A Systematic Approach for Studying How Topological Measurements Respond to Complex Networks Modifications

Alexandre Benatti; Luciano da F. Costa; Roberto M. Cesar Jr.

REVIEW 5 major objections 3 minor 64 references

A Systematic Approach for Studying How Topological Measurements Respond to Complex Networks Modifications

T0 review · 5 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Progressive network changes sort topological measurements into three response types, with Erdős–Rényi and Barabási–Albert networks behaving alike while geographical networks stand apart.

desk verdict A useful but under-validated empirical taxonomy of how network measurements respond to perturbations; the central claims survive the weak spots only partially. read the letter →

arxiv 2505.22345 v1 pith:BZR5E6ZB submitted 2025-05-28 cs.SI physics.soc-ph

Alexandre Benatti , Roberto M. Cesar Jr. , Luciano da F. Costa This is my paper

classification cs.SIphysics.soc-ph MSC 05C8205C8062H30

keywords topologicalmeasurementscoincidencesimilarityindexnetworkmodificationedgeremovalrewiringsizehierarchicalclusteringnetworks

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 asks what happens to standard topological measurements of a network when the network itself is progressively altered, and whether those responses cluster into recognizable types. It studies three network models — Erdős–Rényi, Barabási–Albert, and geographical — under three modification protocols: growing size, edge removal, and edge rewiring. By normalizing each measurement's change curve and comparing curves with a strict coincidence similarity index, the authors identify three response categories: measurements that mostly rise (A), mostly fall (B), and everything else (C). They further report that Erdős–Rényi and Barabási–Albert networks respond to these modifications in mutually similar ways, while geographical networks behave differently and show more heterogeneous topological features. The claimed payoff is a systematic template for deciding which measurements are reliable indicators under network change and how measurement responses are interrelated.

What carries the argument

The argument runs on three linked objects. First, each measurement's change is turned into a normalized signature $c_{i,j} = (x_{i,j} - \min_i x_{i,j})/\sigma_j$, so that only the shape of the response curve survives. Second, pairs of signatures are compared with the coincidence similarity index, $C(\vec{x},\vec{y}) = J_D(\vec{x},\vec{y})\,I_E(\vec{x},\vec{y})$, the Jaccard-style multiset similarity multiplied by an interiority factor, with strictness parameters $D=5$ and $E=1$; this index supplies both the edge weights of a similarity network (nodes are measurements) and the input to agglomerative clustering. Third, the resulting dendrograms and network modules are read as evidence that measurements in the same module respond to perturbations in a common way, with module labels A, B, and C encoding mostly-increasing, mostly-decreasing, and other trajectories.

What would settle it

Run the same pipeline on ER networks under edge rewiring, where the paper itself states the measurement changes are statistical fluctuations: if the A/B/C categorization is genuine, shuffling the order of the rewiring steps (or randomly relabeling the measurement curves) should destroy the modules; if the modules survive permutation, they are produced by the normalization and similarity procedure rather than by real network response types.

Watch

Extended reading notes

Core claim

The central claim is that the response of a topological measurement to progressive network change can be classified into one of three types — mostly increasing (A), mostly decreasing (B), or other, including oscillation and near-constancy (C) — and that these types appear as well-separated modules in coincidence similarity networks and as branches in dendrograms built from the same similarities. For the size-growth experiment, most of the fourteen measurement variants increase across all three network models, with clustering coefficient and assortativity as the main decreasing exceptions. For edge removal, measurements split into a larger decreasing group and a smaller increasing group, with the increasing set containing average shortest path, betweenness centrality, assortativity, and accessibility at level $h=5$. For edge rewiring, the ER curves are statistical fluctuations and fall into category C, whereas BA and GEO networks show defined A/B/C structure. Across experiments, the measurement-change patterns of ER and BA networks resemble each other more than they resemble GEO, whose modules are more compact and whose responses are more heterogeneous, indicating a distinctive spatial finite-size component in the geographical model.

Load-bearing premise

The load-bearing premise is that the chosen normalization (subtracting each curve's minimum and dividing by its standard deviation) together with the coincidence similarity parameters $D=5$ and $E=1$ makes the computed similarities reflect genuine shape relationships, so that the A/B/C modules are properties of the networks rather than artifacts of the analysis pipeline.

