On Connectivity and Robustness in Random Intersection Graphs

Random intersection graphs have received much attention recently and been used in a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks. For these graphs, each node is equipped with a set of objects in a random manner, and two nodes have an undirected edge in between if they have at least one object in common. In this paper, we investigate connectivity and robustness in a <italic>general random intersection graph</italic> model. Specifically, we establish sharp asymptotic zero-one laws for <inline-formula> <tex-math notation="LaTeX">$k$</tex-math></inline-formula>-connectivity and <inline-formula><tex-math notation="LaTeX"> $k$</tex-math></inline-formula>-robustness, as well as the asymptotically exact probability of <inline-formula> <tex-math notation="LaTeX">$k$</tex-math></inline-formula>-connectivity, for any positive integer <inline-formula> <tex-math notation="LaTeX">$k$</tex-math></inline-formula>. The <inline-formula><tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-connectivity property quantifies how resilient is the connectivity of a graph against node or edge failures, while <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-robustness measures the effectiveness of <italic>local-information-based</italic> consensus algorithms (which do not use global graph topology information) in the presence of adversarial nodes. In addition to presenting the results under the general random intersection graph model, we consider two special cases of the general model, a <italic>binomial</italic> random intersection graph and a <italic>uniform</italic> random intersection graph, which both have numerous applications as well. For these two specialized graphs, we present asymptotically exact probabilities of <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-connectivity and asymptotic zero-one laws for <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-robustness.

[1]  Mindaugas Bloznelis,et al.  Degree and clustering coefficient in sparse random intersection graphs , 2013, 1303.3388.

[2]  Katarzyna Rybarczyk.  The Coupling Method for Inhomogeneous Random Intersection Graphs , 2017, Electron. J. Comb..

[3]  Nicolas W. Hengartner,et al.  Component Evolution in General Random Intersection Graphs , 2010, WAW.

[4]  Shreyas Sundaram,et al.  Robustness of complex networks with implications for consensus and contagion , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[5]  Shreyas Sundaram,et al.  A simple median-based resilient consensus algorithm , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

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

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

[8]  Virgil D. Gligor,et al.  On asymptotically exact probability of $k$-connectivity in random key graphs intersecting Erdős-Rényi graphs , 2014, ArXiv.

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

[10]  Virgil D. Gligor,et al.  On the strengths of connectivity and robustness in general random intersection graphs , 2014, 53rd IEEE Conference on Decision and Control.

[11]  Adrian Perrig,et al.  On the distribution and revocation of cryptographic keys in sensor networks , 2005, IEEE Transactions on Dependable and Secure Computing.

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

[13]  Mindaugas Bloznelis,et al.  K-connectivity of Uniform S-intersection Graphs , 2014, Discret. Math..

[14]  Osman Yagan.  Zero-One Laws for Connectivity in Inhomogeneous Random Key Graphs , 2016, IEEE Transactions on Information Theory.

[15]  Erhard Godehardt,et al.  Two Models of Random Intersection Graphs for Classification , 2003 .

[16]  Willemien Kets,et al.  RANDOM INTERSECTION GRAPHS WITH TUNABLE DEGREE DISTRIBUTION AND CLUSTERING , 2009, Probability in the Engineering and Informational Sciences.

[17]  Shreyas Sundaram,et al.  Robustness of information diffusion algorithms to locally bounded adversaries , 2011, 2012 American Control Conference (ACC).

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

[19]  Shreyas Sundaram,et al.  A Notion of Robustness in Complex Networks , 2015, IEEE Transactions on Control of Network Systems.

[20]  Katarzyna Rybarczyk,et al.  Sharp threshold functions for the random intersection graph via coupling method , 2009, 0910.0749.

[21]  Béla Bollobás,et al.  Random Graphs , 1985 .

[22]  Danny Dolev,et al.  The Byzantine Generals Strike Again , 1981, J. Algorithms.

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

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

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

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

[27]  Jun Zhao,et al.  On k-Connectivity and Minimum Vertex Degree in Random s-Intersection Graphs , 2015, ANALCO.

[28]  Roberto Di Pietro,et al.  Sensor Networks that Are Provably Resilient , 2006, 2006 Securecomm and Workshops.

[29]  P. Erd Os,et al.  On the strength of connectedness of random graphs , 1961 .

[30]  Maurizio Porfiri,et al.  Consensus Over Numerosity-Constrained Random Networks , 2011, IEEE Transactions on Automatic Control.

[31]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[32]  Shreyas Sundaram,et al.  Resilient Asymptotic Consensus in Robust Networks , 2013, IEEE Journal on Selected Areas in Communications.

[33]  Douglas R. Stinson,et al.  On the complexity of the herding attack and some related attacks on hash functions , 2012, Des. Codes Cryptogr..

[34]  Shreyas Sundaram,et al.  Resilient continuous-time consensus in fractional robust networks , 2013, 2013 American Control Conference.

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