Modeling Topological Impact on Node Attribute Distributions in Attributed Graphs

TL;DR

This paper presents an algebraic framework to understand how graph topology influences node attribute distributions, offering a new perspective on attributed graphs and their analysis.

Contribution

It introduces a categorical and algebraic approach to model the interaction between topology and attributes, and provides a formal framework for topology-conditioned attribute distributions.

Findings

On complete graphs, the model recovers original attribute distributions.

The approach effectively captures topological effects on attribute distributions.

An anomaly detection testbed demonstrates the framework's applicability.

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.