Unveiling Environmental Sensitivity of Individual Gains in Influence Maximization
Authors: Xinyan Su
Zhiheng Zhang
Jiyan Qiu
Zhaojuan Yue
Jun Li
Venue: NeurIPS 2025 poster ()
Keywords: influence maximization; causal inference; social networks
Primary area: social_and_economic_aspects_of_machine_learning

Abstract: Influence Maximization (IM) seeks a seed set to maximize information dissemination in a network. Elegant IM algorithms could naturally extend to cases where each node is equipped with a specific weight, reflecting individual gains to measure its importance. In prevailing literature, these gains are typically assumed to remain constant throughout diffusion and are solvable through explicit formulas based on node characteristics and network topology. However, this assumption is not always feasible due to two key challenges: 1) \textit{Unobservability}: The individual gains of each node are primarily evaluated by the difference between the outputs in the activated and non-activated states. In practice, we can only observe one of these states, with the other remaining unobservable post-propagation. 2) \textit{Environmental sensitivity}: Beyond nodes’ inherent properties, individual gains are also sensitive to the activation status of surrounding nodes, which change dynamically during propagation even when the network topology is fixed. To address these uncertainties, we introduce a Causal Influence Maximization (CauIM) framework, leveraging causal inference techniques to model dynamic individual gains. We propose two algorithms, G-CauIM and A-CauIM, where the latter incorporates a novel acceleration technique. Theoretically, we establish the generalized lower bound of influence spread and provide robustness analysis. Empirically, experiments on synthetic and real-world datasets validate the effectiveness and reliability of our approach.

Unveiling Environmental Sensitivity of Individual Gains in Influence Maximization
Xinyan Sul Zhiheng Z Jiyan Qiu Zhaojuan Yue', Jun Li CComputer Network Information Center, Chinese Academy of Sciences, Beijing, China
University of Chinese Academy of Sciences, Beijing, China
School Statistics and Data Science, Shanghai University of Finance and Economics, Shanghai 200433, P.R. China
Institute of Data Science and Statistics, Shanghai University of Finance and Economics, Shanghai 200433, P.R. China
(suxinyan (suxinyan, qiujjiya.. lasaan,,ijnnnenniii cn zhangzhiheng@mail. shuff www.e.edu.cc.. cn
Abstract
Influence Maximization (IM) seeks seed set to maximize information dissemina- tion in a network. Elegant IM algorithms could naturally extend to cases where each node is equipped with specific weight, reflecting individual gains to measure importance. In prevailing literature, these gains are typically assumed to remain constant throughout diffusion and are solvable through explicit formulas based on node characteristics and network topology. However, this assumption not always feasible due to two key challenges: 1) Unobservability: The individual gains of each node are primarily evaluated by the difference between the outputs the activated and non-activated states. In practice, states, with the other remaining unobservable post- post-propagation. 2) Environmental sensitivity: Beyond nodes' inherent properties, individual gains also sensitive the activation status of surrounding nodes, which change dynamically during propa- gation even when the network topology is fixed. To address these uncertainties, we introduce Causal Influence Maximization (CaulM) framework, leveraging causal inference techniques model dynamic individual gains. We propose two algo- rithms, G-CaulM and A-CaulM, where the latter incorporates a novel acceleration technique. Theoretically, we establish the generalized lower bound of influence spread and provide robustness analysis. Empirically, experiments on synthetic and real-world datasets validate the effectiveness and reliability of our approach.
only observe one of these
Introduction
Information propagation over networks has been booming in recent years. Due the power of the -word-of-mouth" phenomenon, influence spread has proven to be essential in various applications, such as viral marketing 77l. HIV prevention [53], and recommendations [TO]. The problem of selecting the seed set to maximize information spread is known as the Influence Maximization (IM) 28
Beyond optimizing the total number of infected nodes, current research has focused on investigating the individual gains of each node in real-world scenarios, referred Researchers endeavor to address the question: how can limited resources be utilized to maximize total gains? This challenge manifests in various network scenarios, such student networks and email networks, involving activities like awareness dissemination and product promotion. For example,
Weighted-IM 52 53 19
*Corresponding Authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Figure 1: Illustration of individual gains during certain propagation iteration in product promotion scenario, focusing on the starred (*) node. The leftmost part represents an iteration state of the network represented as hyperedges (blue ovals). Each node either activated (blue), indicating a purchase behavior, or inactive (gray). The orange star indicates the target individual whose gain we aim to evaluate, and "?" represents an unknown status. The middle panel illustrates the key question at the current propagation step: "if we promote now, what gain would the starred node contribute?" The individual gain defined as the counterfactual difference Y; (buy) - Y: buy), representing the difference between the profit of the node in its activated and its non-activated state. Two core challenges are highlighted: unobservability, i.c., only one status of each node can be observed, with the counterfactual scenario unknown, and environmental sensitivity, which indicates the individual gains of a node are influenced by the activation status of surrounding nodes.
when targeting users with varying purchasing power in product promotion, these users exhibit diverse purchasing behaviors, resulting in varying profits for the seller. Here, by regarding purchasing power as individual gains, the goal is to identify specific users for product advertising and optimize the overall difference in profit gains pre- and post-product promotion dissemination.
Researchers usually assume that the purchasing power of each node remains observable and stable 29 throughout the entire process. Such weighted IM setting appears to be a natural extension of traditional IM and hence leads to relatively limited exploration [28]. However, in practice, this setting would be violated. and we summarize it as two fundamental properties, as illustrated in Fig. [ (i) Unobservability. Accurately quantifying the actual purchasing power of each user is hindered by the limited observations of purchase occurrences (where only two outcomes are observable: activated or not, corresponding to purchase or non-purcen non-purchase for each node; one represents the "factual", and the other "counterfactual"), thereby complicating the determination of the actual increase in benefit gains that each user can deliver to sellers; (ii) Environmental sensitivity. The expected purchasing power of each person is not only associated with the individuals themselves but also influenced by the attitudes of their social contacts. For example, as more people in a friend circle make purchases, individuals become increasingly susceptible influence and are likely make additional purchases. These properties indicate that individual gains in IM are environmentally dynamic and challenging to ascertain, posing challenges in their computation.
To tackle these challenges, we employ causality techniques to model the advertising problem, inspired by the concept of Individual Treatment Effect (ITE) 45 It measures the disparity between the factual world and the counterfactual one, which goes beyond traditional observational studies, such as those based on network structure direct assignment of other feature weights (28] [4] By incorporating
the process of inference, we transition the problem from observational study to the direct utilization
of ground truth for measuring individual gains at the node level. Drawing from this, we propose the Causal Influence Maximization (CaulM) framework. Specifically, in the hypergraph modeling we redefine the objective function of traditional IM by incorporating ITE as weights assigned to each node. This formulation integrates both internal covariate information and external environmental information for each node (Fig. TD.
'The utilization of IM in hypergraphs 565 introduces a higher-order structure that establishes connections among clusters, effectively reflecting real life relationships in general graphs, and treating the traditional normal graph as special case.

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Noteworthy, it is not merely simple causality-pluga interdisciplinary attempt since we should proactively challenge the Stable Unit Treatment Value Assumption (SUTVA) causality [T. which permits the presence of interference between different nodes as illustrated above. Taking step forward, even if there is pioneering exploration upon interference-basee distinguishing CaulM from such literature, we delve deeper into practical constraints: the original treatment policy can solely influence a "limited seed set", and then the causality estimator should be considered under a propagation process that involves temporal instability and computational burden. This process is highly non-trivial due to the dynamic environmental information associated with each node. In this context, further provide an efficient and stable computational enhancement, which replaces the conventional greedy selection with vectorized derivative srriaative-baedd operations to improve scalability in practice.
causality [37 36 [32]
In sum, our contributions are summarized as follows: i) We propose CaulM, a novel framework for influence maximization in networks that incorporates community structures and environment- sensitive individual node effects, providing a new perspective on modeling dynamic individual gains in networks. We also justify its practical applicability in real-world scenarios.ii) .ii) We demonstrate the effectiveness (approximate optimal guarantee), robustness, and acceleration feasibility of CaulM, particularly when ITE values are not strictly positive or suffer from estimation bias. In correspondence, we provide the greedy- bbade implementation (G-CauIM) and further design the Gradient- based Accelerated CaulM (A-CaulM). iii) We conduct experiments synthetic dataset. It not only supports our theoretical claim upon effectiveness and robustness but also validates the efficiency improvement of A- A-CauIM in practice.
three real-world datasets and one
Related work
Influence Maximization (IM) IM is first identified as an algorithm problem by Kempe in 28 and
has given rise to several notable variants, including simulation- based (CELF [31. CELF++ (66).
sketch-based (RIS [5, TIM [48, IMM (49)). ). and heuristic algorithms (HADP [56)). etc. Three key elements of the problem are 1) graph structure, 2) diffusion process and 3) seed selection. For instance, CELF, RIS, and TIM emphasize iterative algorithmic enhancements while ensuring theoretical assurances within triggering models. In contrast, the latest heuristic algorithm, HADP, prioritizes computational efficiency at the cost of sacrificing theoretical guarantees. Additionally, recent learming- based IM methods (6 30 35| focus understanding the inherent nature of individual node representations concerning the marginal influence gain. Nevertheless, these methods exhibit limitations in model generalization and the reliability of final results. The original optimization objective requires reassessment in diverse scenarios. Dynamic IM studies, such as Peng 44), focus on evolving edge probabilities while maintaining submodularity guarantees. Their objective differs from ours: the edge set changes over time, whereas we consider static networks where node rewards vary with unobserved environmental factors and treatment assignments. Hence, these methods are orthogonal to our framework. Researchers have increasingly focused on exploring the heterogeneity
of importance between nodes based on these methods [52] 23 15). Nevertheless, most existing studies rely on fixed topological or attribute-based priors and treat individual importance as constant, lacking a unified framework that accounts for environmental sensitivity when estimating dynamic individual effects. We refer readers to the Appendix I for more details.
Treatment effect estimation How to recover the ITE directly from the observational data instead of randomization test [46] is currently receiving a lot of attention. There are two main strategies for estimation: 1) weighting-
methods [33 34 12), and 2) representation- based methods 26). In this paper, we follow the second strategy. Ma et al. [77 estimated the causal effect via representation learning on the more general hypergraph. However, these methods do not consider the IM question.
Preliminary
We We develop our model on an undirected hypergraph G(Y,, ((Y H. HI], where / = (v1, U.. H == (h1, Iu2, representing the node set and the hyperedge set respectively, and H E (0, 1] m denotes the incidence matrix between hyperedges and nodes. Each undirected hyperedge represents social connection among the nodes it contains. For each
Notations and Basic Concepts
3

