  In this paper, we formulate a novel problem for finding blackhole and volcano
patterns in a large directed graph. Specifically, a blackhole pattern is a
group which is made of a set of nodes in a way such that there are only inlinks
to this group from the rest nodes in the graph. In contrast, a volcano pattern
is a group which only has outlinks to the rest nodes in the graph. Both
patterns can be observed in real world. For instance, in a trading network, a
blackhole pattern may represent a group of traders who are manipulating the
market. In the paper, we first prove that the blackhole mining problem is a
dual problem of finding volcanoes. Therefore, we focus on finding the blackhole
patterns. Along this line, we design two pruning schemes to guide the blackhole
finding process. In the first pruning scheme, we strategically prune the search
space based on a set of pattern-size-independent pruning rules and develop an
iBlackhole algorithm. The second pruning scheme follows a divide-and-conquer
strategy to further exploit the pruning results from the first pruning scheme.
Indeed, a target directed graphs can be divided into several disconnected
subgraphs by the first pruning scheme, and thus the blackhole finding can be
conducted in each disconnected subgraph rather than in a large graph. Based on
these two pruning schemes, we also develop an iBlackhole-DC algorithm. Finally,
experimental results on real-world data show that the iBlackhole-DC algorithm
can be several orders of magnitude faster than the iBlackhole algorithm, which
has a huge computational advantage over a brute-force method.
