The hierarchical backbone of complex networks

Given any complex directed network, a set of acyclic subgraphs can be extracted that will provide valuable information about its hierarchical structure. This Letter presents how the interpretation of the network weight matrix as a transition matrix allows the hierarchical backbone to be identified and characterized in terms of the concepts of hierarchical degree, which expresses the total weights of virtual edges established along successive transitions. The potential of the proposed approach is illustrated with respect to simulated random and preferential-attachment networks as well as real data related to word associations and gene sequencing.