On Connectivity Thresholds in the Intersection of Random Key Graphs on Random Geometric Graphs
Abstract
In a random key graph (RKG) of $n$ nodes each node is randomly assigned a key ring of $K_n$ cryptographic keys from a pool of $P_n$ keys. Two nodes can communicate directly if they have at least one common key in their key rings. We assume that the $n$ nodes are distributed uniformly in $[0,1]^2.$ In addition to the common key requirement, we require two nodes to also be within $r_n$ of each other to be able to have a direct edge. Thus we have a random graph in which the RKG is superposed on the familiar random geometric graph (RGG). For such a random graph, we obtain tight bounds on the relation between $K_n,$ $P_n$ and $r_n$ for the graph to be asymptotically almost surely connected.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2013
- DOI:
- 10.48550/arXiv.1301.6422
- arXiv:
- arXiv:1301.6422
- Bibcode:
- 2013arXiv1301.6422S
- Keywords:
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- Computer Science - Information Theory;
- Mathematics - Combinatorics;
- Mathematics - Probability;
- G.2.2;
- G.2.3;
- F.2.2
- E-Print:
- Accepted for Publication at ISIT 2013. 5 Pages (main text) + 6 pages (appendix)