The Parameterised Complexity of Computing the Maximum Modularity of a Graph

Meeks, Kitty; Skerman, Fiona

doi:10.1007/s00453-019-00649-7

Cited by 5 publications

(2 citation statements)

References 25 publications

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“…The general problem of correlation clustering (Bansal et al, 2004) and the subproblem of maximizing modularity (Brandes et al, 2007;Meeks and Skerman, 2020) are known to be NPcomplete. However, modularity maximization is known to be Fixed-Parameter Tractable (FPT) when parametrized by the size of the minimum vertex cover of the graph (Meeks and Skerman, Frontiers in Complex Systems frontiersin.org 2020).…”

Section: Exact Optimizationmentioning

confidence: 99%

The projection method: a unified formalism for community detection

Gösgens,

van der Hofstad,

Litvak

2024

Front. Complex Syst.

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We present the class of projection methods for community detection that generalizes many popular community detection methods. In this framework, we represent each clustering (partition) by a vector on a high-dimensional hypersphere. A community detection method is a projection method if it can be described by the following two-step approach: 1) the graph is mapped to a query vector on the hypersphere; and 2) the query vector is projected on the set of clustering vectors. This last projection step is performed by minimizing the distance between the query vector and the clustering vector, over the set of clusterings. We prove that optimizing Markov stability, modularity, the likelihood of planted partition models and correlation clustering fit this framework. A consequence of this equivalence is that algorithms for each of these methods can be modified to perform the projection step in our framework. In addition, we show that these different methods suffer from the same granularity problem: they have parameters that control the granularity of the resulting clustering, but choosing these to obtain clusterings of the desired granularity is nontrivial. We provide a general heuristic to address this granularity problem, which can be applied to any projection method. Finally, we show how, given a generator of graphs with community structure, we can optimize a projection method for this generator in order to obtain a community detection method that performs well on this generator.

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Section: Exact Optimizationmentioning

confidence: 99%

The projection method: a unified formalism for community detection

Gösgens,

van der Hofstad,

Litvak

2024

Front. Complex Syst.

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“…Furthermore it is NP-hard to approximate modularity to within any constant multiplicative factor [12]. Modularity maximisation is also W [1]-hard, a measure of hardness in parameterised complexity, when parameterised by pathwidth; but approximating modularity to within multiplicative error 1 ± ε is fixed parameter tractable when parameterised by treewidth [35]. The reduction in [6] required some properties of optimal partitions; for example it was shown that a vertex of degree 1 will be placed in the same part as its neighbour in every optimal partition.…”

Section: Properties Of Modularitymentioning

confidence: 99%

Modularity of Erdős‐Rényi random graphs

McDiarmid

Skerman

2020

Random Struct Algorithms

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For a given graph G, each partition of the vertices has a modularity score, with higher values taken to indicate that the partition better captures community structure in G. The modularity q * (G) (where 0 ≤ q * (G) ≤ 1) of the graph G is defined to be the maximum over all vertex partitions of the modularity score. Modularity is at the heart of the most popular algorithms for community detection, so it is an important graph parameter to understand mathematically.In particular, we may want to understand the behaviour of modularity for the Erdős-Rényi random graph G n,p with n vertices and edge-probability p. Two key features which we find are that the modularity is 1 + o(1) with high probability (whp) for np up to 1 + o(1) (and no further); and when np ≥ 1 and p is bounded below 1, it has order (np) −1/2 whp, in accord with a conjecture by Reichardt and Bornholdt in 2006 (and disproving another conjecture from the physics literature).

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Heuristic Modularity Maximization Algorithms for Community Detection Rarely Return an Optimal Partition or Anything Similar

Aref

Mostajabdaveh

Hriday

2023

Lecture Notes in Computer Science

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Community detection is a fundamental problem in computational sciences with extensive applications in various fields. The most commonly used methods are the algorithms designed to maximize modularity over different partitions of the network nodes. Using 80 real and random networks from a wide range of contexts, we investigate the extent to which current heuristic modularity maximization algorithms succeed in returning maximum-modularity (optimal) partitions. We evaluate (1) the ratio of the algorithms’ output modularity to the maximum modularity for each input graph, and (2) the maximum similarity between their output partition and any optimal partition of that graph. We compare eight existing heuristic algorithms against an exact integer programming method that globally maximizes modularity. The average modularity-based heuristic algorithm returns optimal partitions for only 19.4% of the 80 graphs considered. Additionally, results on adjusted mutual information reveal substantial dissimilarity between the sub-optimal partitions and any optimal partition of the networks in our experiments. More importantly, our results show that near-optimal partitions are often disproportionately dissimilar to any optimal partition. Taken together, our analysis points to a crucial limitation of commonly used modularity-based heuristics for discovering communities: they rarely produce an optimal partition or a partition resembling an optimal partition. If modularity is to be used for detecting communities, exact or approximate optimization algorithms are recommendable for a more methodologically sound usage of modularity within its applicability limits.

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The Parameterised Complexity of Computing the Maximum Modularity of a Graph

Cited by 5 publications

References 25 publications

The projection method: a unified formalism for community detection

The projection method: a unified formalism for community detection

Modularity of Erdős‐Rényi random graphs

Heuristic Modularity Maximization Algorithms for Community Detection Rarely Return an Optimal Partition or Anything Similar

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