Modelling sequences and temporal networks with dynamic community structures

Network compression with configuration models and the minimum description length

Random network models, constrained to reproduce specific statistical features, are often used to represent and analyze network data and their mathematical descriptions. Chief among them, the configuration model constrains random networks by their degree distribution and is foundational to many areas of network science. However, configuration models and their variants are often selected based on intuition or mathematical and computational simplicity rather than on statistical evidence. To evaluate the quality of a network representation, we need to consider both the amount of information required to specify a random network model and the probability of recovering the original data when using the model as a generative process. To this end, we calculate the approximate size of network ensembles generated by the popular configuration model and its generalizations, including versions accounting for degree correlations and centrality layers. We then apply the minimum description length principle as a model selection criterion over the resulting nested family of configuration models. Using a dataset of over 100 networks from various domains, we find that the classic configuration model is generally preferred on networks with an average degree above 10, while a layered configuration model constrained by a centrality metric offers the most compact representation of the majority of sparse networks.

Higher-order correlations reveal complex memory in temporal hypergraphs

Many real-world complex systems are characterized by interactions in groups that change in time. Current temporal network approaches, however, are unable to describe group dynamics, as they are based on pairwise interactions only. Here, we use time-varying hypergraphs to describe such systems, and we introduce a framework based on higher-order correlations to characterize their temporal organization. The analysis of human interaction data reveals the existence of coherent and interdependent mesoscopic structures, thus capturing aggregation, fragmentation and nucleation processes in social systems. We introduce a model of temporal hypergraphs with non-Markovian group interactions, which reveals complex memory as a fundamental mechanism underlying the emerging pattern in the data.

Patterns in Temporal Networks with Higher-Order Egocentric Structures

The analysis of complex and time-evolving interactions, such as those within social dynamics, represents a current challenge in the science of complex systems. Temporal networks stand as a suitable tool for schematizing such systems, encoding all the interactions appearing between pairs of individuals in discrete time. Over the years, network science has developed many measures to analyze and compare temporal networks. Some of them imply a decomposition of the network into small pieces of interactions; i.e., only involving a few nodes for a short time range. Along this line, a possible way to decompose a network is to assume an egocentric perspective; i.e., to consider for each node the time evolution of its neighborhood. This was proposed by Longa et al. by defining the "egocentric temporal neighborhood", which has proven to be a useful tool for characterizing temporal networks relative to social interactions. However, this definition neglects group interactions (quite common in social domains), as they are always decomposed into pairwise connections. A more general framework that also allows considering larger interactions is represented by higher-order networks. Here, we generalize the description of social interactions to hypergraphs. Consequently, we generalize their decomposition into "hyper egocentric temporal neighborhoods". This enables the analysis of social interactions, facilitating comparisons between different datasets or nodes within a dataset, while considering the intrinsic complexity presented by higher-order interactions. Even if we limit the order of interactions to the second order (triplets of nodes), our results reveal the importance of a higher-order representation.In fact, our analyses show that second-order structures are responsible for the majority of the variability at all scales: between datasets, amongst nodes, and over time.

Compressing the Chronology of a Temporal Network with Graph Commutators

Studies of dynamics on temporal networks often represent the network as a series of "snapshots," static networks active for short durations of time. We argue that successive snapshots can be aggregated if doing so has little effect on the overlying dynamics. We propose a method to compress network chronologies by progressively combining pairs of snapshots whose matrix commutators have the smallest dynamical effect. We apply this method to epidemic modeling on real contact tracing data and find that it allows for significant compression while remaining faithful to the epidemic dynamics.

