---
title: 'HeteroIR: Heterogeneous Influence Recommendation'
url: https://www.emergentmind.com/topics/heteroir
type: topic
---

# HeteroIR: Heterogeneous Influence Recommendation

to=arxiv_search.search  大发快三如何_code:  {"query":"HeteroIR 2508.13517 Heterogeneous Influence-based Recommendation", "max_results": 5}
to=arxiv_search.search  天天中彩票在哪_code:  {"query":"\"Heterogeneous Influence Maximization in User Recommendation\"", "max_results": 10}
to=arxiv_search.search _日本一级特黄大片_code:  {"query":"influence maximization recommendation user recommendation arXiv 2024 2025 secondary spread", "max_results": 10}
HeteroIR, short for **Heterogeneous Influence-based Recommendation**, is a model-agnostic framework for invitation-based user recommendation that augments conventional user-to-user ranking with explicit modeling of information propagation. It was introduced together with HeteroIM in “Heterogeneous Influence Maximization in User Recommendation” [2508.13517] to address a setting in which an inviter recommends a limited set of friends, some invitees accept, and accepted invitees may later become inviters themselves, thereby creating secondary spread. The framework is designed to preserve interaction willingness while valuing dissemination potential, and it does so through a two-stage architecture that predicts inviting and accepting behavior, constructs a heterogeneous spread probability, estimates capacity-limited influence, and ranks candidate invitees by a profit score combining direct and secondary effects [2508.13517].

## 1. Problem setting and motivation

HeteroIR is defined for invitation campaigns in which a platform first selects inviters \(V_i\) and invitees \(V_e\), then uses a recommendation algorithm to present each inviter \(u\in V_i\) with a limited list of candidate friends \(v\in V_e\). During the event, \(u\) may send invitations to recommended \(v\), and a recommendation is valid only if \(u\) invites \(v\) and \(v\) accepts. The central complication is that invitees can later become inviters, so the value of recommending \(v\) depends not only on whether \(u\) and \(v\) interact, but also on whether \(v\) can trigger further spread [2508.13517].

The framework is motivated by a gap between two established families of methods. Standard recommendation models, including CTR-style or link-prediction systems such as AutoInt, FinalNet, and EulerNet, optimize interaction willingness for a pair \((u,v)\), but do not account for the secondary propagation that an accepted invitee may create. Classical influence-maximization methods, including IMM, OPIM-C, and RR-OPIM+, instead seek seeds with high expected spread, but typically ignore personalized invitation and acceptance behavior between specific inviter-invitee pairs [2508.13517].

A key empirical observation in the Tencent game data is that local social connectivity is a poor proxy for actual spreading power. The average user has about \(50\) friends, yet actual spread influence is \(16\) times smaller on average, and the Pearson correlation between friend count and spread influence is only \(0.06\). This observation, stated as **Observation 1 (Low spread influence)**, motivates a recommendation criterion that values realized propagation rather than raw neighborhood size [2508.13517]. A plausible implication is that pairwise recommendation quality and diffusion capacity must be modeled jointly rather than treated as interchangeable proxies.

## 2. Formalization and the meaning of “heterogeneous”

HeteroIR operates on an attributed directed graph
\[
G=(V,E,P,U),
\]
where \(V=\{v_1,\dots,v_N\}\) is the user set, \(E\subseteq V\times V\) is the directed social graph, \(P\in\mathbb{R}^{N\times N}\) is the invite probability matrix, and \(U\in\mathbb{R}^N\) is the accept probability vector [2508.13517]. For a pair \((u,v)\),
\[
P_{uv}\in[0,1]
\]
is the probability that user \(u\) invites \(v\), while
\[
U_v\in[0,1]
\]
is the probability that \(v\) accepts an invitation from any inviter [2508.13517].

The framework defines a **heterogeneous spread probability**
\[
S_{uv}=P_{uv}\cdot U_v.
\]
This multiplicative form reflects a two-stage interaction process: an invitation must first be sent and then be accepted. In the paper’s terminology, the “heterogeneous” aspect is behavioral rather than structural: inviter and invitee roles are distinct, and edges are weighted by personalized, direction-sensitive spread probabilities instead of a single homogeneous propagation parameter [2508.13517].

