Modeling Topological Impact on Node Attribute Distributions in Attributed Graphs

TL;DR

The paper addresses how graph topology modulates the distribution of node attributes and proposes a categorical-algebraic framework to quantify topology-conditioned posteriors $P(\cdot \mid v)$ and $P(\cdot \mid \mathcal{G})$. It builds these posteriors by embedding node viewpoints into under-categories derived from a GGNN-based structure, introducing the POV and DMI representations to fuse topology with attribute priors. A principled sufficiency result is established on complete graphs, and the Induced Distribution (ID) testbed demonstrates the practical utility of topology-aware probabilistic reasoning through a POV-based graph auto-encoder for unsupervised Graph Anomaly Detection across six datasets. The work offers a scalable approach to enrich graph representations with topology-conditioned attribute distributions, with potential impact on anomaly detection and broader graph-learning tasks.

Abstract

We investigate how the topology of attributed graphs influences the distribution of node attributes. This work offers a novel perspective by treating topology and attributes as structurally distinct but interacting components. We introduce an algebraic approach that combines a graph's topology with the probability distribution of node attributes, resulting in topology-influenced distributions. First, we develop a categorical framework to formalize how a node perceives the graph's topology. We then quantify this point of view and integrate it with the distribution of node attributes to capture topological effects. We interpret these topology-conditioned distributions as approximations of the posteriors $P(\cdot \mid v)$ and $P(\cdot \mid \mathcal{G})$. We further establish a principled sufficiency condition by showing that, on complete graphs, where topology carries no informative structure, our construction recovers the original attribute distribution. To evaluate our approach, we introduce an intentionally simple testbed model, $\textbf{ID}$, and use unsupervised graph anomaly detection as a probing task.

Figures (5)

• Figure 1: Illustration of an attributed graph $\mathcal{G} = (V, E, X)$ alongside the probability distribution of node attributes $X$. The inset graph represents the topology of $\mathcal{G}$, while the bar chart compares the prior distribution $P(X)$ (blue), the topology-influenced distribution from the point of view of node $v_4$, $P(X \mid v_4)$ (red), and the overall topology-conditioned distribution $P(X \mid \mathcal{G})$ (green). These distributions highlight how the topology of $\mathcal{G}$ shapes the points of view of $v_4$ regarding $P$.
• Figure 2: Visualization of the point of view of a node $v\in\mathcal{G}$ (highlighted in blue) with respect to the topology of the graph $\mathcal{G}$. The red color spectrum represents the influence or perspective of node $v$ on other nodes, with varying intensities indicating differing levels of topological proximity or relevance.
• Figure 3: Iterative localization of a rumor source via point of view: Starting from a uniform belief, the agent repeatedly applies $\mathsf{pov}$, producing distributions $P_1, P_2, P_3$. At each stage, the agent moves to the maximizer of the current distribution (from $v_{30}$ to $v_9$, then to $v_2$), where the process stabilizes.
• Figure 4: Sensitivity of AUC to $\gamma$ (with $\lambda = 1-\gamma$) for different graph scales
• Figure 5: Increasing $m$ improves AUC-ROC, AP, and robustness to Gaussian noise ($\mu=0$, $\sigma^2=1$) on Disney

Theorems & Definitions (32)

• Definition 2.1
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Definition 3.4
• Theorem 3.5
• Theorem 3.6
• Theorem 3.7
• Definition 4.1
• Definition 4.2