Zero-One Law for Connectivity in Superposition of Random Key Graphs on Random Geometric Graphs

Abstract

We study connectivity property in the superposition of random key graph on random geometric graph. For this class of random graphs, we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here strengthen recent work by the Krishnan et al.

Asymptotic Distribution of Isolated Nodes in Secure Wireless Sensor Networks under Transmission Constraints

“…Connectivity in secure wireless sensor networks under transmission constraints is another important subject. Although the significant improved conditions and results for asymptotic connectivity are presented by Krishnan et al [4], Krzywdziński and Rybarczyk [13], Tang and Li [32], and Zhao et al [33], we want to show that the connectivity in WSN with the EG scheme (i.e., ( , , S)) is exactly analogue of the counterpart in classic random graphs. (ii) If → , then the probability that ( , , S) is connected tends to − − .…”

“…And Krishnan et al [4] demonstrated that if 2 ( 2 / ) ≥ 2 (ln / ) with = (1) and ( 2 / ) = (1), then ( , , S) is almost surely connected. Recently, Tang and Li [32] and Zhao et al [33] presented the first zero-one laws for connectivity in ( , , S); these laws improve the results [4,13] significantly and help specify the critical transmission ranges for connectivity. Also the distribution of isolated nodes in ( , , S) is considered by Yi et al [10] and Pishro-Nik et al [34], where the network with nodes distributed uniformly over a unit disk D or a unit square S.…”