Explicit objective functions in modularity-based community detection on multiplex networks

1. Introduction

• Early fusion, simultaneous fusion, and late fusion respectively combine layers before detection, use all layers simultaneously, or detect per layer before consensus.The taxonomy is extended from node-attributed networks to multiplex networks under the paper’s assumptions.Source: §1 · p. 1
• The analysis compares objective functions because empirical rankings and external quality measures do not establish analytical relationships among fusion methods or general superiority across community-detection problems.It separately tests objective relationships and external label agreement on Cora without using subject labels in graph construction or optimization.Source: §1 · p. 1
• The EF objective equals the SF objective plus a non-negative heterogeneity term measuring variation in community degree volumes across active layers.The term is zero when community-wise layer volumes are identical across active layers.Source: §1 · pp. 1–2
• For a specified LF construction, its objective decomposes into original-layer modularity, a heterogeneity term, and a graph-transformation correction.The correction is non-negative at each layer’s transient partition and vanishes when the transformation preserves within-community edge weight.Source: §1 · p. 2
• The analysis is limited to representative modularity-based EF, SF, and one LF construction on normalized, shared-node multiplexes without interlayer edges and with a common resolution parameter.It does not address arbitrary multilayer networks, alternative LF consensus constructions, or unconstrained node-layer assignments.Source: §1 · p. 2
• On label-free Cora, all three methods with both Louvain and Leiden select citations alone, while fixed equal-weight mixtures achieve higher median ARI and NMI than either single-layer baseline.Subject categories are external annotations, not definitive community ground truth.Source: §1 · p. 2

2.1. Definition of the Multiplex Network

• The framework treats a multiplex as weighted layers defined over the same node set, with non-negative edge weights and a common set of possible edges.Input layers are loop-free; transient self-loops introduced by late fusion are handled separately.Source: §2.1 · p. 2
• A node-attributed network can be represented by a topology layer plus one or more non-negative attribute-similarity layers.The attribute-layer edge weights can be normalized matching coefficients for attribute-vector pairs.Source: §2.1 · p. 2, Figure 1 · p. 3
• The studied multiplex setting requires shared nodes across layers, although multiplex definitions may also permit interlayer links and differing node sets or types.The common node set and absence of interlayer edges are assumptions of this study, not requirements of every multiplex definition.Source: §2.1 · pp. 2–3

2.2. Definition of the Community Detection Problem

• Community detection partitions the common node set into nonempty, disjoint communities, with every node assigned to exactly one community.The formulation considers non-overlapping partitions.Source: §2.2 · p. 3
• Newman’s modularity is used as the common building block for the modularity-based community-detection methods considered, and the EF, SF, and LF objectives are formulated in terms of this single-layer quantity.The resolution parameter γ > 0 enables detection of communities at different scales.Source: §2.2 · p. 3
• Internal objectives and external criteria answer different questions, and annotated categories need not coincide with structural communities.The paper separately evaluates external agreement using ARI and NMI.Source: §2.2 · p. 3
• Each layer’s edge weights are normalized to the same total so that layer weights α represent relative importance rather than differences in layer totals.Each layer must have finite, strictly positive total edge weight.Source: §2.2 · p. 4
• Layer weights are non-negative and normalized, and the analysis uses a common resolution parameter across layers.Layer-specific resolution parameters are outside the paper’s scope.Source: §2.2 · p. 4

3. Related works and classification of CD methods for multiplex networks

• The taxonomy groups representative methods by network setting and objective type, distinguishing modularity-based approaches from other objectives.It classifies methods by how they incorporate information from multiple layers.Source: §3 · p. 4, Table 1 · p. 4
• The study focuses on representative modularity-based EF, SF, and LF formulations for shared-node multiplex networks.The taxonomy is also used in node-attributed-network literature.Source: §3 · p. 4
• Prior analytical work addresses multilayer modularity, interlayer coupling, detectability, and statistical equivalences, rather than explicit relationships among the EF, SF, and LF objectives studied here.The two-layer EF–SF relation for L = 2 and γ = 1 was previously obtained in; this paper extends it to arbitrary L and γ > 0 by a weighted-variance calculation.Source: §3 · p. 4
• The paper isolates differences between the stated fusion objectives from other empirical questions such as link weighting and direction.Source: §3 · p. 5

