[Paper Review] Analysis of data in the form of graphs

Karthikeyan Rajendran, Ioannis G. Kevrekidis|arXiv (Cornell University)|Jun 14, 2013

Complex Network Analysis Techniques1 references3 citations

TL;DR

This paper proposes a data mining framework for ensembles of graphs by introducing two similarity metrics—subgraph density and spectral analysis—for non-linear dimensionality reduction via diffusion maps. It demonstrates that these methods can effectively identify low-dimensional, meaningful representations of graph data, enabling coarse-grained modeling of complex network systems, with validation on synthetic and dynamic network ensembles.

ABSTRACT

We discuss the problem of extending data mining approaches to cases in which data points arise in the form of individual graphs. Being able to find the intrinsic low-dimensionality in ensembles of graphs can be useful in a variety of modeling contexts, especially when coarse-graining the detailed graph information is of interest. One of the main challenges in mining graph data is the definition of a suitable pairwise similarity metric in the space of graphs. We explore two practical solutions to solving this problem: one based on finding subgraph densities, and one using spectral information. The approach is illustrated on three test data sets (ensembles of graphs); two of these are obtained from standard graph generating algorithms, while the graphs in the third example are sampled as dynamic snapshots from an evolving network simulation.

Motivation & Objective

• To address the challenge of applying data mining techniques to datasets where each data point is a graph, rather than a vector in Euclidean space.
• To identify intrinsic low-dimensional structures in ensembles of graphs, which is essential for coarse-graining complex network dynamics.
• To develop and compare two practical similarity metrics—subgraph density and spectral methods—for measuring graph-to-graph similarity.
• To validate the proposed approach on synthetic and dynamic network datasets, comparing results to known parameterizations.
• To support the use of data-mined observables in equation-free modeling frameworks for network evolution.

Proposed method

• Uses diffusion maps for non-linear dimensionality reduction, which embeds graph data into a lower-dimensional space based on a similarity matrix.
• Employs a subgraph density approach to compute similarity by comparing the frequency and distribution of small, connected subgraphs across graphs.
• Applies a spectral method based on the eigenstructure of the graph Laplacian or transition matrix to define similarity via diffusion processes.
• Tunes parameters such as neighborhood size (ε) in diffusion maps and weighting functions (μ(k)) in spectral methods to optimize performance.
• Combines similarity metrics with data mining algorithms to extract reduced sets of observables that capture the essential features of graph ensembles.
• Extends the framework to handle graphs of varying sizes using weighted graph size in the similarity measure.

Experimental results

Research questions

• RQ1 How can graph data be effectively represented in a low-dimensional space suitable for modeling and analysis?
• RQ2 What are suitable pairwise similarity metrics between graphs that preserve structural and topological information?
• RQ3 How do subgraph density and spectral similarity methods compare in their ability to recover known network parameters from graph ensembles?
• RQ4 Can data-mined observables from graph ensembles serve as effective coarse variables in equation-free modeling of network dynamics?
• RQ5 What are the computational trade-offs between subgraph-based and spectral similarity methods in large-scale graph data mining?

Key findings

• The subgraph density method provides a more computationally expensive but potentially more accurate measure of graph similarity by capturing local structural motifs.
• The spectral method offers a faster alternative for similarity computation, especially when large subgraphs are involved, but requires careful tuning of parameters like ε and μ(k).
• Both methods successfully identified low-dimensional embeddings of graph ensembles that correlated with known generating parameters in synthetic models.
• On Erdős–Rényi and Chung–Lu type graph ensembles, the data mining approach recovered the underlying model parameters with high fidelity.
• For dynamic network snapshots, the method revealed slow, coarse-grained variables that tracked the evolution of network structure over time.
• The framework supports the use of data-mined observables as candidates for coarse variables in equation-free modeling, enabling simulation-based coarse-scale computation without explicit macroscopic equations.