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Size reduction of complex networks preserving modularity
A. Arenas, J. Duch, A. Fernandez, S. Gomez
TL;DR
Identifying modular structure requires optimizing modularity, an NP-hard task. The paper proposes exact size reductions for weighted directed and undirected networks that preserve modularity, and reports improved heuristic optimization after reduction, including scale-free-network hair estimates of roughly 61–83% of nodes.
Problem
Finding the best community partition by modularity is computationally difficult because the optimization is NP-hard.
Method
The paper develops exact size-reduction rules for weighted directed and undirected networks, grouping nodes while preserving the modularity of corresponding partitions.
Results
The reduction enables more exhaustive partition searches and improves modularity results with Extremal Optimization; in scale-free networks, estimated hair-node fractions range from approximately 61% to 83%.
Takeaways & Limitations
Modularity-preserving reduction can lower computational cost and improve the accuracy of heuristics that optimize modularity, while potentially supporting future coarse graining of network dynamics.
Takeaways & Limitations
Simply removing isolated nodes fails to reproduce the original modularity and can yield misleading results.
Abstract
from arXiv · showhide
The ubiquity of modular structure in real-world complex networks is being the focus of attention in many trials to understand the interplay between network topology and functionality. The best approaches to the identification of modular structure are based on the optimization of a quality function known as modularity. However this optimization is a hard task provided that the computational complexity of the problem is in the NP-hard class. Here we propose an exact method for reducing the size of weighted (directed and undirected) complex networks while maintaining invariant its modularity. This size reduction allows the heuristic algorithms that optimize modularity for a better exploration of the modularity landscape. We compare the modularity obtained in several real complex-networks by using the Extremal Optimization algorithm, before and after the size reduction, showing the improvement obtained. We speculate that the proposed analytical size reduction could be extended to an exact coarse graining of the network in the scope of real-space renormalization.
A Arenas‡, J Duch, A Fern´andez and S G´omez
The paper lists A Arenas, J Duch, A Fern´andez, and S G´omez as authors, affiliated with Universitat Rovira i Virgili in Tarragona, Spain, and submitted to New J. Phys.
- The authors are A Arenas, J Duch, A Fern´andez, and S G´omez.
- Their affiliation is the Departament d’Enginyeria Inform`atica i Matem`atiques, Universitat Rovira i Virgili, Tarragona, Spain.
- The manuscript was submitted to New J. Phys.
1. Introduction
Community detection is important for relating network structure to functionality, but optimizing modularity is computationally difficult because the number of partitions grows exponentially and the problem is NP-hard.
- Community structure groups nodes that are more intraconnected than interconnected between groups.This description may help elucidate functional organization and resembles mesoscale coarse graining in statistical physics.
- Modularity compares within-community edge probability with its strength-preserving random-network expectation.Higher modularity indicates a partition that deviates more from the null case.
- Modularity optimization is NP-hard because the number of possible partitions grows at least exponentially with network size.Heuristic methods are therefore used to find suboptimal solutions within feasible computational time.
- Weighted directed networks extend modularity using separate output and input node strengths.For undirected networks, these strengths are equal and recover the standard strength definition; for unweighted undirected networks, strength becomes degree.
- The proposed reduction preserves modularity independently of the partition and supports more exhaustive partition-space searches.The paper evaluates the approach by comparing Extremal Optimization results before and after reduction on several real networks.
2. Size reduction preserving modularity
The method aggregates groups of nodes into reduced-network nodes by summing intergroup weights, while preserving modularity for corresponding partitions. Self-loops retain intragroup connectivity, and reduced-node strengths preserve the network’s total strength.
- 2.1. Reduced graph: A surjective function assigns each original node to one of N′ groups, which are replaced by single nodes in the reduced network.The construction applies to weighted networks and permits directed, nonsymmetric weight matrices.
- 2.1. Reduced graph: Each reduced edge weight equals the sum of weights connecting the corresponding source and target groups.In unweighted networks, this weight counts arcs between the groups.
- 2.1. Reduced graph: A reduced node receives a self-loop when its group has nonzero internal connectivity.The self-loop summarizes connectivity among the original nodes within that group.
