HeteroIM: Diffusion-based Recommendation Model

• HeteroIM is a recommendation model that transforms invitation recommendations into a heterogeneous influence maximization problem, integrating both immediate acceptance and secondary spread.
• It employs reverse reachable (RR) sets to measure global influence and shared RR-set counts to rerank inviter–invitee pairs, ensuring reduced redundancy in recommendations.
• Empirical results on Tencent gaming platforms demonstrate that HeteroIM significantly improves secondary invite rates and diffusion efficiency compared to traditional CTR-based and influence methods.

HeteroIM is a model for user recommendation that recasts invitation recommendation as a heterogeneous influence-maximization problem on a social graph with distinct inviter and invitee roles. Introduced in “Heterogeneous Influence Maximization in User Recommendation,” it addresses a gap between conventional recommendation methods, which primarily model interaction willingness, and classical influence-maximization methods, which prioritize spread but ignore whether recommendations are likely to be acted upon. In the Tencent gaming scenario used to motivate the model, invitees can themselves become inviters and continue propagation, so maximizing only click- or acceptance-oriented utility is misaligned with the downstream objective of spread coverage. HeteroIM is designed to improve interaction willingness while maximizing spread coverage through reverse reachable (RR) sets, heterogeneous diffusion probabilities, incremental selection, and reranking (Hou et al., 19 Aug 2025).

1. Problem setting and conceptual motivation

HeteroIM is formulated for invitation-based activities on Tencent’s online gaming platforms. Before each event, the platform selects an inviter set $V_i$ and an invitee set $V_e$. For each inviter $u \in V_i$, the system recommends a small list of invitees. When invitee $v$ receives an invitation from $u$, $v$ decides whether to accept; a valid recommendation occurs if $v$ accepts (Hou et al., 19 Aug 2025).

The model is motivated by two deficiencies in existing approaches. Traditional recommendation models, including CTR-type methods, mainly learn interaction willingness, that is, who is likely to accept, but they do not leverage the invitees’ potential to further propagate information. Conversely, influence-maximization methods focus on selecting influential nodes to maximize spread, but they ignore personalized willingness, including the probability that an invitation is sent and accepted. HeteroIM is therefore situated in what the paper formalizes as recommendation with influence maximization, or RIM, where the recommendation task is coupled to downstream propagation rather than limited to immediate interaction (Hou et al., 19 Aug 2025).

This positioning is central to the method’s interpretation. HeteroIM is not a generic recommender reranker and not a direct transplant of standard IM into recommendation. Its defining premise is that inviter–invitee recommendation should jointly optimize heterogeneous willingness and spread capability. A plausible implication is that the model is especially relevant in social systems where first-order acceptance and second-order diffusion are both operationally consequential.

2. Graph formulation, heterogeneous diffusion, and RR sets

The problem is defined on a directed, weighted, attributed graph $G = (V, E, P, U)$. Here, $V$ is the user set, $E$ is the social edge set, $V_e$0 is the invitation probability matrix with $V_e$1 denoting the probability that inviter $V_e$2 sends an invitation to $V_e$3, and $V_e$4 stores acceptance probabilities, with $V_e$5 the probability that user $V_e$6 accepts an invitation from any inviter (Hou et al., 19 Aug 2025).

The diffusion model is explicitly heterogeneous and two-stage. First, inviter $V_e$7 invites $V_e$8 with probability $V_e$9. Second, invitee $u \in V_i$0 accepts with probability $u \in V_i$1. The paper defines the heterogeneous spread probability as

$u \in V_i$2

This parameter couples inviter-side willingness and invitee-side willingness into a single propagation probability. Under the Independent Cascade model instantiated with these heterogeneous edge probabilities, live-edge sampling independently activates each directed edge $u \in V_i$3 with probability $u \in V_i$4, producing a live-edge graph (Hou et al., 19 Aug 2025).

RR sets provide the model’s core approximation machinery. The paper defines a random RR set $u \in V_i$5 as a set of nodes generated by randomly selecting $u \in V_i$6 as source node and reversely sampling the set of nodes that can spread to $u \in V_i$7 in terms of diffusion model $u \in V_i$8. Operationally, for a target node $u \in V_i$9, the RR set consists of all nodes that can reach $v$0 via directed paths in the live-edge graph, equivalently the nodes reachable from $v$1 by reverse BFS (Hou et al., 19 Aug 2025).