Editorial extensions

If this is right

In the size-growth experiments, most of the fourteen measurement variants behave as type A across ER, BA, and GEO, so growth trends of a network can be expected to push most topological indicators upward; clustering coefficient and assortativity are the visible exceptions.
Under edge removal, average shortest path length, betweenness centrality, assortativity, and accessibility at $h=5$ form a stable increasing group, meaning these measurements track edge density in a predictable direction.
For edge rewiring, ER networks are statistically indistinguishable from their rewired versions, so any apparent response pattern in that setting should be treated as fluctuation rather than topology.
GEO networks, with their compact and densely interconnected modules, can be expected to show stronger and more linear measurement responses to rewiring than ER or BA networks.
The A/B/C labels give a compact vocabulary for comparing how different networks and modifications affect the same measurement, as summarized in the parallel-coordinate diagram.

Reading between the lines

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

A natural extension the paper does not run: apply the same pipeline to node-removal attacks or to sampled subgraphs of a real network and ask whether the A/B/C modules persist or shift.
Because the normalization step subtracts each curve's minimum and divides by its standard deviation, curves with tiny absolute changes are amplified; comparing against non-normalized magnitudes would reveal how much of the taxonomy is produced by the normalization choice.
The reported ER/BA similarity could be a consequence of both models lacking the spatial constraints that organize GEO networks; a parameterized family interpolating between random and geometric topologies could test whether the similarity scales continuously with spatial structure.
If the module structure is stable, it would support measurement substitution: within a module, one measurement's response could be used to impute another's when only a subset of measurements can be computed.

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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

Colleague,

This is a genuinely systematic empirical study that extends Villas Boas et al. (2010) to more measurements, shape-based normalized response curves, and two clustering tools. The raw observations – which measurements tend to rise or fall under size changes, edge removal, and rewiring in ER, BA, and GEO networks – are plausible and useful for reference. The parallel coordinates diagram is a nice summary.

What is actually new: the addition of accessibility measurements (hierarchical and generalized), the emphasis on the shape of the response curve rather than its raw magnitude, and the use of coincidence similarity networks and hierarchical clustering to group measurements by how their changes covary. The methodology is described well enough for a competent group to reimplement, and the dispersion and magnitude barplots are a sensible attempt to separate signal from noise.

The soft spots are concentrated in the analysis pipeline. The normalization – subtract min, divide by standard deviation, then subtract the initial value – can amplify statistical fluctuations into something that looks like structure. The paper admits this, but then feeds those normalized curves into similarity networks and dendrograms and reads modules off them. The clearest instance is the ER rewiring case: the text says all ER measurement changes are category C, i.e., statistical fluctuations, yet Fig. 16(a) is captioned as involving three modules. That is over-reading noise. Relatedly, the coincidence similarity index uses D=5, E=1 with no sensitivity analysis, and there is no null model (e.g., shuffled curves) to show that the resulting A/B/C structure is not what the pipeline would produce by chance. The A/B/C assignment is done by visual inspection, without thresholds or error bars.

Having said that, the paper's main comparative claim – ER and BA responses are more similar to each other than to GEO – is credible for the network-size and edge-removal experiments, where the curves show clear trends and the dendrograms differ visibly. The weak ER-rewiring case does not kill that claim, but it does mean the paper overstates the universality of the taxonomy. The taxonomy is not yet shown to be intrinsic; it is a product of one plausible but unvalidated pipeline.

Who is this for? Network scientists looking for a reference on measurement sensitivity, and methodologists interested in shape-based comparison of response curves. It deserves a serious referee: the empirical scope is real and the method is clear. But the referee should insist on a sensitivity analysis for the normalization and index parameters, a null model for the clustering, and code/data release. I would not cite the taxonomy in its current form.