Hyperedge prediction and the statistical mechanisms of higher-order and lower-order interactions in complex networks

Complex networked systems often exhibit higher-order interactions, beyond dyadic interactions, which can dramatically alter their observed behavior. Consequently, understanding hypergraphs from a structural perspective has become increasingly important. Statistical, group-based inference approaches are well suited for unveiling the underlying community structure and predicting unobserved interactions. However, these approaches often rely on two key assumptions: that the same groups can explain hyperedges of any order and that interactions are assortative, meaning that edges are formed by nodes with the same group memberships. To test these assumptions, we propose a group-based generative model for hypergraphs that does not impose an assortative mechanism to explain observed higher-order interactions, unlike current approaches. Our model allows us to explore the validity of the assumptions. Our results indicate that the first assumption appears to hold true for real networks. However, the second assumption is not necessarily accurate; we find that a combination of general statistical mechanisms can explain observed hyperedges. Finally, with our approach, we are also able to determine the importance of lower and high-order interactions for predicting unobserved interactions. Our research challenges the conventional assumptions of group-based inference methodologies and broadens our understanding of the underlying structure of hypergraphs.

Bayesian inference of transition matrices from incomplete graph data with a topological prior

Many network analysis and graph learning techniques are based on discrete- or continuous-time models of random walks. To apply these methods, it is necessary to infer transition matrices that formalize the underlying stochastic process in an observed graph. For weighted graphs, where weighted edges capture observations of repeated interactions between nodes, it is common to estimate the entries of such transition matrices based on the (relative) weights of edges. However in real-world settings we are often confronted with incomplete data, which turns the construction of the transition matrix based on a weighted graph into an inference problem. Moreover, we often have access to additional information, which capture topological constraints of the system, i.e. which edges in a weighted graph are (theoretically) possible and which are not. Examples include transportation networks, where we may have access to a small sample of passenger trajectories as well as the physical topology of connections, or a limited set of observed social interactions with additional information on the underlying social structure. Combining these two different sources of information to reliably infer transition matrices from incomplete data on repeated interactions is an important open challenge, with severe implications for the reliability of downstream network analysis tasks. Addressing this issue, we show that including knowledge on such topological constraints can considerably improve the inference of transition matrices, especially in situations where we only have a small number of observed interactions. To this end, we derive an analytically tractable Bayesian method that uses repeated interactions and a topological prior to perform data-efficient inference of transition matrices. We compare our approach against commonly used frequentist and Bayesian approaches both in synthetic data and in five real-world datasets, and we find that our method recovers the transition probabilities with higher accuracy. Furthermore, we demonstrate that the method is robust even in cases when the knowledge of the topological constraint is partial. Lastly, we show that this higher accuracy improves the results for downstream network analysis tasks like cluster detection and node ranking, which highlights the practical relevance of our method for interdisciplinary data-driven analyses of networked systems.

Locating community smells in software development processes using higher-order network centralities

Community smells are negative patterns in software development teams' interactions that impede their ability to successfully create software. Examples are team members working in isolation, lack of communication and collaboration across departments or sub-teams, or areas of the codebase where only a few team members can work on. Current approaches aim to detect community smells by analysing static network representations of software teams' interaction structures. In doing so, they are insufficient to locate community smells within development processes. Extending beyond the capabilities of traditional social network analysis, we show that higher-order network models provide a robust means of revealing such hidden patterns and complex relationships. To this end, we develop a set of centrality measures based on the MOGen higher-order network model and show their effectiveness in predicting influential nodes using five empirical datasets. We then employ these measures for a comprehensive analysis of a product team at the German IT security company genua GmbH, showcasing our method's success in identifying and locating community smells. Specifically, we uncover critical community smells in two areas of the team's development process. Semi-structured interviews with five team members validate our findings: while the team was aware of one community smell and employed measures to address it, it was not aware of the second. This highlights the potential of our approach as a robust tool for identifying and addressing community smells in software development teams. More generally, our work contributes to the social network analysis field with a powerful set of higher-order network centralities that effectively capture community dynamics and indirect relationships.

