89 research outputs found
On the strengths of connectivity and robustness in general random intersection graphs
Random intersection graphs have received much attention for nearly two
decades, and currently have a wide range of applications ranging from key
predistribution in wireless sensor networks to modeling social networks. In
this paper, we investigate the strengths of connectivity and robustness in a
general random intersection graph model. Specifically, we establish sharp
asymptotic zero-one laws for -connectivity and -robustness, as well as
the asymptotically exact probability of -connectivity, for any positive
integer . The -connectivity property quantifies how resilient is the
connectivity of a graph against node or edge failures. On the other hand,
-robustness measures the effectiveness of local diffusion strategies (that
do not use global graph topology information) in spreading information over the
graph in the presence of misbehaving nodes. In addition to presenting the
results under the general random intersection graph model, we consider two
special cases of the general model, a binomial random intersection graph and a
uniform random intersection graph, which both have numerous applications as
well. For these two specialized graphs, our results on asymptotically exact
probabilities of -connectivity and asymptotic zero-one laws for
-robustness are also novel in the literature.Comment: This paper about random graphs appears in IEEE Conference on Decision
and Control (CDC) 2014, the premier conference in control theor
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.Comment: Full version of a paper published in Annual Allerton Conference on
Communication, Control, and Computing (Allerton) 201
k-Connectivity in Random Key Graphs with Unreliable Links
Random key graphs form a class of random intersection graphs and are
naturally induced by the random key predistribution scheme of Eschenauer and
Gligor for securing wireless sensor network (WSN) communications. Random key
graphs have received much interest recently, owing in part to their wide
applicability in various domains including recommender systems, social
networks, secure sensor networks, clustering and classification analysis, and
cryptanalysis to name a few. In this paper, we study connectivity properties of
random key graphs in the presence of unreliable links. Unreliability of the
edges are captured by independent Bernoulli random variables, rendering edges
of the graph to be on or off independently from each other. The resulting model
is an intersection of a random key graph and an Erdos-Renyi graph, and is
expected to be useful in capturing various real-world networks; e.g., with
secure WSN applications in mind, link unreliability can be attributed to harsh
environmental conditions severely impairing transmissions. We present
conditions on how to scale this model's parameters so that i) the minimum node
degree in the graph is at least k, and ii) the graph is k-connected, both with
high probability as the number of nodes becomes large. The results are given in
the form of zeroone laws with critical thresholds identified and shown to
coincide for both graph properties. These findings improve the previous results
by Rybarczyk on the k-connectivity of random key graphs (with reliable links),
as well as the zero-one laws by Yagan on the 1-connectivity of random key
graphs with unreliable links.Comment: Published in IEEE Transactions on Information Theor
On Topological Properties of Wireless Sensor Networks under the q-Composite Key Predistribution Scheme with On/Off Channels
The q-composite key predistribution scheme [1] is used prevalently for secure
communications in large-scale wireless sensor networks (WSNs). Prior work
[2]-[4] explores topological properties of WSNs employing the q-composite
scheme for q = 1 with unreliable communication links modeled as independent
on/off channels. In this paper, we investigate topological properties related
to the node degree in WSNs operating under the q-composite scheme and the
on/off channel model. Our results apply to general q and are stronger than
those reported for the node degree in prior work even for the case of q being
1. Specifically, we show that the number of nodes with certain degree
asymptotically converges in distribution to a Poisson random variable, present
the asymptotic probability distribution for the minimum degree of the network,
and establish the asymptotically exact probability for the property that the
minimum degree is at least an arbitrary value. Numerical experiments confirm
the validity of our analytical findings.Comment: Best Student Paper Finalist in IEEE International Symposium on
Information Theory (ISIT) 201
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