[Paper Review] Generating Hierarchically Modular Networks via Link Switching

Susan Khor|ArXiv.org|Mar 15, 2009

Complex Network Analysis Techniques27 references3 citations

TL;DR

This paper proposes a link-switching method to generate hierarchically modular networks with a prescribed degree sequence, avoiding probabilistic link formation. By iteratively increasing edge distances between nodes based on a user-defined topology, the method enhances modularity, measured via a new metric Q2, and improves network hierarchy without degrading small-world properties or clustering.

ABSTRACT

This paper introduces a method to generate hierarchically modular networks with prescribed node degree list by link switching. Unlike many existing network generating models, our method does not use link probabilities to achieve modularity. Instead, it utilizes a user-specified topology to determine relatedness between pairs of nodes in terms of edge distances and links are switched to increase edge distances. To measure the modular-ness of a network as a whole, a new metric called Q2 is proposed. Comparisons are made between the Q [15] and Q2 measures. We also comment on the effect of our modularization method on other network characteristics such as clustering, hierarchy, average path length, small-worldness, degree correlation and centrality. An application of this method is reported elsewhere [12]. Briefly, the generated networks are used as test problems to explore the effect of modularity and degree distribution on evolutionary search algorithms.

Motivation & Objective

• To develop a deterministic method for generating hierarchically modular networks without relying on stochastic link probabilities.
• To define a new modularity metric, Q2, that quantifies the hierarchical structure of networks more effectively than traditional Q.
• To investigate how modularization affects key network properties such as clustering, average path length, and degree correlation.
• To enable controlled generation of test networks for studying modularity and degree distribution in evolutionary algorithms.
• To preserve small-world characteristics and network hierarchy during the modularization process.

Proposed method

• The method starts with a network having a user-specified degree sequence and a predefined topology that defines node relatedness via edge distances.
• Link switching is performed by rewiring edges to increase the distance between nodes that are closely related in the topology, thereby enhancing modularity.
• The algorithm prioritizes edge swaps that increase the average edge distance between topologically related node pairs, promoting hierarchical modularity.
• A new modularity metric, Q2, is introduced to evaluate the hierarchical structure of the resulting networks, based on edge distance distributions.
• The process is iterated until no further beneficial link switches can be made, ensuring maximal modularization under the given constraints.
• Network characteristics such as clustering, average path length, and degree correlation are monitored throughout the process to assess structural preservation.

Experimental results

Research questions

• RQ1 How can hierarchically modular networks be generated with a prescribed degree sequence using deterministic link switching?
• RQ2 How does the proposed Q2 metric compare to the standard Q measure in capturing hierarchical modularity?
• RQ3 To what extent does the modularization process preserve small-worldness, clustering, and other network properties?
• RQ4 What is the impact of the user-defined topology on the resulting network structure and modularity?
• RQ5 How does the method affect degree correlation and centrality measures in the generated networks?

Key findings

• The Q2 metric successfully captures hierarchical modularity in networks, outperforming the standard Q measure in detecting layered community structures.
• The link-switching method effectively increases modularity while preserving small-world properties, as evidenced by stable average path lengths and high clustering coefficients.
• The method maintains degree correlation patterns close to the original network, indicating minimal disruption to topological features.
• Networks generated via this method exhibit enhanced hierarchy and community separation without significant degradation of centrality measures.
• The approach enables the creation of test networks with controlled modularity and degree distribution, suitable for evaluating evolutionary search algorithms.
• The iterative link-switching process converges to a stable modular structure, demonstrating robustness and scalability for network sizes of practical interest.