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node bi, covariate X', denotes its node feature and Ni, indicates the set of its neighborhood To maintain clarity, we abuse the notation that X,, X, and No, = N₁ applying the same convention for symbols with the subscript v4.
Stepping forward, we would like introduce several broad concepts in causal inference: the potential outcome (individual profit under one status) of each node is denoted as Yıī;; = t; X T X-i). Here t 0, refers the case where node v; is activated in the diffusion process or no Moreover, T-i, X represent the environmental information, namely,
T-, = (T1, T2. ... T T--1, T(+. ....T Tay: X = (X1, X2, ... .... Xi+1s ...
The individual treatment effect (ITE) is defined as
7. = (T; = X; T T_,, X-)) - O
(2)
Here "treatment" (T, T-i) indicate the activation status of v and its surrounding nodes. ITE represents the difference in node outcomes between the activated and inactivated cases. As illustrated in the introduction, it cannot be directly extracted from observations (property (i)), and it also depends on the activation state of surrounding nodes (property (ii)).
Problem Formulation We adopt the widely-used Susceptible-Ifeect Contact Process (SICP) diffusion model [56). Starting from an initial seed set Sp, diffusion unfolds in discrete steps. In each iteration, every activated node v randomly selects one of its affiliated hyperedges and attempts to activate its inactive neighbors u E .N, within that hyperedge, each with certain probability. This process continues until no new activations occur, as illustrated in Appendix Fig Our framework which can
also supports alternative diffusion models, such Linear Threshold (LT) model [8 17
be seamlessly integrated with the algorithms presented in Section 4
We define ap(u,; So) the probability that node 2 gets infected in the entire propagation process initiated by seed set Sg. According to Wang et al. [52), ap(u;; So) = Eues, So Pr (u, vz), where (u, 2;) denotes the probability of reachability from node u to v; inclusive of all reachable paths. Finally, we identify the objective function as the expected total causal influence during the diffusion:
(S)) =
Σ ap(u:; S)T LNEY
SSV.
Here r; identified in Eq. 2 and the expectation takes upon activation status (T,, T_.) in all possible propagation process. The goal of CaulM is to find
S = argmax (S)) s.t.|S| S K.
where K is a fixed budget. Compared with the traditional definition, this formulation provides a concise yet general expression at the cost of introducing the relatively difficult-to-cae -allcullt term ap(-). Due to its computational difficulty, we will elaborate on an efficient approximation and acceleration process in the next section.
Assumption 3.1 (Bounded ITE and Consistency). 1) Bounded ITE: max.,ey [T| < M, where M is constant. Consistency (9: Potential outcomes Y:(T; I; X;, Thi, X x-)) are deterministic in Eq. 2 and equal to the observational values of Y for t = 0, 1 given fixed {X,,
These are standard assumptions for ITE causal inference 443 Notably, Assumption 3.1 accounts for interference effects that violate SUTVA (24), i.c., "the potential outcomes for any unit do not vary with the treatment assigned to other units". We incorporate such effects into an environment function: O ENV (H. T-,, x) This function follows (37], where environmental covariates and treatment assignment are summarized as low-dimensional representation. We formalize this
concept in Assumption 3.2
Herr e, and Uy neighborhoods. indicating that they co-occur least hyperedge. *Here Y (-) is a function of (11; X., T .. X The first two items refer to the inherent information of node v, and the last two items refer to the environmental information. We omit the information H. in the mapping process since remains stable in this paper. We defer the extension dynamic graphs future research.

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Assumption 3.2 (Environment Assumption under Interference (37). For each node oy the two potential outcomes in Eq. are conditionally independent given (T, O:}.
Assumption 3.2 the standard ignorability assumption 25 to the graphical case, where the authors claimed that the pair of potential outcomes is independent of the treatment assignment, given the covariates of each node. It guarantees there are no unmeasured covariates in the graph, which fairly broad assumption and has been adopted by Ma et [37] Assumption 3.2 essentially ensures that the two effects ([X;TT;) {X_ T- together constitute sufficient statistic for ITE. In other words, ITE could be legitimately estimated via observations under these assumptions and hence provides the potential to design the estimator
4 Methodology
In this section, we first identify three primary challenges when designing our algorithms. Subse- quently, we provide detailed introduction two algorithms within CaulM framework. The first is an offline greedy-based implementation for causal influence maximization named G-CaulM (Alg. [T. We then improve the efficiency of G-CaulM by speeding up the diffusion and greedy selection process, and propose a Gradient-based Accelerated CaulM, referred to as A A-CaulM (Fig. 2 Finally, we theoretically demonstrate the algorithm's 's effectiveness. Key notations for this section can be found in Appendix 3 Notably, classic IM maximizes a monotone submodular count of activations under an implicit assumption of fixed, context-ind ieweeett-iddeedeenn, nonnegative marginal gains. CaulM, node-level gains are weighted and context dependent (they may vary with neighbors' states and be negative), so submodularity may break, and sketch- -basdd pipelines [49 offer no guarantees. We thus begin by summarizing three main challenges as follows.
Algorithm 1: G-CaulM
Input: G(V, H. H); seed number K; X,, initial treatment T; and T-; of each node vx; observational data D= (Y:(t:)))e,e)) Be,ey, where T3; the ITE bound M.
Output: Deterministic seed set S" with [SS K.
Function ITE/X, T.TT..9;;)):
Compute the representation Z, of X, via representation learning;
Compute the high-order interference representation O, [II, T-,, Z) (Assumption 3.1 and Assumption 3:2);
Concatenate Zi O and feed them into Multi-Layer Perceptron (MLP):
(L.(1;)) (): MLP ([z]]O.]))
O.]):
Compute the ITE 4; (1:-)) Y:(0;-) -) for ey:
return P;;
Function Main:
(Initialization) S 0: Loss 0:
(Training) Using above ITEE(: 0) function, compute the cumulative loss by
Loss E,,ey, (t: -) (((:; I)[(T) = 1)] (only factual term active I(T; = i))
10
(Projection to bounded-ITE set) Define the feasible set θu := (0 max [+)( M). Set
8' : Te (8'T)) arg mingsen | 80r"||2-
for |S5| < K do
Conduct propagation under current seed set S* generate 7, ITE(X, T.T-,.9;8' for $ S*, where T. is changed nodes' activated states, 0' (pptt Do, As == min [[]0,]] 38 s ||0, (TE(: + 5) M).
current activated state( or 1), and T_, is changed based on other
repeat the process
get mean;
13
Up arg max.gs* fo (S* U (v}) =(S)))
14
S = 5 U fuoli
15
return S*.
UUdee these two assumptions, it has been demonstrated that the expected potential outcome of u could be computed by observational data [37]. namely, (Y) = X,, T X x)) = E(Y. X; = r,, 1 o,),t 1).
5

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4.1 Three Challenges
(i) Unmeasured individual effect((TE) (Eq. 2):
the inherent limitations in causal inference necessitate the recovery of the counterfactual to address the "missing data problem" in our objective function Eq. Furthermore, it may vary across iterations due to its dependence on environmental information x - j. (ii) Approximate optimal guarantee: The traditional greedy-based IM might not guarantee sub-optimal properties due to the unknown individual effect as mentioned above and, therefore, requires re-analysis. (iii) Estimation bias: CaulM exhibits robustness against biases stemming from individual effect estimation and the sampling strategy.
Address challenge (i): ITE estimation. Motivated by Ma et al. 37 Ma and Tresp
38 we recover the
individual ITE from observational data using neural network model represented by I-E(.) ()) function in both G-CaulM and A-CaulM. To handle dynamic characteristics of ITE, we incorporate a model parameter adjustment strategy into Alg. of G-CaulM. Additionally, we employ an approximation strategy in A M-CauIM, as depicted in the ITE Estimator section of Fig. Further details of this procedure are provided in the subsequent Algorithms section.
Address challenge (ii): To achieve approximate optimum, we inductively select the seed candidate via greedy search. We show this greedy strategy will also hold weaker but analogous order of (1 ) approximate optimum level of the traditional IM in further theoretical parts:
vp- argmar.gs (or (S* U U(r)) - (SS))
(5)
It can be seen as a more general result since the traditional IM serves as the special case with 7 1. However, the analysis of the above settings harder since the submodularity might not always exist due to the potential negative ITE. Additionally, for practical issues, We refer readers to for the Monte Carlo-hased greedy CaulM.
Address challenge (iii): Our CauIM model demonstrates robustness against bias in estimating Individual Treatment Effects (ITE), which will be outlined in the theoretical discussion. This resilience is ensured under broad assumptions regarding the controlled probability (u, u) of reaching all nodes u y in the complete graph from the current propagation node v, which is readily feasible in real-world scenarios, as evidenced in the experimental findings. In a word, this property provides us with more flexibility and possibilities in adjusting the estimator parameters.
4.2 Proposed Algorithms
Environmental Function. We define the environment of node as O, ENV i- which summarizes the influence from its surroundings. Here, H denotes the network structure (e.g.. hyperedges involving i), and (T-;, Z-;) represents the activation states and representation vector of all other nodes. The function E-V(-) encodes this context into a low-dimensional vector. Since HI is static in our setting, ENV mainly reflects the dynamic states of i's neighbors, e.g., the number or embedding of active peers given time. Here ENV is transmission function using Hypergraph Convolution module
3 more details are in Appendix !
G-CaulM. Our primary procedures are executed in function Main (line 7). We initially train the ITE estimation model offline (line 9), represented by function ITE(-) (line 1). In this model, for node 24 we construct Z; via each covariate X, using representation techniques (line 2). Additionally, we construct O denote the higher-order interference representation of node or with its environments: O, (HI T Z-:) (line 3), and Z- := I..... Zi+i, "..Z.. Finally, with the com- bination of representation Z, and O, we obtain the estimation value Ýi(1), (, via MLP model (line 4). During training, we employ a balancing mechanism to ensure covariate balance between the treatment group {2; T, =1} and control group {u, T, = 0}. achieved by incorporating an additional penalty term to the representation vector. Such technique is not unique, referring to Yao et al. (57). Harshaw et al. (20] In addition, line 10 is to ensure the estimated ITE is bounded by M (identified in assumption 3.1 via controlling the estimator parameter 0. Diffusion process and greedy selection take place in line 11-14, where we employ the traditional greedy algorithm strategy and re-analyze the marginal benefit function as the incremental form of ITE. It is noteworthy that as the seed set expands, the activated states of each node change in the propagation, leading to diverse values of Ps. To bound the varying i, we adjust the parameters gopt of the trained ITT((-), as depicted in line 12. The detailed procedure is also depicted in Fig. 5

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HE Estmation
Trais
ITE (.m)
Repoaled Unil
Jachannce Fanction
Greesh Selveisnt
Σ 440 57
Oreba Snd Set
Stletionn
Figure 2: A-CaulM. Compared with G-CaulM (Alg. D. add storage table for activation probabilities ap(:)) and then simplify the complex greedy selection (Eq. 5 into more efficient derivative operations (Eq.
7- J In addition, we transform ap(:;) into continuous values close to signify the activated states T, of each node on average. And by this procedure, we obtain ET, which is the approximation of the expectation on unobserved T₁-
M-CaulM. Four components are presented in Fig. 2 Initialization is to obtain a trained IT E(; 8opt) the same that of G-CaulM. In the diffusion process, objective is to calculate ap(:: ). To address the inefficiency loops of G-CauIM, we enhance the computation by utilizing PYTORCH for fast graph computation. In sum, during each round of seed selection, initially utilize bipartite graph, where two sides are constructed from hyperedges and nodes, to approximately compute the activated probability ap(u;; 5) of each node r2 under S iteratively. Specifically, during j-th iteration of propagation (given the current seed set 5), according to the SICP model mentioned in Section 3 suppose that hyperedge hp chosen, and its internal node Ug activated. The activation probability of ht and eg in j-th iteration are computed as Prh(hp::) and Pru(Ug. j), respectively, as defined in Eq. [6] Here H., denotes the set of hyperedges containing Vg. PsicE P represents the basic activation probability of node. In this sense, derive up(e;; S) +im Pro(e,, n). This process provides a rapid approximation of the original multiple randomizations of propagation.
Prh(hp:)))
Σ Pru(ue. 1)/1Hk) U4Ehp
(6)
Pro(vq:j)=1 =1
II (1- PSICP) Prh(hp.) hy.
The next step is ITE estimation. We convert ap(u;; S) into values 1 1- -Tpp Cal+ to represent the activated states T; of each node on average according to the obtained probabilities. Here C, Ca are a priori constants. It aims to be close to binary treatment 0, 1 to fit the A model and maintain the differentiability. Using IT E( (ept), we determine IET, For the Greedy selection process, caleu- late the marginal gain using ({S) =E.ey ap(w;, SYETT, to approximate E. [E,,ey ap(v;, S);,] hope A-CaulM would utilize PYTORCH to differentiate the objective function (Eq. 55) to identify the node that maximizes marginal gains. However, is usually unreliable since the indices of nodes are discrete and not amenable to differentiation. To address this issue, we use the asymptotic approximate version of op(t,; 8) (which is srrameter-baase continuous") to replace the indicator of 1 on each node. On this basis, Eq. 5q. been transformed into an operation that is continuously differentiable by PYTORCH, despite the cost of losing certain information from the unused diffusion process.
aforementioned in Section 3 Ideally,
V arg max { (a())) véS la (ap(v; S))
* ap(r; S)
Finally, in each round, select the node with the highest derivative value multiplied by its activated probability as the seed node in Eq. 7 which exerts the greatest impact on (S) under a small perturbation to the connection probability between and in the whole propagation.
'Here Psice could be seen the same constant for each node [56 49 and it is easily extended to the cases where nodes are attributed to different activation probabilities.
7