4.1. EF method

• EF flattens the multiplex by taking a convex combination of corresponding edge weights, then performs community detection on the resulting normalized network.Its objective is Newman modularity on the fused graph, and its relation to SF is derived for arbitrary L ≥ 2 and γ > 0.Source: §4.1 · p. 5

4.2. SF method

• SF maximizes a convex combination of the single-layer modularity values over common candidate partitions for a fixed layer-weight vector.The formulation uses normalized layer weights and a common resolution parameter without interlayer coupling.Source: §4.2 · pp. 5–6

4.3. LF method

• LF first obtains layer-wise partitions using a fixed procedure, whose settings and tie-breaking determine the transient partitions.Heuristic settings and random seeds can affect the procedure’s returned partition.Source: §4.3 · pp. 6–7
• The LF transformation retains within-community edges and redistributes each removed between-community edge’s weight across pairs within its endpoint communities.For singleton communities, redistributed weight is assigned to a self-loop.Source: §4.3 · p. 7
• The absorbing transformation preserves layer normalization and does not decrease modularity at the transient partition by strengthening its within-community structure.It is a modeling choice: redistributed inter-community weight can increase the apparent strength of those communities.Source: §4.3 · pp. 7–8
• After transformation, LF applies EF to the transient multiplex to find a common partition.Its analytical properties are conditional on the specific fixed transformation operator τ, not the entire LF family.Source: §4.3 · p. 8
• The chosen absorbing rule preserves community-wise degree sums and does not decrease community-wise internal edge weight.The paper contrasts it with a normalized cluster-network adaptation for which the stated property need not hold.Source: §4.3 · p. 8

5.1. Explicit objective function of an EF method.

• For a fixed partition, early fusion exceeds simultaneous fusion by γ/4 times the weighted variance of community-wise degree weights across active layers.The variance is zero exactly when those community-wise layer volumes agree across active layers; identical layers are sufficient but not necessary.Source: §5.1 · pp. 8–9

5.2. Explicit objective function of an LF method.

• The late-fusion objective decomposes into the layer modularities, a transformation correction Θ, and γ∆ evaluated on the transformed network.The decomposition is conditional on the fixed transient-construction operator τ.Source: §5.2 · pp. 9–10

5.3. Conclusions on the explicit objective functions.

• Early fusion equals simultaneous fusion plus γ∆, while the specified late-fusion objective adds a transformation correction and a degree-heterogeneity term evaluated on the transformed layers.These identities distinguish changes to the graph from changes to the objective.Source: §5.3 · p. 11
• Equal objective values at a partition do not require identical layers, and equality at one partition does not make the objectives identical.For late fusion, the two correction terms must sum to zero for its value to equal simultaneous fusion at that partition.Source: §5.1 · p. 9, §5.2 · p. 10, §5.3 · p. 11
• Late-fusion absolute objective values should not be ranked against early- or simultaneous-fusion values, and heuristic cross-objective rankings can reflect optimizer dependence.The meaningful comparison evaluates a partition returned by one method under the other objectives.Source: §5.3 · p. 11

6. Finding optimal α

• SF’s global maximum over partitions and normalized layer weights is attained at a simplex vertex, so a single-layer weight suffices to attain it.The maximizing vertex need not be unique; for a fixed optimal partition, convex combinations of globally maximizing vertices may also be optimal.Source: §6 · pp. 11–12
• For a fixed partition, EF is concave in the layer weights and may attain its optimum either on the simplex boundary or in its interior.With two layers, its objective is the weighted sum of the two layer modularities for that fixed partition.Source: §6 · p. 12
• Jointly optimizing EF over partitions and weights need not yield a concave weight profile, and an interior weight can outperform every simplex vertex.A two-layer, four-node example has vertex maxima of 0 but a higher value at equal weights.Source: §6 · p. 12
• For LF, fixing the transient graph gives the final optimization EF’s geometry, but no equally general vertex theorem applies to the original graph.The transient graph depends on layer-wise returned partitions and the absorbing rule.Source: §6 · p. 12

6.1. Interaction with the modularity resolution limit.

• EF, SF, and LF retain modularity’s resolution-limit concern; normalization does not align layers’ detection scales or remove that limitation.The analysis provides no separate resolution-limit theorem for fused networks; resolution-limit-free alternatives are outside its scope.Source: §6.1 · p. 12

