Detecting the community structure and activity patterns of temporal networks: a non-negative tensor factorization approach

The increasing availability of temporal network data is calling for more research on extracting and characterizing mesoscopic structures in temporal networks and on relating such structure to specific functions or properties of the system. An outstanding challenge is the extension of the results achieved for static networks to time-varying networks, where the topological structure of the system and the temporal activity patterns of its components are intertwined. Here we investigate the use of a latent factor decomposition technique, non-negative tensor factorization, to extract the community-activity structure of temporal networks. The method is intrinsically temporal and allows to simultaneously identify communities and to track their activity over time. We represent the time-varying adjacency matrix of a temporal network as a three-way tensor and approximate this tensor as a sum of terms that can be interpreted as communities of nodes with an associated activity time series. We summarize known computational techniques for tensor decomposition and discuss some quality metrics that can be used to tune the complexity of the factorized representation. We subsequently apply tensor factorization to a temporal network for which a ground truth is available for both the community structure and the temporal activity patterns. The data we use describe the social interactions of students in a school, the associations between students and school classes, and the spatio-temporal trajectories of students over time. We show that non-negative tensor factorization is capable of recovering the class structure with high accuracy. In particular, the extracted tensor components can be validated either as known school classes, or in terms of correlated activity patterns, i.e., of spatial and temporal coincidences that are determined by the known school activity schedule

Mitigation of infectious disease at school: targeted class closure vs school closure

School environments are thought to play an important role in the community spread of airborne infections (e.g., influenza) because of the high mixing rates of school children. The closure of schools has therefore been proposed as efficient mitigation strategy, with however high social and economic costs: alternative, less disruptive interventions are highly desirable. The recent availability of high-resolution contact networks in school environments provides an opportunity to design micro-interventions and compare the outcomes of alternative mitigation measures. We consider mitigation measures that involve the targeted closure of school classes or grades based on readily available information such as the number of symptomatic infectious children in a class. We focus on the case of a primary school for which we have high-resolution data on the close-range interactions of children and teachers. We simulate the spread of an influenza-like illness in this population by using an SEIR model with asymptomatics and compare the outcomes of different mitigation strategies. We find that targeted class closure affords strong mitigation effects: closing a class for a fixed period of time -equal to the sum of the average infectious and latent durations- whenever two infectious individuals are detected in that class decreases the attack rate by almost 70% and strongly decreases the probability of a severe outbreak. The closure of all classes of the same grade mitigates the spread almost as much as closing the whole school. Targeted class closure strategies based on readily available information on symptomatic subjects and on limited information on mixing patterns, such as the grade structure of the school, can be almost as effective as whole-school closure, at a much lower cost. This may inform public health policies for the management and mitigation of influenza-like outbreaks in the community

Activity clocks: spreading dynamics on temporal networks of human contact

Dynamical processes on time-varying complex networks are key to understanding and modeling a broad variety of processes in socio-technical systems. Here we focus on empirical temporal networks of human proximity and we aim at understanding the factors that, in simulation, shape the arrival time distribution of simple spreading processes. Abandoning the notion of wall-clock time in favour of node-specific clocks based on activity exposes robust statistical patterns in the arrival times across different social contexts. Using randomization strategies and generative models constrained by data, we show that these patterns can be understood in terms of heterogeneous inter-event time distributions coupled with heterogeneous numbers of events per edge. We also show, both empirically and by using a synthetic dataset, that significant deviations from the above behavior can be caused by the presence of edge classes with strong activity correlations

Estimating the outcome of spreading processes on networks with incomplete information: a mesoscale approach

Recent advances in data collection have facilitated the access to time-resolved human proximity data that can conveniently be represented as temporal networks of contacts between individuals. While this type of data is fundamental to investigate how information or diseases propagate in a population, it often suffers from incompleteness, which possibly leads to biased conclusions. A major challenge is thus to estimate the outcome of spreading processes occurring on temporal networks built from partial information. To cope with this problem, we devise an approach based on Non-negative Tensor Factorization (NTF) -- a dimensionality reduction technique from multi-linear algebra. The key idea is to learn a low-dimensional representation of the temporal network built from partial information, to adapt it to take into account temporal and structural heterogeneity properties known to be crucial for spreading processes occurring on networks, and to construct in this way a surrogate network similar to the complete original network. To test our method, we consider several human-proximity networks, on which we simulate a loss of data. Using our approach on the resulting partial networks, we build a surrogate version of the complete network for each. We then compare the outcome of a spreading process on the complete networks (non altered by a loss of data) and on the surrogate networks. We observe that the epidemic sizes obtained using the surrogate networks are in good agreement with those measured on the complete networks. Finally, we propose an extension of our framework when additional data sources are available to cope with the missing data problem

Compensating for population sampling in simulations of epidemic spread on temporal contact networks

Data describing human interactions often suffer from incomplete sampling of the underlying population. As a consequence, the study of contagion processes using data-driven models can lead to a severe underestimation of the epidemic risk. Here we present a systematic method to alleviate this issue and obtain a better estimation of the risk in the context of epidemic models informed by high-resolution time-resolved contact data. We consider several such data sets collected in various contexts and perform controlled resampling experiments. We show how the statistical information contained in the resampled data can be used to build a series of surrogate versions of the unknown contacts. We simulate epidemic processes on the resulting reconstructed data sets and show that it is possible to obtain good estimates of the outcome of simulations performed using the complete data set. We discuss limitations and potential improvements of our method

Predicting human mobility through the assimilation of social media traces into mobility models

Predicting human mobility flows at different spatial scales is challenged by the heterogeneity of individual trajectories and the multi-scale nature of transportation networks. As vast amounts of digital traces of human behaviour become available, an opportunity arises to improve mobility models by integrating into them proxy data on mobility collected by a variety of digital platforms and location-aware services. Here we propose a hybrid model of human mobility that integrates a large-scale publicly available dataset from a popular photo-sharing system with the classical gravity model, under a stacked regression procedure. We validate the performance and generalizability of our approach using two ground-truth datasets on air travel and daily commuting in the United States: using two different cross-validation schemes we show that the hybrid model affords enhanced mobility prediction at both spatial scales.Comment: 17 pages, 10 figure

Gender homophily from spatial behavior in a primary school: a sociometric study

We investigate gender homophily in the spatial proximity of children (6 to 12 years old) in a French primary school, using time-resolved data on face-to-face proximity recorded by means of wearable sensors. For strong ties, i.e., for pairs of children who interact more than a defined threshold, we find statistical evidence of gender preference that increases with grade. For weak ties, conversely, gender homophily is negatively correlated with grade for girls, and positively correlated with grade for boys. This different evolution with grade of weak and strong ties exposes a contrasted picture of gender homophily