# Time-variant graph learning and classification

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Graph classification is an important tool for analyzing data with structure dependency. In traditional graph classification, graphs are assumed to be independent where each graph represents an object. In a dynamic world, it is very often the case that the underlying object continuously evolves over time. The change of node content and/or network structure, with respect to the temporal order, presents a new time-variant graph representation, where an object corresponds to a set of time-variant graphs (TVG). A time-variant graph can be used to characterize the changing nature of the structured object, including the node attribute and graph topological changing over time. Therefore, the evolution of time-variant graphs could be either network structure or node content over time. In this dissertation, we formulate a new time-variant graph learning and classification (TVGLC) task.
To learn and classify time-variant graphs, the vital steps are feature extraction, modeling and algorithm design. However, for time-variant graph classification, frequent subgraph features are very difficult to obtain. Because one has to consider the graph structure space and the temporal correlations to find subgraph candidates for validation, the search space for finding frequent subgraph features is infinite and unlikely to obtain stable structures. Secondly, graph structures that imply subgraph features may irregularly change over time. Thus, to extract effective and efficient features is a great challenge for TVGLC. In addition, carrying out applicable models and algorithms to cater for the extracted features for TVGLC is also a challenge.
Considering the above challenges, this research aims to extract efficient features and design new algorithms to enable the learning of the time-variant graph. Because time variant graphs may involve changes in the network structures and changes in the node content, which complicate the algorithm designs and solutions, our research employs a divide and conquer principle to first solve a simplified case where (1) network topology is fixed whereas the node content continuously evolves (i.e., networked time series classification). After that, we advance to the setting to (2) evolving network structure and propose solutions to TVGLC with incremental subgraph features. To enhance the subgraph feature exploration for time variant graph classification, we propose (3) graph-shapelet features for TVGLC. Last, but not the least, we study (4) an application of online diffusion provenance detection.
Temporal Feature Selection on Networked Time Series: As the time-variant graph can be graph node content and/or graph structure evolution, we first study a simple case where the structure is fixed but the node content continuously evolves. The problem forms time series data when the node content changes over time, and we combine time series data with a static graph to form a new problem called networked time series. We formulate the problem of learning discriminative features (i.e., segments) from networked time series data considering the linked information among time series (e.g., social users are taken as social sensors that continuously generate social signals (tweets) represented as time series). The discriminative segments are often referred to as shapelets of time series. Extracting shapelets for time series classification has been widely studied. However, existing works on shapelet selection assumes that time series are independent and identically distributed (i.i.d.). This assumption restricts their applications to social networked time series analysis. This thesis proposes a new Network Regularized Least Squares (NetRLS) feature selection model, which combines typical time series data and user network graph data for analysis.
Incremental Subgraph based TVGLC: To learn and classify the time-variant graph with network structure evolve, the key challenges are to extract features and build models. To date, subgraphs are often used as features for graph learning. In reality, the dimension of the subgraphs has a crucial dependency on the threshold setting of the frequency support parameter, and the number may become extremely large. As a result, subgraphs may be incrementally discovered to form a feature stream and require the underlying graph classifier to effectively discover representative subgraph features from the subgraph feature stream. Moreover, we propose a primal-dual incremental subgraph feature selection algorithm (ISF) based on a max-margin graph classifier. The ISF algorithm constructs a sequence of solutions that are both primal and dual feasible. Each primal-dual pair shrinks the dual gap and renders a better solution for the optimal subgraph feature set. To avoid the bias of the ISF algorithm on short-pattern subgraph features, we present a new incremental subgraph join feature selection algorithm (ISJF) by forcing graph classifiers to join short-pattern subgraphs and generate long-pattern subgraph features.
Graph-shapelet based TVGLC: As graph structure continuously evolves over time, the search space for finding frequent subgraph features is infinite and unlikely to obtain stable structures. To tackle this challenge, we formulate a new time-variant graph classification task, and propose a new graph feature, graph-shapelets, for learning and classifying time-variant graphs. Graph-shapelet is compact and discriminative graph transformation subsequences. A graph-shapelet can be regarded as a graphical extension of shapelets – a class of discriminative features designed for vectorial temporal data classification. In order to discover graph-shapelets, we propose to convert a time-variant graph sequence as time-series data, and use shapelets discovered from the time-series data to find graph transformation subsequences as graph-shapelets. By converting each graph-shapelet as a unique tokenized graph transformation sequence, we can use the editing distance to calculate the distance between two graph-shapelets for time-variant graph classification.
Application of Online Diffusion Provenance Detection: In social network analysis, the information propagation graph (i.e., cascade) is a kind of time-variant graph because the information diffusion forms a graph at a certain time and the graph evolves over time. An important application of information diffusion networks (i.e., time-variant graph) is provenances detection. Existing work on network diffusion provenance identification focuses on offline learning where data collected from network detectors are static and a snapshot of the network is available before learning. However, an offline learning model does not meet the needs of early warning, real-time awareness and real-time response to malicious information spreading in networks. In this part, we study a new problem of online discovering diffusion provenances in large networks. To this end, we propose an online regression model for real-time diffusion provenance identification. Specifically, we first use offline collected network cascades to infer the edge transmission weights, and then use an online l₁ nonconvex regression model as the identification model. The proposed methods are empirically evaluated on both synthetic and real-world networks.
Experiments on synthetic and real-world data validate and demonstrate the effectiveness of the proposed methods for time-variant graph learning and classification.

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