A plausible representation of the relational information among entities in dynamic systems such as a living cell or a social community is a stochastic network that is topologically rewiring and semantically evolving over time. While there is a rich literature in modeling static or temporally invariant networks, little has been done toward recovering the network structure when the networks are not observable in a dynamic context. In this paper, we present a new machine learning method called TESLA, which builds on a temporally smoothed l 1 -regularized logistic regression formalism that can be cast as a standard convex-optimization problem and solved efficiently using generic solvers scalable to large networks. We report promising results on recovering simulated timevarying networks, and on reverse engineering the latent sequence of temporally rewiring political and academic social networks from longitudinal data, and the evolving gene networks over more than 4000 genes during the life cycle of Drosophila melanogaster from a microarray time course at a resolution limited only by sample frequency. time varying network | social network | gene network | logistic regression | Lasso | regularization | optimization | Markov random fields I n many problems arising in social, biological, and other fields, it is often necessary to analyze populations of entities (e.g., individuals, genes) interconnected by a set of relationships (e.g., friendship, communication, influence) represented as a network. Real-time analysis of network data is important for detecting anomaly, predicting vulnerability, and assessing the potential impact of interventions in various social, biological, or engineering systems. It is not unusual for network data to be large, dynamic, heterogeneous, noisy and incomplete. Each of these characteristics adds a degree of complexity to the interpretation and analysis of networks. Classical network analyses mostly assume that the networks in question are fully observable, static, and isotropic. Given the heterogeneity and complexity of network data in many domains, these assumptions are limiting. For example, a majority of current investigations of biological regulatory circuitry focus on networks with invariant topology over a given set of molecules, such as a proteinprotein interaction network over all proteins of an organism regardless of the conditions under which individual interactions may take place, or a static gene network inferred from microarray data even though the samples may be collected over a time course or multiple conditions. In reality, over the course of a cellular process, such as a cell cycle or an immune response, there may exist multiple underlying "themes" that determine the functionalities of each molecule and their relationships to each other, and such themes are dynamic and stochastic. As a result, the molecular networks at each time point are context-dependent and can undergo systematic rewiring, rather than being invariant over time. Indeed, in a seminal study by Luscombe et al. (1), it was shown that the "active regulatory paths" in a gene expression correlation network of Saccharomyces cerevisiae exhibit dramatic topological changes and hub transience during a temporal cellular process, and in response to diverse stimuli. Similar phenomena are not uncommon in many other domains, such as the evolution of citation networks in scientific communities, and the emergence/disingetration of cryptic liaisons and communities in society. A key technical hurdle that prevents us from an in-depth investigation of the mechanisms underlying this phenomena is the unavailability of serial snapshots of the rewiring network during the unfolding and progression of the process in question. For example, under a dynamic biological system, usually it is technologically impossible to experimentally determine time-specific network topologies for a series of time points based on techniques such as two-hybrid or ChIP-chip experiments. While there is a rich (and growing) literature on modeling observed network data in either static (2) or dynamic scenarios (3, 4, 5), with a few exceptions (6, 7), much less has been done toward developing efficient inference techniques for recovering unobserved network topologies from nodal (i.e., entity-wise) observations in a dynamic context. Existing methods usually ignore potential network rewiring under different conditions, and infer an invariant network (8, 9, 10). Recently, a new class of models known as the temporal Exponentially Random Graph Model (tERGM) has been proposed for modeling networks evolving over discrete time steps (5). It is based on a log-linear graph transition model P (G t |G t−1 ) defined on a set of temporal potentialsthat capture certain characteristics of the graph rewiring dynamics, such as the "edge-stability", "reciprocity", and "transitivity" statistics over time-adjacent graphs. Based on tERGM, subsequently a hidden tERGM model (htERGM) was developed to impose stochastic constraints on latent rewiring graphs so that given a time series of nodal attributes, one can infer time-specific network topology based on the posterior distribution of G t of every time point (7). However, the MCMC-based algorithm in (7) for posterior inference under htERGM is very inefficient, and up till now, providing an efficient inference algorithms for this model remains an open problem. We propose TESLA, a new machine learning algorithm for recovering the structure of time-varying networks over a fixed set of nodes from time series of nodal attributes. In these networks, an edge between two nodes implies a dependency between their nodal states, such as co-activation in a gene regulatory network. TESLA stems from the acronym TESLLOR, which stands for TEmporally Smoothed l1-regularized LOgistic Regression. It represents a novel extension of the well-known lasso-style sparse structure recovery technique, and is based on a key assumption that temporally adjacent networks are likely not to be dramatically different from each other in topology, and therefore are more likely to share common edges than temporally distal networks. Building on the highly scalable iterative l1-regularized logistic regression algorithm for estimating single sparse networks (11),
Similarly, Ahmed and Xing REF proposed a machine learning algorithm named TESLA for recovering the underlying structure of time-varying networks, which could be also used on temporal community detection.
Recovering time-varying networks of dependencies in social and biological studies
Proc. Nat. Acad. Sci
Computer Science Medicine