Predicting variable-length paths in networked systems using multi-order generative models

Apart from nodes and links, for many networked systems, we have access to data on paths, i.e., collections of temporally ordered variable-length node sequences that are constrained by the system's topology. Understanding the patterns in such data is key to advancing our understanding of the structure and dynamics of complex systems. Moreover, the ability to accurately model and predict paths is important for engineered systems, e.g., to optimise supply chains or provide smart mobility services. Here, we introduce MOGen, a generative modelling framework that enables both next-element and out-of-sample prediction in paths with high accuracy and consistency. It features a model selection approach that automatically determines the optimal model directly from data, effectively making MOGen parameter-free. Using empirical data, we show that our method outperforms state-of-the-art sequence modelling techniques. We further introduce a mathematical formalism that links higher-order models of paths to transition matrices of random walks in multi-layer networks.

Understanding Hazardous Materials Transportation Accidents Based on Higher-Order Network Theory

In hazardous materials transportation systems, accident causation analysis is important to transportation safety. Complex network theory can be effectively used to understand the causal factors of and their relationships within accidents. In this paper, a higher-order network method is proposed to establish a hazardous materials transportation accident causation network (HMTACN), which considers the sequences and dependences of causal factors. The HMTACN is composed of 125 first- and 118 higher-order nodes that represent causes, and 545 directed edges that denote complex relationships among causes. By analyzing topological properties, the results show that the HMTACN has the characteristics of small-world networks and displays the properties of scale-free networks. Additionally, critical causal factors and key relationships of the HMTACN are discovered. Moreover, unsafe tank or valve states are important causal factors; and leakage, roll-over, collision, and fire are most likely to trigger chain reactions. Important higher-order nodes are discovered, which can represent key relationships in the HMTACN. For example, unsafe distance and improper operation usually lead to collision and roll-over. These results of higher-order nodes cannot be found by the traditional Markov network model. This study provides a practical way to extract and construct an accident causation network from numerous accident investigation reports. It also provides insights into safety management of hazardous materials transportation.

Flow stability for dynamic community detection

Many systems exhibit complex temporal dynamics due to the presence of different processes taking place simultaneously. An important task in these systems is to extract a simplified view of their time-dependent network of interactions. Community detection in temporal networks usually relies on aggregation over time windows or consider sequences of different stationary epochs. For dynamics-based methods, attempts to generalize static-network methodologies also face the fundamental difficulty that a stationary state of the dynamics does not always exist. Here, we derive a method based on a dynamical process evolving on the temporal network. Our method allows dynamics that do not reach a steady state and uncovers two sets of communities for a given time interval that accounts for the ordering of edges in forward and backward time. We show that our method provides a natural way to disentangle the different dynamical scales present in a system with synthetic and real-world examples.

Hidden dependence of spreading vulnerability on topological complexity

Many dynamical phenomena in complex systems concern spreading that plays out on top of networks with changing architecture over time-commonly known as temporal networks. A complex system's proneness to facilitate spreading phenomena, which we abbreviate as its "spreading vulnerability," is often surmised to be related to the topology of the temporal network featured by the system. Yet, cleanly extracting spreading vulnerability of a complex system directly from the topological information of the temporal network remains a challenge. Here, using data from a diverse set of real-world complex systems, we develop the "entropy of temporal entanglement" as a quantity to measure topological complexities of temporal networks. We show that this parameter-free quantity naturally allows for topological comparisons across vastly different complex systems. Importantly, by simulating three different types of stochastic dynamical processes playing out on top of temporal networks, we demonstrate that the entropy of temporal entanglement serves as a quantitative embodiment of the systems' spreading vulnerability, irrespective of the details of the processes. In being able to do so, i.e., in being able to quantitatively extract a complex system's proneness to facilitate spreading phenomena from topology, this entropic measure opens itself for applications in a wide variety of natural, social, biological, and engineered systems.

