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Rigidity theory enables us to specify the conditions of unique localizabilityin the cooperative localization problem of wireless sensor networks. This paperpresents a combinatorial rigidity approach to measure (i) generic rigidity and(ii) generalized redundant rigidity properties of graph structures throughgraph invariants for the localization problem in wireless sensor networks. Wedefine the rigidity index as a graph invariant based on independent set ofedges in the rigidity matroid. It has a value between 0 and 1, and it indicateshow close we are to rigidity. Redundant rigidity is required for globalrigidity, which is associated with unique realization of graphs. Moreover,redundant rigidity also provides rigidity robustness in networked systemsagainst structural changes, such as link losses. Here, we give a broaderdefinition of redundant edge that we call the "generalized redundant edge."This definition of redundancy is valid for both rigid and non-rigid graphs.Next, we define the redundancy index as a graph invariant based on generalizedredundant edges in the rigidity matroid. It also has a value between 0 and 1,and it indicates the percentage of redundancy in a graph. These two indicesallow us to explore the transition from non-rigidity to rigidity and thetransition from rigidity to redundant rigidity. Examples on graphs are providedto demonstrate this approach. From a sensor network point of view, these twoindices enable us to evaluate the effects of sensing radii of sensors on therigidity properties of networks, which in turn, allow us to examine thelocalizability of sensor networks. We evaluate the required changes in sensingradii for localizability by means of the rigidity index and the redundancyindex using random geometric graphs in simulations.

Graph invariants for unique localizability in cooperative localization of wireless sensor networks: rigidity index and redundancy index