Network topologies for energy efficient networks

One embodiment identifies all one-hop neighbor nodes and two-hop neighbor nodes of a node; determines an active set of one-hop neighbor nodes for the node, comprising: includes in the active set each one-hop neighbor node that is either an edge node or connected with at least one two-hop neighbor node with which no other one-hop neighbor nodes are connected; and if the active set is not yet complete, then: determine all combinations of one-hop neighbor nodes that are not already in the active set; and tests each combination in order of each combination's total-energy value to determine whether a specific combination is able to complete the active set; if no combination is able to complete the active set, then including all one-hop neighbor nodes in the active set; and communicates a message to each one-hop neighbor node in the active set indicating that it is in the active set.