Quantifying agent impacts on contact sequences in social interactions

Human social behavior plays a crucial role in how pathogens like SARS-CoV-2 or fake news spread in a population. Social interactions determine the contact network among individuals, while spreading, requiring individual-to-individual transmission, takes place on top of the network. Studying the topological aspects of a contact network, therefore, not only has the potential of leading to valuable insights into how the behavior of individuals impacts spreading phenomena, but it may also open up possibilities for devising effective behavioral interventions. Because of the temporal nature of interactions—since the topology of the network, containing who is in contact with whom, when, for how long, and in which precise sequence, varies (rapidly) in time—analyzing them requires developing network methods and metrics that respect temporal variability, in contrast to those developed for static (i.e., time-invariant) networks. Here, by means of event mapping, we propose a method to quantify how quickly agents mingle by transforming temporal network data of agent contacts. We define a novel measure called contact sequence centrality, which quantifies the impact of an individual on the contact sequences, reflecting the individual’s behavioral potential for spreading. Comparing contact sequence centrality across agents allows for ranking the impact of agents and identifying potential ‘behavioral super-spreaders’. The method is applied to social interaction data collected at an art fair in Amsterdam. We relate the measure to the existing network metrics, both temporal and static, and find that (mostly at longer time scales) traditional metrics lose their resemblance to contact sequence centrality. Our work highlights the importance of accounting for the sequential nature of contacts when analyzing social interactions.

Non-Markovian temporal networks with auto- and cross-correlated link dynamics

Many of the biological, social and man-made networks around us are inherently dynamic, with their links switching on and off over time. The evolution of these networks is often observed to be non-Markovian, and the dynamics of their links are often correlated. Hence, to accurately model these networks, predict their evolution, and understand how information and other relevant quantities propagate over them, the inclusion of both memory and dynamical dependencies between links is key. In this article we introduce a general class of models of temporal networks based on discrete autoregressive processes for link dynamics. As a concrete and useful case study, we then concentrate on a specific model within this class, which allows to generate temporal networks with a specified underlying structural backbone, and with precise control over the dynamical dependencies between links and the strength and length of their memories. In this network model the presence of each link is influenced not only by its past activity, but also by the past activities of other links, as specified by a coupling matrix, which directly controls the causal relations, and hence the correlations, among links. We propose a maximum likelihood method for estimating the model's parameters from data, showing how the model allows a more realistic description of real-world temporal networks and also to predict their evolution. Due to the flexibility of maximum likelihood inference, we illustrate how to deal with heterogeneity and time-varying patterns, possibly including also nonstationary network dynamics. We then use our network model to investigate the role that, both the features of memory and the type of correlations in the dynamics of links have on the properties of processes occurring over a temporal network. Namely, we study the speed of a spreading process, as measured by the time it takes for diffusion to reach equilibrium. Through both numerical simulations and analytical results, we are able to separate the roles of autocorrelations and neighborhood correlations in link dynamics, showing that not only is the speed of diffusion nonmonotonically dependent on the memory length, but also that correlations among neighboring links help to speed up the spreading process, while autocorrelations slow it back down. Our results have implications in the study of opinion formation, the modeling of social networks, and the spreading of epidemics through mobile populations.

The shape of memory in temporal networks

How to best define, detect and characterize network memory, i.e. the dependence of a network’s structure on its past, is currently a matter of debate. Here we show that the memory of a temporal network is inherently multidimensional, and we introduce a mathematical framework for defining and efficiently estimating the microscopic shape of memory, which characterises how the activity of each link intertwines with the activities of all other links. We validate our methodology on a range of synthetic models, and we then study the memory shape of real-world temporal networks spanning social, technological and biological systems, finding that these networks display heterogeneous memory shapes. In particular, online and offline social networks are markedly different, with the latter showing richer memory and memory scales. Our theory also elucidates the phenomenon of emergent virtual loops and provides a novel methodology for exploring the dynamically rich structure of complex systems.

