Time-resolved analysis of dynamic graphs: an extended Slepian design (original) (raw)

Components in time-varying graphs

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2012

Real complex systems are inherently time-varying. Thanks to new communication systems and novel technologies, it is today possible to produce and analyze social and biological networks with detailed information on the time of occurrence and duration of each link. However, standard graph metrics introduced so far in complex network theory are mainly suited for static graphs, i.e., graphs in which the links do not change over time, or graphs built from time-varying systems by aggregating all the links as if they were concurrent in time. In this paper, we extend the notion of connectedness, and the definitions of node and graph components, to the case of time-varying graphs, which are represented as time-ordered sequences of graphs defined over a fixed set of nodes. We show that the problem of finding strongly connected components in a time-varying graph can be mapped into the problem of discovering the maximal-cliques in an opportunely constructed static graph, which we name the affine graph. It is therefore an NP-complete problem. As a practical example, we have performed a temporal component analysis of time-varying graphs constructed from three data sets of human interactions. The results show that taking time into account in the definition of graph components allows to capture important features of real systems. In particular, we observe a large variability in the size of node temporal in-and out-components. This is due to intrinsic fluctuations in the activity patterns of individuals, which cannot be detected by static graph analysis.

Inferring dynamic topology for decoding spatiotemporal structures in complex heterogeneous networks

Proceedings of the National Academy of Sciences of the United States of America, 2018

Extracting complex interactions (i.e., dynamic topologies) has been an essential, but difficult, step toward understanding large, complex, and diverse systems including biological, financial, and electrical networks. However, reliable and efficient methods for the recovery or estimation of network topology remain a challenge due to the tremendous scale of emerging systems (e.g., brain and social networks) and the inherent nonlinearity within and between individual units. We develop a unified, data-driven approach to efficiently infer connections of networks (ICON). We apply ICON to determine topology of networks of oscillators with different periodicities, degree nodes, coupling functions, and time scales, arising in silico, and in electrochemistry, neuronal networks, and groups of mice. This method enables the formulation of these large-scale, nonlinear estimation problems as a linear inverse problem that can be solved using parallel computing. Working with data from networks, ICON...

Semi-parametric Methods for Estimating Time-varying Graph Structure

2010

Abstract Stochastic networks are a plausible representation of the relational information among entities in dynamic systems such as living cells or social communities. While there is a rich literature in estimating a static or temporally invariant network from observation data, little has been done towards estimating time-varying networks from time series of entity attributes.

A Distributed Framework for Large-Scale Time-Dependent Graph Analysis

2017

In the last few years, we have seen that many applications or computer problems are mobilized as a graph since this data structure gives a particular handling for some use cases such as social networks, bioinformatics, road networks and communication networks. Despite its importance, the graph processing remains a challenge when dealing with large graphs. In this context, several solutions and works have been proposed to support large graph processing and storage. Nevertheless, new needs are emerging to support the dynamism of the graph (Dynamic Graph) and properties variation of the graph during the time (temporal graph). In this paper, we first present the concepts of dynamic and temporal graphs. Secondly, we show some frameworks that treat static, dynamic and temporal graphs. Finally, we propose a new framework based on the limits of the frameworks study.

Direct Embedding of Temporal Network Edges via Time-Decayed Line Graphs

arXiv (Cornell University), 2022

Temporal networks model a variety of important phenomena involving timed interactions between entities. Existing methods for machine learning on temporal networks generally exhibit at least one of two limitations. First, time is assumed to be discretized, so if the time data is continuous, the user must determine the discretization and discard precise time information. Second, edge representations can only be calculated indirectly from the nodes, which may be suboptimal for tasks like edge classification. We present a simple method that avoids both shortcomings: construct the line graph of the network, which includes a node for each interaction, and weigh the edges of this graph based on the difference in time between interactions. From this derived graph, edge representations for the original network can be computed with efficient classical methods. The simplicity of this approach facilitates explicit theoretical analysis: we can constructively show the effectiveness of our method's representations for a natural synthetic model of temporal networks. Empirical results on real-world networks demonstrate our method's efficacy and efficiency on both edge classification and temporal link prediction.

