Topology Based Scalable Graph Kernels

TL;DR

This paper introduces a scalable graph kernel based on Ollivier Ricci curvature, enabling graph classification and comparison solely from topology, especially useful when node attributes are unavailable.

Contribution

It presents a novel curvature-based graph kernel leveraging Ollivier Ricci curvature for topology-only graph analysis.

Findings

Effective in classifying and clustering graphs based on curvature distributions.

Operates solely on graph topology, no node attributes needed.

Applicable to various graph comparison tasks.

Abstract

We propose a new graph kernel for graph classification and comparison using Ollivier Ricci curvature. The Ricci curvature of an edge in a graph describes the connectivity in the local neighborhood. An edge in a densely connected neighborhood has positive curvature and an edge serving as a local bridge has negative curvature. We use the edge curvature distribution to form a graph kernel which is then used to compare and cluster graphs. The curvature kernel uses purely the graph topology and thereby works for settings when node attributes are not available.

Full text

11institutetext: Stony Brook Uni., Stony Brook NY 11794, USA 11email: {kiliu,jgao}@cs.stonybrook.edu 22institutetext: Yahoo Research, Sunnyvale CA 94089, USA 22email: cni02@oath.com 33institutetext: Intel Inc., Hillsboro OR 97124, USA 33email: yu-yao.lin@intel.com

Topology Based Scalable Graph Kernels

Kin Sum Liu 11   

Chien-Chun Ni 22   

Yu-Yao Lin 33   

Jie Gao 11

1 Introduction

We propose a new graph kernel for graph classification and comparison using Ollivier Ricci curvature. The Ricci curvature of an edge in a graph describes the connectivity in the local neighborhood. An edge in a densely connected neighborhood has positive curvature and an edge serving as a local bridge has negative curvature. We use the edge curvature distribution to form a graph kernel which is then used to compare and cluster graphs. The curvature kernel uses purely the graph topology and thereby works for settings when node attributes are not available. The computation of the curvature for an edge uses only information within two hops from the edge and a random sample of $O(1/{\varepsilon^{2}}\log 1/{\varepsilon}+1/{\varepsilon^{2}}\log 1/{\delta})$ edges in a large graph can produce a good approximation to the curvature distribution with error bounded by $\varepsilon$ with probability at least $1-\delta$. Thus, one can compute the graph kernel for really large graphs that some other graph kernels cannot handle. This Ricci curvature kernel is extensively tested on graphs generated by different generative models as well as standard benchmark datasets from bioinformatics and Internet AS network topologies.

Graph classification and comparison are widely applied in bioinformatics, vision and social network analysis. One of the most popular approaches in practice is using graph kernels which compute the similarity of two graphs in terms of subgraph structures. Many graph kernels have been developed, which differ by the subgraph structures they focus on, such as random walks [2], shortest paths [1], subtrees [5], and cycles [3]. Graph kernels have been extensively tested on benchmark datasets from bioinformatics to chemistry [1, 4].

In our work, we focus on the setting of unlabeled graphs and propose a new graph kernel based on discrete Ricci curvature which takes only the network topology as an input. Our work is motivated by the use of curvature related kernels in shape matching. Curvature on a smooth surface defines the amount by which a geometric object deviates from being flat or straight. It is a local measure at each point but nevertheless has deep connections to global topology and structures. Despite the success in shape matching, curvature has not been used much for comparing graphs. In this paper, we propose to use curvature distribution of graphs to build new graph kernels. The goal is to demonstrate that the Ricci curvature distribution and kernels can be efficiently computed and capture interesting graph properties. They add to the family of graph features and kernels, could be combined with other attributes, and used for other classifiers.

2 Discrete Ricci Curvature

For an edge $\overline{uv}$, define a distribution $m_{u},m_{v}$ on the neighborhood of $u,v$ respectively (such as uniform $m_{u}$ and $m_{v}$). Now compute the Earth Mover Distance $W(u,v)$ from $m_{u}$ to $m_{v}$, where the cost of moving mass from a neighbor $u_{i}$ of $u$ to a neighbor $v_{j}$ of $v$ is the shortest path distance in the graph. Here the edges are unweighted unless they inherit weights arising from the application domain, such as tie strength or distances. For example, $W(u,v)$ will be upper bounded by 2 for unweighted graph if we allocate 50% mass to the node’s neighbor. The Ollivier-Ricci curvature is defined as $w(\overline{uv})=1-W(u,v)/d(u,v)$, where $d(u,v)$ is the length of edge $\overline{uv}$. Intuitively the curvature captures the structural properties of the local neighborhood. If $\overline{uv}$ stays in a well connected, dense neighborhood, the curvature is positive; if $\overline{uv}$ is locally a bridge, its curvature is negative.

3 Ricci Curvature Graph Kernel

We define the Ricci Curvature Kernel as the following. Denote the curvature distribution of all edges in $G$ by $\textit{D}(G)$ and that of $G^{\prime}$ by $\textit{D}(G^{\prime})$. We use the standard Gaussian RBF kernel: $k(G,G^{\prime})=\exp({-||\textit{D}(G)-\textit{D}(G^{\prime})||^{2}_{2}/2\sigma^{2}}),$ where $||\textit{D}(G)-\textit{D}(G^{\prime})||_{2}$ is the $\ell_{2}$ norm of two vectors $\textit{D}(G)$, $\textit{D}(G^{\prime})$. Since the kernel depends on the curvature distribution, the distribution is less robust statistically for small graphs. We could boost up the kernel by considering the curvature distribution for pairs of neighboring edges $\{(w(e),w(e^{\prime}))\}$. It appears to be more effective in practice. See Figure 1 for an example. When the graph is really large, computing the curvature distribution might be costly ($O(|G.E|*n^{3})$ where $n$ is the size of concerned neighborhood). Random sampling can be used to approximate the curvature distribution. Taking $O(1/{\varepsilon^{2}}\log 1/{\varepsilon}+1/{\varepsilon^{2}}\log 1/{\delta})$ edges uniformly at random from the graph $G$, it can be shown that $\hat{\textit{D}}(G)$, the curvature distribution on the sampled edges, is a good approximation of $\textit{D}(G)$ with error bound $\varepsilon$ with probability $1-\delta$. Notice that the running time does not even depend on the size of the graph $n$.