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REVIEW 3 major objections 4 minor 81 references

Centrality-based identification of important edges in complex networks

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read By transferring three standard centrality measures from vertices to edges and peeling away less central edges layer by layer, this paper shows that important connections—not important nodes—are what localize seizure-relevant interactions…

desk verdict A useful edge-centrality toolbox paper whose headline epilepsy finding likely rests on an unjustified zero-distance convention for adjacent edges in closeness centrality. read the letter →

arxiv 1908.10667 v1 pith:RZXW2SXE submitted 2019-08-25 physics.soc-ph cs.SI

Timo Bröhl , Klaus Lehnertz This is my paper · ORCID
classification physics.soc-phcs.SI
keywords edgecentralitynetworkdecompositionwebclosenessbetweennesseigenvectorseizureonsetzoneevolvingfunctionalbrainnetworks
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 claims that the standard question of network science—which constituents really matter—can be answered about connections rather than nodes by applying well-known vertex-centrality formulas to edges, treating edges as the units and measuring path lengths between them. It defines edge versions of betweenness, closeness, and eigenvector centrality, and adds a decomposition that repeatedly removes the least-central edges into a hierarchy of "webs," each corresponding to a different level of importance. Across model networks the authors find that highly ranked edges usually attach to highly ranked vertices, yet the two rankings are not interchangeable: different edge centralities often select different top edges, and the webs are typically star-like clusters around central vertices. In the two real-world applications, the most important edges and webs in Zachary's karate club sit around the club's two leaders, and in an epilepsy patient's evolving brain networks they lie near and inside the seizure onset zone—whereas the most important vertices lie elsewhere. The overall claim is that edge-centric importance is a distinct, operational lens for locating the interactions that govern network dynamics.

What carries the argument

The machinery is the edge adjacency matrix (the line graph whose vertices are the original edges, with two such vertices connected when the original edges share an endpoint) combined with a zero-distance convention: the path length between adjacent edges is set to 0, and for weighted networks path length is the sum of inverse weights. Plugging this adjacency matrix into the standard eigenvector equation, and using the same shortest-path distances in the betweenness and closeness formulas, yields the three edge centralities. The web decomposition then iteratively computes a chosen edge centrality, removes all edges at or below the current minimum into a web, recalculates on the remaining edges, and finally reverses the web order so that the last-removed edges form the rank-1 "most important" web. The decomposition is what turns a one-shot edge ranking into a mesoscopic hierarchy of important connection sets.

What would settle it

A decisive test is to rerun the closeness-based web decomposition on the epilepsy data with the distance between adjacent edges set to 1 instead of 0; if the rank-1 web no longer overlaps the seizure onset zone, the clinical conclusion rests on the zero-distance convention rather than on a network property.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that edge centralities defined through the edge adjacency matrix—two edges adjacent if they share a vertex—behave like vertex centralities in many respects but deliver information vertex rankings do not. In weighted small-world, scale-free, and random networks, the top-ranked edge is very often connected to a top-ranked vertex (up to 90% of scale-free realizations, depending on the measure), and edges ranked high by closeness or betweenness are disproportionately traversed by shortest paths; in scale-free networks they connect core or core-periphery vertices. Yet the three edge centralities identify the same top edge in only roughly 20–50% of weighted model networks, so the concept is not redundant. The edge-centrality-based web decomposition produces a bottom-up hierarchy of edge sets that are usually star-like, and in the epilepsy application the temporally stable edges and the rank-1 web point to interactions near and inside the seizure onset zone, in contrast to vertex centralities, which point outside it. This contrast is the strongest evidence the paper offers that edge-level importance is not a corollary of vertex-level importance.

Load-bearing premise

The load-bearing premise is that two edges sharing a vertex are at distance zero from each other in the path metric; if that convention is changed, edge closeness changes and the star-like webs—including the seizure-zone localization—may not survive.

Editorial extensions

If this is right

  • If edge centralities capture importance that vertex centralities miss, then studies that rank only vertices—for example brain-network hub analyses—are missing a layer of structure that can point to the clinically relevant region.
  • The web decomposition supplies a hierarchy of important edge sets, which is useful exactly when the single most important edge cannot be identified unambiguously.
  • In evolving epileptic brain networks, the most important edge webs localize near and inside the seizure onset zone and in homologous contralateral regions, suggesting a network-based marker for seizure dynamics.
  • Because the rank-1 web is typically star-like, the method connects edge importance to structures relevant to synchronization and percolation, giving dynamics studies a specific subgraph to analyze.
  • The methods apply unchanged to binary and weighted networks of any topology; only the number and composition of webs depend on the centrality chosen.