The overall task is formalized as **Recommendation with Influence Maximization (RIM)**: given \(G=(V,E,P,U)\), produce capacity-limited recommendation lists that maintain interaction willingness and maximize spread coverage. For each inviter \(u\in V_i\) and recommendation length \(k\), HeteroIR returns a ranked subset
\[
\mathrm{TopK}_u\subseteq V_e,\qquad |\mathrm{TopK}_u|=k,
\]
ordered by a profit score that integrates first-order interaction and second-order spread [2508.13517].

The paper also defines **spread influence** as the expected number of users influenced by a user \(u\). In HeteroIR, however, this quantity is not computed through full global diffusion simulation under arbitrary IC or LT cascades. Instead, it is approximated by a capacity-limited estimator, HeteroInf, that intentionally restricts effective propagation to a small number of high-probability neighbors [2508.13517].

## 3. Two-stage architecture and recommendation profit

HeteroIR is explicitly a **two-stage framework**. In the first stage, the system models behavior by estimating
\[
P_{uv}=\Pr(\text{\(u\) invites \(v\)}),\qquad U_v=\Pr(\text{\(v\) accepts any invitation}),
\]
using standard supervised CTR-like models; in the reported implementation, EulerNet is used [2508.13517]. In the second stage, these behavior estimates are transformed into spread-aware recommendation scores.

The central scoring function is the **recommendation profit**
\[
B_{uv}=P_{uv}U_v + P_{uv}U_u I_H(v),
\]
where \(I_H(v)\) is the HeteroInf estimate of user \(v\)’s heterogeneous influence [2508.13517]. The two additive components are named **1st-IF** and **2nd-IF**. The first term, \(P_{uv}U_v\), is the direct profit that the invitation from \(u\) to \(v\) is realized. The second term, \(P_{uv}U_u I_H(v)\), scales the downstream influence of \(v\) by the probability that the initial interaction occurs and by \(U_u\), which encodes the inviter’s own acceptance propensity when acting as an invitee in other contexts [2508.13517].

This additive decomposition is the framework’s core synthesis. It preserves the local pairwise semantics of recommendation while introducing an explicit estimate of secondary spread. The paper characterizes HeteroIR as model-agnostic because the learned behavior components can be supplied by standard recommenders, while the final ranking rule is deterministic once \(P_{uv}\), \(U_v\), and \(I_H(v)\) are available [2508.13517]. It further states that HeteroIR itself has no additional trainable parameters beyond those used in the behavior models.

An important contrast is with HeteroIM. HeteroIR does not use reverse reachable sets or greedy seed selection. Instead, it assigns a pointwise score to each inviter-invitee pair and ranks candidates directly. This design keeps the framework simple and scalable while still injecting an influence-aware objective into recommendation [2508.13517].

## 4. HeteroInf and algorithmic procedure

The spread term \(I_H(v)\) is computed by **HeteroInf**, a capacity-limited influence estimator introduced to reflect the empirical finding that real spreading capacity is much smaller than nominal friend count [2508.13517]. For each user \(u\), let \(N_u\) denote the out-neighbors and let \(N_u^w\subseteq N_u\) be the subset of size \(w\) with the largest spread probabilities \(S_{uv}\). HeteroInf then defines
\[
I_H(u)=\sum_{v\in N_u^w} P_{uv}U_v.
\]
This is a single-step influence measure rather than a multi-hop cascade simulation, and the global interaction capacity \(w\) is chosen by grid search; in deployment, the paper reports \(w=4\), matching the platform’s observed average spread capacity [2508.13517].

