Connecting the Dots: Identifying Network Structure via Graph Signal Processing (1810.13066v1)

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.

• The paper presents novel methods integrating graph signal processing with statistical frameworks to infer network structures from observed signals.
• It leverages spectral decomposition and diffusion models to enable scalable and robust topology inference in high-dimensional settings.
• Applications in social networks, neuroscience, and economics illustrate the practical impact of uncovering hidden network dynamics.

Overview of "Connecting the Dots: Identifying Network Structure via Graph Signal Processing"

The paper "Connecting the Dots: Identifying Network Structure via Graph Signal Processing," explores the intersection of network science and signal processing to tackle the problem of inferring network topologies from observed data. Traditionally, graph signal processing (GSP) has analyzed graph signals under the assumption of a known network structure. This work breaks new ground by shifting focus towards learning this underlying structure from graph signal observations, especially in cases where direct observation of the network is infeasible or inaccurate. By integrating statistical frameworks with emerging GSP techniques, the paper offers a comprehensive survey and new methodologies for graph learning that leverage properties like smoothness, stationarity, and network diffusion processes.

Statistical Frameworks for Network Inference

Initially, the paper explores traditional statistical methods which utilize correlation measures as a proxy for network structure. These methods infer graphs by analyzing sample correlations and partial correlations, laying foundation on hypothesis testing principles and controlled by techniques like false discovery rate (FDR). The constraints of linear and symmetric interactions in such methods are acknowledged, inspiring subsequent forays into Gaussian graphical models (GMRFs). Here, the paper examines state-of-the-art approaches like graphical lasso and its variations that introduce regularization to enable accurate topology inference even in high-dimensional settings.

Graph Signal Processing Based Approaches

With the introduction of GSP into topology inference, the paper presents a paradigm shift towards methods that operate in the graph spectral domain. The GFT is a pivotal concept, allowing graph signals to be decomposed into spectral components, facilitating efficient representation and processing. The paper focuses on stationarity-based models which view signals as outputs of diffusion processes on a graph. Ordinarily, signal models like those assuming smoothness have been inferred through Laplacian frameworks and various optimization techniques. The relationships between signal smoothness and edge sparsity are exploited to propose scalable algorithms that identify graph structures by easing computational demands.

Challenges and Generalizations

The paper also acknowledges the challenges and future directions in the field, particularly emphasizing the need for robust methods that can handle dynamic graphs, address computational complexities, and operate under non-linear interaction paradigms. The potential of these methods is illustrated through applications spanning social networks, neuroscience, and economic systems, affirming their practicality.

Practical and Theoretical Implications

The theoretical implications underscore the value of GSP as a unifying framework for processing complex data, extending the graph Fourier transform beyond traditional signals to offer rich insights into networked systems. Practically, these methodologies enhance our ability to make sense of large-scale datasets prevalent in various domains, ensuring that the inferred networks are not only representative but also conducive to further scientific exploration.

In summary, by bridging the gap between traditional inference methods and contemporary GSP techniques, the paper provides a roadmap for future research in network science, promising advancements in understanding the dynamics and controllability of complex systems. The insights from this research are expected to drive innovative solutions across diverse fields, from biology to cyber-physical systems, as data continues to grow in complexity and volume.