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REVIEW 3 major objections 4 minor 44 references

Importance of Overlapping Network Nodes in Influence Spreading

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Nodes belonging to multiple overlapping circles consistently show higher influence in spreading processes than nodes in a single circle, under both simple and complex contagion, across four real-world networks.

desk verdict Useful descriptive study of overlap and spreading centrality, but the missing degree-matched baseline means the paper cannot yet support the claim that overlap itself drives influence. read the letter →

arxiv 2510.24360 v3 pith:7LU2WBKW submitted 2025-10-28 cs.SI

Kosti Koistinen , Vesa Kuikka , Kimmo Kaski This is my paper · ORCID
classification cs.SI
keywords overlappingnodesinfluencespreadingcomplexcontagionsimplecentralitymatrixcirclessocialnetworks
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

This paper asks whether nodes that sit in several overlapping network circles matter more for spreading influence than nodes that belong to only one. Using a probabilistic influence spreading model on real ego-networks from Facebook, LiveJournal, Orkut, and Wikipedia, the authors compare overlapping and non-overlapping nodes with three centrality measures: In-Centrality (susceptibility), Out-Centrality (spreading power), and Betweenness Centrality (mediating role). They find that overlapping nodes are consistently more influential at every stage of the process, with Out-Centrality differences reaching about 90% in the saturated phase for LiveJournal and Orkut, and geometric-mean ratios exceeding one for Out- and Betweenness Centrality in every network under both contagion mechanisms. The result matters because it identifies a small, recognizable class of nodes that could act as levers for accelerating or containing spread, and it also clarifies the distinction between local, attribute-driven circles and global community structures.

What carries the argument

The central object is the Influence Spreading Matrix (ISM), a matrix $C$ whose entry $C_{ij}$ is the probability that influence originating at node $i$ reaches node $j$ under the probabilistic Influence Spreading Model. From this matrix the paper derives three node-level metrics: In-Centrality (column sums, susceptibility), Out-Centrality (row sums, spreading power), and an ISM-based Betweenness Centrality defined as the relative decrease in total network cohesion when the node is removed. Because the ISM accounts for all propagation paths rather than only shortest paths, these metrics capture probabilistic and temporal features of spreading that conventional centrality measures miss, and the same framework accommodates both simple contagion (single-pass self-avoiding paths) and complex contagion (recurrent interactions and feedback). The key comparison is the relative difference between average metric values for overlapping and non-overlapping nodes, backed by bootstrap geometric-mean ratios.

What would settle it

Repeat the analysis after matching each overlapping node to a non-overlapping node with the same degree, or re-running the simulations on degree-preserving randomizations of circle memberships; if the Out-Centrality advantage and the geometric-mean ratios above one vanish, then overlap per se does not drive the effect.

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Extended reading notes

Core claim

The paper's central claim is that overlapping nodes are not peripheral participants but consistently the strongest drivers of influence spreading. This is demonstrated by computing, from the Influence Spreading Matrix, the average In-, Out-, and Betweenness centrality of the two node classes and comparing them over time. At the start of spreading, overlapping nodes show markedly higher In-Centrality, indicating greater exposure; as the process saturates, the In-Centrality gap narrows while the Out-Centrality gap persists or grows, so overlapping nodes keep spreading power long after the initial wave. The betweenness analysis shows that overlapping nodes retain their mediatory role over a longer period, and bootstrap ratios of geometric means confirm the effect in every dataset and under both contagion models, with the exception of a less decisive In-Centrality ratio hovering near unity. The authors further report that the circle definition matters: when only the largest circles are kept, the overlap advantage shrinks only gradually, implying that the most influential overlapping nodes live in large circles rather than small triads.

Load-bearing premise

The comparison does not control for node degree or other attributes, so the entire case rests on the assumption that the overlap itself, rather than the higher connectivity that often comes with it, is what produces the measured influence advantage.

Editorial extensions

If this is right

  • Immunization and targeted influence campaigns can concentrate on overlapping nodes, since these nodes keep a disproportionate spreading role even after saturation.
  • Out-Centrality computed from the ISM offers a computationally cheaper proxy for Betweenness Centrality when only relative differences between node classes are needed.
  • Circle-definition choices change the measured overlap effect, so any comparison across datasets must state the minimum circle size; restricting to large circles preserves the influence advantage and locates key influencers in the largest circles.
  • The ordering of overlapping over non-overlapping nodes holds under both simple and complex contagion in all four networks, so the effect appears robust to the contagion mechanism.
  • Overlap alone is not enough to identify true influencers; the most influential nodes likely sit at the intersection of circles and community structures.

