A new measure of modularity density for community detection

Gerardo Veltri; Swathi M. Mula

REVIEW 4 major objections 5 minor 22 references

A new measure of modularity density for community detection

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A new modularity density score, $M$, is claimed to be free from both large-cluster and small-cluster bias and to detect weakly separated communities in heterogeneous networks.

desk verdict New modularity-density variant with a clean two-clique comparison and an interesting normalized-cut relation, but the no-bias proofs only cover friendly cases on expected graphs, so the headline claim overreaches. read the letter →

arxiv 1908.08452 v1 pith:ZQFX6AO5 submitted 2019-08-22 cs.SI stat.ML

Swathi M. Mula , Gerardo Veltri This is my paper · ORCID

classification cs.SIstat.ML MSC 05C8268R10

keywords communitydetectionmodularitydensityresolutionlimitheterogeneousnetworksnormalizedcutrandomgraphsunbiasedqualityfunctionweighted

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a new quality function for community detection, a modularity density $M$ defined in Eq. (13), built from the intuitive requirement that a community should have strong internal cohesion and weak external separation. Its central claim is that maximizing $M$ is free from both biases that plague earlier modularity-based scores: it will not split a random graph or clique into pieces, and it will keep communities of very different sizes intact rather than merging them. That property matters for heterogeneous networks, where real communities span a wide range of sizes and earlier scores either miss small clusters or swallow them into large ones. The paper also shows that $M$ can be evaluated in $O(|E|)$ time and that the gain from splitting a cluster under $M$ is a constrained normalized-cut expression.

What carries the argument

The load-bearing object is the normalized degree vector $\vec{d}_c = \hat{n}_c\cdot T$, whose entries are the adjacency-weighted connections of each node to cluster $c$, scaled by $1/\sqrt{n_c}$. Projecting $\vec{d}_c$ onto the cluster's own unit vector $\hat{n}_c$ gives the mean internal degree, while projecting onto another cluster's unit vector gives a normalized measure of external association. Summing internal minus external projections over all clusters yields Eq. (13): $M = \sum_c\left\{\frac{\sum_{i,j\in c} T_{ij}}{n_c} - \sum_{c'\neq c}\frac{\sum_{i\in c,j\in c'}T_{ij}}{\sqrt{n_c n_{c'}}}\right\}$. The no-bias proofs work by comparing $M$ for the true partition with $M$ for merged or split alternatives on Erdős–Rényi graphs, using the assumption that a natural community of $m$ nodes has at least $m$ edges, which fixes the minimum edge probability at $p_{\min}=2/(m-1)$; substituting that bound turns every difference into a positive algebraic expression.

What would settle it

On a single sampled Erdős–Rényi graph with $m$ nodes and edge probability $p=2/(m-1)$, compute $M$ for the whole graph and for every possible two-way split; if any split has $M_{\mathrm{split}} > M_{\mathrm{single}}$, the no-split guarantee fails on that sample. Similarly, on the two-clique network with a single connecting edge, exact maximization of $M$ should always return two clusters; one run that merges them would falsify the claimed improvement over Chen et al.'s modularity density.

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Extended reading notes

Core claim

The central discovery is a score $M$ that combines each community's mean internal degree with its normalized external degree into a single number, and the main result is that this score is size-unbiased in the sense that matters for community detection. On an Erdős–Rényi random graph or a clique, the single-community value of $M$ is algebraically larger than the value of any split, so optimizing the score will not invent clusters. On a network of two or a ring of many random communities of different sizes connected by single edges, the separated value of $M$ is larger than every merged value for every subset of communities, which is exactly the regime where the resolution limit makes modularity and earlier modularity densities fail. In a two-clique network with $w$ connecting edges, the threshold $w_M$ below which the cliques are detected is always at least the threshold $w_D$ for Li et al.'s modularity density, with equality only for equal-sized cliques. Finally, the change in $M$ from a bipartition is shown to be $\hat{f}\cdot D_c\cdot\hat{f}\,[1-\lambda]-\beta$, a local Rayleigh quotient with a non-negative external penalty, which places optimization of $M$ as a constrained version of the normalized cut problem.