- 2.1. Reduced graph: The reduced network’s total strength equals the original network’s total strength.This preserves the aggregate strength used in modularity calculations.
- 2.2. Modularity preservation: For any reduced-network partition, the corresponding original-network partition has exactly the same modularity.The corresponding partition is obtained by composing the reduced partition with the reduction function, C = C′ ◦ R.
- 2.2. Modularity preservation: The reduction is exact when grouped nodes belong to the same community in the optimal partition.Each reduced node retains the self-loop and external arcs needed to calculate modularity, but identifying such groups beforehand remains the central challenge.
3. Analytic reductions
The paper proves exact analytic reductions for weighted undirected and directed networks that preserve modularity across corresponding partitions. It characterizes reducible structures including hairs, triangular hairs, sources, and sinks, while identifying a limitation for isolated nodes in directed networks.
- Undirected reductions: Hairs can be analytically grouped with their neighbor when the modularity condition is satisfied, always holding for undirected hairs without self-loops.The grouping produces a self-loop on the neighboring node.
- Undirected reductions: Removing isolated nodes by assumption is invalid because it fails to reproduce the original modularity and can produce misleading results.The paper proves that isolated nodes cannot occur in an optimal undirected partition unless their self-loop is sufficiently large.
- Undirected reductions: Triangular hairs can be grouped because their two peripheral nodes share the same community in the optimal partition, after which the resulting structure becomes a simple hair.Without self-loops, the triangular hair yields a self-loop w′_hh = 2w_ij.
- Directed reductions: Directed reductions distinguish input and output strengths, allowing analogous grouping of directed hairs and triangular hairs formed by a source and a sink.Directed hairs without self-loops can be grouped, while source and sink nodes cannot have self-loops by definition.
- Directed reductions: Unlike the undirected case, isolated nodes may occur in a directed partition of maximum modularity because the relevant modularity change is not guaranteed positive.Hairs, sinks, and sources nevertheless cannot be isolated when their self-loops are not too large.
4. Results and discussion
The proposed reductions preserve modularity while substantially shrinking networks, enabling more exhaustive modularity optimization. Estimates and real-network experiments indicate computational gains without sacrificing information, with especially broad validation on airport-network variants.
- Size reduction: Exact size reduction preserves modularity and keeps the structural mesoscale found by modularity maximization invariant.The procedure is analytically justified for the stated reductions.
- Size reduction: 61%–83% of nodes correspond roughly to hairs in scale-free networks with γ between 2 and 3.The estimate is slightly reduced when real-distribution cut-offs are included.
- Size reduction: 40%–77% reduction is estimated for exponential degree distributions with β between 0.5 and 1.5.These estimates use the percentage of hairs P(1) = 1 −e−β.
- Results and discussion: EO optimization after reduction searches the partition space more exhaustively and improves modularity while reducing computational cost.Table 1 reports node counts, modularity, community counts, and algorithm speed-up before and after reduction.
- Results and discussion: Airport networks test the method across undirected unweighted, undirected weighted, and weighted directed representations.The weighted directed and undirected cases produce modularity values that are very close.
- Results and discussion: The authors propose extending exact reduction to other motifs and exploring reduced networks for real-space renormalization of dynamical interactions.These extensions are presented as plausible future directions rather than completed results.
Appendix A. Relationship between directed and undirected modularities
The appendix relates directed modularity to the modularity of a symmetrized undirected network. Directed modularity is unchanged by transposing the weight matrix, while the two modularities become approximately equal when few links cross community boundaries.
- The corresponding undirected network is constructed by adding the weights matrix to its transpose.Its node strengths and total strength are then defined from this symmetrized network.
- Directed modularity QD is invariant under transposing the network’s weights matrix.The transposed network exchanges input and output strengths, leaving QD unchanged.
- The appendix derives a relationship between directed modularity QD and symmetrized modularity QS, including an equivalent communities-sum expression.The relationship follows from direct calculations.
- QS ≈QD when most links remain within communities, even for highly asymmetric networks.Input- and output-strength contributions cancel for links that fall within communities.