Reading between the lines

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

  • A degree-matched replication would show whether the overlap advantage is independent of connectivity; the paper does not perform this matching.
  • The same ISM-based machinery could be applied to directed and temporal networks, where path reversal symmetry breaks and In- versus Out-Centrality differences become more informative.
  • The concentration of super-influencers in large circles suggests a practical two-step targeting heuristic: find large circles, then pick nodes that belong to several, which could be tested against standard influence-maximization algorithms.
  • Because the bridging mechanism is largely topological, overlapping communities rather than circles should show an even stronger effect, a prediction the authors leave open.
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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 studies the role of nodes that belong to multiple overlapping circle structures in influence spreading on networks. Using a probabilistic Influence Spreading Model (ISM), the authors define three centrality metrics—In-centrality, Out-centrality, and Betweenness Centrality—and compare their values between overlapping (OL) and non-overlapping (NOL) nodes in four real-world network datasets (Facebook, LiveJournal, Orkut, and Wikipedia categories) under both simple and complex contagion. The main reported finding is that OL nodes exhibit consistently higher Out-centrality and Betweenness Centrality than NOL nodes, with relative differences up to 90% in LiveJournal and Orkut, and geometric-mean ratios exceeding one in all networks. The paper also analyzes how the minimum circle size affects the OL advantage, concluding that influential nodes reside predominantly in larger circles.

Significance. If the central claim were established, the paper would contribute to understanding the role of overlapping substructures in spreading dynamics and would have practical implications for targeted intervention or influence maximization. The study has notable strengths: it uses externally defined ground-truth circles rather than algorithmically detected communities, compares multiple networks with diverse structural properties, tests both simple and complex contagion, and provides bootstrap confidence intervals and a robustness check over edge weights. However, the current analysis does not rule out the simpler explanation that the observed OL advantage is a proxy for higher node degree, and the reliance on the ISM model—referenced but not described in sufficient detail—limits the reader's ability to assess the generality of the findings. The paper's headline claim is also stronger than the presented evidence: the FB dataset and the In-centrality results show only weak or no consistent advantage.

major comments (3)
  1. [Results, Eqs. (3), (8), (10), Figs. 4–8] The comparison between OL and NOL nodes is not adjusted for node degree. Since nodes in multiple circles are likely to have more social connections, and since Out-centrality and Betweenness Centrality derived from the ISM are strongly influenced by the number and length of paths emanating from a node, the reported 90% higher Out-centrality (LJ, ORK) and the geometric-mean ratios R>1 (Figs. 7–8) may reflect the fact that OL nodes are better connected rather than an effect of overlap itself. Appendix B reinforces this concern: when edge weights approach 1, the OL advantage diminishes, consistent with a connectivity-driven mechanism. The authors should provide a degree-matched or degree-stratified comparison (e.g., matching each OL node to a NOL node of the same degree, or including degree as a covariate) to support the claim that overlap has an independent effect on influence.
  2. [Methods, 'Methods' subsection] The Influence Spreading Model is only referenced (ref. 30) rather than described; the reader is told that the model outputs a matrix C of pairwise influence probabilities, but not the update equations, the handling of simple versus complex contagion, or the role of parameters such as the uniform edge weight 0.05 and maximum path length 100. Because all centrality metrics and the betweenness measure are defined on this ISM, the reported results are entirely internal to this model. The authors should either summarize the model's equations in the manuscript or provide a more detailed description in an appendix, and they should discuss how the model's parameters are chosen and whether the qualitative findings are robust to reasonable variations. Ideally, the model should be validated against an external spreading dataset or at least compared with a standard SIR-like simulation to ensure the centralities are not an artifact of the ISM's specific formulation.
  3. [Abstract; Results, Fig. 2, Fig. 7c] The abstract states that 'at each stage of the spreading process the overlapping nodes consistently exhibit greater influence than the non-overlapping ones,' but this is not fully supported by the authors' own results. In the FB dataset, Figure 2 shows OL nodes shifted to lower Betweenness Centrality in the top decile, and the text reports only a 'smaller difference' for FB. Moreover, the In-centrality relative difference decreases smoothly over time (Fig. 4b) and the geometric-mean ratio for In-centrality concentrates around unity (Fig. 7c). The strong claim of consistency should be qualified to specify the metrics (Out-centrality and Betweenness Centrality) and datasets for which the effect is robust, or the analysis should be extended to establish a consistent effect across all three metrics.
minor comments (4)
  1. [Abstract] There is a typo: 'importanc' should be 'importance'.
  2. [Methods, 'Choosing the Edge Weights' (Appendix B)] The parameter choice of edge weight 0.05 is defended in Appendix B, but the discussion of weights appears after the results; consider moving a brief justification of the edge weight and maximum path length to the Methods section for readers who do not read the appendix.
  3. [Results, Fig. 2 caption] The caption says 'Cumulative density' but the figure plots cumulative distribution functions; the wording should be corrected to 'cumulative distribution function'.
  4. [Discussion, 'The Choice of Circles'] The sentence 'Subsequently, only few very peripherial and isolated nodes would be classified as NOL' contains a typo: 'peripherial' should be 'peripheral'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the OL/NOL advantage is an unconstrained model computation on external ground-truth circles, not a fitted or definitionally forced prediction.