Referee Report

5 major / 3 minor

Summary. The paper proposes a methodological pipeline for studying how seven families of topological network measurements (degree, clustering coefficient, betweenness centrality, assortativity, average shortest path length, hierarchical degree at several scales, and accessibility at several scales) respond to three progressive network modifications: changes in network size, edge removal, and edge rewiring. For Erdős–Rényi, Barabási–Albert, and geographical networks, the authors compute measurement-change curves, normalize them using Eq. (1), and use the coincidence similarity index (Eqs. (2)–(4)) to build similarity networks and agglomerative dendrograms. From these, they identify three qualitative categories of measurement response (A: mostly increasing, B: mostly decreasing, C: other) and conclude that ER and BA networks respond more similarly to each other than to GEO networks.

Significance. If robustly established, the proposed taxonomy of topological-response types and the ER/BA-vs-GEO similarity claim would be a useful descriptive addition to network-science methodology, particularly for researchers concerned with how incomplete or noisy network representations affect measured topology. The paper is commendably explicit about experimental settings: it reports numbers of realizations (Q=1000 for the size experiment and Q=50 for edge removal and rewiring), network sizes, average degrees, and the exact indices used. The main weakness is that the central conclusions rest on a normalization and on similarity-index parameters whose effects are not validated with sensitivity analyses, null models, or quantitative classification criteria. The reported observations are plausible, but the paper does not yet rule out that some of the apparent structure, especially for ER rewiring, is produced by the analysis pipeline itself.

major comments (5)

[Sec. 4.3 and Fig. 16(a)] There is a direct contradiction between the text and the figure caption: Sec. 4.3 states that “all measurements changes obtained for the ER networks belong to the category C,” while the caption of Fig. 16(a) describes the ER rewiring coincidence similarity network as “involving three modules.” Because the input to the similarity network is the Eq. (1) normalized curves, and because the paper itself says these ER curves “consist of statistical fluctuations,” the modular structure in Fig. 16(a) appears to be a potential artifact of normalizing noise. This is load-bearing for the A/B/C taxonomy and for the similarity-network-based grouping. Please reconcile the inconsistency and provide a null-model analysis (for example, applying the same pipeline to shuffled or resampled curves) showing that modular structure is not produced by the pipeline on pure noise.
[Sec. 4.1, Eq. (1)] The normalization c_{i,j} = (x_{i,j} - min(x_{i,j}))/sigma_j is used as the direct input for all similarity networks and dendrograms. The paper itself warns in Sec. 4.1 that this normalization “can amplify the dispersion of a set of values originally presenting standard deviations smaller than 1.” Since curves with very small original variance (including the ER rewiring fluctuations) are scaled to unit standard deviation, the subsequent similarity structure may be dominated by noise amplification rather than by meaningful topological response. No alternative normalization (e.g., min-max, variance-stabilizing, or thresholded) is tested. A sensitivity analysis with at least one other normalization would be needed to support the claim that the identified modules reflect intrinsic properties of the measurement responses.
[Sec. 3.4 and Sec. 3.6] The coincidence similarity index parameters are fixed at D=5, E=1, and delta=0 without a justification or a sensitivity analysis. Since larger values of D and E are said to make comparisons stricter, the module structure and dendrogram topology in Figs. 7, 9, 12, 13, 16, and 17 could depend strongly on these arbitrary parameter choices. The paper should report whether the A/B/C taxonomy and the ER/BA-vs-GEO ordering remain stable across a range of D and E values.
[Secs. 4.1–4.3] The assignment of measurements to the categories A, B, and C is performed by visual inspection of the curves in Figs. 3–5, 10, and 14, with no operational definition. Terms such as “mostly monotonical increase,” “mostly monotonical decrease,” and “other types of changes” are not formalized, and no inter-rater reliability, automated rule, or statistical significance testing is reported. Because the central claim is the existence of these three response types, a reproducible decision rule (for example, based on monotonicity, slope sign, or oscillation amplitude) is necessary.
[Sec. 4.4 and Fig. 18] The main comparative claim that ER and BA networks are more similar to each other than to GEO networks is supported by visual inspection of the dendrograms and the parallel-coordinate diagram in Fig. 18, rather than by a quantitative measure. No confidence intervals, permutation tests, or similarity indices between dendrograms or module assignments are provided. Given that the claim is a central conclusion, the authors should quantify the comparison, for example using normalized mutual information between module assignments or cophenetic correlation between dendrograms, with a suitable null model.

