  Dense subgraph discovery is an important primitive in graph mining, which has
a wide variety of applications in diverse domains. In the densest subgraph
problem, given an undirected graph $G=(V,E)$ with an edge-weight vector
$w=(w_e)_{e\in E}$, we aim to find $S\subseteq V$ that maximizes the density,
i.e., $w(S)/|S|$, where $w(S)$ is the sum of the weights of the edges in the
subgraph induced by $S$. Although the densest subgraph problem is one of the
most well-studied optimization problems for dense subgraph discovery, there is
an implicit strong assumption; it is assumed that the weights of all the edges
are known exactly as input. In real-world applications, there are often cases
where we have only uncertain information of the edge weights. In this study, we
provide a framework for dense subgraph discovery under the uncertainty of edge
weights. Specifically, we address such an uncertainty issue using the theory of
robust optimization. First, we formulate our fundamental problem, the robust
densest subgraph problem, and present a simple algorithm. We then formulate the
robust densest subgraph problem with sampling oracle that models dense subgraph
discovery using an edge-weight sampling oracle, and present an algorithm with a
strong theoretical performance guarantee. Computational experiments using both
synthetic graphs and popular real-world graphs demonstrate the effectiveness of
our proposed algorithms.
