Neural Influence Ranking Model (NIRM)

• NIRM is a family of neural ranking architectures that model mutual influences among items and network nodes to achieve globally optimized ranking objectives.
• It integrates feature extraction, sequential modeling (via DNN and RNN with attention), and graph attention networks to enhance both e-commerce GMV and network dismantling effectiveness.
• Empirical evaluations and ablation studies show that fusing local and global features significantly improves performance while introducing trade-offs in computational latency.

The Neural Influence Ranking Model (NIRM) is a family of neural ranking architectures designed to incorporate mutual influence effects among entities—either items in e-commerce result lists or nodes in complex networks—enabling globally optimized ranking for downstream objectives such as purchase-driven Gross Merchandise Volume (GMV) or network dismantling efficiency. NIRM achieves this by integrating neural feature extraction, context-dependent modeling, and sequential/contextual decision processes, leading to empirical advances in both e-commerce search and network robustness analysis (Zhuang et al., 2018, Zhang et al., 2022).

1. Mathematical Formulations

NIRM addresses fundamentally different ranking settings for e-commerce (maximizing GMV) and network dismantling (minimizing the size of a residual component). Both rely on context-sensitive scoring:

• E-Commerce Ranking Given items $S = \{1, \ldots, N\}$ and a permutation $o = (o_1, \ldots, o_N)$, the expected GMV is

$$E[\mathrm{GMV} \mid o] = \sum_{i=1}^N v(i)\;p\bigl(i\mid c(o, i)\bigr)$$

where $v(i)$ is item price, $p(i\mid c(o,i))$ is purchase probability conditioned on item context $c(o,i)$. The global optimization objective is

$$o^* = \arg\max_{o \in \mathrm{Perm}(S)}\ \sum_{i=1}^N v(i)\,p(i \mid c(o, i))$$

• Network Dismantling For network $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ and removal set $\mathcal{V}_t$, normalized residual component size is

$$\mathrm{NGCC}(\mathcal{V}_t) = \frac{|\mathcal{G} \setminus \mathcal{V}_t|}{|\mathcal{V}|}$$

The problem is

$o = (o_1, \ldots, o_N)$0

NIRM instead learns node dismantling scores $o = (o_1, \ldots, o_N)$1 for greedy ranking.

This unifying principle positions NIRM as a context-aware global ranking method, with neural architectures defining context computation in application-specific ways.

2. Neural Mutual Influence Modeling

2.1 E-Commerce Mutual Influence Estimation

Two neural models are described:

• Global-Feature DNN (miDNN):

Concatenates local features $o = (o_1, \ldots, o_N)$2 (e.g., price, relevance, historical CTR, user-specific preferences) with normalized global-context features

$o = (o_1, \ldots, o_N)$3

yielding $o = (o_1, \ldots, o_N)$4. The DNN (three ReLU layers—50, 50, 30 units; sigmoid output) estimates independent purchase probabilities $o = (o_1, \ldots, o_N)$5 trained with cross-entropy loss.

• Sequence-Aware RNN (miRNN/miRNN+Attention):

Models left-to-right ranking as a sequence, using LSTM recurrence

$o = (o_1, \ldots, o_N)$6

For miRNN+attention, position embeddings and attention weights model dependencies on previous states:

$o = (o_1, \ldots, o_N)$7

Context vector $o = (o_1, \ldots, o_N)$8 and final prediction use both $o = (o_1, \ldots, o_N)$9 and the attention-aggregated context.

2.2 Graph Structure Modeling for Dismantling

NIRM for network dismantling constructs node features

$$E[\mathrm{GMV} \mid o] = \sum_{i=1}^N v(i)\;p\bigl(i\mid c(o, i)\bigr)$$0

where $$E[\mathrm{GMV} \mid o] = \sum_{i=1}^N v(i)\;p\bigl(i\mid c(o, i)\bigr)$$1 is degree, local clustering coefficient $$E[\mathrm{GMV} \mid o] = \sum_{i=1}^N v(i)\;p\bigl(i\mid c(o, i)\bigr)$$2, and additional contextual statistics. The architecture comprises:

• Initial feature scoring: $$E[\mathrm{GMV} \mid o] = \sum_{i=1}^N v(i)\;p\bigl(i\mid c(o, i)\bigr)$$3
• Multi-layer multi-head GAT encodes node representations.
• Local and global scores ($$E[\mathrm{GMV} \mid o] = \sum_{i=1}^N v(i)\;p\bigl(i\mid c(o, i)\bigr)$$4, $$E[\mathrm{GMV} \mid o] = \sum_{i=1}^N v(i)\;p\bigl(i\mid c(o, i)\bigr)$$5) are computed by aggregation and dot-product operations involving local neighborhoods and a global trainable projector $$E[\mathrm{GMV} \mid o] = \sum_{i=1}^N v(i)\;p\bigl(i\mid c(o, i)\bigr)$$6.
• The final dismantling score is $$E[\mathrm{GMV} \mid o] = \sum_{i=1}^N v(i)\;p\bigl(i\mid c(o, i)\bigr)$$7.

