During the past decades significant efforts have been made to propose data structures for answering connectivity queries on fully dynamic graphs, i.e., graphs with frequent insertions and deletions of edges. However, a comprehensive understanding of how these data structures perform in practice is missing, since not all of them have been implemented, let alone evaluated experimentally. We provide reference implementations for the proposed data structures and experimentally evaluate them on a wide range of graphs. Our findings show that the current solutions are not ready to be deployed in systems as is, as every data structure has critical weaknesses when used in practice. Key limitations that must be overcome are the space and time overhead incurred by balanced data structures, the degeneration of the runtime of space-efficient data structures in worst case scenarios, and the maintenance costs for balanced data structures. We detail our findings in the experimental evaluation and provide recommendations for implementing robust solutions for answering connectivity queries on dynamic graphs.

Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices are inserted and deleted frequently. Existing work proposes data structures and algorithms with worst case guarantees. We propose a new data structure, the dynamic tree (D-tree), together with algorithms to construct and maintain it. The D-tree is the first data structure that scales to fully dynamic graphs with millions of vertices and edges and, on average, answers connectivity queries much faster than data structures with worst case guarantees.

A graph stream, which refers to the graph with edges being updated sequentially in a form of a stream, has important applications in cyber security and social networks. Due to the sheer volume and highly dynamic nature of graph streams, the practical way of handling them is by summarization. Given a graph stream G, directed or undirected, the problem of graph stream summarization is to summarize G as SG with a much smaller (sublinear) space, linear construction time and constant maintenance cost for each edge update, such that SG allows many queries over G to be approximately conducted efficiently. The widely used practice of summarizing data streams is to treat each stream element independently by e.g., hash- or sample-based methods, without maintaining the connections (or relationships) between elements. Hence, existing methods can only solve ad-hoc problems, without supporting diversified and complicated analytics over graph streams. We present TCM, a novel generalized graph stream summary. Given an incoming edge, it summarizes both node and edge information in constant time. Consequently, the summary forms a graphical sketch where edges capture the connections inside elements, and nodes maintain relationships across elements. We discuss a wide range of supported queries and establish some error bounds. In addition, we experimentally show that TCM can effectively and efficiently support analytics over graph streams, which demonstrates its potential to start a new line of research and applications in graph stream management.