On the Role of Expander Graphs in Key Predistribution Schemes for Wireless Sensor Networks

Abstract

Abstract. Providing security for a wireless sensor network composed of small sensor nodes with limited battery power and memory can be a nontrivial task. A variety of key predistribution schemes have been proposed which allocate symmetric keys to the sensor nodes before deployment. In this paper we examine the role of expander graphs in key predistribution schemes for wireless sensor networks. Roughly speaking, a graph has good expansion if every 'small' subset of vertices has a 'large' neighbourhood, and intuitively, expansion is a desirable property for graphs of networks. It has been claimed that good expansion in the product graph is necessary for 'optimal' networks. We demonstrate flaws in this claim, argue instead that good expansion is desirable in the intersection graph, and discuss how this can be achieved. We then consider key predistribution schemes based on expander graph constructions and compare them to other schemes in the literature. Finally, we propose the use of expansion and other graph-theoretical techniques as metrics for assessing key predistribution schemes and their resulting wireless sensor networks.

Graph-theoretic design and analysis of key predistribution schemes

“…We say that two nodes N i and N j can communicate securely if (N i , N j ) ∈ E 1 ∩ E 2 , that is if they are adjacent in the intersection graph G 1 ∩ G 2 = (V, E 1 ∩ E 2 ). If nodes are to be scattered such that there will be no control over the communication graph, the only way of affecting the intersection graph is through careful design of the key graph [38]. It is therefore the key graph which we design and analyse in this paper.…”

“…For a thorough survey of expander graphs and their applications, see [25,36,38]. Here we introduce only the aspects of expander graphs which are relevant to our study, and in particular we restrict our attention to edge expansion and finite graphs.…”

“…A graph with large ε will also have low diameter, logarithmic in the size of the network [36] and contain multiple short, disjoint paths between nodes [40], the many benefits of which were discussed in [38], where it is shown that ε ≤ min x∈V d(x).…”

“…In addition to the three parameters introduced above, it has been recognised that the topology of the key graph is also an important design consideration. In particular, it is desirable for the key graph to have good expansion [38], which in essence means that all subsets of vertices are well-connected to the rest of the graph. The papers [17,49] use expander graph constructions to produce KPSs.…”

“…In [38] it was shown that many existing KPSs with less-than-perfect resilience produce highly connected key graphs with good expansion, thereby providing a more commonly desirable trade-off between connectivity and resilience than is afforded by the strong condition of perfect resilience. We develop this analysis, proving that the key graphs of many existing classes of KPSs have good expansion, which provides further support and justification for their use.…”