Abstract
Graph rigidity provides the conditions of unique localizability for cooperative localization of wireless ad hoc and sensor networks. Specifically, redundant rigidity and 3-connectivity are necessary and sufficient conditions for unique localizability of generic configurations. In this article, we introduce a graph invariant for 3-connectivity, called 3-connectivity index. Using this index along with the rigidity and redundancy indices provided in previous work, we explore the rigidity and connectivity properties of two classes of graphs, namely, random geometric graphs and clustered graphs. We have found out that, in random geometric graphs and clustered graphs, it needs significantly less effort to achieve 3-connectivity once we obtain redundant rigidity. In reconsidering the general conditions for unique localizability, the most striking finding in random geometric graphs is that it is unlikely to observe a graph, in which 3-connectivity is satisfied before the graph becomes redundantly rigid. Therefore, in random geometric graphs, it is more likely sufficient to test only 3-connectivity for unique localizability. On the contrary to random geometric graphs, our findings indicate that 3-connectivity may be satisfied before the graph becomes redundantly rigid in clustered graphs, which means that, in clustered graphs, we have to test both redundant rigidity and 3-connectivity for unique localizability.
Highlights
Locations of sensor nodes are often required in several applications of wireless sensor networks because information gathered or communicated by wireless sensor nodes is often meaningful with knowledge of the locations of the nodes
It may be expensive to equip all sensor nodes with global positioning system (GPS) receivers, and manual configuration of each sensor node may be impractical
We summarize as follows: [1] in random geometric graphs, a 3-connected graph is more likely to be redundantly rigid, that is, Kc = 1 more likely implies Ku = 1; (b) in clustered graphs, a 3-connected graph is not necessarily redundantly rigid, in other words, Kc = 1 does not imply Ku = 1, so we need to check both 3-connectivity and redundant rigidity to make sure that they are both satisfied for global rigidity
Summary
Locations of sensor nodes are often required in several applications of wireless sensor networks because information gathered or communicated by wireless sensor nodes is often meaningful with knowledge of the locations of the nodes. It may be expensive to equip all sensor nodes with global positioning system (GPS) receivers, and manual configuration of each sensor node may be impractical. It is not clear how much increase in sensing radii is needed to obtain 3-connectivity once redundant rigidity is satisfied, or vice versa, because neighborhood (and unit disk graph structure) does not play a role in general settings. We know that neighborhood is important in random geometric graphs and clustered graphs; how much increase in sensing radii is needed to obtain 3-connectivity once redundant rigidity is satisfied (vice versa) is another open question. It is not even known whether redundant rigidity or 3-connectivity is satisfied first in a random geometric graph or clustered graph.
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