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REVIEW 4 major objections 5 minor 79 references

Dynamics of temporal influence in polarised networks

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that temporal degree centrality and a modified temporal independent cascade model can reliably assign users to influence bands in polarised temporal networks, while eigenvector and closeness centralities cannot.

desk verdict A useful empirical comparison of temporal centralities in polarized networks, but the headline conclusion is not independently tested because both synthetic and real-world ground truths are defined by degree or T-ICM. read the letter →

arxiv 2507.17177 v1 pith:EHPXWYYA submitted 2025-07-23 cs.SI

Caroline B. Pena , David J.P. O'Sullivan , Pádraig MacCarron , Akrati Saxena This is my paper · ORCID
classification cs.SI
keywords temporalnetworkspolarisationcentralitymeasuresinfluencebandsindependentcascademodelcommunitystructurebalancedaccuracyTwitterpolitics
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 who the influential users are in a social network split into opposing communities, and whether their influence persists as the network changes over time. It proposes grouping users into three influence bands — high, mid, and low — and tracking how users move between bands across time-slices. Using synthetic networks with known bands, the authors show that temporal degree centrality and a modified temporal independent cascade model recover the bands with balanced accuracy around 0.8 to 0.9, while eigenvector and closeness centrality score lower and are distorted by community structure. On a Twitter network about the 2018 Irish abortion referendum, the same two methods give band assignments that agree, and eigenvector centrality again behaves poorly. The practical consequence is that simple degree-based measures can serve as reliable proxies for influence ranking in polarised online conversations.

What carries the argument

The argument is carried by three constructions. First, a layered temporal unfolding of the network — each node at time $t$ linked to itself and to neighbours at time $t+1$ — supplies temporal degree and closeness centrality, while a supra-centrality matrix with identity inter-layer links supplies eigenvector, PageRank, and Katz scores. Second, the benchmark is a modified temporal independent cascade model encoded as an infection-probability matrix $W(\rho)$ whose diagonal blocks add the identity, so an infected node remains able to spread in later time-slices instead of returning to the susceptible state. Third, influence bands are produced by hierarchical clustering with complete linkage on the centrality scores, with the number of clusters chosen by the elbow method. Performance is measured by balanced accuracy against known bands on synthetic networks and against the cascade-model bands on the real network. The construction of $W(\rho)$ is the key adaptation: it turns cascade spreading into a weighted multilayer matrix while keeping content persistent across time-slices.

What would settle it

Build ground-truth influence from observed diffusion in the RT8 data itself — for example, by measuring how many distinct users eventually mention or retweet content originated by each seed user — and compare the resulting band assignments with those of temporal degree centrality. If degree centrality's balanced accuracy against this empirical ground truth is not in the 0.8–0.9 range, or if using a different infection probability in the cascade model changes which method matches, the claim that degree centrality reliably isolates influence bands in real polarised networks fails.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that in polarised temporal networks, influence ranking should be built on local temporal connectivity rather than global spectral properties. Temporal degree centrality, defined as the normalised number of edges incident to a node over a time interval, and a modified temporal independent cascade model that keeps infected nodes active across time-slices both assign nodes to the correct influence bands with balanced accuracy near 0.9 in the simplest synthetic network and near 0.85 when degrees are Poisson-distributed. Eigenvector centrality consistently over-scores one community because its eigenvectors localise in modular networks, and closeness centrality produces unstable bands. The same pattern appears in the real RT8 Twitter network, where the modified temporal independent cascade model is used as the benchmark: degree, Katz, and closeness agree with it, while PageRank diverges. The paper concludes that the cascade model and degree centrality reliably isolate nodes into influence bands, and that eigenvector and closeness centralities are not appropriate for polarised networks.

Load-bearing premise

The load-bearing premise is that the modified temporal independent cascade model with infection probability $\rho=0.02$ correctly represents influence spread on the RT8 Twitter network, so its band assignments can serve as ground truth; if that model is wrong, the real-world conclusion that degree centrality performs best has no support.

