Bounds on Minimum Number of Anchors for Iterative Localization and its Connections to Bootstrap Percolation

Iterated localization is considered where each node of a network needs to get localized (find its location on 2-D plane), when initially only a subset of nodes have their location information. The iterated localization process proceeds as follows. Starting with a subset of nodes that have their location information, possibly using global positioning system (GPS) devices, any other node gets localized if it has three or more localized nodes in its radio range. The newly localized nodes are included in the subset of nodes that have their location information for the next iteration. This process is allowed to continue, until no new node can be localized. The problem is to find the minimum size of the initially localized subset to start with so that the whole network is localized with high probability. There are intimate connections between iterated localization and bootstrap percolation, that is well studied in statistical physics. Using results known in bootstrap percolation, we find a sufficient condition on the size of the initially localized subset that guarantees the localization of all nodes in the network with high probability.