Influence Strength Estimation in Hyperbolic Space for Social Influence Maximization

Fan Li; Hongliang Qiao; Min Zhou; Shanshan Feng; Shuo Shang; Xutao Li; Yew-Soon Ong; Yunming Ye

arxiv: 2502.13571 · v2 · submitted 2025-02-19 · 💻 cs.SI · cs.LG

Influence Strength Estimation in Hyperbolic Space for Social Influence Maximization

Hongliang Qiao , Shanshan Feng , Min Zhou , Xutao Li , Yunming Ye , Fan Li , Shuo Shang , Yew-Soon Ong This is my paper

Pith reviewed 2026-05-23 02:31 UTC · model grok-4.3

classification 💻 cs.SI cs.LG

keywords influence maximizationhyperbolic embeddingsocial networksrepresentation learningdiffusion model agnosticseed selectiongraph representation

0 comments

The pith

Hyperbolic embeddings learn user influence strength directly from propagation data without assuming a diffusion model.

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

The paper establishes that social influence exhibits latent hierarchical structure that Euclidean embeddings miss, so influence magnitude cannot be read off from learned representations in flat space. It introduces HIM, which encodes network structure and historical activations into hyperbolic user vectors where stronger influence appears as proximity to the origin. An adaptive seed selector then uses these positions to pick seeds. A sympathetic reader would care because most real networks lack known diffusion parameters, making model-agnostic methods necessary for practical influence maximization.

Core claim

HIM consists of a hyperbolic influence representation module that encodes influence spread patterns from network structure and historical influence activations into expressive hyperbolic user representations, where the influence magnitude of users can be reflected through the geometric properties of hyperbolic space with highly influential users tending to cluster near the space origin, together with a novel adaptive seed selection module developed to flexibly and effectively select seed users using the positional information of learned user representations.

What carries the argument

Hyperbolic influence representation module that maps propagation data into hyperbolic embeddings so that influence magnitude appears as geometric clustering near the origin.

If this is right

The method works on networks where the diffusion model parameters are unknown.
Influence magnitude becomes directly readable from embedding position without additional simulation.
Adaptive selection based on hyperbolic position yields higher spread than fixed-model baselines.
The approach scales to large real-world networks while remaining diffusion-model agnostic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the origin-clustering property holds, the same hyperbolic encoder could be reused for related tasks such as link prediction or community detection that also involve hierarchy.
Testing the method on synthetic networks with explicitly planted hierarchies would isolate whether the geometric signal is the decisive factor.
Dynamic networks where influence evolves could be handled by updating the hyperbolic embeddings incrementally rather than retraining from scratch.

Load-bearing premise

That the hierarchical features of social influence are faithfully captured by hyperbolic geometry and cannot be captured by Euclidean geometry.

What would settle it

On the same five datasets, replace the hyperbolic module with a Euclidean counterpart and measure whether seed sets achieve equal or higher spread under the unknown diffusion setting.

Figures

Figures reproduced from arXiv: 2502.13571 by Fan Li, Hongliang Qiao, Min Zhou, Shanshan Feng, Shuo Shang, Xutao Li, Yew-Soon Ong, Yunming Ye.

**Figure 1.** Figure 1: The overall framework of HIM. representations in the hyperbolic space. (2) Adaptive Seed Selection selects target seed users based on learned hyperbolic representations via an adaptive algorithm. 4.2 Hyperbolic Influence Representation We first encode influence spread features from social influence data to construct user representations in hyperbolic space. The social influence data includes social networ… view at source ↗

**Figure 2.** Figure 2: Illustration of the influence propagation learning. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Results on Youtube under IC (left) and WLT (right). [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Distance-influence relation on Power Grid in hy [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: LDO-degree relations on Power Grid (left) and Cora [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

read the original abstract

The Influence Maximization (IM) problem aims to find a small set of influential users to maximize their influence spread in a social network. Traditional methods rely on fixed diffusion models with known parameters, limiting their generalization to real-world scenarios. In contrast, graph representation learning-based methods have gained wide attention for overcoming this limitation by learning user representations to capture influence characteristics. However, existing studies are built on Euclidean space, which fails to effectively capture the latent hierarchical features of social influence distribution. As a result, users' influence spread cannot be effectively measured through the learned representations. To alleviate these limitations, we propose HIM, a novel diffusion model agnostic method that leverages hyperbolic representation learning to estimate users' potential influence spread from social propagation data. HIM consists of two key components. First, a hyperbolic influence representation module encodes influence spread patterns from network structure and historical influence activations into expressive hyperbolic user representations. Hence, the influence magnitude of users can be reflected through the geometric properties of hyperbolic space, where highly influential users tend to cluster near the space origin. Second, a novel adaptive seed selection module is developed to flexibly and effectively select seed users using the positional information of learned user representations. Extensive experiments on five network datasets demonstrate the superior effectiveness and efficiency of our method for the IM problem with unknown diffusion model parameters, highlighting its potential for large-scale real-world social networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

HIM tries hyperbolic embeddings for model-free influence maximization but the origin-clustering claim has no visible loss or derivation to back it.

