[Paper Review] Clustering as a measure of the local topology of networks

Alexandre Hannud Abdo, Alessandro P. S. de Moura|ArXiv.org|May 26, 2006

Complex Network Analysis Techniques2 references7 citations

TL;DR

This paper introduces the clustering profile—a multi-order extension of the traditional clustering coefficient that quantifies local network topology by measuring the fraction of neighboring node pairs at increasing distances from a given vertex. It reveals that high-degree nodes in real networks like the metabolic network and the World Wide Web exhibit distinct clustering decay patterns (exponential vs. power-law), exposing structural differences and constraining network growth models.

ABSTRACT

Usual formulations of the clustering coefficient can be shown to be insufficient in the task of describing the local topology of very simple networks. Motivated by this, we review some alternatives in order to present an extension, the clustering profile. We show, both conceptually and through applications to well studied networks, that this measure is a more complete and robust measure of clustering. It imposes stringent constraints on theoretical growth models, specially on aspects of the network structure that play a central role in dynamics on networks. In addition, we study how it provides a richer perspective of phenomena such as hierarchy, small-worlds and clusterization.

Motivation & Objective

• Address the limitations of the standard clustering coefficient in capturing nuanced local network structures.
• Overcome the insensitivity of traditional clustering measures to local topology in networks like lattices and bipartite graphs.
• Develop a more comprehensive and robust measure of local topology that applies uniformly across different network types.
• Provide a framework to evaluate and improve network growth models by revealing structural constraints in local organization.
• Investigate how clustering behavior varies with vertex degree and network type, especially in highly connected nodes.

Proposed method

• Define higher-order clustering coefficients as the fraction of neighboring node pairs at distance d from each other, excluding paths through the central vertex.
• Construct the clustering profile C^d(v) as a function of distance d for each vertex v, capturing multi-scale local connectivity.
• Aggregate the clustering profile across vertices by degree to analyze degree-dependent clustering behavior.
• Use graph-tool software to compute higher-order clustering coefficients efficiently on real-world networks.
• Rebin and log-rebin data to produce stable, interpretable visualizations of clustering decay across distances.
• Compare clustering decay patterns (exponential vs. power-law) in high-degree nodes of the metabolic network and the WWW.

Experimental results

Research questions

• RQ1 Why do traditional clustering coefficients fail to distinguish between structurally different networks like a square lattice and a single cycle?
• RQ2 How can clustering be generalized beyond the first-order measure to capture richer local topological features?
• RQ3 What structural differences emerge when analyzing clustering decay across multiple distance orders in real networks?
• RQ4 How does the clustering profile constrain the design of network growth models, especially those involving community structure or preferential attachment?
• RQ5 Why do high-degree nodes in the WWW and metabolic networks show different clustering decay behaviors (power-law vs. exponential)?

Key findings

• The clustering profile reveals that the metabolic network exhibits exponential clustering decay for high-degree nodes, while the WWW shows a power-law decay, indicating a structural shift in the most connected vertices.
• For medium-degree nodes, both the metabolic network and the WWW display exponential decay of clustering coefficients with increasing distance, suggesting a common local organization at intermediate scales.
• The first and second orders of clustering in the metabolic network are consistently lower than expected from extrapolation, indicating a suppression effect possibly due to selective pressure against pathway congestion.
• In the WWW, only the first-order clustering is suppressed at high degrees, suggesting a degree-dependent dynamic rule that affects early-order clustering more strongly.
• The clustering profile provides a more robust and comprehensive characterization of local topology than the standard clustering coefficient, especially for highly connected nodes.
• The method exposes hidden structural constraints in network growth models, particularly those relying on preferential attachment or community formation, by revealing deviations in clustering decay patterns.