Minimizing Eccentricity in Composite Networks via Constrained Edge Additions

In practical military or first responder deployment scenarios, information flows need to adhere to specified policies regardless of the physical connectivity of nodes. Nodes in such networks are associated with various levels in a command-and-control hierarchy, and therefore typically form a logical hierarchical tree network that is used to route both command and data traffic. Associated with this logical hierarchical network is a communication network that represents the connectivity of these nodes in the deployed scenario. Such composite networks introduce constraints that can result in information flows having to traverse much longer paths in the underlying communication network. In this paper, we look at the problem of adding edges to a logical hierarchical network (or any other social network) so as to minimize the number of hops required to route data traffic in the underlying communication network from a node to other specified nodes. The edges added are a subset of all possible edges in the complementary logical hierarchical graph and have to satisfy specified hierarchical constraints. First, we consider the general problem of minimizing the eccentricity of a source node 's' (where eccentricity of 's' is the maximum of the shortest paths from 's' to all other nodes) in a metric graph on adding upto 'B' unequal cost metric edges from the set of all edges in the complementary graph. We develop an efficient constant factor approximation algorithm for this case that outperforms existing constant factor algorithms for eccentricity minimization. Here the added edge metric cost as well as the graph edge metric cost correspond to the number of hops in the shortest path required to route traffic in the actual deployed topology (i.e., underlying communication network). Next, we consider the case where the set of possible added edges is a specified subset of the edges in the complementary graph and the set of destinations is a subset of the graph nodes. For this case, we develop heuristic algorithms based on the previous eccentricity minimizing algorithm that show good performance. We validate our algorithms using two realistic military deployment scenarios. We find that adding even a low number of hierarchically constrained edges (of the order of 10) can cause a significant decrease (around 50%) in the eccentricity of a node in the logical hierarchical network and thus can reduce the number of hops required for data traffic traversal.