  Rigidity theory enables us to specify the conditions of unique localizability
in the cooperative localization problem of wireless sensor networks. This paper
presents a combinatorial rigidity approach to measure (i) generic rigidity and
(ii) generalized redundant rigidity properties of graph structures through
graph invariants for the localization problem in wireless sensor networks. We
define the rigidity index as a graph invariant based on independent set of
edges in the rigidity matroid. It has a value between 0 and 1, and it indicates
how close we are to rigidity. Redundant rigidity is required for global
rigidity, which is associated with unique realization of graphs. Moreover,
redundant rigidity also provides rigidity robustness in networked systems
against structural changes, such as link losses. Here, we give a broader
definition 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 generalized
redundant 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 indices
allow us to explore the transition from non-rigidity to rigidity and the
transition from rigidity to redundant rigidity. Examples on graphs are provided
to demonstrate this approach. From a sensor network point of view, these two
indices enable us to evaluate the effects of sensing radii of sensors on the
rigidity properties of networks, which in turn, allow us to examine the
localizability of sensor networks. We evaluate the required changes in sensing
radii for localizability by means of the rigidity index and the redundancy
index using random geometric graphs in simulations.