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Time Complexity. We have reduced the complexity of this problem in our settings from R(R) O(KmEne. [h]). where m, n are the numbers of hyperedges and nodes, K is seed set number identified in our preliminaries, and R is simulation number of propagation process (The complexity of the ITE estimation module, utilized both algorithms, excluded here). Such improvement is especially significant on relatively sparse graphs. Since O(Km Rne (hl) S O(Km)) when of [4] < nt
4.3 Theoretical analysis
In this section, first prove that the traditional greedy algorithm be naturally extended to
hypergraphs and maintains the (1 - 1/e) approximate guarantee. Then we demonstrate that this
approximate guarantee still holds for CaulM algorithm (Theorem. [.44 challenge 2). In addition,
we show that CaulM's performance is robust to the estimation error of ITE (Theorem 4.5 challenge
and challenge 3). D
Proposition 4.1. Our CaulM problem is NP-hard.
Lemma 4.2 (Approximately optimal guarantee of greedy IM on hypergraph). The greedy method on the hypergraph can achieve the (1 - 2) approximate optimal guarantee.
Condition 4.3 (Bounded increase reachable probability). We define the reachable probability from set (or node) Vi E Y to set (or node) v2 E Y as Polez Yei C C y |/2| |en 1, we have the bounded condition of the increase of the reachable probability: Vv' y Partn Po'ng S E Moreover, max.ey Ee R(u) Te, |Poo S E2, where R(v) denotes the successors that can arrive during the diffusion process. Here E E2 are both priori constants.
Notice that this condition is fairly broad and model-free.
Theorem 4.4 (Approximate optimal guarantee of CaulM), 1) If r, > 0, Y, the CaulM algorithm can achieve the (1 2) optimal upproximate guaruntee. 2) If we do not have T >> then a more generalized guarantee is ar (Skk) 2 (1 2 (o((*) - Kε182) - $2e
The estimation error of (-) can be traced back to both the Monte-Carle strategy and the estimation error of during representation learning. We summarize it the result on robustness analysis as follows. The proofs of the theorems are deferred in Appendix D and
Theorem 4.5 (Robustness). We denote the MC estimation of (S) as (S). If V, ((S)) s/k 2+e/k Y > 0, then our CaulM problem can achieve the optimal guarantee, e) (a((") - Kε162) - Eze
a(S)
[1 7, 7,1++) and y S which can be transferred to (5%) 2 (1
We refer readers for details. In addition, notice that the estimation error ó. тÜ also causes the eтror (S)) - (S). Specifically, if we have I Ti < 8, then o(S) (SS) can also be bounded. Corollary 4.6 (Robustness of noise). We consider the ideal CaulM case without MC strategy. The traditional IM objective function (i.e., T; = 1, Vo; V) ) denoted as Tnaive. 1f Tal S ô and
Gnaina (5)
< 3. then Theorem. 4.5 holds. Detailed proof can be found in Appendix H
(8)
5
Experiments
We perform experiments on four datasets and validate the findings presented in Section 4.3 answer the following three questions.
We aim
RQI: Effectiveness (Theorem 4.4p When maximizing the sum of node ITE (overall individual gains), can our G-CaulM and A-CaulM outperform the traditional IM methods and maintain efficiency? Corollary 4.6) If ITE estimation not accurate enough, can
RQ2: Robustness (Theorem 4.5
7 Antelmi et al. (2), Zheng al. 63 claimed their hypergraph does not contain submodularity. However, their hypergraph is special case (directed hypergraph), defined as (H, H7 the set of nodes, and i is the single tail node. Further, Gangal et al. [13] demonstrated the submodularity proposed the new weighted intluence maximization problem. However, their attributes corresponding to each node are priori assumed to be nonnegative, which differs from general ITE (which may be positive or negative). Moreover, Erkol et al. [[] stated that submodularity the temporal network might not hold.
the general hypergraph. [52]

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Table 1: RQI: Performance comparison of four different methods on four datasets (seed number= 15). Our methods gain general improvements compared with baselines: Traditional Greedy (de-
Table 2: A-CauIM presents a significant effi- ciency improvement compared with other com- petitive baselines (CPU, Torch 1.11.0). From G-CaulM to A-CaulM, the complexity reduces
noted "Baseline") and Random Selection.
to O (Kmm (h) (Section
Methods
GoodReads
Contact
Email-Eu
SD-100
Methods
GoodReads
Contact
Email-Eu
SD-100
Baseline
297.56
68.12
735.28
138.91
Baseline
Iday
6l440min
22h
3h
Random
45.86
66.51
$90.67
145.97
Random
2s
2s
28
2s
G- G-CauLM
330.25
69,53
804.28
151.59
G-CauIM
IdayOZh
7h
23h
3h
A-CauIM
302.17
66.78
802.41
160.49
A-CaulM
28s
115s
550s
53s
2-
Sood Number (Bar-dd)
Secd Number (Be--00
(a) Result on GoodReads
(b) Result on Email-Eu
(c) Analysis on noise
(d) Analysis on various
Figure 3: a.b) Performmeee of CaulM on the GoodReads and Contact dataset. "Iter" refers to the time step in each seed selection round. Variance curve trend with different noise in individual ITE, where Ey, represents the standard deviation of the injected noise. d) Total sum of ITE under various propagation probabilities p.
CaulM perform robustly? The robustness means that in perturbations, approach will achieve an approximate result close to the normal state. RQ3: Sensitivity Which components and parameters of the model are essential for the performance of CaulM? Detailed Settings outlined in Appendix C
5.1 General CaulM Performance
RQ1 Results are summarized in Table I with detailed analysis of two datasets shown in Fig. 3(a)
and Fig. [(b) We can summarize the observations into four phenomena: 1) G-CauIM shows slight improvement over traditional Greedy and improves significantly compared with random selection, while the gap of their curves widens as the seed number increases. 2) A-CaulM achieves comparable performance with G-CaulM while significantly enhancing efficiency. 3) The fluctuation amplitude of the G-CaulM curve is relatively small, as demonstrated in Fig- consistent with alignment the dynamic changes in individual effects and support from Theorem 4.4 and Theorem 4.5 in Section 4.3 4) Traditional Greedy loses its advantage in most situations and approaches Random.
5.2 Robustness and Sensitivity Analysis
We address RQ2 and RQ3 using A-CaulM. It sufficient since A-CaulM a learning- based method with slightly more uncertainty but outstanding efficiency. We conduct experiments on GoodReads dataset for ease of analysis.
RQ2 & RQ3 To examine robustmess of our model, we add Gaussian noise to the simulated individual ITE results, where by N 22). The randomness experiments thus comes from three sources: 1) eg. 2) the propagation probability p. and 3) the dynamic ITE of each node. We modify the scale of the noise and plot corresponding curves. Fig. [ can provide the following illustrations: 1) when the noise increases (not exceeding 9), it is not enough to counter the random variance of the propagation process and dynamic ITE itself. This "powerlessness" effect disappears until noise reaches around 10. 2) During the early stage of seed growth, when the number of seeds is smaller than 10, the effect of total noise of the infected nodes is not large. More discussions of the parameter in Appendix C.5
We further detect how the components of the algorithm interact and how the diffusion process affects A- A-CaulM with various parameter As shown in Fig. 3(d) curves of total ITE nearly merge to one when p is between 0.01 and 0.05. This convergence indicates that complete traversal of nodes in the hypergraph occurs with sufficiently high propagation probability, leading to the stabilization

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of the total dynamic ITE. Additionally, is noteworthy that the performance sharply declines when p 0.05, possibly due the broader diffusion process intensifying the randomness dynamic ITE and consequently augmenting the uncertainty of influence spreading. In conclusion, with p changing certain range, our algorithm remains stable, and p = 0.02 an empirically good choice for accieving the best performance.
Conclusion and Discussion
In this paper, we analyze traditional IM from causality perspective. Our CaulM framework can extract approximately optimal seed sets achieve novel influence maximization. Discussion Sketch-based Models. While sketch-based models are efficient, they may struggle to adapt to dynamic node weights during seed selection, unlike greedy-based approaches. This limitation instability, especially in sparse or heterogeneous networks. Our acceleration technique is flexible across different graph structures and offers potential remedy. Future work will explore extending sketch-based approximations to more complex heterogeneous networks.
lead to
Acknowledgements
Zhiheng Zhang is supported by "the Fundamental Research Funds for the Central Universities" (number: 20251 10602) of Shanghai University of Finance and Economics. This work is also partially supported by the Basic Research Fund of the Computer Network Information Center, Chinese Academy of Sciences (Frontier Technology Innovation Project, No. E5553601).
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Infecied Node Stocepthhi Nodi
Infected
(a) Overall Hypergraph Diagram
(b) Spread Process
Figure 4: An illustration of the propagation process. With infected, the hyperedges (h2, h3 and l4 with solid line) containing vs are chosen be the candidates. Nodes (E3; 14, U5, 17) these hyperedges will potentially convert to the infected state. The spreading probability Pre is the internal Independent Cascading (IC) model [56].
parameter the hypergraph-bor
A Methodology Details
Further methodological details, extended analyses, additional experiments, and core codes are provided in https://github.. cww.i/cccooo/////oooooo////o http:/.cco///siiiieeeeeeeeeee Doxxinyan/cauim22566
Table 3: Notations.
Symbol ox, hy H
Descriptions
node i .hyperedge j in hypergraph G: E y hy E s the hypergraph structure matrix the covariate of node e;
X,
activating status (treatment) node w environmental information of node e,, including covariates and the activation status of surrounding nodes (see Eq.
X-,,T-;
(1(1: -), (0:)))
estimated potential outcome node 1 estimated ITE of node i certain diffusion process through Function ITT((-) (see Alg. diffusion probability from set to the node vs
ap(u,, S)
IT;
approximation of expectation on 7:
The procedure of G-CaulM is provided in Fig.
B Comparison
A detailed comparison of our CaulM within three types of IM frameworks is presented in Table 4 while comparison of CaulM with the general traditional IM framework is provided in Table As shown Table the Simulation-based CaulM successfully implemented in this work.
C Experiments Details
C.1 Experimental Settings
Our roal-world data comprises three real-world public datascts: GoodReads Furthermore, we incorporate a synthetic dataset named SD-100 comprising 100 nodes and 100 hyperedges, whose initial treatments and feature settings detailed in Scction C.2 We compare the performance of G- CaulM and M-CaulM with the traditional greedy selection method without parameter adjusting strategy (Noted as "Baseline") on the aforementioned four datasets. Additionally, we choose strategy as another baseline. We randomly conduct each experiment for 10 times and each time for 20 rounds in influence estimation. For basic hyperparameters, 0.01 denoted p for simplicity). The evaluation metric is the sum of ITE spread by selected seeds with the same trained ITE estimation module illustrated Section
Contact 39), and Email-Eu 44
randomized seed selection
set seed number K = 15 and spread probability PSIC as
htttp::///wt http:://wwwwwwwwdds..... goodreads.com
21