7. Experimental study

• The study separates exhaustive toy optimization from real-network Louvain and Leiden heuristics; heuristic results illustrate optimizer dependence rather than exact argmax theory.Reported best values are grid-restricted medians, and the bootstrap intervals describe seed variability conditional on the tested settings—not dataset or optimizer uncertainty.Source: §7 · p. 12, §7.2 · pp. 14–15

7.1. Toy examples of multiplex networks and brute-force optimization.

• Figure 9’s exact toy results show that cross-objective rankings depend on the tested weight: at a = 0.70 the stated hierarchy holds, but at a = 0.75 LF scores above SF under QEF.The strict rankings are specific to the examples and weights, not universal across networks and α.Source: Figure 9 · p. 13, §7.1 · p. 14
• In the second toy example, tied layer-wise maximizers yield multiple transient graphs and LF solutions, making LF objective scores depend on the transient choice.The first example also shows that the spread among LF values collapses as transient multiplicity decreases, a pattern specific to that example.Source: §7.1 · p. 14

7.2.1. Historical network comparisons.

• Among historical networks, London Stations has an interior EF winner, whereas the other reported networks attain their highest observed median EF score at a simplex vertex.The experiments use heuristic optimizers, so these are observed medians rather than exact optima.Source: §7.2.1 · p. 15, Figure 10 · p. 16

7.3. External validation on label-free Cora.

• On label-free Cora, subject categories are excluded from graph construction and optimization, then retained for postoptimization evaluation.The attribute layer uses word indicators, while citation edges are unchanged; no setting is selected using subject labels.Source: §7.3 · p. 15
• Figure 11 shows that all six method–optimizer combinations select citation-only weight a = 1, while equal weighting yields higher median ARI than both that choice and words-only.Equal-weight median ARI ranges from 0.381 to 0.448, versus 0.228–0.259 at objective-selected weights and 0.301–0.311 for words only; NMI has the same direction of comparison.Source: §7.3 · pp. 15–16, Figure 11 · p. 17

Interpretation and limits

• For the studied formulations, EF, SF, and LF objectives make fusion effects explicit, but optimizing them does not establish scientifically meaningful communities.Higher EF values can reflect the heterogeneity bonus, and absolute LF values use a modified graph.Source: §8 · pp. 17–18
• SF has an optimal weight vector at a simplex vertex, whereas EF can improve with a nontrivial layer mixture; LF preferences depend on its fixed construction operator.For a fixed partition, EF is concave in layer weights; this does not establish concavity of the empirical profile when heuristic partitions change.Source: §8 · pp. 17–18

Transport networks

• On London Stations, EF’s highest median occurs at an interior Tube–Overground–DLR mixture while SF selects Tube; on Major Airlines, EF and SF select British Airways and LF selects Lufthansa.No tested interior mixture beats the best single-layer EF configuration on Major Airlines; these preferences concern the normalized modularity objective, not traffic volume.Source: §8 · p. 17

Node-attributed networks

• On Political Blogs, EF and SF favor the attribute layer that retains same-leaning links; on historical Cora, citations give the highest observed EF median and adding attributes does not improve it.The Political Blogs result is a consistency check, not independent validation; historical Cora used subject-label features and cannot independently validate recovery of subject categories.Source: §8 · p. 17

Label-free external validation

• Equal-weight fusion achieves higher median subject agreement by both ARI and NMI than either single-layer baseline across both optimizers and all three methods, while objectives select citations alone.This result concerns one fixed network and tested construction, weight grid, and optimization settings; subject categories are not definitive ground truth.Source: §8 · pp. 17–18

Clinical-trial networks

• For clinical-trial networks, EF and SF favor intervention similarity in the two-layer case and shared sponsors or collaborators in the three-layer case, while LF favors intervention in both.The two networks cannot isolate the effect of adding a layer because their layer content differs; exploratory winner-gap intervals include zero.Source: §8 · p. 18

• The research received no external funding.Source: Funding · p. 18

• Source code, software requirements, numerical checks of the EF–SF and LF identities, and materials for reproducing the historical analysis are available in a public repository.Source: Data and code availability · p. 18
• The separate label-free Cora validation is documented in an electronic supplement with its analysis contract, executable code snapshot, input checksums, partitions, and per-run scores.The supplement distinguishes it from the historical Cora experiment, which included subject-label features.Source: Data and code availability · p. 18
• Clinical-trial data sources and reuse terms are given in Appendix C.Source: Data and code availability · p. 19

A.1. Proof of Theorem 1.

