Credit Attribution in Timestamped Soccer Networks Using Multivariate Hawkes Processes

arXiv:2610.12775 · cs.SI, cs.LG, physics.soc-ph, stat.AP · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Credit Attribution in Timestamped Soccer Networks Using Multivariate Hawkes Processes".

Jane: The gist The two main traditional player statistics used in soccer are goals scored and assists, with an assist credited when a player’s pass immediately leads to a teammate scoring.

Tom: First, who's behind it and why it matters.

Paper summary: Tom: So we're looking at this paper, "Credit Attribution in Timestamped Soccer Networks Using Multivariate Hawkes Processes." Basically, they're tackling the old way of counting stats—goals and assists—because that just doesn't show how much work a player does in building an attack.

Jane: Right. The traditional way gives you a goal if you score, and an assist if your pass leads to a goal immediately after. But what about all those passes leading up to it? That paper says we need something better to see the whole picture of involvement, not just that one final touch.

Lu: Exactly. They use multivariate Hawkes processes, or MHPs, to model the sequence of passes and shots as continuous-time networks. The core idea is using a branching mechanism where one event can cause another event to happen next in time, which lets them quantify each pass's contribution to the next pass or shot>

Meng: From a production standpoint, modeling sequences like this sounds complex. How does that actually translate into something we can use reliably for performance metrics? What’s the actual data structure they are working with?

Tom: Well, they model it by making each variable in the MHP correspond to a specific pair of players, which is a sender and a receiver. The excitation matrix, alpha, captures how much one pass between two pairs influences another pass between different pairs in the near future>

Jane: That sounds like they are trying to capture that influence structure. But what's the big difference here compared to just looking at who made the last pass before a goal?

Lu: The paper proposes computing shot responsibilities using the value of beta along with the estimated alpha and mu, which is computed using equation one> This lets them examine the six preceding passes for each shot to calculate how much each pass contributed, plus a probability that a shot happened even without any specific pass>

Meng: That sounds like a lot of calculation. How do they handle the fact that you can't just assign credit globally across all pairs? There are so many sender-receiver pairs in a game.

Tom: They don't use one fixed alpha for everyone, which is important because it accounts for temporal proximity and learned influence between specific pairs> They compute this using equation one which lets them look at the last six passes for each shot to calculate that contribution probability>

Jane: So it’s not just about the final pass; it’s about the whole sequence leading up to a shot. What does that mean for how we see player performance in a game?

Lu: They build on those shot responsibility values to create expected goal involvements, or xGI. This metric measures a player's involvement in creating goal scoring opportunities based on their passes> It distributes the expected goal value of a shot among all players who were involved in the attacking sequence>

Paper summary: Meng: So this means a player who makes an early, dangerous pass gets credit even if they aren't the one who made the final assist or scored themselves? That seems like it could change how we evaluate playmakers.

Tom: It does. It allows players who initiate or advance dangerous plays but don't make the final pass to receive credit for their involvement in creating those opportunities> The analysis uses data from La Liga during the two thousand fifteen/two thousand sixteen season, focusing on players like Lionel Messi and Neymar Junior who showed high xGI per game>

Jane: I see. And what were the key findings there regarding team dynamics? Were there any patterns in how different teams behaved when you looked at this metric?

Lu: The analysis showed that Barcelona had a mean alpha ij value of zero point zero zero five zero compared to Levante UD's zero point zero zero four zero, which indicates a stronger core influence network for Barcelona> They also demonstrated that passes not directly involving the target player can still contribute indirectly to future shots through subsequent passes>

Meng: How stable is this model when you try to deploy it in a live, high-speed environment? What are the limitations they flagged in the paper?

Tom: The authors noted a few things. They assume that a frequently leading pass should receive more credit than an infrequent one, and they exclude passes not among the top one hundred most frequent sender-receiver pairs> Plus, they had to manually select beta which impacts how the alpha matrix and shot responsibilities end up>

Jane: So it's not a perfect solution yet. What’s the bigger picture here for people who just follow soccer stats?

Lu: The paper suggests that developing models like the multivariate community Hawkes, or MULCH, could help address some of those limitations regarding node grouping in the network structure> It points toward refining how we group these player interactions>

Meng: From a deployment view, if you were running this at scale with millions of passes per second, that alpha estimation needs to be very robust. The reliance on the top one hundred pairs shows a vulnerability there because rare interactions get filtered out>

Tom: It’s about moving beyond the simple final pass metric to account for the entire buildup, and it uses these specific mathematical tools to give credit where it should be distributed across multiple passes> That's what this paper is all about.

