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arxiv: 1906.12204 · v1 · pith:QYG7QP3Snew · submitted 2019-06-27 · 💻 cs.SI · physics.data-an· physics.soc-ph

Modularity in Multilayer Networks using Redundancy-based Resolution and Projection-based Inter-Layer Coupling

Alessia Amelio , Giuseppe Mangioni , Andrea Tagarelli This is my paper

Pith reviewed 2026-05-25 13:59 UTC · model grok-4.3

classification 💻 cs.SI physics.data-anphysics.soc-ph
keywords multilayer networksmodularitycommunity detectionresolution factorinter-layer couplingdynamic networksmultislice modularity
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The pith

A revised multilayer modularity sets its resolution and coupling factors from community structures and layer orderings rather than arbitrary choices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a new modularity formulation for multilayer networks that derives the resolution factor from layer-specific relevance in the within-layer community structure and the inter-layer coupling factor from connections across layers. It extends the standard multislice modularity to explicitly handle orderings among layers and the resulting constraints on how layers couple. This addresses the limitation that prior parameters could be chosen without regard to network topology or any available community information. A reader would care because the approach aims to make modularity more directly reflective of the multilayer topology, particularly for ordered cases such as dynamic networks.

Core claim

The proposed multilayer modularity revises the semantics of the resolution and inter-layer coupling factors using information from the within-layer and inter-layer structures of the multilayer communities, and is general enough to incorporate orderings of the network layers along with the constraints those orderings impose on layer coupling.

What carries the argument

The revised modularity function whose resolution and inter-layer coupling terms are defined directly from within-layer and inter-layer community structure, extended to accept layer orderings as constraints on coupling.

If this is right

  • Experiments on synthetic and real-world multilayer networks can reveal the effects of different combinations of the resolution and inter-layer coupling functions.
  • The formulation can be paired with existing state-of-the-art multilayer community detection methods.
  • The work supplies a foundation for developing new optimization algorithms that exploit the revised modularity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the parameters are now tied to observed community structure, the approach may reduce the sensitivity of detected communities to manual parameter tuning.
  • The explicit handling of layer orderings could improve community detection accuracy on networks whose layers represent sequential time steps.
  • The same structure-based definition of coupling might be adapted to other multilayer settings that contain natural orderings, such as hierarchical or spatial networks.

Load-bearing premise

The within-layer and inter-layer structures of the multilayer communities supply independent information that can be used to define the resolution and coupling functions without introducing circular dependence on the communities the modularity is meant to discover.

What would settle it

On synthetic multilayer networks with known ground-truth communities, if the new structure-derived resolution and coupling functions produce partitions whose agreement with the ground truth is no higher than partitions obtained with arbitrary parameter values, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 1906.12204 by Alessia Amelio, Andrea Tagarelli, Giuseppe Mangioni.