read the letter

The paper introduces HIM, which learns user embeddings in hyperbolic space from network structure and past activations, then uses their positions for seed selection in influence maximization when the diffusion model is unknown. That combination is the main new piece: prior graph-learning approaches stayed in Euclidean space, and this one claims hyperbolic geometry better captures the hierarchical spread patterns so that influence magnitude shows up in the embedding norm or distance to origin. The adaptive seed module built on those positions is a straightforward addition that could be useful in practice. Experiments are reported on five datasets with claims of better effectiveness and efficiency than baselines. That is the extent of what is shown to work. The soft spot is exactly the one in the stress-test note. Nothing in the abstract or description indicates a term in the objective that pushes high-influence nodes toward the origin, nor a proof that the learned geometry must correlate with actual spread. If that property does not emerge reliably from standard hyperbolic embedding objectives, the method reduces to using positional features whose link to influence is unestablished, which undercuts the model-agnostic selling point. The citation pattern looks standard for the area and the work is internally consistent on its own terms. Readers working on influence maximization or geometric graph methods would get value from the experiments once the full numbers and training details are checked. It is worth sending to peer review so referees can examine the loss functions, the actual results with error bars, and whether the hyperbolic advantage holds under scrutiny.

Referee Report

2 major / 2 minor

Summary. The paper proposes HIM, a diffusion-model-agnostic approach to influence maximization that learns hyperbolic user embeddings from network structure and historical propagation data. It claims that the resulting geometry encodes influence magnitude (high-influence users cluster near the origin), enabling an adaptive seed-selection module that uses only positional information. Experiments on five datasets are reported to demonstrate superior effectiveness and efficiency compared with existing methods.

Significance. If the geometric correlation between hyperbolic norm and influence spread can be shown to emerge reliably from standard representation-learning objectives, the method would supply a concrete way to estimate spread without committing to a particular diffusion model or its parameters, addressing a recognized limitation of both classical IM algorithms and Euclidean embedding baselines.

major comments (2)

[Abstract / §3] Abstract / Hyperbolic influence representation module: the assertion that 'highly influential users tend to cluster near the space origin' is stated as a direct consequence of the embedding procedure, yet no loss term, hyperbolic-norm regularizer, or derivation is supplied that would enforce or guarantee this property. Without such a mechanism the adaptive seed-selection module reduces to using arbitrary positional features whose relation to actual spread is unestablished.
[Experiments] Experiments section: the abstract states superior performance on five datasets, but the provided description supplies neither quantitative metrics, baseline comparisons, nor error bars. If these results exist in the full manuscript they must be reported with statistical tests; otherwise the central empirical claim cannot be evaluated.

minor comments (2)

[§3] Notation for the hyperbolic distance and the origin-centered norm should be defined explicitly before the claim about clustering is made.
[§3] The manuscript should clarify whether the hyperbolic representation module is trained solely on the observed activation sequences or also incorporates the underlying graph adjacency; the current description leaves this ambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We respond to each major comment below, indicating where revisions will be made.

read point-by-point responses

Referee: [Abstract / §3] Abstract / Hyperbolic influence representation module: the assertion that 'highly influential users tend to cluster near the space origin' is stated as a direct consequence of the embedding procedure, yet no loss term, hyperbolic-norm regularizer, or derivation is supplied that would enforce or guarantee this property. Without such a mechanism the adaptive seed-selection module reduces to using arbitrary positional features whose relation to actual spread is unestablished.

Authors: We acknowledge the point. The manuscript presents the clustering as an observed geometric property that arises when hyperbolic embeddings are learned from propagation data, capitalizing on hyperbolic geometry's suitability for hierarchical structures. However, no explicit loss term or derivation is supplied to guarantee the property. In revision we will expand §3 with additional analysis explaining the emergence of this behavior from the training objective and will include supporting empirical checks. revision: partial
Referee: [Experiments] Experiments section: the abstract states superior performance on five datasets, but the provided description supplies neither quantitative metrics, baseline comparisons, nor error bars. If these results exist in the full manuscript they must be reported with statistical tests; otherwise the central empirical claim cannot be evaluated.

Authors: The full manuscript reports quantitative results, baseline comparisons, and evaluations across the five datasets in the Experiments section. To strengthen the presentation we will add error bars and the results of statistical significance tests in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation self-contained with no reductions to inputs by construction

full rationale

The provided abstract and description contain no equations, fitting procedures, or self-citations. The central claim that hyperbolic geometry reflects influence magnitude via clustering near the origin is presented as an emergent property of the representation module without any shown loss term, norm constraint, or derivation that reduces the estimate to a fitted parameter or prior result by construction. No load-bearing step matches the enumerated circularity patterns, as the method is described at a high level without visible self-referential definitions or imported uniqueness theorems. The derivation chain remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based solely on abstract; no explicit free parameters, axioms, or invented entities are detailed beyond the high-level claim that hyperbolic geometry captures hierarchical influence features.

axioms (1)

domain assumption Hyperbolic space captures latent hierarchical features of social influence distribution better than Euclidean space, allowing influence magnitude to be read from geometric position near the origin.
Invoked in the abstract to justify the switch from Euclidean methods and to explain why influence spread can be estimated from representations.

pith-pipeline@v0.9.0 · 5793 in / 1203 out tokens · 32395 ms · 2026-05-23T02:31:53.163751+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean; Cost/FunctionalEquation.lean (Jcost, cosh-cost) reality_from_one_distinction; dAlembert_cosh_solution_aczel; alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

highly influential users tend to cluster near the space origin... proactive influence regularization term: I_GD = sum α_u log σ(d²_L(x_u, o_L))... LDO_u = d²_L(x_u, o_L)... user nodes with smaller LDO values are more likely to be influential

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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