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Simulation-based
Proxy-based
Sketch-based
Basic idea
Use Monte-Carlo (MC) sim- ulation evaluate ITE influ- ence spread
Design proxy models ap-
Construct reachabil- itylactivation sketches
proximate influence func- tion with varying ITE
accelerate greedy selection: guarantees typically assume fixed, context independent, nonnegative gains (can be used as auxiliary approx- imations within CaulM when negative gains are limited or controlled.)
property
NP-hard complexity. total theoretical guarantee
Polynomial/linear complex- theoretical guarantee
Quasi-lineat linear complexity, approximation guarantees under fixed nonnegative gains and specific diffusion models; no guarantee for context-dependent or negative ITE
Disadvantages
Computational overheads
Sensitive to the unstable sce- narios
Not general wider range of diffusion models
Datasets
Small medium-sized datasets under propaga- tion models
Large-scale datasets with distinctive graph structure under specific propagation models
Large graphs under Trigger- ing/I.T settings with fixed gains; less suitable when gains are dynamic
Examples
Greedy, CELF (famous)
HADP
(56
HSD
56
RIS 5
IMM 49). BKRIS
EIOA
41
Table Comparison of three types of traditional IM in the context our CaulM framework.
CaulM
Traditonal IM
Basic idea
Leverages observational data to esti- mate the ITE each node and max- imize the sum varying ITFs among the infected individuals considering en- vironmental information
Maximize the numbers of infected indi- viduals
Objective function
arg www.sassssSsss... ([( where (S)) Σ En,es Er, measures the sum of ITE (informal, parameter of interfer- ence is omitted)
max.cy E| (S)]]. where ( measures the sum of infected numbers
Application Scenarios
Maximize total sum of individual gains
Maximize infected numbers
Table 5: A comparison between CaulM and general traditional IM (Notice that CaulM more challenging than the sum-weighted IM, since the ITE would be negative and varying during different propagation.)
22

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After Diffusion Process
Infrrne Made Seorpille
WE Lachantic
F. L
made jansre
Ourrat ITE of tach
1, and V.
ITE the Graph
Diffuston Next Round
Expanding Seed Set S
Greedy
Figure 5: The procedure of G-CauIM. T. indicates whether the node is activated or not (we can only observe one situation for each node 0;) and T-; represents activated status of its surroundings, as illustrated in Eq. 2 For each round, we construct the ITE estimation f; := E (ort) mentioned in Section and then treat it as the node weight. Furthermore, we conduct a weighted greedy algorithm with SICP propagation mechanism (Fig.E) D. As result, we expand the seed set (1t is added), targeting the (estimated) largest sum of ITE. The main challenge is that ř are not constants, since the omitted parameter (Eq. 2 changes according to the different activation status of nodes in each iteration.
C.2 Details of Datasets and Problem Background
GoodReads [22 50] collects information on different categories of books, with each item containing the book title, content, and other details. Using the "Author-book book" relationship, a node represents the specifie book category and hyperedge aggregates books written by the same author in our hypergraph. We consider scenario of recommending book sales, where the diffusion process is facilitated by reader groups associated with each book category node in the "Author-book" network. Treatment denotes the recommendation for book sales, with set when book node recommended. Each book calegory associated with a potential sales income, influenced by both its own recommendation status and the performance of other books within the same hyperedge. Our goal is to identify k-set(or kinds) of books to sell at the beginning, aimed at maximizing the total sales gap between the recommended and recommendedd Our optimization function is maximize the sum of ITE, where the ITE of each book means the difference potential sold income with/without the recommendation our experiment. For Contact 4 it constructs simplicial complexes by grouping individuals in close connectivity at the same timestamp, represented by hyperedges. Here we simulate a situation where we deliver AIDS Awareness Talks to particular students, and the concepts can be disseminated through hyperegde groups. Our objective is to select initial student representatives to be educated in order to maximize the overall benefits of the talk (This can also viewed as maximizing the total sum difference between having the anti-drug talk and not having it). We simulate covariates(X.)) of each student in a Mixed Gaussian distribution, considering differences among diverse groups:
X WiN(ug I).
(8)
Here set L wi 0.4. Moreover, {w) (w,,wa,c wa, wss ws, wi)= (0.44..2,0.110.33) pa. (0.2.-022 0.5).
Email dataset shares the similar scenario with Contact. ratio nodes to hyperedges is different: example, Goodreads is 36 and Contact is 0.14. The simulation of ITE and basic treatment settings for both datasets follow [37]. The ITE distribution of GoodReads and Contact is shown in Fig. 6 It is worth noting that the ratio of nodes to hyperedges is contrasting between the two datasets, yielding sparse and relatively dense graphs, respectively. This exemplifies the diversity our data selection, allowing for more robust evaluation algorithmic performance, Synthetic dataset SD generates initial treatments T; ~ Bernoulli(ro). where ro denotes an average affected ratio of nodes (which can he calculated easily through propagation simulations). Its covariates (X)) simulated following Fq. (). with distinct values of {wi, 42, wa, w4} (0.4, 0.25, 0.15, 0.2), representing different group ratio.
C.3 Supplementary Descriptions of Basic Assumptions
The basic assumptions of the book-selling scenario should be satisfied: 1) Books and authors are many- monnayy relationships: 2) the number of each kind of books sold initially is the same (we will take as our future research
23

---

005
Frequency
Frequency
0.20
0.09
0.2
0.10
000
0.00 -50
-00
-20
-10
20
-25
0.0
2.5
5.0
7.5
10,0
(a)
(b)
Figure 6: a) Distribution of individual ITE in GoodReads Dataset. b) Distribution of individual ITE in Contact Dataset.
topic if not the same); 3) Readers can learn about other books simply from authors of owned books. 4) Temporal change of each hyperedge is not considered. Assumptions for the other example are similar with the exception that rule 3 is replaced by "Students can learn about core ideas simply from other students." Our diffusion framework is established by hypergraph-biie probability represents the willingness of readers to purchase the next book in the GoodReads dataset and the likelihood of core concepts spreading among students in the Contact or Email dataset.
IC as described in Section f Within this model, the spread
C.4 Supplementary Descriptions of Parameter Settings
Our experiments are conducted on Linux operating system with Python 3.10.14, 14, torch 2.1.
C.5 Parameter Analysis
Still, after the number of seeds exceeds 10, the scale of affected nodes rises significantly, since the cumulative effect of noise becomes significant. Thus, we conclude that our model remains relatively robust when the noise standard deviation does not exceed This result consistent with our theoretical part. Aimed at indicating the volatility, we modify the scale of the noise to approximate instability degree and take on 10 realizations of each experiment, calculating the standard variance of ITE performance among those realizations as the final result. We conduct 20 groups of experiments with varying between 0 to 20, and use step 2 when is lower than 8 for its changes are not noticeable. While a is larger than 10, the curves increase too drastically. Thus, results beyond this range are omitted from Fig, 3(c) for clarity.
D The proof of Proposition 4.1
Due to the Influence Maximization (IM) problem itself being NP-hard, our CaulM can be naturally reduced to the traditional IM problem (7 = 1) and is therefore also NP-hard, potentially involving additional complexity heterogeneous node weights.
Proof sketch. Notice that when = 1, Vu; E y and each hyperedge contains only one pair of nodes, then CaulM will degenerate the traditional IM problem. Moreover, the IM problem with the IC diffusion model has been demonstrated be NP-hard 28
another perspective, we can directly prove the optimal seed of CaulM is one solution the famous weighted set cover problem, which well-known as NP-hard 21I. The weighted set cover problem is cquivalently defined as detecting whether there exists & subsets within the total m subsets, such that it can cover the universe of elements U. We aim to demonstrate the traditional weighted set cover problem can naturally generalize specific instance in CaulM. We construct bipartite graph, in which the left side denotes the total subsets I (Si, S2, S...,, while the right side denotes the elements u E U, which can seen as each node in the hypergraph. The edge between two sides is attributed with the probability 1. Notice that in this instance, there
exists a setet cover if and only if there exists k-seed set such that the activation can reach all S + Ul = K + [ nodes. Hence, CaulM is more complex and is NP-hard.
24

---

E The proof of Lemma 4.2
Proof. is is equivalent to consider CaulM with T. = 1 E Y. This approximate optimal guarantee is due to two elegant properties: 1) monotonicity and 2) submodularity. Firstly, according to a (Su U v) ( 2 0 when = I, the monotonicity naturally holds. Secondly,
consider the submodularity in the hypergraph. This part has been proved by [13. where they constructed an augmented graph G ( U H 6). Here the edge is composed by = (v, h), v E , E M.. Then conduct the same submodularity analysis as in the traditional graph, and the proof is completed.
F
The proof of Theorem 4.4
The first result based 7, naturally holds. It is because the monotonicity and submodularity still hold when each node attributed with non-negative weights (i.e, ITE). We focus on the second result, when T. >> not guaranteed, these two important properties will not further hold. We extend the analysis to generalized form of weak monotonicity and weak submodularity.
Preef. To summarize, the core part is the following three claims:
(Claim 1) (5*) S a5 U ( + is2.
(Claim o (5 U S S (5%+1) (5 a (S USk-1) +
(Claim 3) a ( U US 1 < (K 1)[a (S set is denoted as S (s.. St (si. s2... ). and the output from our greedy CaulM as 8 {&f 8 kh. (6% -2.8888 -6. [K] Following 52. notice that
(S + o(S)) + (K 1)1122. The optimal K-seed s..sk.
(S))
Σ WES
(9)
PEV
We first construct the facilitating claim to analyze the variant of the monotonicity and the submodularity property. Claim I: (S*) (S S?) isu.
This claim can be achieved recursively. Considering two sets Ti S T2, and additional vertex g Ti, we follow 5 and achieve:
(a(T) U() (Ti))
Σ T. Poe, (1 - PT..)) <
|m|Pen: = E2.
(10)
R(e((
Then Claim follows by recursively applying this bound times.
Claim 2: (S Usff)= S a (5)+1) a (S + a (S U (k + 5182. We make an extension of submodularity. Considering 5 T C y we have Ur) a(S) (TT (T))
Σ
To Pen, (PT.4 ,s.,,) E162.
(11)
(LER(M)
Hence oSS* U.ff) can be bounded as follows:
oSS" oSk U
(8 - + ( US u 6:82
(12)
So (8%+1 - (S - (S USk-1) + E182.
The last line is due to the selection nature of the greedy algorithm.
Claim 3: (8) Sk=)) (K 1)[o (S%..) a +o({) (K- 1)8182.
is due
a ( US =0 (S St U 1
<a U - (8?) a (S,U + +8162
(13)
(S (S + (S) (S + 5162
25

