A Regularized Graph Layout Framework for Dynamic Network Visualization

Many real-world networks, including social and information networks, are dynamic structures that evolve over time. Such dynamic networks are typically visualized using a sequence of static graph layouts. In addition to providing a visual representation of the network structure at each time step, the sequence should preserve the mental map between layouts of consecutive time steps to allow a human to interpret the temporal evolution of the network. In this paper, we propose a framework for dynamic network visualization in the on-line setting where only present and past graph snapshots are available to create the present layout. The proposed framework creates regularized graph layouts by augmenting the cost function of a static graph layout algorithm with a grouping penalty, which discourages nodes from deviating too far from other nodes belonging to the same group, and a temporal penalty, which discourages large node movements between consecutive time steps. The penalties increase the stability of the layout sequence, thus preserving the mental map. We introduce two dynamic layout algorithms within the proposed framework, namely dynamic multidimensional scaling (DMDS) and dynamic graph Laplacian layout (DGLL). We apply these algorithms on several data sets to illustrate the importance of both grouping and temporal regularization for producing interpretable visualizations of dynamic networks.Comment: To appear in Data Mining and Knowledge Discovery, supporting material (animations and MATLAB toolbox) available at http://tbayes.eecs.umich.edu/xukevin/visualization_dmkd_201

Higher-Order Aggregate Networks in the Analysis of Temporal Networks: Path structures and centralities

Recent research on temporal networks has highlighted the limitations of a static network perspective for our understanding of complex systems with dynamic topologies. In particular, recent works have shown that i) the specific order in which links occur in real-world temporal networks affects causality structures and thus the evolution of dynamical processes, and ii) higher-order aggregate representations of temporal networks can be used to analytically study the effect of these order correlations on dynamical processes. In this article we analyze the effect of order correlations on path-based centrality measures in real-world temporal networks. Analyzing temporal equivalents of betweenness, closeness and reach centrality in six empirical temporal networks, we first show that an analysis of the commonly used static, time-aggregated representation can give misleading results about the actual importance of nodes. We further study higher-order time-aggregated networks, a recently proposed generalization of the commonly applied static, time-aggregated representation of temporal networks. Here, we particularly define path-based centrality measures based on second-order aggregate networks, empirically validating that node centralities calculated in this way better capture the true temporal centralities of nodes than node centralities calculated based on the commonly used static (first-order) representation. Apart from providing a simple and practical method for the approximation of path-based centralities in temporal networks, our results highlight interesting perspectives for the use of higher-order aggregate networks in the analysis of time-stamped network data.Comment: 27 pages, 13 figures, 3 table

Statistical clustering of temporal networks through a dynamic stochastic block model

Statistical node clustering in discrete time dynamic networks is an emerging field that raises many challenges. Here, we explore statistical properties and frequentist inference in a model that combines a stochastic block model (SBM) for its static part with independent Markov chains for the evolution of the nodes groups through time. We model binary data as well as weighted dynamic random graphs (with discrete or continuous edges values). Our approach, motivated by the importance of controlling for label switching issues across the different time steps, focuses on detecting groups characterized by a stable within group connectivity behavior. We study identifiability of the model parameters, propose an inference procedure based on a variational expectation maximization algorithm as well as a model selection criterion to select for the number of groups. We carefully discuss our initialization strategy which plays an important role in the method and compare our procedure with existing ones on synthetic datasets. We also illustrate our approach on dynamic contact networks, one of encounters among high school students and two others on animal interactions. An implementation of the method is available as a R package called dynsbm

