processes (PPPs). The legitimate nodes follow the half-duplex mode of operation
employing the slotted ALOHA protocol for transmission. For such a network, a
novel communication scheme that induces a time-varying secure connectivity
graph (SCG) is proposed, and the connectivity behavior of this induced SCG is
studied. In particular, for a legitimate node in the network, we analyze (i)
the average number of incoming edges and the average number of outgoing edges;
(ii) the time to nearest-neighbor secure connectivity; and (iii) a condition on
the EDs' density that allows information percolation, {\ie}, a condition for
the existence of a `giant' component. The average time for secure connectivity
among the nodes in this giant component is shown to scale linearly with the
Euclidean distance. Further, we show that by splitting the packets into two
sub-packets and routing each sub-packet along paths that are sufficiently far
apart can (a) potentially improve secure connectivity and (b) reduce the
overall delay incurred in exchanging packets between any two legitimate nodes
in the giant component.