Load-bearing premise

The load-bearing premise is that an Erdős–Rényi random graph can be treated by its expected edge weights and that a natural community of $m$ nodes must contain at least $m$ edges; if real sampled graphs do not honor those assumptions, the inequalities proving $\Delta M>0$ may not hold for individual networks.

Editorial extensions

If this is right

A community-detection algorithm that optimizes $M$ would not need a resolution parameter and could return both small and large clusters in a single run, addressing the main failure mode of modularity in heterogeneous networks.
For two connected cliques, $M$ separates them for a wider range of connecting-edge counts than Li et al.'s modularity density, so weakly separated communities are detectable in a strictly larger regime.
Because evaluating $M$ costs $O(|E|)$, it can be plugged into greedy or spectral search heuristics without an asymptotic slowdown relative to the earlier modularity density, and it avoids the $O(|C|^2)$ split-penalty cost of Chen et al.'s variant.
The bipartition formula shows that recursive community detection by maximizing $M$ is a constrained normalized-cut problem, linking a parameter-free quality score to a well-studied spectral clustering objective.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The unbiasedness proofs operate on expected edge counts of Erdős–Rényi graphs; a natural testable extension is to check on sampled graphs and on synthetic networks with power-law community sizes whether the inequalities survive finite-sample fluctuations.
The normalized-cut connection suggests a spectral relaxation of $M$-maximization could be derived, in which leading eigenvectors of a modified Laplacian would approximate optimal partitions; the paper leaves this approximation as future work.
The comparison with Li et al. is derived for unweighted cliques; extending the threshold calculation to weighted edges and to communities with internal edge-probability gradients would show whether $w_M \ge w_D$ persists in weighted heterogeneous networks.
If $M$ really is bias-free, it offers a single-objective alternative to multiresolution methods, so practitioners would not need to scan a resolution parameter to find communities at all scales.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper offers a new modularity-density objective M (Eq. 13) and demonstrates on two stylized networks that it behaves sensibly; the wM/wD comparison with Li et al. is clean, and the normalized-cut relation in Section IV is a genuinely new observation. The stress-test note is on point: the central claim that maximizing M is 'proven to be free from bias' is not actually proven. Section III only checks selected partitions — one bisection of a random graph, merging two communities, merging consecutive communities in a ring. Three-or-more splits, non-adjacent merges, and boundary shifts are not analyzed, and the dismissed cases are waved off with 'it is not hard to show.' Moreover, every inequality is computed on the expected adjacency matrix (all off-diagonal entries equal to p), so the proofs concern E[M], not M on a realized graph; because M is linear in T, the expectation of the maximizer is not the maximizer of the expectation. The authors themselves state in Section IV that they have not yet developed an approximation algorithm, so there is no empirical validation either. What is genuinely useful: the metric is a reasonable variant in the cohesion-separation family, with an interpretable normalization; the analytical comparison showing wM >= wD for weakly separated cliques is a legitimate result; and the connection to normalized cut is worth preserving. The citation pattern is fine and there is no circularity. This is a promising candidate objective, not a validated method. If the authors soften the 'proven' language, fill the obvious gaps or state the restricted scope, and add at least one small empirical test, it could become a reasonable contribution. I would not desk-reject it: a serious referee could check the algebra and push back on the proof gaps. In its current form I wouldn't cite it as a strong result.

