Generating Hierarchically Modular Networks via Link Switching

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.

💡 Research Summary

The paper presents a novel algorithm for generating hierarchically modular networks while preserving a prescribed degree sequence, without relying on probabilistic link formation. The core idea is to define a “edge distance” between any two nodes based on a user‑specified binary tree decomposition topology (T). In this tree, internal nodes represent modules and leaf nodes correspond to actual network nodes. The edge distance of an edge (x, y) is the length of the longest shared internal path of x and y in T; larger distances indicate that the two nodes are more closely related (i.e., likely belong to the same module).

The algorithm proceeds in six steps: (1) create a random simple graph G₀ that exactly matches the given degree list (ndl); (2) randomize G₀ by uniformly swapping edge pairs to eliminate construction bias, yielding Gᵣ; (3) construct the decomposition topology T according to user‑defined parameters such as minimum module size; (4) compute edge distances for all edges in Gᵣ and obtain the average edge distance aed(Gᵣ); (5) modularize Gᵣ by repeatedly selecting edge pairs from a list weighted by the complementary edge distance (ced = ed_max – ed + 1). For each selected pair, the algorithm evaluates the four possible rewiring outcomes and chooses the one that maximizes the product of the two new edge distances (ped). This process preferentially creates edges with larger edge distances, thereby strengthening intra‑module connections and weakening inter‑module links. The number of iterations is controlled by a parameter P_g; the total number of swaps is P_g·