Community search is important in graph analysis and can be used in many applications. In the literature, various community models have been proposed. However, most of them cannot well identify the overlaps between communities which is an essential feature of real graphs. To address this issue, $k$-clique percolation community model was proposed and has been proven effective in many applications. Motivated by this, in this paper, we adopt the $k$-clique percolation community model and study the densest clique percolation community search problem which aims to find the $k$-clique percolation community with the maximum k value that contains a given set of query nodes. We adopt an index-based approach to solve this problem. Based on the observation that a $k$-clique percolation community is a union of maximal cliques, we devise a compact index, DCPC-Index, to preserve the maximal cliques and their connectivity information of the input graph. With DCPC-Index, we can answer the densest clique percolation community query efficiently. Besides, we also propose an index construction algorithm based on the definition of DCPC-Index and further improve the algorithm in terms of efficiency and memory consumption. The experimental results demonstrate the efficiency of our index-based query processing algorithm and index construction algorithm.

In indexing a graph for distance queries, distance labeling is a common practice; in particular, 2-hop labeling which guarantees the exactness of the query results is widely adopted. When it comes to a massive real graph with a relatively large treewidth such as social networks and web graphs, however, 2-hop labeling can hardly be constructed due to the oversized index. This paper discloses the theoretical relationships between the graph treewidth and 2-hop labeling's index size and query time. To scale up distance labeling, this paper proposes Core-Tree (CT) Index to facilitate a critical and effective trade-off between the index size and query time. The reduced index size enables CT-Index to handle massive graphs that no existing approaches can process while the cost in the query time is negligible: the query time is below 0.4 milliseconds on all tested graphs including one graph with 5.5 billion edges.

K-defective cliques relax cliques by allowing up-to k missing edges from being a complete graph. This relaxation enables us to find larger near-cliques and has applications in link prediction, cluster detection, social network analysis and transportation science. The problem of finding the largest k-defective clique has been recently studied with several algorithms being proposed in the literature. However, the currently fastest algorithm KDBB does not improve its time complexity from being the trivial O(2n), and also, KDBB's practical performance is still not satisfactory. In this paper, we advance the state of the art for exact maximum k-defective clique computation, in terms of both time complexity and practical performance. Moreover, we separate the techniques required for achieving the time complexity from others purely used for practical performance consideration; this design choice may facilitate the research community to further improve the practical efficiency while not sacrificing the worst case time complexity. In specific, we first develop a general framework kDC that beats the trivial time complexity of O(2n) and achieves a better time complexity than all existing algorithms. The time complexity of kDC is solely achieved by our newly designed non-fully-adjacent-first branching rule, excess-removal reduction rule and high-degree reduction rule. Then, to make kDC practically efficient, we further propose a new upper bound, two new reduction rules, and an algorithm for efficiently computing a large initial solution. Extensive empirical studies on three benchmark graph collections with 290 graphs in total demonstrate that kDC outperforms the currently fastest algorithm KDBB by several orders of magnitude.