full rationale

The paper's central comparison—OL vs NOL centrality—is computed from the Influence Spreading Matrix (Eqs. 1-2), the Betweenness metric (Eqs. 4-6), and the aggregate differences (Eqs. 3, 8, 10). The overlap labels are external ground-truth circle memberships, not quantities constructed from the centrality values. The ISM parameters are fixed uniformly (edge weight 0.05, path length 100, node probabilities 1) and are nowhere fitted to reproduce the reported 90% Out-centrality gap or the R-ratios; the sign of Eq. 3 and the value of Eq. 10 are not forced by the definitions and could, in principle, have gone either way. The only self-citation touchpoints are the adoption of the Influence Spreading Model from [30] and the ISM-based Betweenness Centrality from [32], both by the present authors; these are methodological citations rather than load-bearing uniqueness theorems, and they are supported by the model's prior use and by the paper's own robustness check in Appendix B (weight sweep). The skeptic's degree-confound concern is a covariate-control limitation of the empirical comparison, not a circular reduction; Appendix B even reports that the effect diminishes at high edge weights, which is a substantive falsifiable feature rather than an artifact of the definition. Therefore no equation equates the target result to an input by construction, and the appropriate finding is no significant circularity, with only minor methodological self-citation. Degree confounding is better treated as a correctness or confounding risk than as circularity, because the paper does not fit its parameters to the OL/NOL gap and does not define overlap in terms of influence.

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

The central claim depends on the ISM model from the authors' prior work, ground-truth circle quality, and several hand-set parameters (edge weight, path length, circle size threshold). None of these are fitted to reproduce the OL/NOL effect, which keeps circularity burden low, but the model is not independently validated.

free parameters (4)
  • Uniform edge weight = 0.05
    Set to 0.05 for all edges to capture gradual saturation; sensitivity analysis in Appendix B shows the qualitative effect persists across 0.001 to 1.0, so it is not fitted to the target result.
  • Maximum path length = 100
    Chosen to account for far-reaching influence without excessive computational cost; no sensitivity check is reported.
  • Minimum circle size = 10 (for LJ and ORK)
    Circles with fewer than 10 nodes are excluded for LiveJournal and Orkut to make OL/NOL comparison reliable. The Discussion and Figure 9 show the effect varies with this threshold.
  • Subnetwork size range = 500 to 1500 neighbor nodes
    Ego-network extraction selects nodes between 500 and 1500 neighbors; the sampling procedure is not fully specified.
assumptions (3)
  • domain assumption The probabilistic Influence Spreading Model (ISM) from Kuikka (2018) faithfully represents simple and complex contagion in real networks.
    The entire analysis uses the ISM to compute centrality; the paper references Refs [30,32,39] for definitions and does not validate the model against empirical spreading data.
  • domain assumption The ground-truth circles in the SNAP datasets are meaningful cohesive subgroups representing actual social circles.
    OL/NOL labels are derived from these circles; if the circles are noisy or inconsistent, the comparison is compromised.
  • domain assumption The edge-weight parameter w=0.05 is a reasonable uniform interaction strength for all networks.
    Used for all datasets; sensitivity is checked only for ORK (Appendix B).