Jane: So we’ve seen how they use multivariate Hawkes processes to model soccer action sequences, how they calculate shot responsibilities using alpha and beta, and what that means for expected goal involvements> It really shifts the focus from single actions to the whole attacking sequence>

Conclusion: Tom: So we're wrapping up this discussion on "Credit Attribution in Timestamped Soccer Networks Using Multivariate Hawkes Processes" by Phillip W. Kornberg and Kevin S. Xu at Case Western Reserve University> Basically, they built a model to figure out who really deserves credit in soccer goals and shots>

Jane: Right, so the main point here is moving away from just counting goals or assists to understanding the actual flow of play throughout an attack> It’s about seeing how every pass contributes, not just the last one>

Lu: The authors use multivariate Hawkes processes to model passes and shots as a continuous-time network> This gives them a branching mechanism where one event can cause another event in time, which helps quantify the contribution of each pass to future action>

Meng: But how do you handle all those interactions between different players in a game? It sounds like the complexity of the data structure is massive when you're trying to run this thing in production>

Tom: They address that by making each variable in their MHP correspond to a specific pair of players, a sender and a receiver> The excitation matrix, alpha, then captures how much one pass between two pairs influences another pass between different pairs in the near future>

Jane: That means they aren't looking at one big picture; they are focusing on the local influence between specific player pairings> But what does that actually tell us about how a team functions as a whole?

Lu: They compute shot responsibilities using the value of beta along with the estimated alpha and mu, computed using equation one> This lets them examine those six preceding passes for each shot to calculate the probability that each pass contributed to it> plus the probability that a shot happened independently of any pass>

Tom: So it’s not just about who made the final touch; it’s about looking back at the sequence leading up to every single goal or shot> It’s a way to distribute credit across multiple passes in an attacking sequence>

Jane: That distribution of credit is what they call expected goal involvements, or xGI> This metric measures a player's involvement in creating those scoring opportunities based on their passes> It lets players who initiate dangerous plays but don't make the final pass receive credit for that buildup>

Lu: The analysis on La Liga data showed that players like Lionel Messi and Neymar Junior had the highest xGI per game, and Barcelona had a higher mean alpha ij value compared to Levante UD> This suggests a stronger core influence network in top teams>

Meng: If I were to deploy this, I'd worry about the computational cost of calculating those shot responsibilities for every single shot in real-time across millions of events> What breaks at scale is usually the estimation of that alpha matrix when you have too many sender-receiver pairs>

Tom: The authors did acknowledge limitations on that, though> They mentioned they only trained their alpha on the one hundred most frequent sender-receiver pairs to keep things stable and then smoothed it with a scaled all-ones matrix> And they also had to manually select a value for beta which impacts the results>

Jane: So while the concept of xGI is really compelling because it gives credit where it’s earned, you have to be careful about those assumptions in the model structure itself> It's not just a magic formula for stats>

Lu: Exactly. They suggested that developing models like the multivariate community Hawkes, or MULCH, could help alleviate some of those limitations regarding how the network is grouped> It points toward refining how we understand these player interactions structurally>

Tom: So it’s a lot of math and data processing to get this result, but the core idea is a fairer way to value attacking contribution than just traditional stats> It’s about seeing the whole sequence instead of just that one pass>

Jane: It really shifts how we evaluate player performance in soccer because it makes you see the entire buildup as valuable work, not just the final action> And that leads us to thinking about what these kinds of sequence models could do for other complex temporal data sets next.

Phillip W. Kornberg, Kevin S. Xu

Case Western Reserve University

cs.SI, cs.LG, physics.soc-ph, stat.AP

Submitted: 2026-10-08

Updated: 2026-10-08

Code: https://github.com/hudl/opendata

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 72/100

The gist: The gist The two main traditional player statistics used in soccer are goals scored and assists, with an assist credited when a player’s pass immediately leads to a teammate scoring.

Key concepts

Multivariate Hawkes Processes (MHPs): MHPs are used to model sequences of events like passes and shots over time. They create a branching mechanism where one event triggers the probability of another event occurring shortly after. This allows researchers to track how every pass influences subsequent actions within the game's timeline.
Shot Responsibilities: This metric calculates the probability that a specific preceding pass contributed to a particular shot. It uses estimated influence parameters derived from the MHP model, accounting for both how close in time passes occurred and the learned influence of different player pairings on goal-scoring outcomes.
Expected Goal Involvements (xGI): xGI is a metric that measures a player's involvement in creating goal-scoring chances based on their passes. It distributes the expected goal value of a shot among all players who participated in the attacking sequence, giving credit to players who set up opportunities even if they didn't make the final pass.
Excitation Matrix ($\alpha$): The excitation matrix captures how a pass between two specific players influences future passes between other player pairs. This matrix is central to the MHP model, as it quantifies the learned influence of different player interactions on each other in the near future.

Summary

The gist The two main traditional player statistics used in soccer are goals scored and assists, with an assist credited when a player’s pass immediately leads to a teammate scoring.

Modeling Passing Networks with Multivariate Hawkes Processes

The authors use multivariate Hawkes processes (MHPs) to model timestamped sequences of passes and shots as continuous-time networks. MHPs provide a branching mechanism where an event can be generated as a child of a previous event, allowing the quantification of each pass’s contribution to a subsequent pass or shot. In their model for soccer action sequences, each variable in the MHP corresponds to a specific pair of players: a sender and receiver. The excitation matrix α captures how a pass between sender-receiver pair j influences a pass between sender-receiver pair i in the near future.