Referee Report

4 major / 5 minor

Summary. The paper proposes a new modularity density measure M, defined in Eq. (13), for detecting non-overlapping communities in undirected, weighted, non-negatively weighted networks. The measure is built from projections of a normalized degree vector onto community indicator vectors, combining internal cohesion and external separation terms. The authors claim that maximizing M is free from the two biases that affect other modularity density variants (favoring small clusters and favoring large clusters), that it detects weakly separated communities in heterogeneous networks better than the modularity densities of Li et al. (2008) and Chen et al. (2013), that it can be computed in O(|E|), and that its optimization is mathematically related to normalized cut minimization. The theoretical analysis consists of algebraic comparisons on small model networks: a random graph or clique, two connected random communities, a ring of communities, and two cliques connected by w edges. Section IV derives an expression for the change in M under bi-partitioning a cluster and relates it to the normalized cut criterion.

Significance. If the paper's central claims were established, the proposed measure M would be a useful parameter-free objective for community detection, particularly for networks with heterogeneous community sizes. The manuscript has genuine strengths: the derivation of M from cohesion and separation is self-contained, the algebra on the chosen examples is mostly explicit, the comparison with Li et al. on the two-clique example gives a concrete threshold inequality, and the normalized-cut connection in Section IV is a nontrivial and interesting observation. The measure has no fitted parameters, and the computational complexity argument is straightforward. However, the significance is substantially weakened by two gaps: the 'free from bias' theorem is not actually proven for the global maximizer of M, and the claimed superiority over prior modularity densities is not tested on any real or synthetic benchmark network. The contribution is therefore at the level of a promising proposal whose advertised properties require further proof and empirical validation.

major comments (4)

[Section III-A(a)-(c), Eqs. (17), (34), (38)] The no-bias claim is about the global maximizer of M over all possible partitions, but the proofs only compare a small set of selected partitions. In III-A(a), only the single-community partition is compared with one arbitrary bisection; splits into three or more clusters are dismissed with 'it is not hard to show' and no argument is given. In III-A(b) and III-A(c), the only alternatives considered are the true partition and the partition obtained by merging adjacent communities into one cluster. Non-contiguous merges, partial boundary shifts, and partitions that simultaneously split some communities and merge others are never analyzed. Without excluding all such partitions, the statements 'optimizing M does not split a random graph' and 'M is free from bias' are not established.
[Section III-A, Eqs. (15)-(25)] The computations replace the Erdős-Rényi random graph G(m,p) by the deterministic weighted complete graph in which every edge has weight p, i.e., by the expected adjacency matrix. Since M is linear in T, the expressions in Eqs. (15), (16), (18), and (19) are expected values E[M]. Inequalities such as Eq. (17) and Eq. (25) therefore concern E[M], not the realized M of an actual sampled graph. The expectation of the maximizer is not the maximizer of the expectation, so the sampled-graph behavior is genuinely left open. Additionally, Eq. (14) defines p_min = 2/(m-1) by asserting that a 'natural community' needs at least m edges; the footnote excludes path graphs, but no proof is given that a path graph is not a natural community. These assumptions are load-bearing for the claimed no-bias theorem.
[Section III-B(a), Eqs. (44)-(50)] The claimed superiority over Li et al. (2008) is established only for the two-clique network of Fig. 5 connected by w edges; the analysis shows w_M >= w_D for that single family. Chen et al. (2013) is not directly compared in this section, and the statement that it suffers from the resolution limit problem relies entirely on prior work. More importantly, the abstract's claim that M 'better detect[s] weakly-separated communities particularly in heterogeneous networks' is not supported by any empirical evaluation on real or synthetic benchmark networks. The paper contains no experiments with community-size heterogeneity, no LFR-type benchmarks, and no comparison of detection accuracy with existing algorithms.
[Section III-B(b)] The statement that the 'computational running time of our modularity density is found to be on par or faster' is not an empirical finding; Section III-B(b) provides only an asymptotic complexity argument (O(|E|) for M versus O(|E|) and O(|C|^2) for previous metrics). No algorithm for actually maximizing M is proposed, so the cost of optimization is not assessed and no runtime measurements are reported. As written, the claim overstates what has been shown.

minor comments (5)