Figure 1
Figure 1. Figure 1: Example multilayer network. The ordering over the set of layers [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Multilayer network for our running example [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Multilayer modularity Q on PMM community structure solutions nodes (resp. edges) in a layer Li that belong to C, averaged over all layers in the network [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Pearson correlation coefficient between average path length (APL), [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Pearson correlation coefficient between average path length (APL), [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Multilayer modularity Q of PMM solutions with layer ordering. APL CC NC EC RED 0 0.25 0.5 0.75 1 Correlation ICia Adj IC oa Adj ICia Suc IC oa Suc [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Pearson correlation coefficient between average path length (APL), [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Mucha et al.’s modularity (Qms) by varying γ with ω = 1 − γ 6.3 Analysis of Qms and comparison with Q We discuss here performance results obtained by the community detection al￾gorithms with Qms as assessment criterion. We will refer to the default setting of unordered set of layers as stated in [15]. Using GL, Qms tends to decrease as γ increases (while ω decreases, as it was varied with γ as ω = 1−γ). Th… view at source ↗
Figure 9
Figure 9. Figure 9: Mucha et al.’s modularity (Qms) by varying ω with γ = 1 to -0.05 on Flickr, from 0.854 to -4 on AUCS). Remarkably, the simultaneous effect of γ and ω = 1 − γ on Qms leads on some datasets (Obama, EU-Air, London) to a drastic degradation of modularity (down to much negative values) for some γ > 1, followed by a rapid increase to modularity of 1 as γ increases closely to 2. Analogous considerations hold for … view at source ↗
Figure 10
Figure 10. Figure 10: Mucha et al.’s modularity (Qms) by varying γ with ω = 1 − γ and by varying ω with γ = 1, on the ground-truth community structure of AUCS. stable at 1 for ω > 0.8. Variations are always on positive intervals (e.g., from 0.248 to 0.621 on Flickr, from 0.305 to 0.541 on FF-TW-YT, from 0.136 to 0.356 on Higgs-Twitter ). 6.3.1 Comparison between Q and Qms: qualitative evaluation on the solutions generated by t… view at source ↗
Figure 11
Figure 11. Figure 11: Computation time (in seconds) of the multilayer modularity [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Computation time (in seconds) of the multilayer modularity [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Pearson correlation coefficient between average path length (APL), [PITH_FULL_IMAGE:figures/full_fig_p044_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Computation time (in seconds) of the multilayer modularity [PITH_FULL_IMAGE:figures/full_fig_p045_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Computation time (in seconds) of the multilayer modularity [PITH_FULL_IMAGE:figures/full_fig_p046_15.png] view at source ↗
read the original abstract

The generalized version of modularity for multilayer networks, a.k.a. multislice modularity, is characterized by two model parameters, namely resolution factor and inter-layer coupling factor. The former corresponds to a notion of layer-specific relevance, whereas the inter-layer coupling factor represents the strength of node connections across the network layers. Despite the potential of this approach, the setting of both parameters can be arbitrarily selected, without considering specific characteristics from the topology of the multilayer network as well as from an available community structure. Also, the multislice modularity is not designed to explicitly model order relations over the layers, which is of prior importance for dynamic networks. This paper aims to overcome the main limitations of the multislice modularity by introducing a new formulation of modularity for multilayer networks. We revise the role and semantics of both the resolution and inter-layer coupling factors based on information available from the within-layer and inter-layer structures of the multilayer communities. Also, our proposed multilayer modularity is general enough to consider orderings of network layers and their constraints on layer coupling. Experiments were carried out on synthetic and real-world multilayer networks using state-of-the-art approaches for multilayer community detection. The obtained results have shown the meaningfulness of the proposed modularity, revealing the effects of different combinations of the resolution and inter-layer coupling functions. This work also represents a starting point for the development of new optimization methods for community detection in multilayer networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a new formulation of multilayer modularity that redefines the resolution factor (layer-specific relevance) and inter-layer coupling factor using information from within-layer and inter-layer structures of the multilayer communities, while also incorporating orderings over layers and their constraints on coupling. It claims this addresses arbitrary parameter selection in multislice modularity and demonstrates the approach via experiments on synthetic and real-world networks with state-of-the-art detection methods, showing effects of different resolution/coupling function combinations.

Significance. If the circularity concern is addressed, the work could provide a more topology-informed alternative to fixed-parameter multislice modularity, particularly for ordered layers such as dynamic networks. The experiments indicate that varying the proposed functions yields distinguishable outcomes, but the significance hinges on whether the parameter definitions supply independent information rather than fitting the communities they help discover.