Credit Attribution via Shot Responsibilities

The paper proposes computing shot responsibilities using the value of β along with the estimated α and µ, which is computed using equation (1). This allows for examining the six preceding passes for each shot to calculate the probability that each pass contributed to the shot, along with the probability that a shot occurred independently of any pass. The model accounts for both temporal proximity and learned influence of each sender-receiver pair, providing a more nuanced attribution than a fixed global α shared across all sender-receiver pairs.

Expected Goal Involvements (xGI)

The shot responsibility values form the basis for the expected goal involvements (xGI) metric, which measures a player’s involvement in creating goal scoring opportunities based on their passes. This metric distributes the expected goal (xG) value of a shot among all players who contributed to the attacking sequence. This allows players who initiate or advance dangerous plays but do not provide the final pass to receive credit for their involvement.

Data and Model Refinements

The analysis uses publicly available data from StatsBomb for all of the teams in La Liga over the 2015/2016 season. To address issues with rare interactions producing unstable alpha estimates, the authors select only the 100 most frequent sender-receiver pairs for training. The estimated α is smoothed by taking a convex combination of the MLE and a scaled all-ones matrix to ensure stability, setting ε = 0.3.

Key Findings in La Liga Analysis

Analysis of Barcelona's 2015/2016 season shows that players like Lionel Messi and Neymar Junior have the highest xGI per game, consistently leading players in xA and key passes. Barcelona has the highest mean αij value of 0.0050 compared to Levante UD's 0.0040, indicating a stronger core influence network. The model demonstrates that passes not directly involving the target player can still contribute indirectly to future shots through subsequent passes.

Limitations of the Approach

The model has limitations, including the basic assumption that a frequently leading pass should receive more credit than an infrequent one. Furthermore, infrequent passes not among the top 100 pairs are excluded and assigned a minimum αij ≈ 0.001. The manual selection of β also impacts the resulting alpha matrix and shot responsibilities. The authors suggest that developing models like the multivariate community Hawkes (MULCH) model could alleviate some limitations regarding node grouping.

Acknowledgments

This material is based upon work supported by the National Science Foundation grant IIS-2318751 and a summer research award provided by the CWRU Undergraduate Research Office. The references cited are listed in the References section. The paper is arXiv:2610.12775v1 [cs.SI] 8 Oct 2026. The paper is by Phillip W. Kornberg and Kevin S. Xu from Case Western Reserve University. The paper is published on page 2610.12775v1 [cs.SI] 8 Oct 2026. The paper is published on page 10 Phillip W. Kornberg and Kevin S. Xu. The paper is published on page 11 Credit Attribution in Soccer Networks Using Multivariate Hawkes Processes. The paper is published on page 12 Phillip W. Kornberg and Kevin S. Xu. The paper is published on page 7 Grieshop et al., A continuous-time stochastic process designed to model different aspects of a soccer match, including ball control, passing, and pace. The paper is published on page 8 Phillip W. Kornberg and Kevin S. Xu. The paper is published on page 3 Dataset Description and Preprocessing. The paper is published on page 4 Phillip W. Kornberg and Kevin S. Xu. The paper is published on page 5 Credit Attribution in Soccer Networks Using Multivariate Hawkes Processes. The paper is published on page 6 Phillip W. Kornberg and Kevin S. Xu. The paper is published on page 12 Phillip W. Kornberg and Kevin S. Xu. The paper is published on page 7 Grieshop et al., A continuous-time stochastic process designed to model different aspects of a soccer match, including ball control, passing, and pace. The paper is published on page 11 Credit Attribution in Soccer Networks Using Multivariate Hawkes Processes. The paper is published on page 2 Phillip W. Kornberg and Kevin S. Xu. The paper is published on page 3 Dataset Description and Preprocessing. The paper is published on page 4 Phillip W. Kornberg and Kevin S. Xu. The paper is published on page 5 Credit Attribution in Soccer Networks Using Multivariate Hawkes Processes. The paper is published on page 6 Phillip W. Kornberg and Kevin S. Xu <ref:Phillip W.

Improvements for AI systems

Bold header: Ability to attribute credit across multiple passes

This system can move beyond assigning credit solely to the player who makes the final pass before a goal by using multivariate Hawkes processes (MHPs) to quantify each pass’s contribution to a subsequent pass or shot, thereby distributing credit across multiple players in the buildup.

Bold header: Creation of Expected Goal Involvements (xGI)

The system can generate the xGI metric, which measures a player’s involvement in creating goal scoring opportunities beyond the final pass based on their passing history, allowing identification of players who play key roles in creating goal scoring opportunities even if they do not make the final pass.

Bold header: Nuanced Shot Responsibility Attribution

The system can compute shot responsibilities using the MHP model, which accounts for both temporal proximity and the learned influence of each sender-receiver pair, providing a more nuanced attribution of credit than a fixed global α shared across all sender-receiver pairs.