Motif-Based Spectral Clustering of Weighted Directed Networks (2004.01293v2)

Abstract: Clustering is an essential technique for network analysis, with applications in a diverse range of fields. Although spectral clustering is a popular and effective method, it fails to consider higher-order structure and can perform poorly on directed networks. One approach is to capture and cluster higher-order structures using motif adjacency matrices. However, current formulations fail to take edge weights into account, and thus are somewhat limited when weight is a key component of the network under study. We address these shortcomings by exploring motif-based weighted spectral clustering methods. We present new and computationally useful matrix formulae for motif adjacency matrices on weighted networks, which can be used to construct efficient algorithms for any anchored or non-anchored motif on three nodes. In a very sparse regime, our proposed method can handle graphs with a million nodes and tens of millions of edges. We further use our framework to construct a motif-based approach for clustering bipartite networks. We provide comprehensive experimental results, demonstrating (i) the scalability of our approach, (ii) advantages of higher-order clustering on synthetic examples, and (iii) the effectiveness of our techniques on a variety of real world data sets; and compare against several techniques from the literature. We conclude that motif-based spectral clustering is a valuable tool for analysis of directed and bipartite weighted networks, which is also scalable and easy to implement.

• The paper introduces a novel motif-based spectral clustering method that enhances detection of latent structures in weighted directed networks.
• It develops weighted motif adjacency matrices using mean-weighted edges to capture higher-order connections and improve clustering accuracy.
• The approach is validated on both synthetic and real-world datasets, demonstrating efficient performance on large, sparse graphs.

Motif-Based Spectral Clustering of Weighted Directed Networks

Introduction

The paper, "Motif-Based Spectral Clustering of Weighted Directed Networks," presents a significant extension to spectral clustering techniques by incorporating motif-based methods for weighted directed networks. Traditional spectral clustering often struggles with directed edges and higher-order network structures. This work addresses these limitations by introducing motif adjacency matrices (MAMs) tailored to capture motif-driven structures in weighted networks.

Methodology

Weighted Motif Adjacency Matrices

The core contribution of the paper is the development of motif-based methods that generalize traditional spectral clustering to harness both directed edges and edge weights. The authors extend the concept of MAMs to weighted contexts leveraging mean-weighted edges. This approach allows them to capture not just the presence but the significance of connections in terms of their weight, thus enabling a more nuanced clustering strategy.

The MAM for a weighted network, denoted as $M_{ij}$, is derived from counting the instances of motifs (pre-defined structures) that connect nodes $i$ and $j$, factoring in their weights.

Algorithmic Framework

The paper's algorithms are designed to scale efficiently, particularly in sparse scenarios common in large networks. The approach involves:

• Calculating the MAM using new matrix-based formulae.
• Performing spectral clustering on this MAM.

For computation, the authors leverage existing linear algebraic libraries and optimize their formulations to be near linear in the number of edges for sparse graphs.

Applications

Synthetic Examples

The authors validate their methods across synthetic datasets demonstrating:

1. The advantage of considering weights in detecting high-order clusters.
2. Differentiated insights yielded by various motifs even within similar datasets.
3. Enhanced performance in bipartite scenarios using specialized motifs.

Real-World Data

The paper applies its methodologies to several real-world networks, such as the US-Migration and US-Political-Blogs networks. These applications reveal how motif-based methods uncover underlying structures often missed by traditional approaches. For instance, migration patterns specific to weighted directed edges are more accurately captured by motifs that consider both direction and weight.

Figure 1: Motif-based clusterings of the US-Migration network.

Conclusions

The paper concludes by emphasizing motif-based spectral clustering as a versatile and scalable method, especially useful in networks where directionality and weight of edges are crucial. These methods facilitate uncovering hidden structures within complex datasets, making them invaluable in fields like biology, social science, and communication networks.

The authors also provide comprehensive software implementations that adhere to their proposed methodologies, available in both R and Python, supporting broader adoption and application across various domains.

Future Work

Future research directions include further optimizing the computational aspects of their methods, exploring motifs of larger cardinality, and applying the framework to a broader array of network types and structures. Integrating these methods into deep learning models remains an open and intriguing possibility, potentially enhancing the interpretability and performance of graph neural networks.