Having insight into the causal associations in a complex system facilitates decision making, e.g., for medical treatments, urban infrastructure improvements or financial investments. The amount of observational data grows, which enables the discovery of causal relationships between variables from observation of their behaviour in time. Existing methods for causal discovery from time series data do not yet exploit the representational power of deep learning. We therefore present the Temporal Causal Discovery Framework (TCDF), a deep learning framework that learns a causal graph structure by discovering causal relationships in observational time series data. TCDF uses attention-based convolutional neural networks combined with a causal validation step. By interpreting the internal parameters of the convolutional networks, TCDF can also discover the time delay between a cause and the occurrence of its effect. Our framework learns temporal causal graphs, which can include confounders and instantaneous effects. Experiments on financial and neuroscientific benchmarks show state-of-the-art performance of TCDF on discovering causal relationships in continuous time series data. Furthermore, we show that TCDF can circumstantially discover the presence of hidden confounders. Our broadly applicable framework can be used to gain novel insights into the causal dependencies in a complex system, which is important for reliable predictions, knowledge discovery and data-driven decision making.

Identifying important nodes for disease spreading is a central topic in network epidemiology. We investigate how well the position of a node, characterized by standard network measures, can predict its epidemiological importance in any graph of a given number of nodes. This is in contrast to other studies that deal with the easier prediction problem of ranking nodes by their epidemic importance in given graphs. As a benchmark for epidemic importance, we calculate the exact expected outbreak size given a node as the source. We study exhaustively all graphs of a given size, so do not restrict ourselves to certain generative models for graphs, nor to graph data sets. Due to the large number of possible nonisomorphic graphs of a fixed size, we are limited to ten-node graphs. We find that combinations of two or more centralities are predictive (R² scores of 0.91 or higher) even for the most difficult parameter values of the epidemic simulation. Typically, these successful combinations include one normalized spectral centrality (such as PageRank or Katz centrality) and one measure that is sensitive to the number of edges in the graph. A central challenge in network epidemiology is to find nodes that are important for disease spreading. Usually, one starts from a certain graph, and tries to rank the nodes in a way that correlates as strongly as possible with measures of importance estimated using simulations of outbreaks. A more challenging prediction task, and the one we take, is to ask if one can guess, from measures of the network structure alone, the values of quantities describing the outbreak. Having this predictive power is important: one can then target nodes that are more important than a certain threshold, rather than just a top fraction of nodes. By exhaustively studying all small graphs, we show that such prediction is possible to achieve with high accuracy by combining standard network measures.

Information flow, opinion, and epidemics spread over structured networks. When using node centrality indicators to predict which nodes will be among the top influencers or superspreaders, no single centrality is a consistently good ranker across networks. We show that statistical classifiers using two or more centralities are instead consistently predictive over many diverse, static real-world topologies. Certain pairs of centralities cooperate particularly well in drawing the statistical boundary between the superspreaders and the rest: a local centrality measuring the size of a node’s neighbourhood gains from the addition of a global centrality such as the eigenvector centrality, closeness, or the core number. Intuitively, this is because a local centrality may rank highly nodes which are located in locally dense, but globally peripheral regions of the network. The additional global centrality indicator guides the prediction towards more central regions. The superspreaders usually jointly maximise the values of both centralities. As a result of the interplay between centrality indicators, training classifiers with seven classical indicators leads to a nearly maximum average precision function (0.995) across the networks in this study.

Data measured and collected from embedded sensors often contains faults, i.e., data points which are not an accurate representation of the physical phenomenon monitored by the sensor. These data faults may be caused by deployment conditions outside the operational bounds for the node, and short- or long-term hardware, software, or communication problems. On the other hand, the applications will expect accurate sensor data, and recent literature proposes algorithmic solutions for the fault detection and classification in sensor data. In order to evaluate the performance of such solutions, however, the field lacks a set of \\emph{benchmark sensor datasets}. A benchmark dataset ideally satisfies the following criteria: (a) it is based on real-world raw sensor data from various types of sensor deployments; (b) it contains (natural or artificially injected) faulty data points reflecting various problems in the deployment, including missing data points; and (c) all data points are annotated with the \\emph{ground truth}, i.e., whether or not the data point is accurate, and, if faulty, the type of fault. We prepare and publish three such benchmark datasets, together with the algorithmic methods used to create them: a dataset of 280 temperature and light subsets of data from 10 indoor \\emph{Intel Lab} sensors, a dataset of 140 subsets of outdoor temperature data from SensorScope sensors, and a dataset of 224 subsets of outdoor temperature data from 16 \\emph{Smart Santander} sensors. The three benchmark datasets total 5.783.504 data points, containing injected data faults of the following types known from the literature: random, malfunction, bias, drift, polynomial drift, and combinations. We present algorithmic procedures and a software tool for preparing further such benchmark datasets.