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Pith. "Pith review of Importance of Overlapping Network Nodes in Influence Spreading." pith.science (2026). https://pith.science/paper/7LU2WBKW

@misc{pith2026251024360,
  author       = {Pith},
  title        = {Pith review of: Importance of Overlapping Network Nodes in Influence Spreading},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7LU2WBKW}},
  note         = {Machine review of arXiv:2510.24360}
}
read the original abstract

In complex networks there are overlapping substructures or "circles" that consist of nodes belonging to multiple cohesive subgroups. Yet the role of these overlapping nodes in influence spreading processes remains underexplored. In the present study, we analyse networks with circle structures using a probabilistic influence spreading model for processes of simple and complex contagion. We quantify the roles of nodes using three metrics, i.e., In-Centrality, Out-Centrality, and Betweenness Centrality that represent the susceptibility, spreading power, and mediatory role of nodes, respectively, and find that at each stage of the spreading process the overlapping nodes consistently exhibit greater influence than the non-overlapping ones. Furthermore, we observe that the criteria to define circles shape the overlapping effects. When we restrict our analysis to only largest circles, we find that circles reflect not only node-level attributes but also of topological importance. These findings clarify the distinction between local attribute-driven circles and global community structures, thus highlighting the strategic importanc of overlapping nodes in spreading dynamics. This provides foundation for future research on overlapping nodes in both circles and communities.

Figures

Figures reproduced from arXiv: 2510.24360 by the authors.

Figure 1
Figure 1. Illustration of a differences between circles and communities. A small network of 12 nodes with three circles (left) and two communities (right). Nodes 5 and 8 lie in Overlapping Circle regions, while the nodes 6 and 7 are in the intersection of two overlapping communities. hubs, and accelerate the spread of information or contagion10, 12. Neglecting the overlapping structure in spreading models can result in undere… view at source ↗
Figure 2
Figure 2. Cumulative density of Betweenness Centrality (BC) of both OL and NOL nodes in CC-model. The first shaded areas from left represent the majority (80%) of nodes, the latter the 91%–99% decile. A small amount of uniform jitter between classes has been added to distinguish these two percentile groups in the plot. with smaller contributions (≲ 5%) in other datasets. The upper tails of the Betweenness Centrality distribut… view at source ↗
Figure 3
Figure 3. The Lorenz curves of Betweenness Centrality distributions. The dashed grey lines mark the 10th and 90th percentiles. The dashed coloured lines represent the proportion of Betweenness Centralities that fall in the bulk. Dataset Total Top 10% Top 1% N10% N1% Lorenz10% Lorenz1% LJ 77.1 87.8 89.2 5913 591 35.9 4.18 ORK 79.3 88.2 92.9 2722 27 41.1 5.16 wiki 27.3 30.3 31.2 14680 1468 43.8 8.00 FB 11.2 10.3 15.1 316 32 20.… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: CC: Comparison of OL and NOL nodes’ relative difference. Out-Centrality (left) and In-Centrality (right) plotted with standard error mean (shaded) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: SC: The network’s relative In- and Out-Centrality differences with standard error means. LJ datasets, which are absent in the CC case. Otherwise, the models show only minor variations within the standard error of the mean (SEM). Two key reasons explain this similarity:…
Figure 6
Figure 6. Figure 6: Temporal evolution of relative Betweenness Centrality for CC (left) and SC (right). Simulations were run on a subsample of networks: all FB networks were included, while for the remaining datasets we randomly sampled 20 networks for CC and 10 for SC. Ratio of Geometric…
Figure 7
Figure 7. Figure 7: Geometric mean–ratio distributions via bootstrapping for T = 50. real-world cross-group "broker" are insufficient or peripheral. Such scenarios result in uneven diffusion, even if average homophily measures suggest otherwise. A similar indirect observation emerges in o…
Figure 8
Figure 8. Figure 8: Geometric mean–ratio distributions via bootstrapping for T = 30 of Betweenness Centrality. would be classified as NOL and the whole comparison would become irrelevant. We therefore restricted the minimum size of a circle to 10 nodes. Indeed, the importance of selecting…
Figure 9
Figure 9. Figure 9: The OL and NOL relative difference in Out- and In-Centrality as a function of circle size for T = 100 for the LJ dataset. The increase of circle’s minimum size reduces the proportion of OL nodes. The most influential OL nodes reside in largest circles. The analysis of …
Figure 10
Figure 10. Figure 10: Complex Contagion with Pokec data. No relative difference in centralities when deriving synthetic circles. Appendix B. Choosing the Edge Weights It is well-known edge weights influence the most to information passing in networks46. Too small edge weights hold the spre…
Figure 11
Figure 11. Figure 11: The OL and NOL relative difference in both complex (top) and simple contagion (bottom) as a function of time for ORK networks with various weights. In- and Out-Centralities are, again, equivalent for SC, which is a property for undirected networks. 17/17 [PITH_FULL_I…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.