Editorial extensions

If this is right

  • On synthetic polarised networks with known ground truth, temporal degree centrality and the modified temporal independent cascade model achieve overall balanced accuracy near 0.9 in the simplest network and near 0.85 when degree distributions overlap.
  • Eigenvector centrality systematically attributes the highest scores to a single community in modular networks, and randomising the network to remove modularity makes it behave better, confirming that community structure is the cause of its poor performance.
  • Community-level influence can be tracked over time by aggregating marginal node centrality into marginal community centrality, which gives comparable per-community influence scores for the cascade model, degree, and PageRank in balanced synthetic networks.
  • In the real RT8 network, all methods place only one or two nodes in the top influence band, and the nodes identified differ across methods: the PageRank hub is a highly active canvasser that no other method places in the top band.
  • The decay of joint community-time scores over early time-slices is shared by the cascade model, degree, closeness, and Katz, while eigenvector-based methods favour the central time-slices instead.

Reading between the lines

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

  • Because the temporal independent cascade model is both the benchmark and one of the evaluated methods on the real network, the real-world comparison is only as strong as the assumption that cascade dynamics with $\rho=0.02$ mirror true influence; testing with other infection probabilities or with observed retweet cascades as ground truth would settle whether degree centrality genuinely outperforms
  • The synthetic band-swapping dynamics are restricted to moves between neighbouring bands with fixed band sizes; allowing node turnover, more distant band jumps, or time-varying infectivity could reveal which centrality methods generalise to more volatile settings.
  • The result suggests a low-cost monitoring tool in practice: streaming temporal degree centrality alone could flag users who cross influence bands in polarised conversations, with expensive cascade simulations reserved for periodic recalibration.
  • The band-aggregation view could be extended to forecast influence shifts around external events, since the real-network time-slices are already defined by televised debates and the referendum day, making those natural change points for testing predictive accuracy.
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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

4 major / 5 minor

Summary. The paper studies how to identify influential users in polarised temporal social networks. It compares temporal centrality measures (degree, closeness, eigenvector, PageRank, Katz) and a modified Temporal Independent Cascade Model (T-ICM) on synthetic 'BandNet' networks with known influence bands, on the RT8 Twitter conversation around the 2018 Irish abortion referendum, and on a configuration-model randomisation of RT8. Centrality scores are clustered into three influence bands using hierarchical clustering, and a Marginal Community Centrality (MCC) is introduced to track community-level influence over time. The paper's central claim is that temporal degree centrality and the modified T-ICM 'reliably isolate nodes into their bands', while eigenvector and closeness centrality do not perform well in polarised networks.

Significance. If the validation were sound, the paper would offer a practical framework for tracking influence in polarised temporal networks: the influence-band aggregation is interpretable, the MCC provides a clean community-level summary, and the systematic comparison across synthetic and real networks is well structured. The authors also correctly identify that temporal centrality in modular networks is understudied relative to static community-aware centrality. However, the current evidence does not independently establish the headline claim: the synthetic ground truth is defined by degree classes, and the empirical benchmark is the very T-ICM method under evaluation. The significance is therefore contingent on replacing or supplementing these validation steps with an independent measure of influence.

