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A time-dependent graph is, informally speaking, a graph structure dynamically changes with time. In such graphs, the weights associated with edges dynamically change over time, that is, the edges in such graphs are activated by sequences of time-dependent elements. Many real-life scenarios can be better modeled by time-dependent graphs, such as bioinformatics networks, transportation networks, and social networks. In particular, the time-dependent graph is a very broad concept, which is reflected in the related research with many names, including temporal graphs, evolving graphs, time-varying graphs, historical graphs, and so on. Though static graphs have been extensively studied, for their time-dependent generalizations, we are still far from a complete and mature theory of models and algorithms. In this paper, we discuss the definition and topological structure of time-dependent graphs, as well as models for their relationship to dynamic systems. In addition, we review some classic problems on time-dependent graphs, e.g., route planning, social analysis, and subgraph problem (including matching and mining). We also introduce existing time-dependent systems and summarize their advantages and limitations. We try to keep the descriptions consistent as much as possible and we hope the survey can help practitioners to understand existing time-dependent techniques.

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Classification over data streams is a crucial task of explosive social stream mining and computing. Efficient learning techniques provide high-quality services in the aspect of content distribution and event browsing. Due to the concept drift and concept evolution in data streams, the classification performance degrades drastically over time. Many existing methods utilize supervised and unsupervised learning strategies. However, supervised strategies require labeled emerging records to update the classifiers, which is unfeasible to work in the practical social stream applications. Although unsupervised strategies are popularly applied to detect concept evolution, it takes tremendous run-time computation cost to run online clustering. To this end, in this paper, we address these major challenges of social stream classification by proposing an efficient incremental semi-supervised classification method named CODES (Classification Over Drifting and Evolving Stream). The proposed CODES method consists of an efficient incremental semi-supervised learning module and a dynamic novelty threshold update module. Thus, in the drifting and evolving social streams, CODES is able to provide: 1) semi-supervised learning ability to eliminate dependency on the labels of emerging records; 2) fast incremental learning with real-time update ability to tackle concept drift; 3) efficient novel class detection ability to tackle concept evolution. Extensive experiments are conducted on several real-world datasets. The results indicate a higher performance than several state-of-the-art methods. CODES achieves efficient learning performance over drifting and evolving social streams, which improves practical significance in the real-world social stream applications. INDEX TERMS Social stream, incremental learning, semi-supervised learning, extreme learning machine.

The constrained shortest path (CSP) query over static graphs has been extensively studied, since it has wide applications in transportation networks, telecommunication networks and etc. Such networks are dynamic and evolve over time, being modeled as time-dependent graphs. Therefore, in this paper, we study the CSP query over a large time-dependent graph. Specifically, we study the point CSP (PCSP) query and interval CSP (ICSP) query. We formally prove that it is NP-complete to process a PCSP query and at least EXPSPACE to answer an ICSP query. We propose approximate sequential algorithms to answer the PCSP and ICSP queries efficiently. We also develop parallel algorithms for the queries that guarantee to scale with big time-dependent graphs. Using real-life graphs, we experimentally verify the efficiency and scalability of our algorithms.