Influencers in Dynamic Financial Networks

To monitor risk in temporal financial networks, an understanding of how individual behaviours affect the temporal evolution of networks is needed. This is typically achieved using centrality and importance metrics, which rank nodes in terms of their position in the network. This approach works well for static networks, that do not change over time, but does not consider the dynamics of the network. In addition to this, current methods are often unable to capture the complex, often sparse and disconnected structures of financial transaction networks. This thesis addresses these gaps by considering importance from a dynamical perspective, first by using spectral perturbations to derive measures of importance for nodes and edges, then adapting these methods to incorporate a structural awareness. I complement these methods with a generative model for transaction networks that captures how individual behaviours give rise to the key properties of these networks, offering new methods to add to the regulatory toolkit. My contributions are made across three studies which complement each other in their findings. Study 1: \begin{itemize} \item I define a structural importance metric for the edges of a network, based on perturbing the adjacency matrix and observing the resultant change in its largest eigenvalues. \item I combine this with a model of network evolution where this metric controls the scale and probabilities of subsequent edge changes. This allows me to consider how edge importance relates to subsequent edge behaviour. \item I use this model alongside an exercise to predict subsequent change from edge importance. Using this I demonstrate how the model parameters are related to the capability of predicting whether an edge will change from its importance. \end{itemize} Study 2: \begin{itemize} \item I extend my measure of edge importance to measure the importance of nodes, and to capture complex community structures through the use of additional components of the eigenspectrum. \item While computed from a static network, my measure of node importance outperforms other centrality measures as a predictor of nodes subsequently transacting. This implies that static representations of temporal networks can contain information about their dynamics. \end{itemize} Study 3: \begin{itemize} \item I contrast the snapshot based methods used in the first two studies by modelling the dynamic of transactions between counterparties using both univariate and multivariate Hawkes processes, which capture the non-linear `bursty’ behaviour of transaction sequences. \item I find that the frequency of transactions between counterparties increases the likelihood of them to transact in the future, and that univariate and multivariate Hawkes processes show promise as generative models for transaction sequences. \item Hawkes processes also perform well when used to model buys and sells through a central clearing counterparty when considered as a bivariate process, but not as well when these are modelled as individual univariate processes. This indicates that mutual excitation between buys and sells is present in these markets. \end{itemize} The observations presented in this thesis provide new insights into the behaviour of equities markets, which until now have mainly been studied via price information. The metrics I propose offer a new potential to identify important traders and transactions in complex trading networks. The models I propose provide a null model over which a user could detect outlying transactions and could also be used to generate synthetic data for sharing purposes

European Gas Markets, Trading Hubs, and Price Formation: A Network Perspective

We apply network theory to analyse the interactions of trading hub prices, and to assess the harmonisation of the European gas market. We construct dynamic networks, where the nodes correspond to the twelve EU trading hubs, and where the edges weight the causality between the variations of the respective gas prices. Network density dynamically calculates the aggregate quantity of causal interactions recorded within the system, which provides information pertaining to the integration of the European gas network. We document a number of spikes in network density, suggesting short periods of improved connectivity of European gas markets. We argue that these results appear to be driven by exogenous factors, such as unseasonal weather patterns, seismic activity and pipeline capacity reductions or outages. The findings elucidate the time varying nature of European gas market dynamics, and the importance of continual monitoring of market evolution

Detecting Community Structure in Dynamic Social Networks Using the Concept of Leadership

Detecting community structure in social networks is a fundamental problem empowering us to identify groups of actors with similar interests. There have been extensive works focusing on finding communities in static networks, however, in reality, due to dynamic nature of social networks, they are evolving continuously. Ignoring the dynamic aspect of social networks, neither allows us to capture evolutionary behavior of the network nor to predict the future status of individuals. Aside from being dynamic, another significant characteristic of real-world social networks is the presence of leaders, i.e. nodes with high degree centrality having a high attraction to absorb other members and hence to form a local community. In this paper, we devised an efficient method to incrementally detect communities in highly dynamic social networks using the intuitive idea of importance and persistence of community leaders over time. Our proposed method is able to find new communities based on the previous structure of the network without recomputing them from scratch. This unique feature, enables us to efficiently detect and track communities over time rapidly. Experimental results on the synthetic and real-world social networks demonstrate that our method is both effective and efficient in discovering communities in dynamic social networks

Fast filtering and animation of large dynamic networks

Detecting and visualizing what are the most relevant changes in an evolving network is an open challenge in several domains. We present a fast algorithm that filters subsets of the strongest nodes and edges representing an evolving weighted graph and visualize it by either creating a movie, or by streaming it to an interactive network visualization tool. The algorithm is an approximation of exponential sliding time-window that scales linearly with the number of interactions. We compare the algorithm against rectangular and exponential sliding time-window methods. Our network filtering algorithm: i) captures persistent trends in the structure of dynamic weighted networks, ii) smoothens transitions between the snapshots of dynamic network, and iii) uses limited memory and processor time. The algorithm is publicly available as open-source software.Comment: 6 figures, 2 table