major comments (4)
  1. [§2.3.1, Tables 2–4] The synthetic ground truth is defined directly from node degree: band 1 nodes have degree 30 or Poisson(40), band 2 nodes have degree 10 or Poisson(20), and band 3 nodes have degree 2 or Poisson(5), with band swaps implemented as swaps of degree. Degree centrality is monotone in degree, so the high balanced accuracy of degree centrality (and of T-ICM, which is degree-aligned as discussed below) is guaranteed to a large extent by construction. The tables therefore measure whether degree-based clustering recovers degree classes, not whether those classes are influence bands. Moreover, the claim that degree and T-ICM are the best is not uniform in these tables: PageRank scores 0.88 in BandNet3 versus 0.78 for degree and 0.81 for T-ICM, and PageRank ties degree and T-ICM in BandNet1. The conclusion in Section 4 that 'T-ICM and degree centrality perform the best' needs to be reconciled with these numbers or replaced by a more nuanced statement.
  2. [§2.4, §3.2.1, Fig. 14] The real-world validation is circular. Section 2.4 states that T-ICM is used 'as the true in the empirical networks', and Section 3.2.1 treats T-ICM as the benchmark for the other methods. Yet T-ICM is also one of the methods being evaluated, so the balanced accuracy reported in Fig. 14 is pairwise agreement between candidate methods, not agreement with an independent ground truth. The statement in Section 4 that T-ICM is 'a good benchmark' and that degree centrality 'reliably isolates nodes into their bands' on RT8 is therefore unsupported. The paper should either obtain an independent standard (e.g., observed retweet/cascade data on the same network) or explicitly reframe the empirical contribution as a consistency analysis rather than validation.
  3. [Eq. (3), §2.2] The modified T-ICM in Eq. (3) inserts the identity matrix I into every temporal block, so an infected node remains infected in all subsequent time-slices and attempts to infect neighbours in each later layer. Under this construction, the expected cascade size is heavily aligned with the summed temporal degree of the seed node: a high-degree node in any time-slice gets many infection attempts, and persistence amplifies this effect. This makes the observed agreement between T-ICM and degree centrality structurally unsurprising rather than an independent confirmation. The authors should test a variant without indefinite persistence (e.g., the original Haldar et al. model without inter-layer self-links) and report whether the ranking and band assignments change.
  4. [§2.3, §3.1, §3.2, Tables 2–4] The infection probability ρ is hand-set to different values per network (ρ = 0.1 for BandNet1, 0.08 for BandNet2/3, 0.02 for RT8) with no sensitivity analysis, so it is not shown that the conclusions are robust to ρ. In addition, the synthetic networks are generated stochastically (configuration model and random rewiring), yet Tables 2–4 report only point estimates of balanced accuracy with no error bars, confidence intervals, or repeated runs. The differences between methods in BandNet3 (e.g., 0.88 vs 0.81 vs 0.78) may be within the sampling noise of a single realisation. The authors should provide ensemble results over repeated network generations and a sensitivity analysis over ρ.
minor comments (5)
  1. [§3.2.2, Fig. 16] The caption of Fig. 16 refers to the 'original RT8 network', but the surrounding text and the discussion of the configuration model indicate that the figure reports results for the randomised RT8 network; the caption should be corrected.
  2. [§3.1.3, near Fig. 10] The text contains the broken phrase 'This is due toThis causes the localization of eigenvector centrality...'; the sentence should be rewritten.
  3. [§2.2] There is a duplicated word in 'This can can be interpreted as analogous to the matrix M defined in Eq. 1.'
  4. [Abstract, §1] The phrase 'information becomes soiled within communities' should probably be 'siloed' or 'confined'; as written, 'soiled' is likely a typo.
  5. [Table 1] The row labels 'RT8: config.' and 'RT8: original' are terse and should be expanded to 'RT8: configuration model' and 'RT8: original network' to avoid ambiguity.

Circularity Check

3 steps flagged · score 6.0 of 10

Central validation is partially circular: synthetic 'true' influence bands are defined by degree, and the real-world benchmark T-ICM is also a candidate method whose construction aligns it with temporal degree.

  1. self definitional [Section 2.3.1 (BandNet construction) and Section 3.1.1 / Tables 2-4]
    "Create two networks where a small number of nodes with a high degree (band 1), a moderate amount of nodes with a moderate degree (band 2), and a large amount of nodes with a low degree (band 3). ... Here the true bands are tracked over time from the initial setup in time-slice t1, i.e., nodes that swap bands are tracked over time."

    The synthetic 'true' influence bands are, by construction, degree classes: band 1 is high degree, band 2 moderate degree, band 3 low degree. Balanced accuracy in Tables 2-4 then measures how well each centrality clustering recovers degree-defined classes. Temporal degree centrality is monotone in node degree, so its high scores (overall 0.90, 0.85, 0.78) are guaranteed up to clustering noise rather than being evidence that degree identifies influence. Calling these classes 'influence bands' presupposes the relation the paper claims to establish.

  2. other [Section 2.2, Section 2.4, Section 3.2.1, Fig. 14]
    "As a benchmark to the centrality methods studied, we use the Temporal Independent Cascade Model ... with T-ICM being used as the true in the empirical networks ... (please note that when we lack ground true we rely on the T-ICM as the benchmark for the other methods)."

    T-ICM is declared the benchmark, but it is also one of the methods being evaluated: it appears alongside the centralities in Figs 5, 8, 10, 13 and in the BandNet balanced-accuracy tables. On the real RT8 network there is no external ground truth, so the empirical validation reduces to pairwise agreement among candidate methods, with T-ICM acting as both referee and player. The reported agreement between degree centrality and T-ICM is therefore consistency, not independent confirmation that either method identifies influence.