major comments (3)
  1. [§3] §3 (Proposed Formulation): the resolution and inter-layer coupling factors are defined from the within-layer and inter-layer structures of the multilayer communities; because these communities are recovered by optimizing the very modularity that employs the factors, the construction is at risk of circular dependence unless an independent estimator, fixed functional form, or convergent iterative procedure is supplied and validated.
  2. [§4] §4 (Experiments): the reported results on synthetic and real networks compare combinations of the new functions but do not include a control that isolates whether the community-derived parameters improve detection over fixed or externally estimated values, leaving the central claim that the formulation is 'general enough' and 'meaningful' without direct support.
  3. [Abstract and §2] Abstract and §2: the claim that the new modularity 'explicitly model[s] order relations over the layers' is not accompanied by a derivation showing how the ordering constraints are enforced in the objective without reducing to the standard multislice form when the ordering is ignored.
minor comments (2)
  1. [§3] Notation for the new resolution and coupling functions is introduced without an explicit equation number or comparison table to the original multislice parameters, making it difficult to verify the claimed revision of semantics.
  2. [Abstract] The abstract states that experiments 'reveal the effects of different combinations,' but the manuscript does not report quantitative metrics (e.g., NMI, modularity values, or statistical significance) that would allow readers to assess the magnitude of those effects.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and indicate where revisions will be made to the manuscript.

read point-by-point responses

  1. Referee: §3 (Proposed Formulation): the resolution and inter-layer coupling factors are defined from the within-layer and inter-layer structures of the multilayer communities; because these communities are recovered by optimizing the very modularity that employs the factors, the construction is at risk of circular dependence unless an independent estimator, fixed functional form, or convergent iterative procedure is supplied and validated.

    Authors: We acknowledge the validity of this concern about circular dependence. The manuscript defines the factors from community structure without specifying the initialization or convergence mechanism. In the revised version we will add an explicit iterative procedure to §3: initialize with fixed parameters from the standard multislice formulation, obtain communities, recompute the resolution and coupling factors from those communities, re-optimize, and iterate until the detected communities stabilize. We will also report convergence behavior and stability on the synthetic networks. revision: yes

  2. Referee: §4 (Experiments): the reported results on synthetic and real networks compare combinations of the new functions but do not include a control that isolates whether the community-derived parameters improve detection over fixed or externally estimated values, leaving the central claim that the formulation is 'general enough' and 'meaningful' without direct support.

    Authors: We agree that a control isolating the contribution of the community-derived parameters is missing. In the revision we will augment the experimental section with comparisons against (i) fixed parameter values drawn from the multislice literature and (ii) parameters estimated from layer topology independently of the detected communities. Performance will be quantified using NMI on synthetic data and modularity on real data to directly test whether the proposed factors yield distinguishable improvements. revision: yes

  3. Referee: Abstract and §2: the claim that the new modularity 'explicitly model[s] order relations over the layers' is not accompanied by a derivation showing how the ordering constraints are enforced in the objective without reducing to the standard multislice form when the ordering is ignored.

    Authors: The manuscript states that the formulation is general enough to incorporate layer orderings but does not supply the requested derivation. We will expand §2 with a mathematical derivation of the modified objective that shows how ordering constraints modulate the inter-layer coupling terms (e.g., distance-based weighting along an ordered sequence of layers). The derivation will also demonstrate that uniform coupling recovers the standard multislice objective when ordering information is omitted. revision: yes

Circularity Check

1 steps flagged

Resolution/coupling factors defined from community structures that modularity is optimized to discover

specific steps
  1. self definitional [Abstract]
    "We revise the role and semantics of both the resolution and inter-layer coupling factors based on information available from the within-layer and inter-layer structures of the multilayer communities."

    The factors are explicitly defined from the community structures, yet the modularity (which uses those factors) is the function optimized to identify the communities. This makes the parameters a function of the partition they help produce, with no independent topological summary or external estimator stated to ground them.

full rationale

The paper's central innovation revises the resolution and inter-layer coupling factors using within-layer and inter-layer structures of the multilayer communities. Because the modularity incorporating these factors is the objective being optimized to recover those communities, the construction reduces to a self-referential dependence unless an independent estimator or fixed non-community-dependent form is supplied. The abstract provides no such grounding and instead presents the revision as the solution to arbitrary parameter choice. This matches the self-definitional pattern exactly, with no equations or external benchmarks shown to break the loop. No self-citation load-bearing or other patterns are evidenced in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents enumeration of specific free parameters or axioms; the approach appears to rest on the domain assumption that community topology supplies usable signals for parameter setting and on the mathematical assumption that a modularity function can be rewritten to incorporate layer ordering without breaking existing optimization guarantees.

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