The canonical RR-set estimator for a seed set $v$2 is written as

$v$3

where $v$4 and $v$5 are $v$6 RR sets. HeteroIM does not introduce a new closed-form estimator; instead, it reuses RR sets to quantify both influence coverage and inviter–invitee alignment (Hou et al., 19 Aug 2025).

3. Selection objective, shared RR sets, and reranking

HeteroIM uses RR sets in two distinct ways. The first is a coverage score for candidate invitees:

$v$7

This score counts how many RR sets contain user $v$8, and thereby identifies high-spread candidates. The second is the shared RR-set count between inviter $v$9 and invitee $u$0:

$u$1

The paper terms the subset of RR sets containing both nodes the shared RR sets, or SRR, of the pair. HeteroIM uses $u$2 to select influential invitees and $u$3 to rerank inviter–invitee pairs according to interaction alignment, with the interpretation that user pairs sharing more RR sets exhibit higher interaction probability (Hou et al., 19 Aug 2025).

The algorithm operates in three stages for an inviter set $u$4. First, RR sets $u$5 are generated under the IC model using heterogeneous probabilities $u$6. The paper emphasizes uniform sampling, in which each node is used as a source uniformly for a fixed number of RR-set generations, to reduce source-node bias. Second, over the candidate pool $u$7, defined as first-order neighbors of inviters $u$8, the algorithm computes $u$9 for each candidate $v$0, selects the most influential candidate $v$1, appends it to per-inviter lists $v$2 when $v$3 is a neighbor of inviter $v$4, and then removes all RR sets containing $v$5 to mitigate overlap. This process continues until no candidates remain or recommendation lists reach capacity $v$6. Third, for each inviter $v$7, the method computes $v$8 for recommended $v$9 and reranks $v$0 primarily by $v$1, then by $v$2 (Hou et al., 19 Aug 2025).

This architecture separates global spread control from personalized pair ranking. The RR-set removal step addresses redundancy in spread coverage, while SRR-based reranking reintroduces inviter-specific heterogeneity after the global greedy phase. This suggests that HeteroIM is best understood as a hybrid of RIS-style coverage maximization and pairwise recommendation reranking, rather than as a monolithic propagation estimator.

4. Relationship to HeteroIR and HeteroInf

HeteroIM is presented alongside HeteroIR, and the distinction between them is substantive. HeteroIR is described as a model-agnostic recommendation framework that integrates spread profits with interaction willingness. It first estimates spread influence through HeteroInf and then scores inviter–invitee pairs with

$v$3

and

$v$4

In this formulation, $v$5 denotes the top-$v$6 out-neighbors of $v$7 ranked by $v$8, and $v$9 combines first-order and second-order influence (Hou et al., 19 Aug 2025).

HeteroIM differs in two ways. First, it uses RR sets rather than only pairwise profit scoring, which enables overlap control during incremental selection. Second, it reranks by SRR counts $G = (V, E, P, U)$0, which ties the recommendation to the inviter’s reachable region. The paper states this contrast directly: HeteroIR focuses on profit scoring per pair without RR-set overlap correction, whereas HeteroIM uses RR sets to control overlap and then reranks by SRR to align with willingness (Hou et al., 19 Aug 2025).

A common misconception would be to treat HeteroIM as merely a more expensive implementation of HeteroIR. The paper does not support that view. The two models instantiate different operational trade-offs: HeteroIR emphasizes spread-aware scoring, whereas HeteroIM adapts IM machinery itself to the recommendation setting.

5. Computational properties, guarantees, and implementation

Under IC with heterogeneous probabilities, generating $G = (V, E, P, U)$1 RR sets typically costs $G = (V, E, P, U)$2 expected time, where $G = (V, E, P, U)$3, and memory is proportional to $G = (V, E, P, U)$4. Coverage computation over all candidates can be done with inverted indices from RR sets to nodes in $G = (V, E, P, U)$5, and greedy removal of RR sets containing selected candidates has the same aggregate order if RR sets are removed or flagged once. Computing $G = (V, E, P, U)$6 requires counting RR sets containing both $G = (V, E, P, U)$7 and $G = (V, E, P, U)$8, which can be implemented as intersection counting in time proportional to $G = (V, E, P, U)$9, where $V$0 is the number of RR sets containing $V$1 (Hou et al., 19 Aug 2025).