minor comments (3)

[Fig. 13 caption] The caption of Fig. 13 refers to “Fig. 7” but the actual reference should be to Fig. 12, since the dendrograms correspond to the edge-removal similarity networks.
[Sec. 4.2] The discussion of the edge-removal experiment is inconsistent about the direction of the free variable: the text mentions “measurement changes increasing with the average node degree” and then says “the average shortest path length decreases with the number of edges,” which is difficult to reconcile with the same paragraph’s list. Please clarify whether curves are plotted against the number of remaining edges or the number of removed edges, and check the associated A/B assignments.
[Sec. 3.1 and throughout] There are several typographical issues that should be corrected, including “S˜ao” for “São” in the author affiliations and the possessive “complex networks measurements” in Sec. 4.2. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the A/B/C taxonomy and ER/BA-vs-GEO comparison are descriptive summaries of simulated measurement-change curves, not predictions derived from fitted parameters or from self-cited uniqueness claims.

full rationale

The paper fits no parameter to the quantities it later reports. Eq. (1) normalizes each measurement-change curve by its standard deviation, and Eqs. (2)-(4) define the coincidence similarity index with fixed, stated values (delta=0, D=5, E=1; Sec. 3.6). The A/B/C grouping and the ER/BA/GEO comparisons are read from coincidence-similarity networks and dendrograms computed on those normalized curves. Because the formulas are given in the paper and the strictness parameters are not tuned against the reported modules (no fitting or calibration step is described), the central taxonomy is not forced by construction. The method citations [32,33,57,58] are to the authors' own index and clustering framework, but the index is explicitly re-derived in Eqs. (2)-(4) and the clustering is standard agglomerative merging; no uniqueness theorem or prior result is invoked to forbid alternative groupings, so the self-citations are not load-bearing in a circular sense. The paper itself flags the main robustness limitations: Sec. 4.1 warns that the normalization 'can amplify the dispersion of a set of values originally presenting standard deviations smaller than 1,' and Sec. 4.3 states that the ER rewiring curves 'consist of statistical fluctuations and are shown here only for reference.' These passages identify correctness/interpretability risks for the rewiring experiment, and the choice D=5/E=1 lacks sensitivity analysis, but these are validity concerns, not a circular reduction of the claimed result to its inputs. No equation in the paper is shown to equal the target conclusion by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the choice of similarity metric and its parameters, the normalization scheme, the representativeness of the three network models, and a subjective categorization of curves. These are not fitted from data but are chosen by the authors, and the paper does not test the sensitivity of the conclusions to these choices.

free parameters (3)

Coincidence similarity exponent D = 5
Chosen ad hoc to make comparisons strict; the modular structure of the similarity networks depends on D.
Interiority exponent E = 1
Chosen ad hoc; together with D it controls the strictness of the coincidence similarity index and affects the separation of modules.
Geographic network perturbation epsilon = 0.001
Controls the regularity of the GEO networks; not varied or justified, and the paper's conclusions about GEO networks rely on this value.

assumptions (4)

domain assumption The coincidence similarity index (Eq. 2-4) is an appropriate measure for comparing measurement-change curves.
The paper adopts this index from prior work and assumes it captures meaningful similarity; no comparison to alternative similarity measures is provided.
domain assumption The normalization in Eq. 1 (subtract min, divide by standard deviation) preserves the shape of the curves while removing magnitude.
The entire analysis is based on shape; if the normalization distorts relative changes or amplifies noise, the resulting groups may be artifacts.
domain assumption The three network models (ER, BA, GEO) are representative of complex networks.
The paper generalizes findings from these models, but real-world networks may have different properties and the conclusions are specific to the chosen parameter configurations.
ad hoc to paper The visual classification into A (increase), B (decrease), C (other) reflects objective properties of the curves.
The classification is performed by inspection of the curves, not by a formal algorithmic rule, introducing subjectivity that is not acknowledged as such.

