[Paper Review] The Importance of Being Correlated: Implications of Dependence in Joint Spectral Inference across Multiple Networks

Konstantinos Pantazis, Avanti Athreya+4 more|PubMed|Aug 1, 2020

Functional Brain Connectivity Studies83 references4 citations

TL;DR

This paper introduces a generalized omnibus embedding method for joint spectral inference across multiple networks, explicitly modeling both inherent network correlation (from data) and induced correlation (from embedding). It proves consistency and a central limit theorem, demonstrating that accounting for both correlation types improves detection of subtle changes in biological network time series beyond prior methods.

ABSTRACT

Spectral inference on multiple networks is a rapidly-developing subfield of graph statistics. Recent work has demonstrated that joint, or simultaneous, spectral embedding of multiple independent networks can deliver more accurate estimation than individual spectral decompositions of those same networks. Such inference procedures typically rely heavily on independence assumptions across the multiple network realizations, and even in this case, little attention has been paid to the induced network correlation that can be a consequence of such joint embeddings. In this paper, we present a <i>generalized omnibus</i> embedding methodology and we provide a detailed analysis of this embedding across both independent and correlated networks, the latter of which significantly extends the reach of such procedures, and we describe how this omnibus embedding can itself induce correlation. This leads us to distinguish between <i>inherent</i> correlation-that is, the correlation that arises naturally in multisample network data-and <i>induced</i> correlation, which is an artifice of the joint embedding methodology. We show that the generalized omnibus embedding procedure is flexible and robust, and we prove both consistency and a central limit theorem for the embedded points. We examine how induced and inherent correlation can impact inference for network time series data, and we provide network analogues of classical questions such as the effective sample size for more generally correlated data. Further, we show how an appropriately calibrated generalized omnibus embedding can detect changes in real biological networks that previous embedding procedures could not discern, confirming that the effect of inherent and induced correlation can be subtle and transformative. By allowing for and deconstructing both forms of correlation, our methodology widens the scope of spectral techniques for network inference, with import in theory and practice.

Motivation & Objective

• To address the limitations of existing joint spectral embedding methods that assume independence across network realizations, which fails to account for natural (inherent) and methodological (induced) correlation.
• To develop a flexible, robust generalized omnibus embedding framework that can handle both independent and correlated networks, including time series data with growing correlation.
• To formally distinguish and analyze the impact of inherent correlation (from data) versus induced correlation (from joint embedding), showing their transformative effects on inference.
• To extend classical statistical concepts—like effective sample size—to correlated network data by quantifying the influence of both correlation types on inference accuracy.
• To validate the method on real biological network time series, demonstrating improved detection of network changes compared to prior embedding techniques.

Proposed method

• Proposes a generalized omnibus embedding matrix constructed as a weighted sum of network adjacency matrices, with a structured mean matrix $\widetilde{P} = J_m \otimes P$ to preserve low-rank structure and enable consistent estimation.
• Uses a Kronecker product framework $E \otimes P$ with $E$ having $\vec{1}$ as a leading eigenvector to maintain low-rank properties and allow for eigenspace alignment across networks.
• Derives a central limit theorem for embedded points under the generalized omnibus framework, establishing asymptotic normality of the estimator.
• Introduces alternative constructions like the forward omnibus matrix $\mathfrak{M}_{\text{for}}$ to model increasing correlation over time, enabling flexible correlation structure modeling.
• Develops a framework for out-of-sample embedding by jointly embedding core aligned vertices and extending to unaligned vertices, supporting applications like seeded graph matching.
• Explores extensions to sparser graphs and asymmetric OMNI matrices, with preliminary results suggesting potential for broader asymptotic validity.

Experimental results

Research questions

• RQ1 How does inherent correlation in network time series—arising naturally from temporal dependence—affect spectral inference accuracy?
• RQ2 To what extent does the joint embedding process itself induce artificial correlation, and how does this impact downstream inference?
• RQ3 Can a generalized omnibus embedding framework consistently estimate latent positions under both independent and correlated network settings?
• RQ4 How can classical concepts like effective sample size be redefined in the context of correlated network data?
• RQ5 Can the proposed method detect biologically meaningful changes in network structure that standard embedding techniques miss?

Key findings

• The generalized omnibus embedding is consistent and satisfies a central limit theorem, ensuring reliable asymptotic inference under both independent and correlated network settings.
• The method successfully detects subtle changes in real biological network time series that were undetectable using previous embedding procedures, demonstrating practical superiority.
• Induced correlation from joint embedding can mask or amplify true network differences, making its explicit modeling essential for accurate inference.
• The forward omnibus model produces increasing correlation across network pairs, enabling modeling of time series where correlation grows over time.
• The Kronecker structure $J_m \otimes P$ ensures that the leading $d$-dimensional eigenspace of the omnibus matrix aligns with the true latent space, preserving interpretability.
• Extensions to out-of-sample embedding and sparser graph regimes are feasible and show promise, though asymptotic results remain under exploration for general cases.