The overall time complexity is therefore stated as

$V$2

with constants dependent on candidate pool size and $V$3. The method also admits parallelization across RR sets for sampling and coverage counting, and across inviters for SRR computation. The paper notes that compressed RR-set representations, streaming removal by flagging, and node-to-RR-set inverted indices help on large graphs (Hou et al., 19 Aug 2025).

The theoretical status of HeteroIM is qualified. The method adopts RR-set machinery and parameter settings from RR-OPIM+ and related RIS/IMM literature, with experimental settings $V$4 and $V$5, but the paper does not state a new formal approximation bound for HeteroIM. It explicitly notes that exact analytical guarantees for the heterogeneous reranking step are not provided (Hou et al., 19 Aug 2025). This is a significant limitation in formal terms: the greedy coverage component inherits the scalability and near-optimal coverage properties of RR-set selection, but the end-to-end heterogeneous reranking stage is justified empirically rather than by a new proof.

6. Empirical results, deployment, and limitations

The experimental study uses three Tencent game datasets, TXG-A/B/C, and Twitter. The TXG datasets have hundreds of millions of nodes and edges; for example, TXG-B has $V$6M, $V$7M, and $V$8M, while Twitter has $V$9M, $E$0M, and $E$1M. On TXG, two EulerNet models predict $E$2 and $E$3, with TXG-A split by $E$4 and validation on TXG-B/C. On Twitter, the setup uses DeepWalk 64-dim embeddings and EulerNet to fit $E$5 (Hou et al., 19 Aug 2025).

The evaluation uses two task families. For spread, the metric is NSpread@K, defined as a normalized measure comparing actual secondary spread to ideal spread coverage ISpread@K, where ISpread@K is the deduplicated union from best $E$6 neighbors per inviter and $E$7. For recommendation, the metrics are Recall@K and NDCG@K on valid interactions, defined as invite plus accept for TXG and mentions for Twitter. Baselines include IM methods such as IMM, OPIM-C, and RR-OPIM+, recommendation methods such as PPR, AutoInt, FinalNet, and EulerNet, and influence-estimation baselines such as Degree, Coreness, Windex, Monte Carlo, LR-influence, TOPSIS, and DeepInf (Hou et al., 19 Aug 2025).

The paper reports that HeteroIM consistently achieves the best spread and strong recommendation metrics, and that HeteroIR also improves over CTR baselines by modeling second-order influence. Many improvements are statistically significant with $E$8. In the HeteroIM ablation, reranking by SRR greatly improves performance, and using heterogeneous spread probabilities and uniform sampling further boosts results. HeteroInf achieves the highest Hit@K across TXG-B/C, and when plugged into recommendation scoring, HeteroInf-based $E$9 yields the best NSpread@K and NDCG@K against Monte Carlo and LR baselines (Hou et al., 19 Aug 2025).

The deployment evidence is unusually large-scale. HeteroIR and HeteroIM were deployed in two large-scale in-game propagation events, X1 and X2, on Tencent’s FPS game, with treatment groups of 15.6M and 16.4M users, respectively. The deployment pipeline predicts $V_e$00 and $V_e$01 via EulerNet, aggregates HeteroInf with $V_e$02, generates RR sets under IC with heterogeneous $V_e$03 using uniform sampling, and constructs recommendations via HeteroIM. In X2, HeteroIM achieved relative improvements up to 10% in Secondary Invite Rate, 9.64% in Secondary Invite Times, and 14.83% in Reach Retain Rate compared with the baseline intimacy model. The abstract also reports online A/B test improvements of 10% for HeteroIM and 8.5% for HeteroIR (Hou et al., 19 Aug 2025).

The paper also identifies several limitations. The diffusion model assumes independent invitation and acceptance with multiplicative probability $V_e$04, whereas real-world behavior may exhibit dependencies such as network effects and temporal correlations. HeteroIM reduces overlap via RR-set removal but does not provide a formal approximation bound under heterogeneous reranking. HeteroInf uses a fixed $V_e$05 as interaction capacity, and the authors indicate future work on personalized interaction capacity modeling and more efficient end-to-end algorithms for invitation and acceptance probabilities (Hou et al., 19 Aug 2025).

In that sense, HeteroIM marks a specific synthesis within social recommendation research: it adapts influence maximization to inviter–invitee recommendation by making heterogeneous willingness part of the propagation model itself, then uses RR-set coverage and SRR-based reranking to connect diffusion efficiency to personalized interaction structure. Its principal significance lies less in a new diffusion theorem than in an engineering reformulation of recommendation as spread-aware, heterogeneous propagation.