Ellipsoidal embeddings of graphs

TL;DR

The paper addresses the problem of embedding graphs in a geometric space by introducing ellipsoidal embeddings derived from trace-optimization with a descriptor matrix $M$ such as the modularity matrix $Q$ or (normalized) Laplacian. The approach computes node coordinates by solving $\max_H \mathrm{Tr}(H^\top M H)$ with row-norm constraints $\|H_{i\ast}\|_2=1$, leveraging a generalized power method on a shifted descriptor to obtain $H_\star$ and its effective dimension $d_{\rm eff}$. A key contribution is the embed-and-partition framework that uses the ellipsoidal embedding for graph clustering via a vector-partitioning algorithm, yielding competitive results on synthetic LFR benchmarks and real networks, with an automatically inferred embedding dimension. The work links ellipsoidal embeddings to spectral methods, offers a scalable, low-dimensional representation, and provides public Julia code, highlighting practical impact for modularity-based clustering and graph analysis.

Abstract

Due to their flexibility to represent almost any kind of relational data, graph-based models have enjoyed a tremendous success over the past decades. While graphs are inherently only combinatorial objects, however, many prominent analysis tools are based on the algebraic representation of graphs via matrices such as the graph Laplacian, or on associated graph embeddings. Such embeddings associate to each node a set of coordinates in a vector space, a representation which can then be employed for learning tasks such as the classification or alignment of the nodes of the graph. As the geometric picture provided by embedding methods enables the use of a multitude of methods developed for vector space data, embeddings have thus gained interest both from a theoretical as well as a practical perspective. Inspired by trace-optimization problems, often encountered in the analysis of graph-based data, here we present a method to derive ellipsoidal embeddings of the nodes of a graph, in which each node is assigned a set of coordinates on the surface of a hyperellipsoid. Our method may be seen as an alternative to popular spectral embedding techniques, to which it shares certain similarities we discuss. To illustrate the utility of the embedding we conduct a case study in which we analyse synthetic and real world networks with modular structure, and compare the results obtained with known methods in the literature.

Figures (6)

• Figure 1: Modularity-based spherical embedding. Left: a spherical embedding -- see \ref{['Def:Embedding']} -- on the depicted graph (black lines), which yields a coordinate vector $h_i$ for each node $i$, interpreted as a 'spin' attached to each node (blue). Right: each of these spins $h_i$ is drawn on the same hypersphere, giving rise to the depicted spherical embedding. The alignment of the spins $h_i$ reflects some of the neighbourhood structure in the graph. Here we chose $d_0 = 2$ and the graph descriptor matrix is the modularity $M=Q$; see \ref{['e:Q_matrix']}.
• Figure 2: Ellipsoidal vs spectral embedding. Laplacian-based ellipsoidal embedding (top left, $d_0 = 10$, $M = \mathcal{L}$, see \ref{['eq:NormLap']}) and the spectral embedding (top right) of the PowerEU graph -- see Table \ref{['Table:RealNetworks']} -- thanks to the three leading eigenvectors of the Laplacian $\mathcal{L}$. On the bottom left, the spectrum of $\frac{1}{n}H_\star H^\top_\star$ associated to the embedding with $d_{\rm eff} = 3$. The spectrum of this normalized Laplacian matrix $\mathcal{L}$ is given on the botton right.
• Figure 3: Modularity-based ellipsoidal embeddings for graphs with community structure. A vizualization of the embedding of two LFR benchmark graphs with $2000$ nodes with $d_0=10$; LFR1 (left, $4$ planted communities and ${ d}_{\rm eff}=3$) and LFR2 (right, $8$ planted communities, a larger mixing parameter and ${ d}_{\rm eff}=5$), see \ref{['a:fig_embed_toy']} for details. The colors indicate the true community structure. On the bottom, the eigenvalues of $\frac{1}{n}H_\star H^\top_\star$. Our embed-and-partition retrieves the planted communities in both cases.
• Figure 4: NMI vs mixing parameter $\texttt{mu}$ of LFR benchmark graphs with $n=1000$ nodes. These graphs were generated with different mixing parameters ranging from $0.1$ to $1$; see \ref{['a:nmi_vs_mixing']} for the numerical setting. An ellipsoidal embedding was computed with $d_0 = 30$ and communities were retrieved thanks to \ref{['Alg:VecPart']} with $k= 100$ initialized centroids. The NMI between the planted and retrieved community structure is here displayed as a function of the mixing parameter. The whole procedure was repeated independently $3$ times and averages as well as standard deviations are reported. We refer to \ref{['Fig:nmi_vs_mu_ablation']} for a study of the sensitivity to the choice of $k$ and $d_0$.
• Figure 5: Generalized Power Method (GPM) compared to Generalized Power Method with Momentum (GPMM) applied to the ellipsoidal embedding of PowerEU illustrated in Figure \ref{['Fig:Power']}.

Theorems & Definitions (14)

• Definition 1: Spherical and Ellipsoidal embeddings
• Remark 1: Eigenvalue thresholding
• Remark 2: Orientation ambiguity
• Proposition 1
• Proposition 2: Equivalence with a nuclear norm minimization
• Lemma 1: Effect of diagonal dominance
• proof
• Lemma 2: Criterion for criticality
• proof
• Proposition 3: Monotonicity of the objectives