3. Training Methodology

3.1 E-Commerce Purchase Probability Models

Models are optimized with cross-entropy loss over observed click/purchase labels. Optionally, the loss is weighted by item price to focus on high-GMV items:

$$E[\mathrm{GMV} \mid o] = \sum_{i=1}^N v(i)\;p\bigl(i\mid c(o, i)\bigr)$$8

Training uses mini-batch SGD variants (Adam/Adagrad), learning rate tuning by holdout set, early stopping, and regularization (dropout, $$E[\mathrm{GMV} \mid o] = \sum_{i=1}^N v(i)\;p\bigl(i\mid c(o, i)\bigr)$$9).

Training data consists of one month of query logs, with session balancing to reduce negative-sample bias.

3.2 Network Dismantling Model

Training is performed on 4,000 synthetic networks of 20–30 nodes, sampled from Erdős–Rényi, Watts–Strogatz, Barabási–Albert, and Power-law Cluster models. Exhaustive enumeration of minimal attack sets provides ground-truth node importance scores, normalized and propagated to neighbors.

Optimization employs mean-squared error loss:

$v(i)$0

Hyperparameters: Adam optimizer (learning rate $v(i)$1, $v(i)$2 weight decay $v(i)$3), $v(i)$4 GAT layers, embedding dimensions 32–16–8, dropout ($v(i)$5/$v(i)$6), LeakyReLU negative slope 0.2, and early stopping.

4. Inference and Deployment

• For e-commerce, purchase probabilities or sequential probabilities $v(i)$7 are obtained per candidate ranking, supporting beam search for globally optimized reranking in $v(i)$8 (reranking top $v(i)$9 candidates with beam width $p(i\mid c(o,i))$0).
• For network dismantling, inference requires only the graph adjacency and the computed five-dimensional feature matrix. GAT encoding and dismantling score computation scale as $p(i\mid c(o,i))$1 plus sorting, with model weights fixed post-training.

No item- or network-specific tuning is needed for deployment; inference proceeds inductively from pretrained parameters.

5. Empirical Performance and Ablation Analysis

5.1 E-Commerce GMV and Probability Estimation

• Offline AUC/Relative Information Gain (purchase probability): | Model | AUC | Rel. Info Gain | |----------------------|-------|---------------| | Baseline DNN | 0.724 | 0.094 | | miDNN | 0.747 | 0.119 | | miRNN | 0.765 | 0.141 | | miRNN+attention | 0.774 | 0.156 |
• Online A/B test (reranking top 50 candidates, beam size 5): | Model | GMV Lift | Latency Δ | |-----------------|-----------|-----------| | miDNN | +2.91% | +9% | | miRNN | +5.03% | +58% | | miRNN+attention | +5.82% | +401% |

Ablation: adding global feature extension yields significant GMV gain; sequential modeling (RNN) doubles the relative gain; attention provides marginal further GMV increases but large latency costs (Zhuang et al., 2018).

5.2 Network Dismantling Effectiveness

• On 15 real networks (sizes 241–4,941 nodes), NIRM achieves lower required removal fractions $p(i\mid c(o,i))$2 than baselines:
  • One-pass: NIRM $p(i\mid c(o,i))$3, 2nd best $p(i\mid c(o,i))$4 (12/15 wins)
  • Adaptive: NIRM $p(i\mid c(o,i))$5, 2nd best $p(i\mid c(o,i))$6 (14/15 wins)
• Area under NGCC–$p(i\mid c(o,i))$7 curve is consistently lowest for NIRM on most graphs.

Ablation:

• Only using global score ($p(i\mid c(o,i))$8) is ineffective ($p(i\mid c(o,i))$9), while local score alone is more robust ($c(o,i)$0). Full fusion reduces $c(o,i)$1 by 5–10%.
• Varying GAT layers, heads, embedding dimensions, or learning rate yields minor $c(o,i)$23% change in average $c(o,i)$3 (Zhang et al., 2022).

6. Broader Implications and Insights

NIRM illustrates that context-dependent neural ranking approaches can efficiently generalize across domain boundaries—leveraging mutual influence, sequential dependencies, and structural graph signals—not only for maximizing monetary objectives in commercial search but also for resilience analysis in complex networked systems. The effectiveness of pretraining NIRM on small synthetic graphs suggests that relevant local/global signal fusion encodes transferable dismantling cues. A plausible implication is the feasibility of applying NIRM paradigms to broader classes of combinatorial or influence-aware ranking problems without exhaustive domain-specific retraining (Zhang et al., 2022).

7. Representative Algorithms and Schematic

NIRM for Network Dismantling (Inference Pseudocode)

$c(o,i)$4

This architecture is extensible to arbitrary graphs, leveraging fixed-layer GAT inference plus posthoc scoring and ranking (Zhang et al., 2022).

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Key references:

• E-commerce search: (Zhuang et al., 2018)
• Network dismantling: (Zhang et al., 2022)