Deconstruct Densest Subgraphs

In this paper, we aim to understand the distribution of the densest subgraphs of a given graph under the density notion of averagedegree. We show that the structures, the relationships and the distributions of all the densest subgraphs of a graph G can be encoded in O(L) space in an index called the ds-Index. Here L denotes the maximum output size of a densest subgraph of G. More importantly, ds-Index can report all the minimal densest subgraphs of G collectively in O(L) time and can enumerate all the densest subgraphs of G with an O(L) delay. Besides, the construction of ds-Index costs no more than finding a single densest subgraph using the state-of-the-art approach. Our empirical study shows that for web-scale graphs with one billion edges, the ds-Index can be constructed in several minutes on an ordinary commercial machine.