• The weighted variance term is zero exactly when community-wise degree volumes are constant across layers with positive weight.The proof also applies to normalized layers with self-loops under the stated adjacency convention, and the theorem applies to the transient multiplex.Source: §A.1 · pp. 24–25

A.2. Proof of Lemma 1.

• The LF correction is defined by separating modularity on the original and transformed graphs, and it is zero exactly when every p_k is zero.The stated inequality follows from the correction definition and LF properties for a single-layer network.Source: §A.2 · p. 25

A.3. Proof of Theorem 3.

• For a fixed partition, EF is concave in the layer weights because each community’s variance contribution is affine minus a convex quadratic term.For γ = 0, EF is affine in the weights directly.Source: §A.3 · p. 25

B. Additional exact toy results

• QLF in Figure 12 is evaluated on transient layers, so its values are not directly comparable with QEF/QSF; Figure 13 reports a maximum modularity of 0.The figures supplement the second toy-network discussion.Source: Figure 12 · p. 26, Figure 13 · p. 26

C.1. Real-world two-layer networks and modularity optimization by Louvain and Leiden.

• Real-world experiments use cleaned, undirected, synchronized shared-node networks, retaining nodes isolated in one layer when present in another.The analyzed common node set includes vertices incident to at least one retained layer edge.Source: §C.1 · p. 26
• Political Blogs tests a node-attributed transformation whose attribute layer retains same-leaning structural links, while Cora builds an attribute layer from textual and categorical features.Political Blogs retains about 91% of structural edges in its attribute layer; Cora uses sparse cosine nearest neighbors.Source: §C.1 · pp. 26–28
• The two Immune Trials layers are attribute-derived similarity graphs with only 13.5% edge Jaccard overlap, and binary-feature ties make exact neighbor selection implementation-sensitive.The canonical processed layers are used; alternative implementations may differ slightly in edge counts without changing component structure.Source: §C.1 · pp. 27–28
• Louvain and Leiden optimize the three fusion methods with shared optimizer families; LF reuses the same transient construction across its weight grid within each run.For SF, the Louvain and Leiden multiplex extensions are specified separately.Source: §C.1 · p. 28

C.2. Real-world multiplex networks and modularity optimization by Leiden.

• The three-layer benchmarks include London transit, three major European airlines, and Immune Trials; their layers represent distinct routes, operators, or trial annotations.The airline and Immune Trials layers have low pairwise edge overlap; the two- and three-layer Immune Trials datasets are not a controlled layer-count comparison because their contents differ.Source: §C.2 · pp. 28–29
• The QLF row uses transient layers and is not on the raw scale of QEF/QSF, so QLF should be compared across α within its panel rather than across objective panels.Source: Figure 14 · p. 29
• Because every layer is normalized to total weight one, the three-layer experiments compare normalized signals rather than raw edge or traffic volume.The correction-term tables summarize auxiliary quantities, not main-objective winners.Source: §C.2 · p. 30, Table 5 · p. 33
• 0.8220 is the best median QSF and QEF on Cora, both at α = (1.00, 0.00), with SF partitions attaining each objective’s reported best.The corresponding Cora QLF maximum is evaluated on the transient-graph scale.Source: §C.2 · p. 30, Table 3 · p. 31
• Most reported best median QSF and QEF configurations select a simplex vertex, while London Stations’ best median QEF is 0.8282 at the interior weight α = (0.40, 0.40, 0.20).For London Stations, the best median QSF is 0.8175 at α = (1.00, 0.00, 0.00).Source: §C.2 · pp. 30, 35
• The bootstrap intervals summarize seed-restart variation and do not establish statistical equivalence between configurations.A dagger marks winner–runner-up median-difference intervals that include zero.Source: §C.2 · p. 32

E. Cora Sensitivity Audit

• In the historical Cora sensitivity audit, all four preprocessing variants retain the best-median QEF winner at the pure-structural vertex α = (1, 0).The audit uses 10 seeds per grid point and point estimates only, so it is a qualitative robustness check rather than a statistically powered comparison.Source: E. Cora Sensitivity Audit · p. 44, Table 8 · p. 44

F. A normalized cluster-network alternative inspired by [60]

• The normalized cluster-network adaptation applies EF to layer-wise partition-derived clique networks, but it excludes all-singleton partitions and need not satisfy condition (6).Because its edge weighting can differ from the absorbing construction, the monotonicity argument in Lemma 1 does not directly carry over.Source: F. A normalized cluster-network… · p. 45