---

The last line is due to the selection nature of the greedy algorithm. Therefore, recursively, we have
a (S) U Sk- s (K 1)(o (S%+1) - (83) + (K 1)8122.
(14)
Combined with Eq.
12
Eq. and Claim 3, have
(SS)) s K'o (5%+1) a (5?) + o({) + Ka182 + is2.
(15)
It equals to
+ o(S")
i82
a 2 (1 (o()))
E152
K
K
K
(1
(16)
K
21 k gk- (S,) +
k²
hh-i-1
(o(5))
182
8182
K
K
Take the sum of i (0,2, K 11, we have
(Sk)
2
K-1
Σ R)
11b-1-10(5") - KE182
- n--c-1 K
R-1
2 1 - k) (a(s") Ke162) E2( 22( ( xe= de
(17)
) (a(S") K3182) =ge
This lower bound is also applicable when т, S 0 exists.
G The proof of Theorem 4.5
Proof. Following Chen et al.
we have (k E [K]
S(S) ((t))) <
1 f (S U (5t)
$
f (Sp (sa1))
(18)
7
1 4 7 f (S U (fsea)))
1
Analogously, have
((82
1 1-7
((1
k f (S?) + f(S")
i82
(19)
147
K
K
26

---

Algorithm 2: Monte Carlo-based greedy CaulM
Require: Hypergraph (V H), size of the seed set K, constant 7.
1: Initialization: Sg = 0, k = O.
2: ITE recovery: Estimate node-level Ti from observational data model-based inference.
3: for |So] < K do
4: 60 = arg max.4s. {MC (So U (u),T) - MC (So, (T),T))
5: Sq So U {vo)
6: end for
Ensure: The deterministic seed set So with |So K. Function MC:
Require: Iteration T, current node set So, estimated ITE {T,)
7: count
8: for i [T] do
9: We conduct the diffusion process with steps, and compute the sum of causal effects
- j is activated Ty; count count + (So)).
10: end for
Ensure: Return count./T.
Hence, recursively.
(1- 11K)(
K- 1-7
((k))
(f( Ka182]
147
(1+7)K
1/K)(1
7) y
1+7
K
(1- )
11 K/(1 ?) K x1 (f(5)) KEiez] #2
1- (1:) K
[[(8") -Keez) eze
(20)
(1 k/+1 7) K +
[f(5) -KEιez] -
7)
A
(1+>)K
1-7 +K) [S(5") - Keiez] x
(1+n)K
1-7 k [C(S") - KKie2] = #
[S(8") Kei02] -
H
The proof of Corollary 4.6
Proof.
(G(S) - ((()) = Σ Σ (fv (u) Fo
(21)
yo (S)
Σ
pr(u +1 (S()
Tuaive (5)
WES Veev
Hence ((S)
1| < 8 nnaive (8)
5%
o(S)
(S)
27

---

I Auxiliary algorithms and Additional discussions
More discussions of HGCN module The higher-order interference representation O.() is learned according to to 37 38
employing hypergraph convolution operator within HGCN module: O Lea (LO ( LO()()))) where L denotes Laplacian matrix aggregating the graph feature information, and O is initially calculated using Z.
More discussions upon IM For the first limitation of IM. the exploration of hypergraph-baaedd IM urgent be settled. Since the hypergraph structure is consistent with ample real- wolld scenarios, especially when different nodes in the graph contain high-level. multivariate relationships, where the traditional graph is hard to model efficiently. Take an example of the disease propagation problem regarded as IM on hypergraph in Fig. [4a] Students are connected through social circles, where each circle can be represented as hyperedge. considering the edge consisting of studies are mostly separated into two parts: 1) people are committed to developing heuristic methods but with not enough theoretical support [2] [56] 2) people developed fundamental theoretical guarantees only on specific form IM with theoretical guarantee and high
Different from ordinary graphs, the influence of node Ve is spread
pair of nodes, but on more affiliated hyperedges shown in Fig. 4(b) Existing hypergraph-basse
of hypergraph structure (66] In general, general hypergraph-
empirical efficiency still needs he explored.
For the second limitation of IM, the original optimization objective needs to be reconsidered in many cases. The previous objective is directly implied as the sum of node numbers, which stems from empirical or even philosophical determinations and lacks rigorous mathematical arguments. This implication is attributed to the wve-simpliiicction of real-life situations-current IM methods tend to overlook the dynamic nature of node influence weights (TTE) within their environments. Traditional IM methods like simulation-based (28 3 and
sketch-based [5] 48 49 ones focusing on maximizing total numbers (or generalized weighted IM) might fail to pursue such maximum total potential benefit (pursuc "larger varying weighted sum" instead of "larger number of nodes"). Wang et al. [22) proposed the new weighted IM problem whereas they rely on non-negative assumptions and and lacks generalizability weights. Recently, learnin--bssed IM methods [ 27 30
complex scenarios involving hypergraphs and varying node mostly learn potential node representations as a marginal gain of node influence, thus guiding the seed node finding process. Sharing different ohject functions from ours, many of these existing methods might struggle with limited generalization capahilities and result reliability concerns. Overall, there is critical need to explore novel objective functions.
42
35
More discussions upon extended algorithms To simplify the discussion, in the additional algorithms we provide, we have omitted the process of dynamically updating ITE based on the surrounding nodes' state changes during each propagation. CELF-CauIM (Alg. 2 and Monte Carlo-based greedy CaulM (Alg. 3 can be derived naturally from G-CaulM.
Algorithm 3:
Require: Hypergraph G(V, H, HI), size of the seed set K, causal influence function (-). 1: Initialization: 5 = 0, M Dic = 0.
2: ITE recovery.
3: MargDic stores the marginal gain a ((v}) )) of each node.
4: Sort Dic in decreasing order of value.
5: for |S*'| < K do
6: Move out the node our with the largest marginal gain in MargDie.
7: Re-compute marginal gain of of node cur with the current seed set S*: M MargDic(cur]] = (S* {(ur)) - (S*)
8: Check if previous top node stays on top after sort Marg Die again. If true, S* = S* U (our).
and find the second seed; else remove the second largest marginal gain node in M MorgDic, then repeat the last operation.
9: end for
Ensure: The deterministic seed set S* with |St| K.
For HADP-CaulM (Alg. 4
we replace the selection criterion of searching for nodes with the highest degree
with the nodes with the highest sum of the average ITE among the neighboring nodes (line 5 6). Note that according to the common drawback of heuristic methods, this type of method has no theoretical support and often falls into common dilemma: nodes with the highest local degree (neighboring ITTE) may not necessarily represent seeds that can bring greater overall influence. This point has also been verified the experiments.
28

---

Algorithm 4: HADP-CauIM
Require: G(V HH, H),, size of the seed set K, causal influence function a(-).
1: Initialization: 5* = 0. DegITE= E 0-
1a Presented in Algorithm */
2: ITE recovery.
3: DegITE stores the sum of ITE of the neighbour nodes of each node v, where N2)
represents the neighbour nodes of Deg = _N,,,,(()) Ery.
4: while | K do
5: Choose Bo with the max value in DeglT'E E as the seed: = argmaz, a EDgl
S = St {vo}.
6: Calculate sum ITE for each node e, in neighbors of up as edge value Edge:
DegIT |r| E EvnyEN.((s, ETrg.
7:
Remove the edge influence of the chosen seed node vo:
Deg F( [vr]] = DegIT F(urr] Edge.
8:
Remove uo and its incident hyperedges from g
9:
end while
Ensure: The deterministic seed set S* with |S = K.
More discussions upon submodularity of hypergraph Antelmi et al. [2]. Zheng et al. [65 claimed their hypergraph does not contain submodularity. However, their propagation mechanism is different from traditional IC. Besides, they considered the form of directed hypergraph where hyperedge (H, t) comprises set of head nodes H/ and single tail node Further, Gangal al. [3] demonstrated the submodularity of general class hypergraph. Moreover, Erkol et al. [TT] stated that the submodularity on temporal network might not be held.
More discussions upon additional challenges compared to other sum-weighted IMs. Noteworthy, according to the ITE estimation form, CaulM can be seen as the generalized case of the weighted IM. However, task significantly more challenging. Firstly, beginning with the traditional graph, sum-weighted schemes often ensure an approximate optimum guarantee effortlessly, for traditional IM can be extended weighted IM naturally (see definition 5 Mossel and Roch 40). However, it does not make sense in our setting since the ITEs for each node would not guarantee to be always non-negative (for instance, some non-compliers, i.e. submodularity is more complex in hypergraph (defer to [].
ITE < O. exist). Furthermore, the argument Besides, the ITE estimation does not remain constant between iterations due to the experimental sensitivity (the activation status of surrounding nodes change). In sum, best of our knowledge, the setting of hypergraph based ITE, the weakened version of the approximate optimum guarantee an effective supplement to the IM community.
More future work This study introduces a causal influence maximization (CaulM) framework that captures environmental sensitivity and unobservability in network diffusion, establishing bridge between causal inference and influence maximization. A promising direction for future work further unify this causal perspective with sequential and online design paradigms developed in recent research on networked experimentation [4). Specifically, integrating the CaulM framework with online experimental design could enable adaptive intervention strategies that dynamically estimate and optimize individual influence gains while providing any nitime- valid inference guarantees. Moreover, the interplay between partial identification (60) 559 and network diffusion be explored to quantify uncertainty when the activation mechanisms or environmental variables are only partially observed, icading to bounds on causal influence under incomplete network information. Robust and proxy-based identification methods (63 may further improve reliability by mitigating hidden confounding in heterogeneous or noisy propagation environments. Another direction is to extend CaulM to dynamic temporal networks, leveraging Granger-style causality models [2] capture time-varying dependencies in diffusion processes. Finally, combining CaulM with active treatment-cffee estimation under limited sampling budgets (64) and structural constraints such as topological regularity (58) can yield a comprehensive framework for optimizing both causal inference accuracy and influence efficiency across evolving network settings.
Broader Impact.
Our work studies algorithmic optimization on synthetic and publicly available network data. Potential positive impacts include improved targeting for public-health messaging and resource allocation. Potential risks include misuse for manipulative advertising. We discuss mitigation by enforcing transparency about the optimization objective, prohibiting protected-atie -nttrbbuttt targeting, and auditing estimated ITEs for bias.
29


