HARNESSING SPECTRAL REPRESENTATIONS FOR SUBGRAPH ALIGNMENT

Abstract

With the rise and advent of graph learning techniques, graph data has become ubiquitous. However, while several efforts are being devoted to the design of new convolutional architectures, pooling or positional encoding schemes, less effort is being spent on problems involving maps between (possibly very large) graphs, such as signal transfer, graph isomorphism and subgraph correspondence. With this paper, we anticipate the need for a convenient framework to deal with such problems, and focus in particular on the challenging subgraph alignment scenario. We claim that, first and foremost, the representation of a map plays a central role on how these problems should be modeled. Taking the hint from recent work in geometry processing, we propose the adoption of a spectral representation for maps that is compact, easy to compute, robust to topological changes, easy to plug into existing pipelines, and is especially effective for subgraph alignment problems. We report for the first time a surprising phenomenon where the partiality associated to the subgraph is manifested as a special structure of the map coefficients, even in the absence of exact subgraph isomorphism, and which is consistently observed over different families of graphs up to several thousand nodes.

1. INTRODUCTION

The ability to align data is at the heart of many successful techniques in machine learning and related areas. In its most abstract form, the problem has a straightforward formulation: Given two generic domains D 1 and D 2 , find a transformation T such that T D 1 ≈ D 2 according to some approximation metric that depends on the task. Examples of such problems are found in numerous applications, including molecular docking (Gainza et al., 2020) , image-based rendering (Fachada et al., 2021 ) , 3D reconstruction (Zhao et al., 2022) , generative models (Dai & Hang, 2021) and style transfer (Zhang et al., 2022) , in addition to countless others. Recent remarkable examples include CLIP Meila & Zhang (2021) , where images are associated to corresponding captions by aligning their learned embeddings, or MaSIF (Gainza et al., 2020) , where the interaction site between protein structures (i.e., the surface patches where the proteins geometrically align) is predicted by a geometric deep learning pipeline. Perhaps the most challenging setting for alignment problems arises whenever the two domains only correspond partially, for example due to the lack of observations or noise in the data. In this case, one is not only interested in aligning the two domains, but also in discovering which portions of the domains actually align. The problem is particularly hard if an exact alignment does not even exist, requiring additional robustness to local perturbations in the data. In this paper, we focus on the general problem of subgraph alignment, as it is representative of a broad spectrum of applications including those mentioned above. We assume to be given two graphs G 1 and G 2 , where G 2 appears within G 1 , possibly up to topological changes. A special case appears when G 2 is isomorphic to a subgraph of G 1 , which is referred to as subgraph isomorphism (see (ii) in Figure 1 ). This case is included in our treatment, but we also consider noisier settings where a subgraph isomorphism does not exist (see (iii) in Figure 1 ), yet a semantic correspondence can still be defined.

Contribution.

In this paper, we focus in particular on the choice of a representation for the correspondence. That is, instead of introducing a new matching pipeline to solve subgraph alignment, we show an alternative way of representing maps between a graph and its subgraphs. In cases where the map is unknown and must be sought for, the new representation makes the inference problem easier to solve; while if the map is given, the new representation is more compact, has a regularizing effect, and bears a natural structure that is missing from classical representations such as node-to-node binary correspondence matrices. From a technical perspective, the map representation is defined with respect to a spectral basis; namely, the eigenvectors of the graph Laplacian. This idea, introduced a decade ago in the geometry processing area (Ovsjanikov et al., 2012) , brought significant progress to several tasks in graphics and vision -yet, its application to graphs has been largely overlooked. We claim that part of the reason is a common misconception. The lack of a smooth metric (i.e., a smooth manifold underlying the graph) leads to the assumption that key properties of the spectral representation of the maps, such as those observed in Ovsjanikov et al. ( 2012); Rodolà et al. ( 2017), only exist for surface domains. With this work, we challenge this view by showing extensive empirical evidence that not only these spectral maps are applicable to general subgraphs, but also that they exhibit robustness properties that go beyond what was shown on surfaces. Therefore, we propose to embrace the spectral representation of maps as compact, efficient, interpretable, robust, and easy to manipulate objects that can be naturally integrated into several pipelines, including but not limited to graph learning models. We summarize our main contributions as follows: • We propose the adoption of spectral representations for maps between graphs and subgraphs. For the first time, we show that such maps exhibit a special structure in their coefficients, capturing the similarity between the Laplacian eigenspaces of the two graphs. • We further show robustness of the representation to topological modifications of the subgraph, due for example to graph rewiring. This leads to well-defined maps even in the absence of exact isomorphism. • We include extensive experiments showing practical applications, such as signal transfer and subgraph matching, on graphs spanning a few dozen to tens of thousands of nodes, and demonstrate key benefits in terms of robustness to noise, interpretability, and computational complexity. All the code and generated data will be publicly released upon acceptance.

2. RELATED WORK

Graph alignment problems are ubiquitous in applications from social network analysis (Liu et al., 2016) to bioinformatics (Singh et al., 2008) . Given its relevance, a rich body of literature is devoted to this problem. A comparative study on several network alignment techniques can be found in Trung et al. (2020) . Though not exhaustive, we discuss the most relevant works in the following.



Figure1: Functional maps of size 50 × 50 between a full graph (depicted on the left) and three different graphs, respectively: an isomorphic graph (i), an isomorphic subgraph containing 80% of the original nodes (ii), and a non-isomorphic subgraph obtained by randomly rewiring the former (iii). The green edges are novel and randomly added (10% of the total), the red edges are randomly removed (10% of the total). The map representation still maintains a visible structure despite the significant changes of connectivity that span several hops.