A statistical framework for streaming graph analysis

Proceedings of the 2013 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining - ASONAM '13, 2013

In this paper we propose a new methodology for gaining insight into the temporal aspects of social networks. In order to develop higher-level, large-scale data analysis methods for classification, prediction, and anomaly detection, a solid foundation of analytical techniques is required. We present a novel approach to the analysis of these networks that leverages time series and statistical techniques to quantitatively describe the temporal nature of a social network. We report on the application of our approach toward a real data set and successfully visualize high-level changes to the network as well as discover outlying vertices. The real-time prediction of new connections given the previous connections in a graph is a notoriously difficult task. The proposed technique avoids this difficulty by modeling statistics computed from the graph over time. Vertex statistics summarize topological information as real numbers, which allows us to leverage the existing fields of computational statistics and machine learning. This creates a modular approach to analysis in which methods can be developed that are agnostic to the metrics and algorithms used to process the graph. We demonstrate these techniques using a collection of Twitter posts related to Hurricane Sandy. We study the temporal nature of betweenness centrality and clustering coefficients while producing multiple visualizations of a social network dataset with 1.2 million edges. We successfully detect vertices whose triangleforming behavior is anomalous.

On time varying undirected graphs

2011

Abstract The time-varying multivariate Gaussian distribution and the undirected graph associated with it, as introduced in Zhou et al.(2008), provide a useful statistical framework for modeling complex dynamic networks. In many application domains, it is of high importance to estimate the graph structure of the model consistently for the purpose of scientific discovery.

Spectral methods for analyzing and visualizing networks: an introduction

Dynamic Social Network Modeling …, 2003

Network analysis begins with data that describes the set of relationships among the members of a system. The goal of analysis is to obtain from the low-level relational data a higher-level description of the structure of the system which identifies various kinds of patterns in the set of relationships. These patterns will be based on the way individuals are related to other individuals in the network. Some approaches to network analysis look for clusters of individuals who are tightly connected to one another; some look for sets of individuals who have similar patterns of relations to the rest of the network. Other methods don't "look for" anything in particular-instead, they construct a continuous multidimensional representation of the network in which the coordinates of the individuals can be further analyzed to obtain a variety of kinds of information about them and their relation to the rest of the network. One approach to this is to choose a set of axes in the multidimensional space occupied by the network and rotate them so that the first axis points in the direction of the greatest variability in the data; the second axis, orthogonal to the first, points in the direction of greatest remaining variability, and so on. This set of axes is a coordinate system that can be used to describe the relative positions of the set of points in the data. Most of the variability in the locations of points will be accounted for by the first few dimensions of this coordinate system. The coordinates of the points along each axis will be an eigenvector, and the length of the projection will be an eigenvalue. The set of all eigenvalues is the spectrum of the network. Spectral methods (eigendecomposition) have been a part of graph theory for over a century. Network researchers have used spectral methods either implicitly or explicitly since the late 1960's, when computers became generally accessible in most universities. The eigenvalues of a network are intimately connected to important topological features such as maximum distance across the network (diameter), presence of cohesive clusters, long paths and bottlenecks, and how random the network is. The associated eigenvectors can be used as a natural coordinate system for graph visualization; they also provide methods for discovering clusters and other local features. When combined with other, easily obtained network statistics (e.g., node degree), they can be used to describe a variety of network properties, such as degree of robustness (i.e., tolerance to removal of selected nodes or links), and other structural properties, and the relationship of these properties to node or link attributes in large, complex, multivariate networks. We introduce three types of spectral analysis for graphs and describe some of their mathematical properties. We discuss the strengths and weaknesses of each type and show how they can be used to understand network structure. These discussions are accompanied by interactive graphical displays of small (n=50) and moderately large (n=5000) networks. Throughout, we give special attention to sparse matrix methods which allow rapid, efficient storage and analysis of large networks. We briefly describe algorithms and analytic strategies that allow spectral analysis and identification of clusters in very large networks (n>1,000,000). 1 We have introduced some notation which will be followed throughout: • matrices are represented by bold capitals: D • (column-)vectors are represented by bold lower case: e • inner products of vectors are represented as e T e = n (a scalar) where e T is the transpose of e. • outer products of vectors are represented as ee T = M (a matrix) • eigenvalues are represented by greek letters, usually with some relationship to the latin letters representing a matrix and an eigenvector. E.g.,(α i , a i) is an eigenpair of adjacency matrix A.