Reading between the lines

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

  • The zero-distance convention for adjacent edges is what makes edge closeness reward edges that share a vertex, so the star-like appearance of webs may be an artifact of the measure; testing with adjacent-edge distance set to 1 would separate metric artifacts from network structure.
  • The epilepsy result rests on a single patient, and the paper does not compare the webs with chance-level surrogates; a multi-patient replication with surrogate networks would be needed before the finding can guide clinical practice.
  • The web decomposition effectively treats communities as groupings of edges rather than vertices; this could yield a new definition of overlapping communities, since an edge belongs to exactly one web but a vertex can be incident to edges in many webs.
  • Because the different edge centralities agree on the top edge in only a minority of model networks, any practical use should probably aggregate across centralities or validate the chosen measure against dynamics.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes edge-level analogues of three standard vertex centralities—betweenness, closeness, and eigenvector centrality—and introduces an iterative 'web decomposition' that partitions the edge set into importance-ranked layers. The methods are tested on synthetic small-world, scale-free, and random networks, on Zachary's karate club, and on evolving functional brain networks of one epilepsy patient. The main reported findings are that important edges tend to connect important vertices, that edge centralities highlight topology-specific structures (including star-like webs), and that in the epilepsy case the most temporally stable important edges and the rank-1 web point to sites near or within the seizure onset zone.

Significance. If the proposed edge centralities and the web decomposition are well defined and reproducible, the paper offers a practical toolbox for edge-level network analysis, and the epilepsy application would be a clinically interesting illustration. The algorithmic decomposition into importance-ranked webs is a useful extension, and the synthetic experiments are systematic across model families and weight settings. However, the central definition of edge closeness is ambiguous or inconsistent as written, and the headline epilepsy conclusion rests on a single patient and on the unexamined zero-distance convention; with these unresolved, the significance of the reported findings is currently uncertain.

major comments (3)
  1. [Sec. II, Eq. (4), definition of d_ij] As written, the convention d_ij := 0 for adjacent edges makes the edge-closeness measure ill-defined. Under the usual additive shortest-path definition, any two edges connected by a chain of adjacent edges then have zero distance, so the denominator in Eq. (4) vanishes for every edge in a connected component. The paper nevertheless reports finite CC_e values, which means that some other computational convention must be used implicitly. Please state precisely how edge distances are computed, and explain the motivation for setting adjacent-edge pairs to zero. This is not a cosmetic point: all closeness-based results, including the web decomposition in Figs. 8, 10, and 13, depend on this choice.
  2. [Sec. IV.B, Fig. 13] The core application claim—that temporally stable edges and the rank-1 CC_e-based web point to the seizure onset zone—is supported by a single patient and by no inferential statistics or surrogate controls. With n=1, the apparent localization could reflect patient-specific electrode placement, the chosen centrality convention, or the zero-distance rule, rather than a property of seizure dynamics. The paper's caveat that the findings 'need to be validated on a larger database' addresses sample size but not this definitional dependency. Please add sensitivity analyses under alternative edge-distance conventions (for example, standard line-graph distance) and a degree/strength-preserving null-model comparison, or substantially soften the conclusion.
  3. [Sec. III.A, Fig. 2, Table II] The correlation analyses between edge ranks and vertex ranks are reported without significance tests, confidence intervals, or null models. Because the centrality definitions share construction principles, part of the observed relationship may be mechanical; for closeness, the zero-distance convention ties edge closeness directly to local degree and strength. A permutation test or a rewiring-based null model is needed before concluding that 'important edges indeed connect important vertices' in a nontrivial sense.
minor comments (4)
  1. [Sec. II, Eq. (2)] The sentence 'excluding the pairs with vertex k' is unclear for edge betweenness, since k is an edge; please specify which endpoint pairs are excluded.
  2. [Sec. II, decomposition algorithm] The threshold rule in step 3 uses the '<' sign for repeated eliminations within the same iteration, but the interplay between re-computation and tie handling is described only through the example; a short pseudo-code or flow chart would remove ambiguity.
  3. [Sec. III, network generation] The expression for the edge density appears to be missing a division sign; it should likely read epsilon = 2E / (V (V - 1)).
  4. [Sec. I, Introduction] The claim that there are 'only a few metrics' for edge importance would be strengthened by a brief comparison with existing edge-centrality measures beyond edge betweenness and bridgeness, especially since edge betweenness is a well-established concept.

Circularity Check

1 steps flagged · score 4.0 of 10

The epilepsy web-localization result is partly a restatement of the zero-distance convention used to define edge closeness centrality.

  1. self definitional [Sec. II (path-length definition and Eq. 4); Sec. III.D; Sec. IV.B (Fig. 13, lower panel)]
    "For i and j being connected to a same vertex, we define dij := 0. ... In case of adjacent edges, i.e., edges connected by a single vertex, we again define dij := 0. ... CC_e(k) = E−1 / Σ_l dkl"

    Setting d_kl=0 for any two edges that share a vertex makes Eq. (4)'s denominator count only non-adjacent edges, so CC_e increases with the number of edges incident to an edge's endpoints. The decomposition removes the lowest-CC_e edges first, so the final rank-1 web is forced to be a star-like set around high-degree/strength vertices. Consequently the paper's reported observation that the most important CC_e/CE_e webs are star-like (Sec. III.D) and the epilepsy claim that the CC_e-based W1 (Fig. 13, lower panel) points to the SOZ are consequences of the d_kl:=0 convention, not independent findings about network topology or seizure dynamics. The stated caveat 'these findings need to be validated on a larger database' addresses sample size, not this definitional dependency.

