Connectivity in Secure Wireless Sensor Networks under Transmission Constraints

In wireless sensor networks (WSNs), the Eschenauer-Gligor (EG) key pre-distribution scheme is a widely recognized way to secure communications. Although connectivity properties of secure WSNs with the EG scheme have been extensively investigated, few results address physical transmission constraints. These constraints reflect real-world implementations of WSNs in which two sensors have to be within a certain distance from each other to communicate. In this paper, we present zero-one laws for connectivity in WSNs employing the EG scheme under transmission constraints. These laws help specify the critical transmission ranges for connectivity. Our analytical findings are confirmed via numerical experiments. In addition to secure WSNs, our theoretical results are also applied to frequency hopping in wireless networks.

💡 Research Summary

The paper investigates connectivity of wireless sensor networks (WSNs) that employ the Eschenauer‑Gligor (EG) key pre‑distribution scheme while also being subject to physical transmission constraints. The authors model the key sharing among sensors as a random key graph (G_{RK G}(n,K_n,P_n)) where each of the (n) nodes receives (K_n) keys drawn uniformly from a pool of size (P_n). Two nodes are linked in this graph if they share at least one key, an event whose probability is asymptotically (p_s\approx K_n^2/P_n) when (K_n^2/P_n=o(1)).

Physical proximity is captured by the classical disk model: nodes are placed independently and uniformly over a region (A) (either a unit torus or a unit square) and two nodes can communicate directly only if their Euclidean distance does not exceed a transmission radius (r_n). This yields a random geometric graph (G_{RGG}(n,r_n,A)). The secure WSN is then the intersection graph
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