The Value of Graph-based Encoding in NBA Salary Prediction

Junhao Su; David Grimsman; Christopher Archibald

cs.LGMar 5, 2026

The Value of Graph-based Encoding in NBA Salary Prediction

Authors: Junhao Su, David Grimsman, Christopher Archibald

Organizations: Department of Computer Science Brigham Young University Provo, UT, USA

Abstract

Market valuations for professional athletes is a difficult problem, given the amount of variability in performance and location from year to year. In the National Basketball Association (NBA), a straightforward way to address this problem is to build a tabular data set and use supervised machine learning to predict a player's salary based on the player's performance in the previous year. For younger players, whose contracts are mostly built on draft position, this approach works well, however it can fail for veterans or those whose salaries are on the high tail of the distribution. In this paper, we show that building a knowledge graph with on and off court data, embedding that graph in a vector space, and including that vector in the tabular data allows the supervised learning to better understand the landscape of factors that affect salary. We compare several graph embedding algorithms and show that such a process is vital to NBA salary prediction.

Figures & tables

Fig. 1 : Schema of the Heterogeneous NBA Knowledge Graph. The graph connects PlayerSeason anchor nodes (center) to diverse entities including Team , Agent , Award , and Injury . Temporal edges (e.g., Won_Previously , Has_Injury_History ) are strictly masked by the admissibility function $A(e,s)$ to prevent look-ahead bias.

Fig. 2 : Tri-State Evaluation on Eligible Outliers. (a) vs. Weak Baseline: Static embeddings provide a favorable rescue–misguidance trade-off. (b) vs. Strong Baseline: Dynamic architectures incur a “Generalization Tax,” reflecting sensitivity to historical networks.

Feature Set	Global RMSE $\downarrow$		Global $R^{2}$ $\uparrow$		Cold-Start $\uparrow$
Feature Set	RF	XGB	RF	XGB	$R^{2}$ (XGB)
Baseline (+Meta)	0.618	0.598	0.717	0.735	0.535
RotatE + Stats	0.654	0.668	0.683	0.670	-0.014
Node2Vec + Stats	0.659	0.665	0.678	0.672	-0.004
V2-Full MG + Stats	0.687	0.695	0.650	0.642	-0.047
V2-Full SG + Stats	0.689	0.702	0.648	0.635	-0.047

TABLE I : Global Predictive Performance. Models ranked by Random Forest (RF) RMSE. Baseline (+Meta) represents the Explicit-ID Upper Bound. The Cold-Start column exclusively evaluates new entrants, demonstrating the collapse of graph models in structural vacuums (

\downarrow

: lower is better,

\uparrow

: higher is better).

Baseline Context	Graph Model	Top Feature	Cliff’s $\delta$	$p$ -value
Weak (Stats+Time)	Node2Vec	round_pick	-0.34	$<0.001$
	Node2Vec	overall_pick	-0.31	$0.001$
	RotatE	round_pick	-0.26	$0.003$
	V2-Ind	TS%_calc	-0.26	$<0.001$
	V2-Trans	overall_pick	-0.28	$<0.001$
Strong (+Meta)	RotatE	draft_year	-0.41	$<0.001$

TABLE II : Top Profiling Traits separating Rescue vs. Misguidance Cohorts (

p\leq 0.10,|\delta|\geq 0.25

). Positive

\delta

indicates the trait is significantly higher in successfully rescued players; negative indicates higher in misguided players.

Baseline	Player (Case)	$\hat{Y}_{\text{base}}$	$\hat{Y}_{\text{graph}}$	$\Delta E$
Weak	F. VanVleet (Rescue)	$7.47M	$17.4M	+$9.97M
Weak	Chris Paul (Misguide)	$13.7M	$23.1M	-$9.43M
Strong	F. VanVleet (Rescue)	$8.08M	$17.4M	+$9.37M
Strong	Chris Paul (Misguide)	$14.7M	$23.1M	-$8.36M

TABLE III : Deterministically selected micro-level corrections for RotatE .

\Delta E=|Y-\hat{Y}_{\text{base}}|-|Y-\hat{Y}_{\text{graph}}|

Graph Model	Weak Baseline (Stats + Time)		Strong Baseline (+ Meta IDs)
Graph Model	Max Rescue ( $\Delta E$ )	Max Misguide ( $\Delta E$ )	Max Rescue ( $\Delta E$ )	Max Misguide ( $\Delta E$ )
RotatE	VanVleet (+10.0)	Paul (-9.4)	VanVleet (+9.4)	Paul (-8.4)
Node2Vec	Oubre Jr. (+5.2)	Quickley (-5.1)	White (+4.6)	Reid (-3.6)
V1	Middleton (+12.1)	Bane (-22.2)	Middleton (+11.7)	Bane (-20.7)
V2-Ind	Beasley (+7.5)	Antetokounmpo (-11.5)	Beasley (+7.5)	Antetokounmpo (-12.2)
V2-Trans	Zubac (+10.0)	Bane (-20.6)	Zubac (+9.8)	Bane (-19.1)

TABLE IV : Model-centric representative cases selected deterministically from the eligible-outlier pool (2024 season). We report the maximum rescue (

\Delta E>0

) and misguidance (

\Delta E<0

) cases, with

\Delta E=|Y-\hat{Y}_{\text{base}}|-|Y-\hat{Y}_{\text{graph}}|

. All values are in $M.

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Probabilistic Salary Prediction with Graph Attention Networks and a Mixture Density Network

Zhipei Qin, Mohammad Shokri, N. van Weeren, F. W. Takes

Leiden Institute of Advanced Computer Science (LIACS) · Randstad N.V.

Mixture Density Networks Tabular Prediction

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Accurate salary prediction is critical for bridging the information gap between employers and job seekers in modern labor markets. Existing approaches predominantly yield a single point estimate and treat job attributes such as location, occupation, and industry as independent categorical features, ignoring both the inherent uncertainty and multi-modality of real-world compensation data and the rich hierarchical and semantic-similarity relationships that govern pay norms. In this paper we propose GAT-MDN, a unified framework that addresses both limitations simultaneously. For each of the three attribute domains we construct a domain-specific graph whose edges encode (i) hierarchical parent-child containment and (ii) weighted similarity links derived from a pre-trained Sentence-Transformer. Parallel Graph Attention Networks (GATs) with edge-feature-aware attention learn rich, context-sensitive node representations from these multi-relational graphs. A priority-based hierarchical selection module then assembles a composite feature vector that gracefully handles missing or coarse attributes, and a Mixture Density Network (MDN) head maps this vector to the parameters of a Gaussian Mixture Model (GMM), yielding a full conditional salary distribution. Extensive experiments on a real-world Dutch job-posting dataset of over 1 million records demonstrate that GAT-MDN significantly outperforms a non-graph MLP-MDN baseline in both Negative Log-Likelihood (NLL) and Mean Squared Error (MSE).

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