1 more flagged steps
  1. other [Section 2.2, Eq. (3)]
    "Matrix W(ρ) ensures that a node infected in time-slice t will still be infected and will attempt to pass the information forward in the subsequent time-slices. Hence, the identity matrix I added to each weighted block matrix in W(ρ)."

    The modified T-ICM inserts I into every temporal block (A(t)ρ + I), so an infected node persists in all later time-slices and attempts to infect neighbours in each time-slice. Expected cascade size is then approximately a temporal sum of the seed node's degrees, making T-ICM's band assignments structurally aligned with temporal degree centrality. Thus the empirical 'agreement' between degree and the T-ICM benchmark is baked into the model construction rather than being an independent validation.

full rationale

The central claim that temporal degree centrality and the modified T-ICM 'reliably isolate nodes into their bands' rests on two contaminated ground truths. In the synthetic BandNets, the true bands are defined as degree classes (Section 2.3.1), so degree centrality's high balanced accuracy is a self-consistency check, not an independent test of influence. In the real RT8 analysis, T-ICM is used as the 'true' benchmark while simultaneously being one of the methods evaluated (Section 2.4 and Section 3.2.1), so the reported accuracies against T-ICM are pairwise agreements among candidates. The construction of W(ρ) with identity blocks in every temporal layer further aligns T-ICM cascade size with temporal degree, explaining why the two agree. The paper contains genuine independent content: the relative comparison among PageRank, Katz, closeness, and eigenvector centrality in synthetic networks, and the randomisation analysis, are not forced. There is no load-bearing self-citation chain: the reuse of the authors' earlier RT8 dataset and community labels is data reuse, not circular argument. However, the headline validation of the two 'best' methods is substantially circular, warranting a score of 6 rather than a higher score.

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

The central claim rests on several hand-chosen parameters (infection probability, Katz alpha offset, epsilon, band count, rewiring fraction) and on benchmark assumptions (T-ICM as ground truth, community labels from prior work). None of these are fitted against independent external data, so the quantitative conclusions should be read as conditional on these choices.

free parameters (5)
  • T-ICM infection probability rho = 0.1 (BandNet1), 0.08 (BandNet2/3), 0.02 (RT8)
    Set by hand to 'ensure the process is subcritical' (Section 2.2, Sections 3.1 and 3.2); the benchmark results depend on this choice, and no sensitivity analysis is given.
  • Katz attenuation alpha = 1/zeta(A) - 1e-2
    Chosen near the convergence limit in Section 2.1.3; the 1e-2 offset is a hand-picked constant that affects the centrality scores.
  • Eigenvector inter-layer coupling epsilon = 1
    Set to 1 in Section 2.1.2 because choosing the best value is non-trivial and case-dependent; the authors note the issue and leave optimization for future work.
  • Number of influence bands = 3
    Fixed at 3 in Section 2.3.1 and used in Section 2.4; the true number of bands in synthetic networks is 3 by construction, and if the clustering optimum is above 3, clusters are merged until 3 remain.
  • Rewiring fraction = 10% per time-slice
    Chosen in Section 2.3.1 for synthetic networks; affects how much temporal noise methods must tolerate, but the authors do not test other values.
assumptions (5)
  • domain assumption RT8 community labels from Pena et al. [13] are correct ground truth for polarized yes/no communities.
    Used in Section 3.2 to compute community-level influence; if these labels are wrong, MCC and band-by-community results are invalid.
  • domain assumption The modified T-ICM with a single infection probability rho accurately models real information spread on Twitter.
    Section 2.2 and Section 3.2.1 use T-ICM as the benchmark for real networks, with rho chosen to keep the process subcritical; no validation against observed diffusion data is provided.
  • domain assumption Temporal slicing into four discrete time-slices based on debate and referendum events captures the relevant dynamics.
    Section 3.2 defines t1-t4 as event intervals; the centrality results depend on this discretization.
  • ad hoc to paper The BandNet evolution rule that nodes only swap to neighbouring bands is a reasonable model of influence change over time.
    Section 2.3.1, step 2(a); this modeling choice is specific to the paper's synthetic design and constrains the dynamics.
  • domain assumption A node's influence band can be recovered by hierarchical clustering with complete linkage and the elbow method, merged to 3 bands.
    Section 2.4; the clustering choices affect how well methods appear to isolate bands.