Operationally, HeteroIR proceeds in five steps. First, historical logs are used to train an invitation model for \(P_{uv}\) and an acceptance model for \(U_v\). Second, the system computes the heterogeneous spread probability \(S_{uv}=P_{uv}U_v\) for candidate edges. Third, for each user \(v\), it selects the top-\(w\) outgoing neighbors by \(S_{vw}\) and aggregates them to obtain \(I_H(v)\). Fourth, it evaluates
\[
B_{uv}=P_{uv}U_v + P_{uv}U_u I_H(v)
\]
for each inviter \(u\) and candidate invitee \(v\). Fifth, it sorts candidates by \(B_{uv}\) and returns the top \(k\) per inviter [2508.13517].

The paper gives the complexity of top-\(w\) selection as \(O(d\log w)\) per node when the average out-degree is \(d\), for a total of \(O(|E|\log w)\) to compute HeteroInf over the graph [2508.13517]. After model inference, the main additional costs are top-\(w\) aggregation and per-user sorting, which is one reason the framework is described as easy to integrate into an existing ranking stack.

The framework’s notion of spread profit can be written equivalently as
\[
\pi(u,v)\equiv B_{uv}=P_{uv}U_v + P_{uv}U_u I_H(v).
\]
The first term represents direct realized interaction; the second is a linear approximation to secondary spread. The paper explicitly notes that HeteroIR does not use RR sets or Monte Carlo simulation; those belong to HeteroIM rather than HeteroIR [2508.13517].

## 5. Training, evaluation, and empirical results

The reported implementation trains its behavior models on two Tencent datasets derived from game logs: an **Exposure-Invitation dataset** for \(P_{uv}\), in which \(P_{u,v}=1\) if \(u\) invited \(v\) at time \(T_{u,v}\), and an **Acceptance dataset** for \(U_v\), in which \(U_v=1\) if \(v\) accepted an invitation from any inviter. TXG-A is split into train, validation, and test by an \(8{:}1{:}1\) ratio, while TXG-B and TXG-C are holdout evaluation sets [2508.13517]. For Twitter, mention events are treated as spread events, and DeepWalk embeddings together with other features are fed into EulerNet to predict spread probability directly [2508.13517].

The experimental datasets are large. TXG-A contains \(109.1\)M nodes, \(177.1\)M edges, and \(62.4\)M spread events; TXG-B has \(131.6\)M nodes, \(211.4\)M edges, and \(43.2\)M spreads; TXG-C has \(115.5\)M nodes, \(179.4\)M edges, and \(35.2\)M spreads. The Twitter dataset has \(0.46\)M nodes, \(14.9\)M edges, and \(0.15\)M spread events [2508.13517].

Evaluation uses both recommendation and spread metrics. The spread metrics are **Spread@K**, defined as total deduplicated spread coverage from direct spreads and secondary spreads, and **NSpread@K**,
\[
\mathrm{NSpread@K}=\frac{\mathrm{Spread@K}}{\mathrm{ISpread@K}},
\]
where \(\mathrm{ISpread@K}\) is the maximal achievable spread under brute-force selection of \(K\) neighbors per user [2508.13517]. Recommendation quality is assessed by Recall@K and NDCG@K. In Tencent data, a correct recommendation requires both invitation and acceptance; on Twitter, a mention from \(u\) to \(v\) is treated as a valid interaction [2508.13517].

Against recommender baselines such as PPR, AutoInt, FinalNet, and EulerNet, and against influence-related baselines such as degree centrality, coreness, Windex, Monte-Carlo influence, LR influence, DeepInf, and TOPSIS, HeteroIR consistently improves both spread-aware and recommendation metrics [2508.13517]. On TXG-B at \(K=1\), the best recommender baseline, EulerNet, attains NSpread@1 \(=0.3949\), Recall@1 \(=0.2021\), and NDCG@1 \(=0.2619\), whereas HeteroIR reaches NSpread@1 \(=0.4253\), Recall@1 \(=0.2303\), and NDCG@1 \(=0.2998\) [2508.13517]. On Twitter at \(K=1\), EulerNet yields NSpread@1 \(=0.2258\), Recall@1 \(=0.2849\), and NDCG@1 \(=0.3302\), while HeteroIR yields \(0.2432\), \(0.2933\), and \(0.3402\), respectively [2508.13517]. The paper reports that the marked improvements for HeteroIR and HeteroIM have \(p\)-value \(<0.05\).

