[Paper Review] Do logarithmic proximity measures outperform plain ones in graph clustering?

Vladimir Ivashkin, Pavel Chebotarev|arXiv (Cornell University)|May 3, 2016

Advanced Clustering Algorithms Research4 citations

TL;DR

This paper investigates whether logarithmic transformations of graph proximity measures improve clustering performance compared to plain measures. It demonstrates that, in most cases—especially on random graphs with homogeneous classes—logarithmic versions of proximity measures like Communicability and Commute Time outperform their plain counterparts due to a fundamental compatibility with the additive nature of clustering algorithms. The logarithmic transformation converts multiplicative kernel properties into additive distance structures, enhancing cluster separation.

ABSTRACT

We consider a number of graph kernels and proximity measures including commute time kernel, regularized Laplacian kernel, heat kernel, exponential diffusion kernel (also called "communicability"), etc., and the corresponding distances as applied to clustering nodes in random graphs and several well-known datasets. The model of generating random graphs involves edge probabilities for the pairs of nodes that belong to the same class or different predefined classes of nodes. It turns out that in most cases, logarithmic measures (i.e., measures resulting after taking logarithm of the proximities) perform better while distinguishing underlying classes than the "plain" measures. A comparison in terms of reject curves of inter-class and intra-class distances confirms this conclusion. A similar conclusion can be made for several well-known datasets. A possible origin of this effect is that most kernels have a multiplicative nature, while the nature of distances used in cluster algorithms is an additive one (cf. the triangle inequality). The logarithmic transformation is a tool to transform the first nature to the second one. Moreover, some distances corresponding to the logarithmic measures possess a meaningful cutpoint additivity property. In our experiments, the leader is usually the logarithmic Communicability measure. However, we indicate some more complicated cases in which other measures, typically, Communicability and plain Walk, can be the winners.

Motivation & Objective

• To evaluate whether logarithmic transformations of graph proximity measures enhance clustering performance compared to plain measures.
• To investigate the underlying reason why logarithmic measures might outperform plain ones in clustering tasks.
• To compare a wide range of proximity measures—including heat kernel, communicability, walk, and commute time kernels—under both plain and logarithmic forms.
• To assess performance across diverse graph structures, including random graphs with varying class sizes and real-world datasets.
• To determine whether the superiority of logarithmic measures holds across different clustering algorithms and parameter settings.

Proposed method

• The study uses a random graph model $G(N, m, p_{\text{in}}, p_{\text{out}})$ with predefined classes and edge probabilities for intra- and inter-class connections.
• It evaluates 13 proximity measure families, including plain and logarithmic variants of Communicability, Walk, Heat, Forest, and Commute Time kernels.
• Performance is assessed via Copeland’s score in pairwise tournaments across 50 generated graphs per setting, using both optimal and 90th/80th percentile parameters.
• Inter-class and intra-class distances are compared using reject curves and ROC curves to assess separability without relying on a specific clustering algorithm.
• Theoretical analysis explores the duality between proximity measures and metrics, particularly focusing on the triangle inequality and the role of logarithmic transformation in aligning multiplicative kernels with additive distance structures.
• The study extends to real-world datasets (e.g., football, polbooks, Zachary) to validate findings beyond synthetic graphs.

Experimental results

Research questions

• RQ1 Do logarithmic proximity measures consistently outperform plain proximity measures in graph clustering tasks?
• RQ2 What is the underlying reason for the observed performance gain of logarithmic measures over plain ones?
• RQ3 How do different proximity measures—especially logarithmic variants—perform across diverse graph structures, including those with heterogeneous class sizes?
• RQ4 Are there specific network types or configurations where plain measures outperform logarithmic ones?
• RQ5 Does the logarithmic transformation effectively reconcile the multiplicative nature of graph kernels with the additive nature of clustering algorithms?

Key findings

• In most cases, particularly on random graphs with homogeneous classes, logarithmic proximity measures outperform plain measures in clustering performance, with log-Communicability emerging as the top performer.
• The logarithmic transformation improves clustering by converting the multiplicative nature of graph kernels into an additive distance structure compatible with the triangle inequality used in clustering algorithms.
• Log-Communicability achieved the highest Copeland score (62) in the best-parameter tournament on random graphs, followed by Sigmoid-Corrected Commute Time (SCCT) with 62 and log-Forest with 59.
• For real-world datasets like football and polbooks, log-Communicability again led with scores of 61 and 59, respectively, in the best-parameter tournament.
• The performance of logarithmic measures is less stable under suboptimal parameters: for example, logHeat and RSP dropped significantly in ranking when using 80th percentile parameters, indicating lower robustness.
• In complex, heterogeneous networks—especially those with unequal class sizes or many classes—plain measures like Comm and pWalk can outperform logarithmic variants, suggesting context-dependent superiority.