The evolution of networks with structure changing in time is dependent on their past states and relevant to diffusion and spreading processes. The authors show that temporal network’s memory is described by multidimensional patterns at a microscopic scale, and cannot be reduced to a scalar quantity.

Analyzing spatial mobility patterns with time-varying graphical lasso: Application to COVID-19 spread

This work applies the time-varying graphical lasso (TVGL) method, an extension of the traditional graphical lasso approach, to address learning time-varying graphs from spatiotemporal measurements. Given georeferenced data, the TVGL method can estimate a time-varying network where an edge represents a partial correlation between two nodes. To achieve this, we use a COVID-19 data set from the Argentine province of Chaco. As an application, we use the estimated network to study the impact of COVID-19 confinement measures and evaluate whether the measures produced the expected result.

State Aggregations in Markov Chains and Block Models of Networks

We consider state-aggregation schemes for Markov chains from an information-theoretic perspective. Specifically, we consider aggregating the states of a Markov chain such that the mutual information of the aggregated states separated by T time steps is maximized. We show that for T=1 this recovers the maximum-likelihood estimator of the degree-corrected stochastic block model as a particular case, which enables us to explain certain features of the likelihood landscape of this generative network model from a dynamical lens. We further highlight how we can uncover coherent, long-range dynamical modules for which considering a timescale T≫1 is essential. We demonstrate our results using synthetic flows and real-world ocean currents, where we are able to recover the fundamental features of the surface currents of the oceans.

Dynamic hidden-variable network models

Models of complex networks often incorporate node-intrinsic properties abstracted as hidden variables. The probability of connections in the network is then a function of these variables. Real-world networks evolve over time and many exhibit dynamics of node characteristics as well as of linking structure. Here we introduce and study natural temporal extensions of static hidden-variable network models with stochastic dynamics of hidden variables and links. The dynamics is controlled by two parameters: one that tunes the rate of change of hidden variables and another that tunes the rate at which node pairs reevaluate their connections given the current values of hidden variables. Snapshots of networks in the dynamic models are equivalent to networks generated by the static models only if the link reevaluation rate is sufficiently larger than the rate of hidden-variable dynamics or if an additional mechanism is added whereby links actively respond to changes in hidden variables. Otherwise, links are out of equilibrium with respect to hidden variables and network snapshots exhibit structural deviations from the static models. We examine the level of structural persistence in the considered models and quantify deviations from staticlike behavior. We explore temporal versions of popular static models with community structure, latent geometry, and degree heterogeneity. While we do not attempt to directly model real networks, we comment on interesting qualitative resemblances to real systems. In particular, we speculate that links in some real networks are out of equilibrium with respect to hidden variables, partially explaining the presence of long-ranged links in geometrically embedded systems and intergroup connectivity in modular systems. We also discuss possible extensions, generalizations, and applications of the introduced class of dynamic network models.

Impact of environmental changes on the dynamics of temporal networks

Dynamics of complex social systems has often been described in the framework of temporal networks, where links are considered to exist only at the moment of interaction between nodes. Such interaction patterns are not only driven by internal interaction mechanisms, but also affected by environmental changes. To investigate the impact of the environmental changes on the dynamics of temporal networks, we analyze several face-to-face interaction datasets using the multiscale entropy (MSE) method to find that the observed temporal correlations can be categorized according to the environmental similarity of datasets such as classes and break times in schools. By devising and studying a temporal network model considering a periodically changing environment as well as a preferential activation mechanism, we numerically show that our model could successfully reproduce various empirical results by the MSE method in terms of multiscale temporal correlations. Our results demonstrate that the environmental changes can play an important role in shaping the dynamics of temporal networks when the interactions between nodes are influenced by the environment of the systems.

