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

Phase diagram of a Schelling segregation model

The collective behavior in a variant of Schelling's segregation model is characterized with methods borrowed from statistical physics, in a context where their relevance was not conspicuous. A measure of segregation based on cluster geometry is defined and several quantities analogous to those used to describe physical lattice models at equilibrium are introduced. This physical approach allows to distinguish quantitatively several regimes and to characterize the transitions between them, leading to the building of a phase diagram. Some of the transitions evoke empirical sudden ethnic turnovers. We also establish links with 'spin-1' models in physics. Our approach provides generic tools to analyze the dynamics of other socio-economic systems

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

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

Schelling segregation in an open city: a kinetically constrained Blume-Emery-Griffiths spin-1 system

In the 70's Schelling introduced a multi-agent model to describe the segregation dynamics that may occur with individuals having only weak preferences for 'similar' neighbors. Recently variants of this model have been discussed, in particular, with emphasis on the links with statistical physics models. Whereas these models consider a fixed number of agents moving on a lattice, here we present a version allowing for exchanges with an external reservoir of agents. The density of agents is controlled by a parameter which can be viewed as measuring the attractiveness of the city-lattice. This model is directly related to the zero-temperature dynamics of the Blume-Emery-Griffiths (BEG) spin-1 model, with kinetic constraints. With a varying vacancy density, the dynamics with agents making deterministic decisions leads to a new variety of "phases" whose main features are the characteristics of the interfaces between clusters of agents of different types. The domains of existence of each type of interface are obtained analytically as well as numerically. These interfaces may completely isolate the agents leading to another type of segregation as compared to what is observed in the original Schelling model, and we discuss its possible socio-economic correlates.Comment: 10 pages, 7 figures, final version accepted for publication in PR

Change points, memory and epidemic spreading in temporal networks

Dynamic networks exhibit temporal patterns that vary across different time scales, all of which can potentially affect processes that take place on the network. However, most data-driven approaches used to model time-varying networks attempt to capture only a single characteristic time scale in isolation --- typically associated with the short-time memory of a Markov chain or with long-time abrupt changes caused by external or systemic events. Here we propose a unified approach to model both aspects simultaneously, detecting short and long-time behaviors of temporal networks. We do so by developing an arbitrary-order mixed Markov model with change points, and using a nonparametric Bayesian formulation that allows the Markov order and the position of change points to be determined from data without overfitting. In addition, we evaluate the quality of the multiscale model in its capacity to reproduce the spreading of epidemics on the temporal network, and we show that describing multiple time scales simultaneously has a synergistic effect, where statistically significant features are uncovered that otherwise would remain hidden by treating each time scale independently.Comment: 9 pages, 6 figure

Wearable proximity sensors for monitoring a mass casualty incident exercise: a feasibility study

Over the past several decades, naturally occurring and man-made mass casualty incidents (MCI) have increased in frequency and number, worldwide. To test the impact of such event on medical resources, simulations can provide a safe, controlled setting while replicating the chaotic environment typical of an actual disaster. A standardised method to collect and analyse data from mass casualty exercises is needed, in order to assess preparedness and performance of the healthcare staff involved. We report on the use of wearable proximity sensors to measure proximity events during a MCI simulation. We investigated the interactions between medical staff and patients, to evaluate the time dedicated by the medical staff with respect to the severity of the injury of the victims depending on the roles. We estimated the presence of the patients in the different spaces of the field hospital, in order to study the patients' flow. Data were obtained and collected through the deployment of wearable proximity sensors during a mass casualty incident functional exercise. The scenario included two areas: the accident site and the Advanced Medical Post (AMP), and the exercise lasted 3 hours. A total of 238 participants simulating medical staff and victims were involved. Each participant wore a proximity sensor and 30 fixed devices were placed in the field hospital. The contact networks show a heterogeneous distribution of the cumulative time spent in proximity by participants. We obtained contact matrices based on cumulative time spent in proximity between victims and the rescuers. Our results showed that the time spent in proximity by the healthcare teams with the victims is related to the severity of the patient's injury. The analysis of patients' flow showed that the presence of patients in the rooms of the hospital is consistent with triage code and diagnosis, and no obvious bottlenecks were found