Researchers use random graph models to analyze complex networks that have no centralized control such as the Internet, peer-to-peer systems, and mobile ad hoc networks. These models explain phenomena like phase changes, clustering, and scaling. It is necessary to understand these phenomena when designing systems where exact node configurations cannot be known in advance. This paper presents a method for analyzing random graph models that combine discrete mathematics and probability theory. A graph connectivity matrix is constructed where each matrix element is the Bernoulli probability that an edge exists between two given nodes. We show how to construct these matrices for many graph classes, and use linear algebra to analyze the connectivity matrix. We present an application that uses this approach to analyze network cluster self-organization for sensor network security. We conclude by discussing the use of these concepts in mobile systems design.
Designing secure sensor networks is difficult. We propose an approach that uses multicast communications and requires fewer encryptions than pairwise communications. The network is partitioned into multicast regions; each region is managed by a sensor node chosen to act as a keyserver. The keyservers solicit nodes in their neighborhood to join the local multicast tree. The keyserver generates a binary tree of keys to maintain communication within the multicast region using a shared key. Our approach supports a distributed key agreement protocol that identifies the compromised keys and supports membership changes with minimum system overhead. We evaluate the overhead of our approach by using the number of messages and encryptions to estimate power consumption. Using data from field tests of a military surveillance application, we show that our multicast approach needs fewer encryptions than pair-wise keying approaches. We also show that this scheme is capable of thwarting many common attacks.
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