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Cite this review

Pith. "Pith review of Dynamics of temporal influence in polarised networks." pith.science (2026). https://pith.science/paper/EHPXWYYA

@misc{pith2026250717177,
  author       = {Pith},
  title        = {Pith review of: Dynamics of temporal influence in polarised networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHPXWYYA}},
  note         = {Machine review of arXiv:2507.17177}
}
read the original abstract

In social networks, it is often of interest to identify the most influential users who can successfully spread information to others. This is particularly important for marketing (e.g., targeting influencers for a marketing campaign) and to understand the dynamics of information diffusion (e.g., who is the most central user in the spreading of a certain type of information). However, different opinions often split the audience and make the network polarised. In polarised networks, information becomes soiled within communities in the network, and the most influential user within a network might not be the most influential across all communities. Additionally, influential users and their influence may change over time as users may change their opinion or choose to decrease or halt their engagement on the subject. In this work, we aim to study the temporal dynamics of users' influence in a polarised social network. We compare the stability of influence ranking using temporal centrality measures, while extending them to account for community structure across a number of network evolution behaviours. We show that we can successfully aggregate nodes into influence bands, and how to aggregate centrality scores to analyse the influence of communities over time. A modified version of the temporal independent cascade model and the temporal degree centrality perform the best in this setting, as they are able to reliably isolate nodes into their bands.

Figures

Figures reproduced from arXiv: 2507.17177 by the authors.

Figure 1
Figure 1. Schematic of the network built for temporal degree and [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Schematic of the multilayer network built for tempora [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Schematic of the process for building a temporal BandN [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Time-slices of BandNet1. Nodes are coloured according to the community they belong to, and the size of the node reflects its degree. Layout produced using the Force Atlas algorithm. We assess how each centrality measure captures the temporal dynamics of the network by …
Figure 5
Figure 5. Figure 5: Results for BandNet1. (a)-(f) how many nodes are in each band in each time-slice, and how nodes move between bands in subsequent time-slices; (g)-(l) normalized influence score for each community over time; (m)-(r) how many nodes of each community are classified in ban…
Figure 6
Figure 6. Figure 6: Time-slices of BandNet2. Nodes are coloured according to the community they belong to, and the size of the node reflects its degree. Layout produced using the Force Atlas algorithm. The results of T-ICM and temporal centrality methods on BandNet2 are shown in [PITH_FU…
Figure 7
Figure 7. Figure 7: Degree distribution of bands in the initial set up of Ba [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Results for BandNet2. (a)-(f) how many nodes are in each band in each time-slice, and how nodes move between bands in subsequent time-slices; (g)-(l) normalized influence score for each community over time; (m)-(r) how many nodes of each community are classified in ban…
Figure 9
Figure 9. Figure 9: Time-slices of BandNet3. Nodes are coloured according to the community they belong to, and the size of the node reflects its degree. Layout produced using the Force Atlas algorithm [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]
Figure 10
Figure 10. Figure 10: Results for BandNet3. (a)-(f) how many nodes are in each band in each time-slice, and how nodes move between bands in subsequent time-slices; (g)-(l) normalized influence score for each community over time; (m)-(r) how many nodes of each community are classified in ba…
Figure 11
Figure 11. Figure 11: Time-slices on the Irish Abortion Referendum mentio [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]
Figure 12
Figure 12. Figure 12: Probability Distribution Function (PDF) of nodes de [PITH_FULL_IMAGE:figures/full_fig_p029_12.png]
Figure 13
Figure 13. Figure 13: Results for the original RT8 network. (a)-(f) how many nodes are in each band in each time-slice, and how nodes move between bands in subsequent time-slices; (g)-(l) normalized influence score for each community over time; (m)-(r) how many nodes of each community are …
Figure 14
Figure 14. Figure 14: Balanced accuracy between pairs of influence method [PITH_FULL_IMAGE:figures/full_fig_p031_14.png]
Figure 15
Figure 15. Figure 15: Results for the randomised RT8 network. (a)-(f) how many nodes are in each band in each time-slice, and how nodes move between bands in subsequent time-slices; (g)-(l) normalized influence score for each community over time; (m)-(r) how many nodes of each community ar…
Figure 16
Figure 16. Figure 16: Balanced accuracy between pairs of influence method [PITH_FULL_IMAGE:figures/full_fig_p034_16.png]

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