Many dynamic networks coming from real-world contexts are link streams, i.e. a finite collection of triplets @math where @math and @math are two nodes having a link between them at time @math . A very large number of studies on these objects start by aggregating the data in disjoint time windows of length @math in order to obtain a series of graphs on which are made all subsequent analyses. Here we are concerned with the impact of the chosen @math on the obtained graph series. We address the fundamental question of knowing whether a series of graphs formed using a given @math faithfully describes the original link stream. We answer the question by showing that such dynamic networks exhibit a threshold for @math , which we call the , beyond which the properties of propagation of the link stream are altered, while they are mostly preserved before. We design an automatic method to determine the saturation scale of any link stream, which we apply and validate on several real-world datasets.
@cite_22 also aims at determining an appropriate time scale for aggregating a link stream into a graph series. Their method does not take into account the loss of information but is instead based on the modes of periodicity and on the self-similarity of the time series of some properties of the snapshots. They observe that the offset time for which the self similarity of these time series is zero is close to half of the period of the highest frequency visible in their spectra, which is the aggregation period suggested as a result of their method. Though this provides a very relevant time scale for analyzing dynamic networks, its meaning is different from the meaning of the saturation scale we are looking for in this paper. Indeed, an important part of the activity of dynamic networks takes place at time scales much smaller than their modes of periodicity. Therefore, using such periods for aggregation usually induces an important loss of information, which we aim at avoiding here. Let us mention that a similar approach based on modes of periodicity of some time series associated to the network was previously used in @cite_6 @cite_31 .
@cite_31 @cite_22 @cite_6
2143554828 1782565250 2107144529
Many dynamic networks coming from real-world contexts are link streams, i.e. a finite collection of triplets (u,v,t) where u and v are two nodes having a link between them at time t. A great number of studies on these objects start by aggregating the data on disjoint time windows of length Δ in order to obtain a series of graphs on which are made all subsequent analyses. Here we are concerned with the impact of the chosen Δ on the obtained graph series. We address the fundamental question of knowing whether a series of graphs formed using a given Δ faithfully describes the original link stream. We answer the question by showing that such dynamic networks exhibit a threshold for Δ, which we call the saturation scale, beyond which the properties of propagation of the link stream are altered, while they are mostly preserved before. We design an automatic method to determine the saturation scale of any link stream, which we apply and validate on several real-world datasets. Catalogs of periodic variable stars contain large numbers of periodic light-curves (photometric time series data from the astrophysics domain). Separating anomalous objects from well-known classes is an important step towards the discovery of new classes of astronomical objects. Most anomaly detection methods for time series data assume either a single continuous time series or a set of time series whose periods are aligned. Light-curve data precludes the use of these methods as the periods of any given pair of light-curves may be out of sync. One may use an existing anomaly detection method if, prior to similarity calculation, one performs the costly act of aligning two light-curves, an operation that scales poorly to massive data sets. This paper presents PCAD, an unsupervised anomaly detection method for large sets of unsynchronized periodic time-series data, that outputs a ranked list of both global and local anomalies. It calculates its anomaly score for each light-curve in relation to a set of centroids produced by a modified k-means clustering algorithm. Our method is able to scale to large data sets through the use of sampling. We validate our method on both light-curve data and other time series data sets. We demonstrate its effectiveness at finding known anomalies, and discuss the effect of sample size and number of centroids on our results. We compare our method to naive solutions and existing time series anomaly detection methods for unphased data, and show that PCAD's reported anomalies are comparable to or better than all other methods. Finally, astrophysicists on our team have verified that PCAD finds true anomalies that might be indicative of novel astrophysical phenomena. The presence of cyclic expansions and contractions in the economy has been known for over a century. The work reported here searches for similar cyclic behavior in stock valuations. The variations are subtle and can only be extracted through analysis of price variations of a large number of stocks. Koopman mode analysis is a natural approach to establish such collective oscillatory behavior. The difficulty is that even non-cyclic and stochastic constituents of a finite data set may be interpreted as a sum of periodic motions. However, deconvolution of these irregular dynamical facets may be expected to be non-robust, i.e., to depend on specific data set. We propose an approach to differentiate robust and non-robust features in a time series; it is based on identifying robust features with reproducible Koopman modes, i.e., those that persist between distinct sub-groupings of the data. Our analysis of stock data discovered four reproducible modes, one of which has period close to the number of trading days year. To the best of our knowledge these cycles were not reported previously. It is particularly interesting that the cyclic behaviors persisted through the great recession even though phase relationships between stocks within the modes evolved in the intervening period. We develop a test of the null hypothesis that an observed time series is a realization of a strictly stationary random process. Our test is based on the result that the kth value of the discrete Fourier transform of a sample frame has a zero mean under the null hypothesis. The test that we develop will have considerable power against an important form of nonstationarity hitherto not considered in the mainstream econometric time-series literature, that is, where the mean of a time series is periodic with random variation in its periodic structure. The size and power properties of the test are: investigated and its applicability to real-world problems is demonstrated by application to three: economic data sets.