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Cite this review

Pith. "Pith review of A Systematic Approach for Studying How Topological Measurements Respond to Complex Networks Modifications." pith.science (2026). https://pith.science/paper/BZR5E6ZB

@misc{pith2026250522345,
  author       = {Pith},
  title        = {Pith review of: A Systematic Approach for Studying How Topological Measurements Respond to Complex Networks Modifications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BZR5E6ZB}},
  note         = {Machine review of arXiv:2505.22345}
}

read the original abstract

Different types of graphs and complex networks have been characterized, analyzed, and modeled based on measurements of their respective topology. However, the available networks may constitute approximations of the original structure as a consequence of sampling incompleteness, noise, and/or error in the representation of that structure. Therefore, it becomes of particular interest to quantify how successive modifications may impact a set of adopted topological measurements, and how respectively undergone changes can be interrelated, which has been addressed in this paper by considering similarity networks and hierarchical clustering approaches. These studies are developed respectively to several topological measurements (accessibility, degree, hierarchical degree, clustering coefficient, betweenness centrality, assortativity, and average shortest path) calculated from complex networks of three main types (Erd\H{o}s-R\'enyi, Barab\'asi-Albert, and geographical) with varying sizes or subjected to progressive edge removal or rewiring. The coincidence similarity index, which can implement particularly strict comparisons, is adopted for two main purposes: to quantify and visualize how the considered topological measurements respond to the considered network alterations and to represent hierarchically the relationships between the observed changes undergone by the considered topological measurements. Several results are reported and discussed, including the identification of three types of topological changes taking place as a consequence of the modifications. In addition, the changes observed for the Erd\H{o}s-R\'enyi and Barab\'asi-Albert networks resulted mutually more similarly affected by topological changes than for the geometrical networks. The latter type of network has been identified to have more heterogeneous topological features than the other two types of networks.

Figures

Figures reproduced from arXiv: 2505.22345 by the authors.

**Figure 2.** Illustration of the similarity comparison implemented by the coincidence sim [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Curves obtained for the measurement changes ∆ in terms of the number of edges [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (15 more)

**Figure 4.** Figure 4: The curve of the measurements changes c in terms of the number of edges respective to the BA network model, containing two main groups as in the ER case. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: The curve of the measurements changes c in terms of the number of edges respective to the GEO network model, containing two main groups characterized by curves respectively increasing and decreasing with the network size. for ∆, as a means of identifying their basic pr…

**Figure 6.** Figure 6: The indices estimating the overall variation [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: The coincidence similarity networks obtained for the experiments involving [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: Diagram indicating the changes of category (respectively to the groups types [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: Dendrograms obtained by hierarchical agglomerative clustering of the coinci [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

**Figure 10.** Figure 10: Curves of measurements changes c in terms of the number of progressively removed edges respectively to the ER (a), BA (b), and GEO (c) types of networks. Markedly distinct results have been obtained for each of these cases. The arrow indicate the direction of change…

**Figure 11.** Figure 11: Barplots of the index estimating the overall variation [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]

**Figure 12.** Figure 12: The coincidence similarity networks obtained for the experiments involving [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]

**Figure 13.** Figure 13: Dendrograms obtained by hierarchical agglomerative clustering of the coinci [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]

**Figure 14.** Figure 14: Curves of measurements changes c in terms of the number of progressively rewired edges respectively to the ER (a), BA (b), and GEO (c) types of networks. Markedly distinct results have been obtained for each of these cases. 4.3 Edge Rewiring This section presents the …

**Figure 15.** Figure 15: The indices quantifying the overall variation [PITH_FULL_IMAGE:figures/full_fig_p028_15.png]

**Figure 16.** Figure 16: The coincidence similarity networks obtained for the experiments involving [PITH_FULL_IMAGE:figures/full_fig_p029_16.png]

**Figure 17.** Figure 17: Dendrograms obtained by hierarchical agglomerative clustering of the coinci [PITH_FULL_IMAGE:figures/full_fig_p029_17.png]

**Figure 18.** Figure 18: Parallel coordinate presenting the membership of the 14 adopted topological [PITH_FULL_IMAGE:figures/full_fig_p030_18.png]

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