A Mixture Model of Truncated Zeta Distributions with Applications to Scientific Collaboration Networks

The degree distribution has attracted considerable attention from network scientists in the last few decades to have knowledge of the topological structure of networks. It is widely acknowledged that many real networks have power-law degree distributions. However, the deviation from such a behavior often appears when the range of degrees is small. Even worse, the conventional employment of the continuous power-law distribution usually causes an inaccurate inference as the degree should be discrete-valued. To remedy these obstacles, we propose a finite mixture model of truncated zeta distributions for a broad range of degrees that disobeys a power-law behavior in the range of small degrees while maintaining the scale-free behavior. The maximum likelihood algorithm alongside the model selection method is presented to estimate model parameters and the number of mixture components. The validity of the suggested algorithm is evidenced by Monte Carlo simulations. We apply our method to five disciplines of scientific collaboration networks with remarkable interpretations. The proposed model outperforms the other alternatives in terms of the goodness-of-fit.

Inference of edge correlations in multilayer networks

Many recent developments in network analysis have focused on multilayer networks, which one can use to encode time-dependent interactions, multiple types of interactions, and other complications that arise in complex systems. Like their monolayer counterparts, multilayer networks in applications often have mesoscale features, such as community structure. A prominent approach for inferring such structures is the employment of multilayer stochastic block models (SBMs). A common (but potentially inadequate) assumption of these models is the sampling of edges in different layers independently, conditioned on the community labels of the nodes. In this paper, we relax this assumption of independence by incorporating edge correlations into an SBM-like model. We derive maximum-likelihood estimates of the key parameters of our model, and we propose a measure of layer correlation that reflects the similarity between the connectivity patterns in different layers. Finally, we explain how to use correlated models for edge "prediction" (i.e., inference) in multilayer networks. By incorporating edge correlations, we find that prediction accuracy improves both in synthetic networks and in a temporal network of shoppers who are connected to previously purchased grocery products.

Dynamics of social network emergence explain network evolution

Networked systems emerge and subsequently evolve. Although several models describe the process of network evolution, researchers know far less about the initial process of network emergence. Here, we report temporal survey results of a real-world social network starting from its point of inception. We find that individuals' ties undergo an initial cycle of rapid expansion and contraction. This process helps to explain the eventual interactions and working structure in the network (in this case, scientific collaboration). We propose a stylized concept and model of "churn" to describe the process of network emergence and stabilization. Our empirical and simulation results suggest that these network emergence dynamics may be instrumental for explaining network details, as well as behavioral outcomes at later time periods.

Predictability of real temporal networks

Links in most real networks often change over time. Such temporality of links encodes the ordering and causality of interactions between nodes and has a profound effect on network dynamics and function. Empirical evidence has shown that the temporal nature of links in many real-world networks is not random. Nonetheless, it is challenging to predict temporal link patterns while considering the entanglement between topological and temporal link patterns. Here, we propose an entropy-rate-based framework, based on combined topological–temporal regularities, for quantifying the predictability of any temporal network. We apply our framework on various model networks, demonstrating that it indeed captures the intrinsic topological–temporal regularities whereas previous methods considered only temporal aspects. We also apply our framework on 18 real networks of different types and determine their predictability. Interestingly, we find that, for most real temporal networks, despite the greater complexity of predictability brought by the increase in dimension, the combined topological–temporal predictability is higher than the temporal predictability. Our results demonstrate the necessity for incorporating both temporal and topological aspects of networks in order to improve predictions of dynamical processes.

weg2vec: Event embedding for temporal networks

Network embedding techniques are powerful to capture structural regularities in networks and to identify similarities between their local fabrics. However, conventional network embedding models are developed for static structures, commonly consider nodes only and they are seriously challenged when the network is varying in time. Temporal networks may provide an advantage in the description of real systems, but they code more complex information, which could be effectively represented only by a handful of methods so far. Here, we propose a new method of event embedding of temporal networks, called weg2vec, which builds on temporal and structural similarities of events to learn a low dimensional representation of a temporal network. This projection successfully captures latent structures and similarities between events involving different nodes at different times and provides ways to predict the final outcome of spreading processes unfolding on the temporal structure.