[References and notation] The reference to Shi and Malik is consistently written 'Shi & Mallik', and the journal name in reference [11] contains typos ('Patter analysis'). The reference list also has inconsistent formatting for arXiv entries and some author names.
[Section III-A(a), Eq. (14)] The text states that for a connected graph at least m-1 edges are required and then asserts that a natural community needs at least m edges, but the distinction between 'connected' and 'natural community' is not formally defined. The footnote about path graphs is not a proof and should be clarified.
[Figure 4 and surrounding text] The caption of Fig. 4 and the text around it use 'm' ambiguously for both the number of nodes and the label of a subgraph; the notation should be made consistent.
[Section III-A(c), Eq. (33)-(34)] The algebraic manipulation in Eqs. (33)-(34) is quite dense and would benefit from an intermediate step that shows how the lower bound 4(k-1)/3 - 2k/[3(k+1)] is obtained from the preceding line. The current presentation is correct but hard to verify.
[Section IV, Eq. (79)] The interpretation of Eq. (79) is clear, but the wording 'maximization of δM with respect to α requires minimization of λ' should be stated more carefully: the expression in parentheses contains a positive factor (1 + 4√(na nb)/nc) that depends on the partition sizes, so the comparison with the standard normalized cut Rayleigh quotient is only qualitative.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the metric is constructed directly and its properties are derived from its definition; no fitted parameter, self-citation chain, or definitional equivalence is present.

full rationale

The paper defines its modularity density M in Eqs. (8)-(13) directly from the cohesion/separation concept (projections of a normalized degree vector), with no parameter fitted to target results and no prior work by the same authors invoked as load-bearing. The subsequent sensitivity analysis compares M for specific partitions of random graphs and of networks of cliques (Eqs. 15-19 and 26-38) by algebraic inequalities; these comparisons use the definition of M and the assumed Erdős-Rényi weights, so the derivations are not equivalent to their inputs by construction. The comparison with Li et al. modularity density (Eqs. 41-50) is an analytic threshold calculation, not a fitted prediction. The normalized-cut relation in Section IV is derived after the fact as a mathematical identity, not used to define M. The main weakness — that the 'no bias' claim only checks selected partitions on expected edge weights rather than all partitions on sampled graphs — is a gap in proof strength or correctness risk, not circularity, because no claim reduces to its own input or to a self-citation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central mathematical claims rest on three axioms: the expectation-based treatment of random graphs, the minimal-edge definition of a natural community, and the network restrictions. The p_min threshold is an ad hoc quantity inside the proofs. No new entities are introduced.

free parameters (1)

p_min threshold for natural community = 2/(m-1)
Chosen by the authors so that a natural community has at least m edges (Eq. 14). It is used to evaluate the minimum of ΔI in the no-bias proofs (Eqs. 22-24, 31-32); a different threshold would change the bounds.

assumptions (4)

domain assumption A connected random graph G(m,p) can be represented by a complete graph with all edge weights p for the purpose of computing M.
Equations (15)-(19) substitute the expected number of edges into M, effectively averaging over the random graph distribution rather than analyzing individual samples.
domain assumption A natural community of m nodes must contain at least m edges, so the minimum edge probability is 2/(m-1).
Stated as a footnote in Section III-A(a); used to derive ΔI_min=2k in the resolution-limit proofs (Eqs. 24, 32).
domain assumption The network is undirected, connected, has non-negative edge weights, and each node belongs to exactly one community.
Stated in Section II and used throughout; the derived properties are not extended to directed, disconnected, or negative-weight networks.
standard math Standard vector/tensor algebra and the graph Laplacian identity for f·L·f are applied.
Used in Section IV to express ΔM in terms of D_c and L_c; these are classical results.

