Sensor Network Self-Organization Using Random Graphs

RANDOM graph theory originated with seminal work by Erdös and Re nyi in the 1950s. Until then, graph theory analyzed either specific graph instances or deterministically defined graph classes. Erdös and Re nyi considered graph classes where the existence of edges between nodes was determined probabilistically. Their results were theoretically interesting and found applications in many practical domains [1]. Erdös and Re nyi used the same probability value to assign edges between any two nodes in the graph. As an extension to this in the 1990s, Strogatz and Watts studied ‘‘small world’’ graphs [2]. The term small world originates with Milgram’s six degrees of separation model of social networks created in the 1960s. Strogatz and Watts’ work considers networks where the probability of edges existing between nodes is not uniform. They were specifically interested in clustered graphs, where edges are more likely to exist between nodes with common neighbors. To study this phenomenon, they defined classes of pseudo-random graphs. These graphs combine a deterministic structure and a limited number of random edges. Their results have been used to analyze both social networks and technical infrastructures. An alternative approach to studying similar systems has been proposed by Barabási [1]. His group considered the probability distributions of graph node degree found in graph models of existing systems. This analysis shows that the probability of a node having degree d follows an inverse power law (i.e., is proportional to d where is a constant). They also explain how this property can emerge from positive feedback in evolving systems. These models appear to be appropriate for studying a wide range of natural and large-scale technical systems. Important results from this model include quantification of the dependability of the Internet [3], and analysis of computer virus propagation [4]. International Journal of Distributed Sensor Networks, 5: 201–208, 2009 Copyright Taylor & Francis Group, LLC ISSN: 1550-1329 print / 1550-1477 online DOI: 10.1080/15501320802498307