Local representatives in weighted networks

The main features of current real-world networks are their large sizes and structures, which show varying degrees of importance of the nodes in their surroundings. The topic of evaluating the importance of the nodes offers many different approaches that usually work with unweighted networks. We present a novel, simple and straightforward approach for the evaluation of the network´s nodes with a focus on local properties in their surroundings. The presented approach is intended for weighted networks where the weight can be interpreted as the proximity between the nodes. Our suggested x-representativeness then takes into account the degree of the node, its nearest neighbors and one other parameter which we call the x-representativeness base. Following that, we also present experiments with three different real-world networks. The aim of these experiments is to show that the x-representativeness can be used to deterministically reduce the network to differently sized samples of representatives, while maintaining the topological properties of the original network.