Loan maturity aggregation in interbank lending networks obscures mesoscale structure and economic functions

Since the 2007-2009 financial crisis, substantial academic effort has been dedicated to improving our understanding of interbank lending networks (ILNs). Because of data limitations or by choice, the literature largely lacks multiple loan maturities. We employ a complete interbank loan contract dataset to investigate whether maturity details are informative of the network structure. Applying the layered stochastic block model of Peixoto (2015) and other tools from network science on a time series of bilateral loans with multiple maturity layers in the Russian ILN, we find that collapsing all such layers consistently obscures mesoscale structure. The optimal maturity granularity lies between completely collapsing and completely separating the maturity layers and depends on the development phase of the interbank market, with a more developed market requiring more layers for optimal description. Closer inspection of the inferred maturity bins associated with the optimal maturity granularity reveals specific economic functions, from liquidity intermediation to financing. Collapsing a network with multiple underlying maturity layers or extracting one such layer, common in economic research, is therefore not only an incomplete representation of the ILN's mesoscale structure, but also conceals existing economic functions. This holds important insights and opportunities for theoretical and empirical studies on interbank market functioning, contagion, stability, and on the desirable level of regulatory data disclosure.

Community discovering in temporal network with spectral fusion

With the deep understanding of the time-varying characteristics of real systems, research studies focusing on the temporal network spring up like mushrooms. Community detection is an accompanying and meaningful problem in the temporal network, but the analysis of this problem is still in its developing stage. In this paper, we proposed a temporal spectral clustering method to detect the invariable communities in the temporal network. Through integrating Fiedler's eigenvectors of normalized Laplacian matrices within a limited time window, our method can avoid the inaccurate partition caused by the mutation of the temporal network. Experiments demonstrated that our model is effective in solving this problem and performs obviously better than the compared methods. The results illustrated that taking the historical information of the network structure into consideration is beneficial in clustering the temporal network.

From networks to optimal higher-order models of complex systems

Rich data is revealing that complex dependencies between the nodes of a network may escape models based on pairwise interactions. Higher-order network models go beyond these limitations, offering new perspectives for understanding complex systems.

Tensorial and bipartite block models for link prediction in layered networks and temporal networks

Many real-world complex systems are well represented as multilayer networks; predicting interactions in those systems is one of the most pressing problems in predictive network science. To address this challenge, we introduce two stochastic block models for multilayer and temporal networks; one of them uses nodes as its fundamental unit, whereas the other focuses on links. We also develop scalable algorithms for inferring the parameters of these models. Because our models describe all layers simultaneously, our approach takes full advantage of the information contained in the whole network when making predictions about any particular layer. We illustrate the potential of our approach by analyzing two empirical data sets: a temporal network of e-mail communications, and a network of drug interactions for treating different cancer types. We find that multilayer models consistently outperform their single-layer counterparts, but that the most predictive model depends on the data set under consideration; whereas the node-based model is more appropriate for predicting drug interactions, the link-based model is more appropriate for predicting e-mail communication.

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.

Foundations of Temporal Text Networks

Three fundamental elements to understand human information networks are the individuals (actors) in the network, the information they exchange, that is often observable online as text content (emails, social media posts, etc.), and the time when these exchanges happen. An extremely large amount of research has addressed some of these aspects either in isolation or as combinations of two of them. There are also more and more works studying systems where all three elements are present, but typically using ad hoc models and algorithms that cannot be easily transfered to other contexts. To address this heterogeneity, in this article we present a simple, expressive and extensible model for temporal text networks, that we claim can be used as a common ground across different types of networks and analysis tasks, and we show how simple procedures to produce views of the model allow the direct application of analysis methods already developed in other domains, from traditional data mining to multilayer network mining.