full rationale

The paper's core methodological development—edge eigenvector, closeness, and betweenness centralities plus the iterative web decomposition—is not circular in the formal sense: the centralities are standard vertex concepts adapted to edges, the decomposition is evaluated on 1000 synthetic small-world/scale-free/random networks and on Zachary's karate club, and no parameter is fitted to a target outcome and then renamed as a prediction. The self-citations (e.g., Refs. 10, 61, 66-70, 80) are used for established methods such as evolving functional networks, mean phase coherence, and the prior SOZ debate, and they are not load-bearing in a way that would force the paper's conclusions. However, one definitional choice is load-bearing for the headline application: setting the distance between adjacent edges to zero in the definition of path length (Sec. II) means Eq. (4)'s closeness centrality is dominated by the number of edges sharing endpoints, and the web decomposition therefore automatically returns star-like structures. The paper transparently notes the star-like character of most important webs (Sec. III.D) but then uses the CC_e-based W1 to conclude that temporally stable important edges point to the seizure onset zone (Sec. IV.B). That conclusion reduces, to a considerable degree, to 'edges incident to dense/high-strength local neighborhoods in the functional network are near the SOZ,' which is an artifact of d_ij:=0 rather than an independent validation of the SOZ's dynamical importance. The paper's own limitation statement only invokes sample size, not this definitional dependency. Overall circularity score 4: the central methodology is independent, but the strongest application-specific claim is substantially built into the measure's definition.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central method rests on a few domain assumptions and one clearly ad hoc convention (zero distance between adjacent edges). There are no fitted parameters; network generation parameters are standard and not fitted to the conclusions.

assumptions (5)
  • ad hoc to paper Distance between adjacent edges is set to zero in all shortest-path computations (Sec. II, definition of dij).
    Not justified in the text; it inflates closeness centrality of edges with many neighbors and shapes the 'star-like' webs.
  • domain assumption Edge centrality measures are valid proxies for importance of edges (Sec. II).
    Standard network-science assumption; the paper does not validate against dynamics or external criteria.
  • domain assumption The network is undirected and either binary or weighted with positive weights; self-loops are excluded (Sec. II).
    Matches many but not all real networks; directed and multiplex networks are not covered.
  • ad hoc to paper Betweenness centrality on disconnected networks treats pairs with no path as contributing zero, although this is not stated (Sec. II, Eq. 2 and example in Table I).
    The decomposition repeatedly disconnects the network; the paper does not define 0/0 cases, yet the example reports a value of 0.1 for a single edge.
  • domain assumption The web decomposition's iterative thresholding at the minimum centrality yields a meaningful importance hierarchy (Sec. II, algorithm steps).
    No theoretical justification is given for why this produces a useful hierarchy.
invented entities (1)
  • Web (edge-centrality-based decomposition layer)
    purpose: To group edges into a hierarchy of importance, from least to most important, analogous to k-core shells but for edges.
    The web is defined entirely by the paper's algorithm; no external validation shows that webs correspond to independently meaningful functional units. The claim that the most important web is star-like is a consequence of the measure, not an independent finding.

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Pith. "Pith review of Centrality-based identification of important edges in complex networks." pith.science (2026). https://pith.science/paper/RZXW2SXE

@misc{pith2026190810667,
  author       = {Pith},
  title        = {Pith review of: Centrality-based identification of important edges in complex networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RZXW2SXE}},
  note         = {Machine review of arXiv:1908.10667}
}
read the original abstract

Centrality is one of the most fundamental metrics in network science. Despite an abundance of methods for measuring centrality of individual vertices, there are by now only a few metrics to measure centrality of individual edges. We modify various, widely used centrality concepts for vertices to those for edges, in order to find which edges in a network are important between other pairs of vertices. Focusing on the importance of edges, we propose an edge-centrality-based network decomposition technique to identify a hierarchy of sets of edges, where each set is associated with a different level of importance. We evaluate the efficiency of our methods using various paradigmatic network models and apply the novel concepts to identify important edges and important sets of edges in a commonly used benchmark model in social network analysis as well as in evolving epileptic brain networks.

Figures

Figures reproduced from arXiv: 1908.10667 by the authors.

Figure 1
Figure 1. Repeating steps during iteration n and resulting from step 4 are marked with *. C B e -values chosen as threshold Θ are shown in red. Column ‘ref.’ refers to subplots (a-d) in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Distributions of correlations between edge ranks and vertex ranks for the investigated topologies of weighted networks [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Left: Fraction of shortest paths in weighted small [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (10 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Percentage [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Means and standard deviations of normalized strength [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Number of webs [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Occurrences of a certain number of substructures within the 10 most important webs (color coded) in 1000 realizations [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Left: Number of edges merged into the most important web [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Zachary’s karate club network. Numbered vertices [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Most important web (rank 1) obtained with an edge [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Same as Fig. 10 but for the binary karate club net [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. (A) Schematics of sensors placed over the left and [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Top: Most important edges and vertices (top) and most important web (rank 1, bottom) projected onto a schematics [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

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Pith tools

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