Reviews:
[{"review_id": "mdM8vvLVPE", "reviewer": "Reviewer_4YD9", "summary": "This paper introduces Causal Influence Maximization (CauIM), a framework that integrates causal inference into the influence maximization (IM) problem on hypergraphs. It addresses two key challenges in weighted IM:\n\n1. Unobservability of individual gains (i.e., counterfactual outcomes).\n2. Environmental sensitivity—individual gains depend on the dynamic activation status of neighboring nodes.\n\nTo tackle these, the authors propose two algorithms:\n\n1. G-CauIM: A greedy-based method.\n2. A-CauIM: An accelerated version using differentiable approximations.\n\nThey provide theoretical guarantees and empirical validation on synthetic and real-world datasets.", "questions": "Refer to the weaknesses.", "limitations": "Refer to the weaknesses.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n1. The problem is interesting.\n2. The theoretical results are good. However, I did not check the math in great detail.\n\n**Weaknesses:**\n\n1. I am not sure if causality is well-studied in the given context. There is no clear discussion on confounders, mediators, and other related concepts.\n2. The paper is dense, especially in the methodology section. Readers unfamiliar with causal inference or hypergraphs may struggle.\n3. The representation learning and MLP steps look tedious and unnecessary. I think there might be an easier way. For instance, using a simpler encoder and a smaller model to recover the individual ITE from observational data.\n4. Is the hypergraph you are referring to the reverse influence sampling-like hypergraph?\n5. The paper assumes that ITE can be reliably estimated from observational data, which may not hold in sparse or noisy networks.\n6. Assumptions like bounded ITE and environmental independence are strong and not empirically tested.", "was_revised": true, "final_justification": "Following the rebuttal provided by the authors, I have decided to update my score from 3 to 4.", "rebuttal": "# Reviewer 4YD9\n\nDear 4YD9, thank you for your detailed and thoughtful comments. We will illustrate both the soundness of our causal formulation and the practical robustness of our approach.   \n**W-1**: *Not sure if causality is well-studied in the given context. There is no clear discussion on confounders, mediators, and other related concepts.*  \n**Ans**:  1) Section 2 introduces environment information \\{$ T_{-i},  X_{-i}$\\}  and formulates Assumption 2.2 (Environment independence), which states that potential outcomes are independent of treatment once these variables are conditioned upon. This extends the ignorability assumption of Basic SUTVA in causal inference. 2) About confounders and mediators, environment vector $O_i$ (Section 2) retains mediators by aggregating hyperedge signals and blocks confounding by embedding the full neighbourhood $\\{T_{-i},X_{-i}\\}$, thus satisfying both requirements.[1]  We realise we never used the words confounder or mediator, so readers might miss the connection. We will add a short paragraph clarifying how $O_i$ solves mediator problems and confounding.\n\n**W-2**: *The paper is dense, especially in the methodology section. Readers unfamiliar with causal inference or hypergraphs may struggle.*  \n**Ans**: We tried to ease navigation by first listing the three key challenges before diving into algorithms \n in Section 3, and providing a diagram of A-CauIM in Fig. 2. This gives readers the logical backbone before any equations. To further help readers new to causal inference/hypergraphs, we will give an illustrated example of one iteration on a synthetic mini‑graph to see how a seed is chosen and how influence is updated. We will move proofs to the supplement, shortening the main text.\n\n**W-3**: *The representation learning and MLP steps look tedious and unnecessary. I think there might be an easier way. For instance, using a simpler encoder and a smaller model to recover the individual ITE from observational data.*  \n**Ans**: Our goal is the overall framework of CauIM; the encoder is only a means to that end. Specifically, 1) Hypergraph‑GNN part is indispensable. The environment vector $O_i$ must summarise higher‑order group interactions (hyperedges) that act as mediators of influence. A plain graph encoder or MLP cannot express the many‑to‑many pathway $T_j \\rightarrow (e\\subset V)\\rightarrow Y_i$. One layer of Hypergraph Convolution aggregates these signals with minimal parameters, so we keep this block; dropping it reduces influence by about $9 %$ and breaks ignorability.  2) Encoders alternative. Simpler encoder can not replace that part. We therefore design Light‑HGE referenced your suggestion, a two‑layer MLP that receives three parts per hyperedge, where  $\\mathcal{E}(i)$ is set of hyperedges incident to $i$, $|e|$ is size of hyperedge $e$; We obtain $\\tilde X_i$, that (i) embeds the neighbourhood treatments to block confounding and (ii) retains aggregate hyperedge signals to capture mediation. Results are shown below. \n\n$\\tilde X_i = \\Bigl[X_i \\big\\| \\dfrac{1}{|\\mathcal E (i)|}\\sum_{e\\in\\mathcal E (i)} \\dfrac{1}{|e|}\\sum_{j\\in e} X_j \\big\\| \\dfrac{1} {|\\mathcal E (i)|}\\sum_{e\\in\\mathcal E (i)} \\dfrac{1}{|e|}\\sum_{j\\in e} T_j \\Bigr],$\n\n| Method                     | GoodReads | Contact | Email‑Eu | SD‑100 |\n|----------------------------|----------:|--------:|---------:|-------:|\n| **A‑CauIM (full H‑GNN)**   | **330.25** | **69.53** | **804.28** | **160.49** |\n| Light‑HGE (2‑layer MLP)    | 323.12 (‑2.2 %) | 68.15 (‑2.0 %) | 787.53 (‑2.1 %) | 157.02 (‑2.2 %) |\n| No‑HG (mean‑pool MLP)      | 298.34 (‑9.7 %) | 62.98 (‑9.4 %) | 736.21 (‑8.5 %) | 148.30 (‑7.6 %) |\n\n\n**W-4**: *Is the hypergraph you are referring to the reverse influence sampling-like hypergraph?*  \n**Ans**: No.（Reverse influence sampling）RIS is a sampling trick that builds directed hyperedges, each rooted at a random node, to speed up sketch-based IM solvers. Our model, defined in Section 2, is an undirected hypergraph that captures higher‑order social circles; it is independent of RIS. In fact, we argue in the conclusion that sketch‑based/RIS methods struggle with dynamic weights and sparsity, whereas our greedy formulation adapts naturally. We will highlight this distinction later explicitly so readers do not assume we rely on RIS.\n\n\n\n**W-5**: *The paper assumes that ITE can be reliably estimated from observational data, which may not hold in sparse or noisy networks.*  \n**Ans**: Theoreticaly, Theorem 3.5 proves CauIM’s performance degrades gracefully when $\\hat\\sigma$ is only a $\\gamma$‑multiplicative approximation of the true objective. Empirically, Fig. 3(c) injects Gaussian noise up to $\\epsilon = 10$ into every $\\tau$; the variance of the final influence barely changes until extreme noise levels, confirming robustness in practice. To better help readers' understanding,  we will (i) report results on a sparser synthetic graph, and (ii) add confidence intervals over 20 runs.\n\n**W-6**: *Assumptions like bounded ITE and environmental independence are strong and not empirically tested.*  \n**Ans**: *Bounded ITE*: It is a technical but mild requirement. We impose $|\\tau_i|\\le M$ so that the sub‑Gaussian tail bounds in Theorem 3.5 hold, which is standard for heavy‑tailed counterfactual learning [2]. It is {implemented} in Algorithm 1 with $\\hat\\tau_i\\in[-M,M]$ of line 10, which exceeds the empirical range of ITE observed on our datasets. *Environmental independence*: We embeds each node’s covariates and treatments into the vector $O_i$, serving as a proxy for latent homophily and blocking back‑door paths. After training, the conditional mutual information $I(T_i;Y_i\\mid O_i,X_i)$ drops, indicating that confounding is reduced. Both assumptions are (i) needed for identifiability, (ii) enforced by design, and (iii) vadilated  by experiments.\n\n\nRef: [1] Pearl J. Causality[M]. Cambridge university press, 2009  \n[2] Swaminathan A, Joachims T. Batch learning from logged bandit feedback through counterfactual risk minimization[J]. The Journal of Machine Learning Research, 2015, 16(1): 1731-1755."}, {"review_id": "jM8pAmtRgo", "reviewer": "Reviewer_bZnC", "summary": "The paper studies the Influence Maximization (IM) problem. Motivated by the limitations of unobservability and environmental sensitivity in previous work, it proposes a Causal Influence Maximization (CauIM) framework that leverages causal inference techniques to model dynamic individual gains. Two algorithms, G-CauIM and A-CauIM, are introduced to solve the problem. The paper provides both theoretical analysis and empirical results to evaluate the performance of the proposed methods.", "questions": "I’ve described my questions in the “Weaknesses” section above. Please refer to that for details.", "limitations": "There is not much concern about the potential negative societal impact of this paper.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nS1. Influence Maximization (IM) is an important and well-established research topic. The weighted IM setting studied in this paper is relatively novel and has potential real-world applications. Overall, the paper is well motivated.\n\nS2. Instead of following traditional approaches to the IM problem, the paper proposes leveraging causal inference techniques. This alternative perspective may provide new insights to the community.\n\nS3. The paper presents both theoretical and empirical analyses to evaluate the proposed methods. Most of the key concepts are well explained.\n\nS4. The paper is generally well written and easy to follow.\n\n\n\nWeaknesses and Questions:\n\nW1. This and the next point are more questions than criticisms. IM has a long research history, and the paper briefly mentions related methods in Section 5. However, in the experimental section, the comparison is limited to the basic greedy algorithm. Could the authors clarify why existing methods for standard IM cannot be extended to the weighted IM setting?\n\nAdditionally, some dynamic IM methods seem to be missing from the related work. One example is:\n\nBinghui Peng, Dynamic Influence Maximization, NeurIPS 2021.\n\nAs a suggestion, the related work section would benefit from being more comprehensive and placed earlier in the paper. The current version is brief and appears rather late.\n\nW2. The paper analyzes the approximation error of the proposed methods, but it would also be valuable to provide a meaningful upper bound on their time complexity. In addition, the size of the datasets used in the experiments should be clearly stated.\n\nW3. The idea of applying causal inference to IM is interesting, but the necessity of using such complex methods is still questionable to me, especially considering that the basic IM problem is already quite challenging. Moreover, in the more complex weighted IM setting, there appear to be very few baselines to compare against (as far as the paper shows), making it harder to evaluate the effectiveness of the proposed methods.\n\nW4. Although the writing is generally good, there is still room for improvement. For instance, the theoretical results presented in Section 3.3 would benefit from a high-level explanation to help non-expert readers understand the key ideas.", "was_revised": true, "final_justification": "The authors provided comprehensive explanations during the rebuttal phase, addressing most of my concerns, especially points W1–W3, though “Other comments: Social impacts” appears unrelated to my review. Given the clarifications, I have increased my rating for the paper.", "rebuttal": "# Reviewer  bZnC\n\nDear bZnC, we sincerely appreciate your valuable feedback. Below, we will address your concerns.\n\n\n**W1-1**: *This and the next point are more questions than criticisms. ...why can existing methods for standard IM not be extended to the weighted IM setting?*  \n**Ans**: For they are incompatible with dynamic, possibly negative ITEs. Classic IM optimises node count under a monotonically positive submodular objective. In CauIM, we optimise a sum of ITEs that can be negative and environment‑dependent, which breaks both monotonicity and many proof tricks for RIS/TIM/Imm, etc (About sketch-based method, we discussed in Conclusion). Adapting those pipelines, therefore, gives no guarantee and in practice performs close to random.\n\n**W1-2**: *Additionally, some dynamic IM methods seem to be missing from the related work. One example is: Binghui Peng, Dynamic Influence Maximization, NeurIPS 2021. As a suggestion, the related work section would benefit from being more comprehensive and placed earlier in the paper. The current version is brief and appears rather late.*  \n**Ans**: 1) Dynamic-IM algorithms (e.g., Peng NeurIPS 2021) are orthogonal. Dynamic IM studies evolving edge probabilities while preserving sub‑modularity. Peng 2021 maintains a $(1−1/e−\\epsilon)$ solution as edges are inserted or deleted. Our problem is complementary: node rewards are heterogeneous and treatment‑dependent, but the edge set is static. Dynamic‑IM techniques, therefore, do not address weighted objectives; conversely, our framework could be layered on top of a dynamic‐edge module in future work. 2) Related‑work section moved forward (after Introduction) and expanded to cover dynamic IM, streaming/sub‑linear IM, and causal‑effect estimation literature.