$k$ -Connectivity in Random $K$ -Out Graphs Intersecting Erdős-Rényi Graphs

We investigate <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-connectivity in secure wireless sensor networks under the random pairwise key predistribution scheme with unreliable links. When wireless communication links are modeled as independent on-off channels, this amounts to analyzing a <italic>random graph model</italic> formed by <italic>intersecting</italic> a random <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-out graph and an Erdős-Rényi graph. We present conditions on how to scale the parameters of this intersection model so that the resulting graph is <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-connected with probability approaching to one (resp. zero) as the number of nodes gets large. The resulting zero-one law is shown to improve and sharpen the previous result on the 1-connectivity of the same model. We also provide numerical results to support our analysis.

[1]  Virgil D. Gligor,et al.  Designing secure and reliable wireless sensor networks under a pairwise key predistribution scheme , 2015, 2015 IEEE International Conference on Communications (ICC).

[2]  K. Joag-dev,et al.  Negative Association of Random Variables with Applications , 1983 .

[3]  Armand M. Makowski,et al.  Zero–One Laws for Connectivity in Random Key Graphs , 2009, IEEE Transactions on Information Theory.

[4]  Donald F. Towsley,et al.  On the diameter of a class of random graphs , 1990, IEEE Trans. Inf. Theory.

[5]  Mukesh Singhal,et al.  Security in wireless sensor networks , 2008, Wirel. Commun. Mob. Comput..

[6]  Virgil D. Gligor,et al.  Secure k-connectivity in wireless sensor networks under an on/off channel model , 2013, 2013 IEEE International Symposium on Information Theory.

[7]  Katarzyna Rybarczyk,et al.  Diameter, connectivity, and phase transition of the uniform random intersection graph , 2011, Discret. Math..

[8]  Virgil D. Gligor,et al.  On secure and reliable communications in wireless sensor networks: Towards k-connectivity under a random pairwise key predistribution scheme , 2014, 2014 IEEE International Symposium on Information Theory.

[9]  Osman Yagan.  Random graph modeling of key distribution schemes in wireless sensor networks , 2011 .

[10]  Armand M. Makowski,et al.  Modeling the Pairwise Key Predistribution Scheme in the Presence of Unreliable Links , 2013, IEEE Transactions on Information Theory.

[11]  Armand M. Makowski,et al.  On the gradual deployment of random pairwise key distribution schemes , 2011, 2011 International Symposium of Modeling and Optimization of Mobile, Ad Hoc, and Wireless Networks.

[12]  Yongdae Kim,et al.  Revisiting random key pre-distribution schemes for wireless sensor networks , 2004, SASN '04.

[13]  Alessandro Panconesi,et al.  Concentration of Measure for the Analysis of Randomized Algorithms , 2009 .

[14]  Peng-Jun Wan,et al.  Asymptotic Distribution of The Number of Isolated Nodes in Wireless Ad Hoc Networks with Unreliable Nodes and Links , 2006, GLOBECOM.

[15]  Armand M. Makowski,et al.  On the resiliency of sensor networks under the pairwise key distribution scheme , 2011, 2011 IEEE 22nd International Symposium on Personal, Indoor and Mobile Radio Communications.

[16]  Dawn Xiaodong Song,et al.  Random key predistribution schemes for sensor networks , 2003, 2003 Symposium on Security and Privacy, 2003..

[17]  D. Walkup,et al.  Association of Random Variables, with Applications , 1967 .

[18]  Bülent Yener,et al.  Key distribution mechanisms for wireless sensor networks : a survey , 2005 .

[19]  Alan M. Frieze,et al.  On the connectivity of random m-orientable graphs and digraphs , 1982, Comb..

[20]  Béla Bollobás,et al.  Random Graphs: Notation , 2001 .

[21]  P. ERDbS.  ON THE STRENGTH OF CONNECTEDNESS OF A RANDOM GRAPH , 2001 .

[22]  Armand M. Makowski,et al.  On the Connectivity of Sensor Networks Under Random Pairwise Key Predistribution , 2013, IEEE Transactions on Information Theory.

[23]  Armand M. Makowski,et al.  On the scalability of the random pairwise key predistribution scheme: Gradual deployment and key ring sizes , 2013, Perform. Evaluation.

[24]  Roberto Di Pietro,et al.  Redoubtable Sensor Networks , 2008, TSEC.

[25]  Virgil D. Gligor,et al.  Toward $k$ -Connectivity of the Random Graph Induced by a Pairwise Key Predistribution Scheme With Unreliable Links , 2014, IEEE Transactions on Information Theory.

[26]  Sun,et al.  Review of Key Management Mechanisms in Wireless Sensor Networks , 2006 .

[27]  Xiaojiang Du,et al.  A survey of key management schemes in wireless sensor networks , 2007, Comput. Commun..

[28]  Virgil D. Gligor,et al.  A key-management scheme for distributed sensor networks , 2002, CCS '02.

[29]  Osman Yagan.  Performance of the Eschenauer–Gligor Key Distribution Scheme Under an ON/OFF Channel , 2012, IEEE Transactions on Information Theory.

[30]  Yong Wang,et al.  A survey of security issues in wireless sensor networks , 2006, IEEE Communications Surveys & Tutorials.

[31]  Armand M. Makowski,et al.  Designing Securely Connected Wireless Sensor Networks in the Presence of Unreliable Links , 2011, 2011 IEEE International Conference on Communications (ICC).

[32]  Mindaugas Bloznelis,et al.  Component evolution in a secure wireless sensor network , 2009, Networks.

[33]  W. T. Gowers,et al.  RANDOM GRAPHS (Wiley Interscience Series in Discrete Mathematics and Optimization) , 2001 .

[34]  Yunghsiang Sam Han,et al.  A pairwise key pre-distribution scheme for wireless sensor networks , 2003, CCS '03.

[35]  Stefanie Gerke,et al.  Connectivity of the uniform random intersection graph , 2008, Discret. Math..

[36]  Jun Zhao,et al.  $k$ -Connectivity in Random Key Graphs With Unreliable Links , 2015, IEEE Transactions on Information Theory.

[37]  Ian F. Akyildiz,et al.  Wireless sensor networks: a survey , 2002, Comput. Networks.

[38]  G. E. Martin.  Counting: The Art of Enumerative Combinatorics , 2001 .