The ablation results clarify the role of the two score components. On Twitter, removing **1st-IF** reduces Recall@1 from \(0.2933\) to \(0.2503\), NDCG@1 from \(0.3402\) to \(0.2933\), and NSpread@1 from \(0.2432\) to \(0.2342\). Removing **2nd-IF** reduces NSpread@1 from \(0.2432\) to \(0.2258\) and NSpread@2 from \(0.3425\) to \(0.3166\) [2508.13517]. This suggests that direct interaction willingness is indispensable for recommendation quality, while secondary influence is the principal source of spread improvement.

The influence-estimation comparison further supports HeteroInf. The paper states that HeteroInf achieves the highest Hit@K at finding truly influential spreaders on TXG-B and TXG-C, especially for small \(K\), and that plugging HeteroInf into the HeteroIR score yields better NSpread@K and NDCG@K than using Monte-Carlo influence or linear-regression influence [2508.13517].

## 6. Deployment, limitations, and scope of the term

HeteroIR and HeteroIM were deployed in two large Tencent FPS-game propagation events, X1 and X2, with treatment groups of \(15.6\)M and \(16.4\)M users, respectively [2508.13517]. The deployment used EulerNet for the base probability estimates, set the interaction capacity to \(w=4\), and partitioned users into communities following Saveski et al. to randomize treatment while controlling network effects [2508.13517]. Online evaluation used three metrics: **Secondary Invite Rate (Sec-IR)**, the fraction of invitees who themselves send invitations; **Secondary Invite Times (Sec-IT)**, the average number of secondary invitations per invitee; and **Reach Retain Rate (RRR)**, the fraction of invitees who log in the next day [2508.13517].

In event X1, the baseline “Intimacy” achieved Sec-IR \(=8.095\%\), Sec-IT \(=0.184\), and RRR \(=17.109\%\), whereas HeteroIR achieved Sec-IR \(=11.029\%\), Sec-IT \(=0.249\), and RRR \(=25.018\%\) [2508.13517]. In event X2, the baseline achieved Sec-IR \(=5.911\%\), Sec-IT \(=0.197\), and RRR \(=30.644\%\), while HeteroIR achieved Sec-IR \(=6.414\%\), Sec-IT \(=0.210\), and RRR \(=35.160\%\) [2508.13517]. The abstract summarizes these results as an online A/B-test improvement of about \(8.5\%\) for HeteroIR and \(10\%\) for HeteroIM [2508.13517].

The framework also has clear limitations. HeteroInf is single-step and capacity-limited, so it does not explicitly model multi-hop IC-style cascades. The propagation model assumes
\[
\Pr(u\to v\text{ spreads})=P_{uv}U_v
\]
and treats \(U_v\) as independent of inviter identity. The interaction capacity \(w\) is a single global parameter rather than a personalized \(w_u\). HeteroIR is model-agnostic but not end-to-end: invite and accept models are trained separately from the spread-aware ranking rule, and the optimization objective of the learned models is not NSpread@K itself [2508.13517]. The paper identifies personalized interaction capacity and richer spread-aware learning objectives as directions for future work.

A common misconception is that “HeteroIR” is a universally standardized term. In fact, the label is polysemous across arXiv. In recommendation, it denotes **Heterogeneous Influence-based Recommendation** [2508.13517]. In wireless communications, closely related terminology appears in the form of a **heterogeneous intelligent reflecting surface (HE-IRS)**, also described as HeteroIR in context [2506.10350]. In oxide spintronics and condensed-matter literature, “HeteroIR” is also used as a shorthand for **iridate-based heterostructures** [1711.07609]. Within the recommendation literature, however, HeteroIR specifically refers to the spread-aware invitation-recommendation framework built around the score
\[
B_{uv}=P_{uv}U_v + P_{uv}U_u I_H(v),
\]
its HeteroInf influence estimator, and its objective of reconciling interaction willingness with spread coverage [2508.13517].

Source: https://www.emergentmind.com/topics/heteroir