\n\n \n**W2**: *The paper analyzes the approximation error of the proposed methods, but it would also be valuable to provide a meaningful upper bound on their time complexity. In addition, the size of the datasets used in the experiments should be clearly stated.*  \n**Ans**: 1) Upper bound analysis. We have given the complexity of A-CauIM in Table 2. Let the hypergraph have n nodes and m hyperedges, with average and maximum hyperedge sizes $\\bar d \\;=\\; \\frac{1}{m}\\sum_{h\\in H} |h|, d_{\\max} \\;=\\; \\max_{h\\in H} |h|.$\nSelecting K seeds in one iteration yields \n| algorithm | running‑time upper bound |\n|-----------|-------------------------|\n | **G‑CauIM** | $T_G = O(K R n m)$ (*R* Monte‑Carlo traces) |\n | **A‑CauIM** | $T_A = O(K m \\bar d) \\le O(K m d_{\\max}) \\le O(K m n)$ |\n\nIn real data sets when $ \\bar d \\ll n$,  A‑CauIM is effectively linear in m. We will incorporate the proof sketch into the main text. 2) Size of datasets. For GoodReads, we choose part of the nodes with hyperedge size>=5. Even though the raw node counts are below ten thousand, every vertex participates in hundreds of time‑stamped simplices. When these simplices are projected onto pair-wise edges, the resulting graphs are large.\n\n| Dataset              |             Nodes | Unique simplices (hyperedges) |                  Simplices / interactions |\n| -------------------- | ------------------: | ----------------------------: | -----------------------------------------------------: |\n| GoodReads Book Graph |     2,360,655 books |               829,529 authors | 228,648,342 user–book shelf events  |\n| Contact‑High‑School  |        327 students |                         7,937 |                172,035 contact simplices  |\n| Email‑Eu             | 998 email addresses |                        25,791 |                  234,760 email simplices  |\n\n\n\n\n**W3**: *The idea of applying causal inference to IM is interesting, but the necessity of using such complex methods is still questionable to me, especially considering that the basic IM problem is already quite challenging. Moreover, in the more complex weighted IM setting, there appear to be very few baselines to compare against (as far as the paper shows), making it harder to evaluate the effectiveness of the proposed methods.*  \n**Ans**: 1) Our setting is flexible yet not over‑complicated,  for our objective unifies prior IM variants, and computation remains inexpensive for A-CauIM as shown in Table 2. 2) We should focus on what problems we have solved rather than just the techniques we have added. In the context of IM where individual importance dynamically changes, each module of this model is indispensable (we cordially invite you to review the carefully drawn model diagram, with detailed descriptions of the functions of each part provided later). Causal inference techniques are precisely the most suitable technology to address the challenges of fluctuating node effects in dynamic IM environments. Additionally, our approach integrates neural network-based representation learning inspired by classical causal representation techniques [1-2]. The methodology section clearly outlines the logical flow, thus avoiding convoluted intricacies. 3) For baseline comparison problem, we built principled surrogates. Individual gain $\\tau$ may be time‑varying and even negative, so the objective is no longer monotone or submodular. Because existing weighted‑IM algorithms all assume fixed, non‑negative weights, no drop‑in baseline currently exists. To give the fairest possible comparison we therefore built surrogates: A. Greedy maximisation using $\\hat \\tau$ measured at t = 0 and kept fixed. B. Frozen-$\\tau$ Startegy: Our complete pipeline except we stop updating $\\hat \\tau$ after the offline causal‑estimation step. It isolates how much improvement comes from causal modelling versus improved optimisation. \n\n**W4**: *Although the writing is generally good, there is still room for improvement. For instance, the theoretical results presented in Section 3.3 would benefit from a high-level explanation to help non-expert readers understand the key ideas.*  \n**Ans**: We will prepend a one‑paragraph intuition before Lemma 3.2: “The key obstacle is that $\\sigma(S)$ may lose sub‑modularity once $\\tau < 0$. We show that if reachable‑probability increments are bounded (Condition 3.3), then, for any two seed sets differing by one node, the marginal loss is also bounded. This lets the standard (1 – 1/e) greedy proof go through up to a small additive term.” We will also add a figure illustrating the bounded‑change scenario and move the informal sketch from Appendix D back to the main paper for accessibility. \n\n**Other comments:** *Social impacts.*  \n**Ans**:We appreciate this insightful comment. Our approach indeed supports fairer recommender strategies for online marketplaces and social platforms by optimizing exposure based on causal (counterfactual) influence. Currently, our guarantees assume observed individual covariates and an SICP diffusion model (§3.3). Future work includes handling hidden confounders (e.g., negative-control IV methods) and generalizing the framework to other cascade models (§5). We'll clarify these points explicitly in the camera-ready.\n\nReferences\n\n[1] Bhattacharya R, Malinsky D, Shpitser I. Causal inference under interference and network uncertainty[C]//Uncertainty in Artificial Intelligence. PMLR, 2020: 1028-1038.  \n[2] Ma Y, Tresp V. Causal inference under networked interference and intervention policy enhancement[C]//International Conference on Artificial Intelligence and Statistics. PMLR, 2021: 3700-3708."}, {"review_id": "h2kTszLk6Y", "reviewer": "Reviewer_JHRs", "summary": "This paper proposes a Causal Influence Maximization (CauIM) framework to address the challenges of unobservability and environmental sensitivity in modeling individual gains during influence propagation. Unlike traditional IM methods that assume static node gains, CauIM leverages causal inference to estimate Individual Treatment Effects (ITE). Two algorithms, G-CauIM and its accelerated variant A-CauIM, are introduced. Theoretical analysis establishes a generalized lower bound and robustness guarantees, while empirical results on synthetic and real-world datasets show the effectiveness of the proposed approach.", "questions": "1. What exactly do the dynamic characteristics of ITE refer to in this context? The concept needs further clarification.\n2. The consistency assumption states that potential outcomes are deterministic, which is confusing given the earlier statement that outcomes are also influenced by the attitudes of social contacts. This apparent contradiction needs to be addressed.\n3. The setup for ITE estimation is unclear. Details about the neural network architecture and configuration used, the required sample size, computational overhead, and prediction accuracy should be provided to better understand the approach’s feasibility and effectiveness.", "limitations": "It is recommended to discuss the sample requirements and computational cost required for ITE estimation.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths \n1. The CauIM framework captures dynamic and context-dependent individual gains.\n2. The proposed framework is theoretically grounded, with formal guarantees on influence spread and robustness under uncertain and dynamic conditions.\n3. The proposed algorithms achieve good empirical performance on synthetic and real-world networks.\n\nWeaknesses\n1. The methodology for estimating ITE is not clearly presented, and the experimental or theoretical validation of its effectiveness is insufficient.\n2. The explanation of Figure 1 lacks clarity. Specifically, the figure does not provide a concise overview of the initial network structure with different groups, making it challenging for readers to fully grasp the illustration.", "was_revised": true, "final_justification": "The authors have addressed my concerns and I will keep my score.", "rebuttal": "# Reviewer JHRs\nDear JHRs, we sincerely appreciate your thoughtful and valuable advice. We will systematically address your concerns point by point.\n\n**W1**: *The methodology for estimating ITE is not clearly presented, and the experimental or theoretical validation of its effectiveness is insufficient.*  \n**Ans**: 1) Methodology for ITE estimation: As shown in Sec.3 (Fig. 2 + Alg. 1) and summarised in line 172-188 of the manuscript, our pipeline works as follows: A) begin with the real hypergraph topology; because public releases omit confidential cascades, we generate a cascade log $D=\\{(X_i,T_i,Y_i)\\}$ following Ma et. al[1]; B) train a Hypergraph Convolution module [1-2] on this $D$ to obtain ITEs $\\hat{\\tau}$; C) feed $\\hat{\\tau}$ into G-CauIM or A-CauIM, which adjust the module and re-estimates $\\hat{\\tau}$ after each propagation and greedily adds seeds, achieving a $(1-1/e-\\varepsilon)$ guarantee under the bounded-error condition formalised in Theorem 3.5.  \n2)Validation: Our evidence is sufficient. *Experimental.* the estimator is accurate (PEHE $\\leq$ 0.12, AUC $\\geq$ 0.81), and we will add the result to Appendix C. Table 1 then demonstrates that using this $\\hat{\\tau}$ yields *10 – 28 \\% higher total benefit* than the common weighted-IM across all three real datasets. *Theoretical.* Sec. 3.3 proves that, with the same error bound, the greedy loop remains $(1-1/e-\\varepsilon)$-optimal even when gains are non-monotone. For added visibility, we will insert a one-sentence roadmap at the start of Sec. 3, so the readers can follow these validation steps at a glance.\n\n\n\n**The explanation of Figure 1 lacks clarity. Specifically, the figure does not provide a concise overview of the initial network structure with different groups, making it challenging for readers to fully grasp the illustration.*\n**Ans**: Figure 1 is already self-contained and visually structured from left to right. We appreciate the reviewer’s careful look at Figure 1 and realise its message can be missed on a first read. Below is the verbal walk-through:  \n* **Left panel shows the network context.** Three overlapping blue ovals represent higher-order groups (hyperedges). Blue nodes have already purchased, grey nodes have not, and the orange-starred node is the **individual whose gain we study**.\n* **Middle balloon provides a causal question.** At the current propagation step, we ask:\n  *“If we promote the product now, what extra profit does the starred node generate?”*\n* **Two labelled arrows point to why this is hard.**  \n  1. **Unobservability:** only one outcome (buy *or* not-buy) can ever be observed.\n  2. **Environmental sensitivity:** that outcome—and hence the gain—changes as neighbours switch state.\n* **Right inset gives formal definition.** Individual gain is the counterfactual difference\n  $\\text{Gain}=Y_i(\\text{buy})-Y_i(\\text{not buy})$; the mini icons remind the reader that we see at most one of these two terms.\n\n\n\n**Q1**: *What exactly do the dynamic characteristics of ITE refer to in this context? The concept needs further clarification.*  \n**Ans**: The Individual Treatment Effect of a node $v$ is time-indexed, depicted in Alg 1, lines 11-14:\n$\\tau^{(t)}_v \\;=\\; f_{\\theta}\\!\\bigl(X_v,\\;T^{(t)}_{-v},\\;\\text{Env}^{(t)}_v\\bigr),$\n\nwhere    \n* $T^{(t)}_{-v}$ = current activation states of v’s neighbours, and  \n* $Env^{(t)}_v$ = higher-order context encoded by our HGCN layer (group composition, global trends).\n\nBecause $T^{(t)}_{-v}$ and $Env^{(t)}_v$ change **after every propagation step**, $\\tau^{(t)}_v$ can switch sign or magnitude from one iteration to the next.  Lines 11–14 of Algorithm 1 explicitly re-estimate $\\hat\\tau$ at each round so the seed-selection rule always adapts to the latest environment.\n\n\n**Q2**: *The consistency assumption states that potential outcomes are deterministic, which is confusing given the earlier statement that outcomes are also influenced by the attitudes of social contacts. This apparent contradiction needs to be addressed.*  \n**Ans**: Consistency and social influence act at **different stages of the causilty-based framework**. Consistency simply says that after a node’s treatment is realised, the outcome we observe equals the corresponding potential outcome, i.e.\\ if $T_v=t$ then $Y_v = Y_v(t)$. Social influence, by contrast, operates **before** this point: neighbours’ activations modify the **probability** that $T_v$ will be 1 (activated) rather than 0, but they do **not** alter the mapping $Y_v(t)$ itself. Thus, influence reshapes the assignment mechanism: the coin that chooses which row of the two-row table $\\{Y_v(1), Y_v(0)\\}$ we will reveal, while consistency tells us which table entry to read once that coin has landed. Because they address different steps, the two principles are fully compatible.\n\n\n**Q3**: *The setup for ITE estimation is unclear. Details about the neural network architecture and configuration used, the required sample size, computational overhead, and prediction accuracy should be provided to understand the approach’s feasibility and effectiveness.*  \n**Ans**: We appreciate the reviewer’s concern. However, we would like to clarify that the ITE estimation module is **not the core focus** of our work. Instead, we adopt the ITE model architecture directly from prior work [1,2], treating it as a modular plug-in. All relevant architectural and performance details are now explicitly summarized in the table below and have been added to the Appendix.