  Community search over large graphs is a fundamental problem in graph
analysis. Recent studies propose to compute top-k influential communities,
where each reported community not only is a cohesive subgraph but also has a
high influence value. The existing approaches to the problem of top-k
influential community search can be categorized as index-based algorithms and
online search algorithms without indexes. The index-based algorithms, although
being very efficient in conducting community searches, need to pre-compute a
special-purpose index and only work for one built-in vertex weight vector. In
this paper, we investigate on-line search approaches and propose an
instance-optimal algorithm LocalSearch whose time complexity is linearly
proportional to the size of the smallest subgraph that a correct algorithm
needs to access without indexes. In addition, we also propose techniques to
make LocalSearch progressively compute and report the communities in decreasing
influence value order such that k does not need to be specified. Moreover, we
extend our framework to the general case of top-k influential community search
regarding other cohesiveness measures. Extensive empirical studies on real
graphs demonstrate that our algorithms outperform the existing online search
algorithms by several orders of magnitude.