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Pith. "Pith review of A new measure of modularity density for community detection." pith.science (2026). https://pith.science/paper/ZQFX6AO5

@misc{pith2026190808452,
  author       = {Pith},
  title        = {Pith review of: A new measure of modularity density for community detection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZQFX6AO5}},
  note         = {Machine review of arXiv:1908.08452}
}

read the original abstract

Using an intuitive concept of what constitutes a meaningful community, a novel metric is formulated for detecting non-overlapping communities in undirected, weighted heterogeneous networks. This metric, modularity density, is shown to be superior to the versions of modularity density in present literature. Compared to the previous versions of modularity density, maximization of our metric is proven to be free from bias and better detect weakly-separated communities particularly in heterogeneous networks. In addition to these characteristics, the computational running time of our modularity density is found to be on par or faster than that of the previous variants. Our findings further reveal that community detection by maximization of our metric is mathematically related to partitioning a network by minimization of the normalized cut criterion.

Figures

Figures reproduced from arXiv: 1908.08452 by the authors.

**Figure 1.** Projecting the normalized degree vector d~c on the unit vector nˆc of cluster c. To obtain a measure of internal associations (cohesion) within cluster c, we take the concept of ~dc further by projecting this vector on nˆc, which is a unit vector representing cluster c as in (1). This projection, as illustrated in figure 1, is determined by the dot product: ~dc · nˆc = X j dcjncj = X j∈c dcj √ nc . (4) Using equatio… view at source ↗

**Figure 2.** Network of communities of different sizes. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** (a) Ring of cliques. (b) Network with two cliques and a random subgraph. In the current section, we show that optimizing our metric M successfully identifies all the true communities of this generic heterogeneous network as separate clusters. If Msep represents the modularity density of the heterogeneous network when all the natural communities in figure 3 are identified as separate clusters, then using equation (1… view at source ↗

Figures from the paper (3 more)

**Figure 3.** Figure 3: Heterogeneous modular network. corresponding network G(mi , pmi ) becomes a clique. Note that the sample networks presented in figure 4 are all specific examples of the generic heterogeneous network in figure 3. m m m m m m m m m G(n, pn) m m (a) (b) [PITH_FULL_IMAGE:…

**Figure 5.** Figure 5: Network of weakly separated communities of different sizes. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Relation between the limiting values wD and wM i.e. m = n. In the case of heterogeneous networks, i.e. m 6= n, the limiting value wD is always less than wM. Additionally, figure 6 also depicts that larger the heterogeneity between the communities, greater is the differ…

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Analysis of the structure of complex networks at different resolution levels,

A. Arenas, A. Fern ´andez and S. G ´omez, “Analysis of the structure of complex networks at different resolution levels,” New Journal of Physics, vol. 10, 053039, May 2008

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[2] [2]

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M. Chen, K. Kuzmin and B.K. Szymanski, “Community detection via maximization of modularity and its variants,” IEEE Transactions on Computational Social Systems, vol. 1(1), pp. 46–65, March 2014

work page 2014

[3] [3]

A new metric for quality of network community structure,

M. Chen, T. Nguyen and B. K. Szymanski “A new metric for quality of network community structure,” ASE Human Journal, vol. 2(4), pp. 226–240, 2013. 11

work page 2013

[4] [4]

Network community detection using modularity density measures,

T. Chen, P. Singh and K. E. Bassler “Network community detection using modularity density measures,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2018, May 2018

work page 2018

[5] [5]

Effect of size heterogeneity on community identiﬁcation in complex networks,

L. Danon, A. D ´ıaz-Guilera and A. Arenas, “Effect of size heterogeneity on community identiﬁcation in complex networks,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2006, November 2006

work page 2006

[6] [6]

On Random Graphs,

P. Erdos and A. R ´enyi, “On Random Graphs,” Publicationes Mathemat- icae (Debrecen), vol. 6, pp. 290–297, 1959

work page 1959

[7] [7]

Resolution limit in community detection,

S. Fortunato and M. Barth ´elemy, “Resolution limit in community detection,” Proc. Natl. Acad. Sci. U.S.A., vol. 104(1), pp. 36–41, 2007

work page 2007

[8] [8]

Community detection in graphs,

S. Fortunato, “Community detection in graphs,” Physics Reports, vol. 486, issues 3-5, pp. 75–176, 2010

work page 2010