Graph representations offer powerful and intuitive ways to describe data in a multitude of application domains. Here, we consider stochastic processes generating graphs and propose a methodology for detecting changes in stationarity of such processes. The methodology is general and considers a process generating attributed graphs with a variable number of vertices edges, without the need to assume a one-to-one correspondence between vertices at different time steps. The methodology acts by embedding every graph of the stream into a vector domain, where a conventional multivariate change detection procedure can be easily applied. We ground the soundness of our proposal by proving several theoretical results. In addition, we provide a specific implementation of the methodology and evaluate its effectiveness on several detection problems involving attributed graphs representing biological molecules and drawings. Experimental results are contrasted with respect to suitable baseline methods, demonstrating the effectiveness of our approach.
The sample mean is one of the most fundamental concepts in statistics. Properties of the sample mean that are well-defined in Euclidean spaces become unclear in graph spaces. This paper proposes conditions under which the following properties are valid: existence, uniqueness, and consistency of means, the midpoint property, necessary conditions of optimality, and convergence results of mean algorithms. The theoretical results address common misconceptions about the graph mean in graph edit distance spaces, serve as a first step towards a statistical analysis of graph spaces, and result in a theoretically well-founded mean algorithm that outperformed six other mean algorithms with respect to solution quality on different graph datasets representing images and molecules. HighlightsA first step towards a theory of statistical graph space analysis is proposed.MMM-algorithm is proposed that outperformed six other mean algorithms.Necessary conditions of optimality are proved.Convergence of MMM-algorithm is shown.Basic statistical and geometrical properties are shown. Anomaly detection is an important problem with multiple applications, and thus has been studied for decades in various research domains. In the past decade there has been a growing interest in anomaly detection in data represented as networks, or graphs, largely because of their robust expressiveness and their natural ability to represent complex relationships. Originally, techniques focused on anomaly detection in static graphs, which do not change and are capable of representing only a single snapshot of data. As real-world networks are constantly changing, there has been a shift in focus to dynamic graphs, which evolve over time. The power of any kind of network approach lies in the ability to simplify a complex system so that one can better understand its function as a whole. Sometimes it is beneficial, however, to include more information than in a simple graph of only nodes and links. Adding information about times of interactions can make predictions and mechanistic understanding more accurate. The drawback, however, is that there are not so many methods available, partly because temporal networks is a relatively young field, partly because it is more difficult to develop such methods compared to for static networks. In this colloquium, we review the methods to analyze and model temporal networks and processes taking place on them, focusing mainly on the last three years. This includes the spreading of infectious disease, opinions, rumors, in social networks; information packets in computer networks; various types of signaling in biology, and more. We also discuss future directions. Detecting anomalies in data is a vital task, with numerous high-impact applications in areas such as security, finance, health care, and law enforcement. While numerous techniques have been developed in past years for spotting outliers and anomalies in unstructured collections of multi-dimensional points, with graph data becoming ubiquitous, techniques for structured graph data have been of focus recently. As objects in graphs have long-range correlations, a suite of novel technology has been developed for anomaly detection in graph data. This survey aims to provide a general, comprehensive, and structured overview of the state-of-the-art methods for anomaly detection in data represented as graphs. As a key contribution, we give a general framework for the algorithms categorized under various settings: unsupervised versus (semi-)supervised approaches, for static versus dynamic graphs, for attributed versus plain graphs. We highlight the effectiveness, scalability, generality, and robustness aspects of the methods. What is more, we stress the importance of anomaly attribution and highlight the major techniques that facilitate digging out the root cause, or the why', of the detected anomalies for further analysis and sense-making. Finally, we present several real-world applications of graph-based anomaly detection in diverse domains, including financial, auction, computer traffic, and social networks. We conclude our survey with a discussion on open theoretical and practical challenges in the field.