On extracting consistent graphs in wireless sensor networks

Robustness and security of services like localisation, routing and time synchronisation in Wireless Sensor Networks (WSNs) have been critical issues. Efficient mathematical (graph-theoretic) models for these services exist. Since, these services were not designed with robustness and security in mind, new mathematical models are needed to address these issues. In this paper, we propose a practical approach for modelling these services using weighted undirected graphs called Partially Consistent Grounded Graphs (PCGG). In such graphs, malicious behaviour or inaccurate information reporting is modelled as a function that assigns incorrect or inconsistent values (weights) to the edges connecting the corresponding nodes, called inconsistent edges. We formulate two optimisation problems, namely MAX-CON and LARGEST-CON. Given a PCGG, these are the problems of determining the largest induced subgraph (obtained by eliminating a subset of vertices) containing only consistent edges. MAX-CON maximises the number of vertices, while LARGEST-CON maximises the number of consistent edges of the resulting subgraph. We also discuss the practical implications of solving these problems, analyse their computational hardness and provide a comparitive analysis of heuristics-based algorithms for them.