Beyond symmetrization: effective adjacency matrices and renormalization for (un)singed directed graphs

TL;DR

Directed and signed graphs challenge standard spectral tools, motivating deformed Laplacians such as the magnetic and dilation variants. The authors define effective adjacency matrices via edge discrepancies and generalized frustrations, yielding graphs $G_e^{(f,\beta)}$ that map directed graphs to unsigned-like representations and permit the use of undirected graph techniques. They leverage the Hodge-Helmholtz decomposition to dissect how gradient, curl-free, and harmonic components interact with deformations, and demonstrate an effective-graph Renormalization Group (RGEG) that reinforces block structure and directionality, outperforming simple symmetrization on both synthetic and real networks (e.g., polblogs). The work thus provides a principled framework to preserve directional information in graph analysis, enabling richer insights and potential improvements for downstream tasks such as clustering, centrality, and coarse-graining.

Abstract

To address the peculiarities of directed and/or signed graphs, new Laplacian operators have emerged. For instance, in the case of directionality, we encounter the magnetic operator, dilation (which is underexplored), operators based on random walks, and so forth. The definition of these new operators leads to the need for new studies and concepts, and consequently, the development of new computational tools. But is this really necessary? In this work, we define the concept of effective adjacency matrices that arise from the definition of deformed Laplacian operators such as magnetic, dilation, and signal. These effective matrices allow mapping generic graphs to a family of unsigned, undirected graphs, enabling the application of the well-explored toolkit of measures, machine learning methods, and renormalization groups of undirected graphs. To explore the interplay between deformed operators and effective matrices, we show how the Hodge-Helmholtz decomposition can assist us in navigating this complexity.

Figures (13)

• Figure 1: The left image shows a signal on a geometric graph. The center image displays the same graph with the signal contaminated by salt & pepper noise. The right image shows the result of filtering using the aggregation of discrepancies between the signals, that is, $f(u) = f(u) + \frac{1}{D(u)}\sum_{v\in Nei(u)} (f(v) - f(u))$.
• Figure 2: Helmholtz-Hodge decomposition of a directed graph and analysis of their respective magnetic and dilation Laplacians for each component. The top-left corner image presents a directed graph. The sequences illustrated in b, c, d, and e in the top row display the results of the HodgeRank decomposition of the directed graph, represented in curl-free, divergence-free, harmonic, and rotational components respectively. The second row portrays the magnetic specific heat for the original graph and each of its components. Lastly, the score obtained by the dilation Laplacian is displayed in the final row, both for the original graph and for each of its components. Interestingly, in some components, the result of the magnetic Laplacian is informative while the dilation Laplacian is not, and vice versa.
• Figure 3: Gradient approach for image segmentation using $q=1/10$ and $\eta =0.5$. The first row represents the six intensity fields. The second and third rows depict the frustration of the $x$ axis ($\cos \Phi$) and the $y$ axis ($\sin \Phi$) respectively. The fourth row displays the absolute value of the first eigenvector of the magnetic Laplacian, while the final row illustrates the result of the K-means algorithm for $2$ clusters.
• Figure 4: Gradient approach for image segmentation using $q=1/10$ and $\eta =0.5$ applied to the rotational component of the directed graph associated with each image. The first row presents the six intensity fields. The second and third rows represent the $x$-axis frustration ($\cos \Phi$) and the $y$-axis frustration ($\sin \Phi$), respectively. The fourth row illustrates the absolute value of the first eigenvector of the magnetic Laplacian. The last row displays the result of the K-means algorithm for $2$ clusters. The results in (d), (e), and (f) remain equivalent to those obtained in figure \ref{['fig:imgSegGrad']}.
• Figure 5: Histograms of the edge weights for the circular components (a) and the gradient components (b) for the Cora, Squirrel, and Citeseer networks. The discrepancy in the behavior of the gradient component in the Squirrel network stands out, encouraging further studies that could aid in understanding the behavior of GCNs in specific directed networks.

Theorems & Definitions (5)

• Definition 1: Generalized Degree Function
• Definition 2: Deformed Laplacian
• Definition 3: Edge Discrepancy
• Definition 4: Generalized frustration
• Definition 5: Effective Weight