node fig2 b shows that at least degree 5 e g 5 antennas is required to maintain two node connectivity of the graph shown again the 5 nodes along the fringe of each unit disk ud 2 ud 3 are even distributed two nodes are connected if and only if their euclidean distance is at most 1 the central nodes of the unit disks ud 2 ud 3 both have degree 5 in this example the removal of any link results in a subgraph that is not two node connected having presented a problem statement and examples of both a two edge connected and two node connected nwnm we now turn our discussion to an explanation of the methods discovered by the inventors which were used to determine the minimal number of antennas needed in both a two edge and two node connected based nwnm in accordance with the present invention the first step is to build two connected sparse subgraphs to this end the inventors discovered two methods of doing so the first referred to as a simple backlink based algorithm u201c bba u201d builds a subgraph g that preserves the two connectivity of an original full graph either two edge connectivity or two node connectivity from which the subgraph is selected if the full graph is a udg the maximal node degree of the subgraph g is at most 10 e g 10 directional antennas as we have discussed before this is the minimal node degree that is required to maintain the two edge connectivity of udgs i e maintain an operational network when a link.