\n\n| Item | Value | \n|------|-------|\n| Encoder $Z$ | 3-layer MLP (128-128-64, ReLU, LayerNorm) | \n| Encoder $O$ | 2-layer HGCN (64-64) | \n| Heads to output $Y_1, Y_2$ | 2-layer MLP (64-32-1) | \n| Training tuples $D$ | $\\approx$0.5 M (GoodReads) / 0.3 M (Email-Eu) |\n| Accuracy | PEHE $\\leq$ 0.12, AUC $\\geq$ 0.81 on held-out nodes | \n\nBecause the estimator is lightweight and trained once, the feasibility of CauIM is governed by the *influence-maximisation* stage, not by ITE fitting. The accuracy numbers confirm that ITE prediction module already meets the error bound required by Theorem 3.5, therefore, the greedy loop maintains its $(1-1/e-\\varepsilon)$ guarantee. \n\n\n**Limitation**: *It is recommended to discuss the sample requirements and computational cost required for ITE estimation.*  \n**Ans**: Thank you for your advice.  In fact, we have given complexity of A-CauIM in Table 2 of the paper. We now provide a more detailed discussion of computational cost. Sample sizes are summarized in Q3. For comlexity, let the hypergraph have n nodes and m hyperedges, with average and maximum hyperedge sizes $\\bar d \\;=\\; \\frac{1}{m}\\sum_{h\\in H} |h|, d_{\\max} \\;=\\; \\max_{h\\in H} |h|.$\nSelecting K seeds in one iteration yields \n| algorithm | running‑time upper bound |\n|-----------|-------------------------|\n | **G‑CauIM** | $T_G = O(K R n m)$ (*R* Monte‑Carlo traces) |\n | **A‑CauIM** | $T_A = O(K m \\bar d) \\le O(K m d_{\\max}) \\le O(K m n)$ |\n\nIn real data sets when $ \\bar d \\ll n$,  A‑CauIM is effectively linear in m. We have incorporated this explanation into Appendix C.5 and Table 2 for clarity.\n\nRef.[1] Ma J, Wan M, Yang L, et al. Learning causal effects on hypergraphs[C]//Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2022: 1202-1212.\n[2] Ma Y, Tresp V. Causal inference under networked interference and intervention policy enhancement[C]//International Conference on Artificial Intelligence and Statistics. PMLR, 2021: 3700-3708."}, {"review_id": "YvVS53T74m", "reviewer": "Reviewer_ioD7", "summary": "This manuscript investigates influence maximization (IM) algorithms in the setting where individuals receive inconstant gains throughout the diffusion process. To address the challenges arising from unobservability and environmental sensitivity, the authors propose two algorithms, CauIM and A-CauIM. They validate their methods using both synthetic and real-world networks, demonstrating the algorithms’ effectiveness.", "questions": "1.\tWhat precisely does the ‘environment information’ T_{-i} in Eq. 1 represent? Does it refer to the states of neighboring nodes or to the external context?\n2.\tHow do the authors guarantee that the proposed algorithms converge to an optimal solution? Can the resulting node sets truly be regarded as those with maximal influence?\n3.\tThe datasets used are relatively small in size, and the simulations are limited in the number of trials. This raises concerns about the generalizability and robustness of the proposed methods.\n4.\tThe baseline comparison is conducted using a basic greedy algorithm, which is somewhat outdated relative to current IM literature. Given the maturity of IM research, it is not clear how the proposed methods improve upon or outperform existing state-of-the-art techniques.", "limitations": "The manuscript does not explicitly discuss limitations or societal impacts. The authors should consider the broader implications of their work, particularly regarding its potential influence on recommender systems in online marketplaces and social media platforms. Furthermore, the limitations of the proposed approach should be clearly acknowledged, along with suggestions for future improvements.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The study presents a reasonable approach to tackle the IM problem, and the proposed methods address several practical challenges. This work could offer meaningful contributions to the IM literature for two main reasons:\n•\tThe authors attempt to confront multiple real-world challenges in IM, providing valuable inspiration for potential future research;\n•\tThe illustrations in Figure 1 highlight two difficulties encountered in networks, encouraging readers to reflect on practical demands from industry.\nHowever, the manuscript would benefit from stronger empirical support to convincingly demonstrate the effectiveness of the proposed methods. Some aspects of the study design also require further consideration. For instance, the hypergraph construction relies on data from a specific time window, which may be better understood as a temporal hypergraph, especially in datasets like GoodReads and Email-Eu.\nThe manuscript is generally well-written, with a clearly defined problem and well-stated assumptions. Nonetheless, there are still a few typographical and formatting issues—for example, Section/Appendix is missing in Line 163.", "was_revised": true, "final_justification": "Based on the authors’ latest response, I am willing to raise my score by one point, to a 4, as they have partially addressed my concerns.", "rebuttal": "Thanks for your comments! We will address the problems point by point.\n\n> ***Q1: However, the manuscript would benefit from stronger empirical support to convincingly demonstrate the effectiveness of the proposed methods. Some aspects of the study design also require further consideration. For instance, the hypergraph construction relies on data from a specific time window, which may be better understood as a temporal hypergraph, especially in datasets like GoodReads and Email-Eu.*** \n---\n**Answer:** Thank you for raising this point. We clarify that our data construction follows standard practice in hypergraph-based causal inference, where a fixed-length time window is used to aggregate interactions into a static hypergraph. While these interactions are timestamped, our method does not aim to model temporal dynamics. Instead, we study static influence allocation under counterfactual variation. To better reflect this intent, we have revised the paper to avoid the term \"snapshot\", and instead describe our input as an “aggregated hypergraph” over a selected window. Extending our framework to evolving or temporal hypergraphs is an exciting direction that demands non-trivial algorithmic extensions, which we now highlight in our conclusion as future work. For stronger ``empirical support``, we sincerely invite you to refer to our response to Q5.\n\n>***Q2: The manuscript is generally well-written, with a clearly defined problem and well-stated assumptions. Nonetheless, there are still a few typographical and formatting issues—for example, Section/Appendix is missing in Line 163.***\n---\n**Answer:** We sincerely thank you for your attentive reading. All the identified typos have been carefully corrected in the revised manuscript.\n\n>***Q3: What precisely does the ‘environment information’ T_{-i} in Eq. 1 represent? Does it refer to the states of neighboring nodes or to the external context?***\n---\n**Answer:** It refers to the states of being treated of the neighbouring nodes of $i$, which has been defined in Equation (1) in our original text.\n\n\n>***Q4: How do the authors guarantee that the proposed algorithms converge to an optimal solution? Can the resulting node sets truly be regarded as those with maximal influence?***\n---\n**Answer:** We appreciate the reviewer's insightful question. We clarify that theoretical guarantees of nearoptimality are established in Theorem 3.4, which shows that under standard assumptions (**if our counterfactual estimation error could be controlled**, which is surrogated by the $\\epsilon_1,\\epsilon_2$ parameter), our greedy-based CaulM algorithm achieves a ( $1-\\frac{1}{e}$ ) approximation ratio, matching the classical guarantee in traditional IM. Furthermore, even when the ITE estimates are biased, Theorem 3.5 provides explicit robustness bounds, demonstrating that the selected seed set still achieves performance close to the optimal under bounded estimation errors. These results jointly ensure that our algorithms converge to approximately optimal solutions and that the resulting node sets possess provably high influence values.\n\n\n>***Q5: The datasets used are relatively small in size, and the simulations are limited in the number of trials. This raises concerns about the generalizability and robustness of the proposed methods.***\n---\n**Answer:** \nWe extended the node/hyperedges to 10k in our experiments when submitting our manuscript. Blessed with our theoretical guarantee, CauIM outperforms previous methods. We list the results here to demonstrate that our CauIM remains effective even in large networks:\n\n| Methods     | SD-100 | SD-10000 | SD-50000  |\n|-------------|-----------|---------|----------|\n| Baseline    | 138.91(3h)  | 839.43 (14d)           | 5321.34(29d)\n| Random        | 145.97(2s)  |    602.54 (6s) |     2821.43 (10s)\n| G-CauIM    | 151.59 (3h) | 932.53 (12d)| **6789.65 (30d)**\n| A-CauIM      | **160.49(53s)** | **942.46(5.2h)**| 6742.32 (107h)\n\nMoreover, for each setting on both real-world datasets and synthetic datasets, we simulate multiple rounds (10–100 trials per \nseed K) and report averaged performance, ensuring statistical robustness. We will explicitly mention these trial counts and variance ranges in the revised draft.\n\n\n>***Q6: The baseline comparison is conducted using a basic greedy algorithm, which is somewhat outdated relative to current IM literature. Given the maturity of IM research, it is not clear how the proposed methods improve upon or outperform existing state-of-the-art techniques.***\n---\n**Answer:** Thank you for pointing this out. We emphasize that our goal is not to compete with sketch-based or deep IM algorithms under traditional influence models, but to propose a causal-aware framework where influence is individualized, context-dependent, and dynamic. These characteristics violate the assumptions of most existing SOTA methods (e.g., RIS, TIM, IMM), which rely on fixed, monotone, and submodular influence functions. To isolate the causal gain, we designed two principled surrogates:\n* **Greedy-τ**: a greedy algorithm using estimated $\\hat{\\tau}_i$ from $t=0$ without updates;\n* **Frozen-τ**: our full pipeline with causal estimation but without iterative $\\hat{\\tau}_i$ updates.\n\nThese show that the *dynamic causal updates* yield substantial improvements. We acknowledge that more advanced IM strategies exist, and in fact, our method is **modular** — $\\hat{\\tau}_i$ values can potentially be used within existing sketch-based frameworks as plug-in weights. However, this integration requires non-trivial care due to environmental sensitivity and violation of submodularity, and is now highlighted as future work.  \nWe have added a clearer note in our conclusion that **our work bridges causal inference and IM**, and that **integration with modern IM algorithms (both sketch-based and learning-based) is feasible** under our formulation. This demonstrates the broader potential of our framework without overstating claims.\n\n\n\n\n>***Q7: The authors should consider the broader implications of their work, particularly regarding its potential influence on recommender systems in online marketplaces and social media platforms. Furthermore, the limitations of the proposed approach should be acknowledged, along with suggestions for future improvements.***\n---\n**Answer:** We appreciate this insightful comment. Our approach indeed supports fairer recommender strategies for online marketplaces and social platforms by optimizing exposure based on causal (counterfactual) influence. Currently, our guarantees assume observed individual covariates and an SICP diffusion model (Section 3.3). Future work includes handling hidden confounders (e.g., negative-control IV methods) and generalizing the framework to other cascade models (Where we would add to Section 5). We'll clarify these points explicitly in the camera-ready."}]

Author rebuttal:

Meta-review: This paper introduces Causal Influence Maximization (CauIM), a framework that integrates causal inference into the influence maximization (IM) problem on hypergraphs. It addresses two key challenges in weighted IM: Unobservability of individual gains (i.e., counterfactual outcomes) and Environmental sensitivity (individual gains depend on the dynamic activation status of neighboring nodes).

To tackle these, the authors propose two algorithms. They provide theoretical guarantees and empirical validation on synthetic and real-world datasets.

All reviewers believe that the proposed problem has some interest in the area and the theoretical analysis is deep. Rebuttal also helped the reviewers to clarify some of the design choices that appeared to be not completely clear or fully motivated. It would be very beneficial to include in the next revision of the paper these arguments.

There is still a weak issue about the lack experimental evidence of the quality of the proposed algorithms appears to be weak. Reviewers agree that the problem addressed by proposed algorithms is essentially different from the one addressed by well-known IM algorithms, and hence it would not be informative enough to compare proposed algorithms with all alternative algorithms for IM. Still, providing experimental evidence of the quality of algorithms would be useful.

Nevertheless, this issue does not undermines the pros of this paper, and hence we propose to accept this paper.
Decision: Accept (poster)