Convex skeletons of complex networks

A convex network can be defined as a network such that every connected induced subgraph includes all the shortest paths between its nodes. A fully convex network would therefore be a collection of cliques stitched together in a tree. In this paper, we study the largest high-convexity part of empirical networks obtained by removing the least number of edges, which we call a convex skeleton. A convex skeleton is a generalization of a network spanning tree in which each edge can be replaced by a clique of arbitrary size. We present different approaches for extracting convex skeletons and apply them to social collaboration and protein interactions networks, autonomous systems graphs and food webs. We show that the extracted convex skeletons retain the degree distribution, clustering, connectivity, distances, node position and also community structure, while making the shortest paths between the nodes largely unique. Moreover, in the Slovenian computer scientists coauthorship network, a convex skeleton retains the strongest ties between the authors, differently from a spanning tree or high-betweenness backbone and high-salience skeleton. A convex skeleton thus represents a simple definition of a network backbone with applications in coauthorship and other social collaboration networks.

Nonparametric weighted stochastic block models

We present a Bayesian formulation of weighted stochastic block models that can be used to infer the large-scale modular structure of weighted networks, including their hierarchical organization. Our method is nonparametric, and thus does not require the prior knowledge of the number of groups or other dimensions of the model, which are instead inferred from data. We give a comprehensive treatment of different kinds of edge weights (i.e., continuous or discrete, signed or unsigned, bounded or unbounded), as well as arbitrary weight transformations, and describe an unsupervised model selection approach to choose the best network description. We illustrate the application of our method to a variety of empirical weighted networks, such as global migrations, voting patterns in congress, and neural connections in the human brain.

Epidemic spreading in modular time-varying networks

We investigate the effects of modular and temporal connectivity patterns on epidemic spreading. To this end, we introduce and analytically characterise a model of time-varying networks with tunable modularity. Within this framework, we study the epidemic size of Susceptible-Infected-Recovered, SIR, models and the epidemic threshold of Susceptible-Infected-Susceptible, SIS, models. Interestingly, we find that while the presence of tightly connected clusters inhibits SIR processes, it speeds up SIS phenomena. In this case, we observe that modular structures induce a reduction of the threshold with respect to time-varying networks without communities. We confirm the theoretical results by means of extensive numerical simulations both on synthetic graphs as well as on a real modular and temporal network.

The many facets of community detection in complex networks

Community detection, the decomposition of a graph into essential building blocks, has been a core research topic in network science over the past years. Since a precise notion of what constitutes a community has remained evasive, community detection algorithms have often been compared on benchmark graphs with a particular form of assortative community structure and classified based on the mathematical techniques they employ. However, this comparison can be misleading because apparent similarities in their mathematical machinery can disguise different goals and reasons for why we want to employ community detection in the first place. Here we provide a focused review of these different motivations that underpin community detection. This problem-driven classification is useful in applied network science, where it is important to select an appropriate algorithm for the given purpose. Moreover, highlighting the different facets of community detection also delineates the many lines of research and points out open directions and avenues for future research.

Using higher-order Markov models to reveal flow-based communities in networks

Complex systems made of interacting elements are commonly abstracted as networks, in which nodes are associated with dynamic state variables, whose evolution is driven by interactions mediated by the edges. Markov processes have been the prevailing paradigm to model such a network-based dynamics, for instance in the form of random walks or other types of diffusions. Despite the success of this modelling perspective for numerous applications, it represents an over-simplification of several real-world systems. Importantly, simple Markov models lack memory in their dynamics, an assumption often not realistic in practice. Here, we explore possibilities to enrich the system description by means of second-order Markov models, exploiting empirical pathway information. We focus on the problem of community detection and show that standard network algorithms can be generalized in order to extract novel temporal information about the system under investigation. We also apply our methodology to temporal networks, where we can uncover communities shaped by the temporal correlations in the system. Finally, we discuss relations of the framework of second order Markov processes and the recently proposed formalism of using non-backtracking matrices for community detection.