Connecting the Dots: Identifying Network Structure via Graph Signal Processing

Abstract

Network topology inference is a prominent problem in Network Science. Most graph signal processing (GSP) efforts to date assume that the underlying network is known, and then analyze how the graph's algebraic and spectral characteristics impact the properties of the graph signals of interest. Such an assumption is often untenable beyond applications dealing with e.g., directly observable social and infrastructure networks; and typically adopted graph construction schemes are largely informal, distinctly lacking an element of validation. This tutorial offers an overview of graph learning methods developed to bridge the aforementioned gap, by using information available from graph signals to infer the underlying graph topology. Fairly mature statistical approaches are surveyed first, where correlation analysis takes center stage along with its connections to covariance selection and high-dimensional regression for learning Gaussian graphical models. Recent GSP-based network inference frameworks are also described, which postulate that the network exists as a latent underlying structure, and that observations are generated as a result of a network process defined in such a graph. A number of arguably more nascent topics are also briefly outlined, including inference of dynamic networks, nonlinear models of pairwise interaction, as well as extensions to directed graphs and their relation to causal inference. All in all, this paper introduces readers to challenges and opportunities for signal processing research in emerging topic areas at the crossroads of modeling, prediction, and control of complex behavior arising in networked systems that evolve over time. †

Time-varying Graph Learning Based on Sparseness of Temporal Variation

“…Several approaches address graph learning problem, and two overview papers about graph learning have been published recently [14], [15]. Among the techniques for learning timevarying graphs, the Kalofolias et al method, where constraints are introduced so that the edge weights change smoothly over time, is close to ours [33].…”

“…Note that graphs estimated with this approach often have negative edge weights. In contrast, our proposed method constrains to have non-negative edges in the estimated graph, because such graphs are often desired for many applications [14], [15]. Furthermore, our method has a lower computation complexity, as TVGL requires eigendecompositions of target matrices to calculate a proximal operator of a logarithm determinant term, whereas our approach is an eigendecomposition-free algorithm.…”

Estimating Centrality Blindly From Low-Pass Filtered Graph Signals

“…We treat the observations as filtered graph signals and learn the graph topology as a blind deconvolution problem. Several inference methods have been proposed, e.g., based on smoothness [14], spectral template [15], structures of topology [16] etc., see the overview in [17,18]. Meanwhile, for practitioners of graph learning, obtaining the graph topology is only the first step, and oftentimes the end goal is to deduce interpretable features from the graph topology such as node centrality, communities.…”

Network Weight Estimation for Binary-Valued Observation Models

“…They can be used for online tasks, such as adaptive control and decision-making, and thus have attracted much interest in the control community. More attention has been paid, however, on batch algorithms for network estimation, e.g., [31], [35]. Quantized data are ubiquitous across domains, for example, active/inactive states of a gene [2], ordinal rating of an individual [16], and failure conditions of an infrastructure [5].…”

“…Quantized data are ubiquitous across domains, for example, active/inactive states of a gene [2], ordinal rating of an individual [16], and failure conditions of an infrastructure [5]. Network estimation problems based on quantized timeseries data need to be investigated in more depth with rigorous performance analysis [2], [31].…”