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# Measures on spatio-temporal graph-based models for team sports analysis

Camille Grange, Rodolphe Charrier, Géraldine Del Mondo, Antoine Dutot, Éric Sanlaville, Pierre-Marie Stevenin

► To cite this version:

Camille Grange, Rodolphe Charrier, Géraldine Del Mondo, Antoine Dutot, Éric Sanlaville, et al.  
Measures on spatio-temporal graph-based models for team sports analysis. ROADEF 2026, Université  
de Tours [UT], Feb 2026, Tours, France. hal-05534859

HAL Id: hal-05534859

https://hal.science/hal-05534859v1

Submitted on 3 Mar 2026

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# Measures on spatio-temporal graph-based models for team sports
## analysis

Camille Grange¹, Rodolphe Charrier², Géraldine Del Mondo³,
Antoine Dutot², Eric Sanlaville², Pierre-Marie Stevenin³

¹ LIPN, Paris, France

grange@lipn.univ-paris13.fr

² LITIS, Le Havre, France

{rodolphe.charrier, antoine.dutot, eric.sanlaville}@univ-lehavre.fr

³ INSA, Rouen, France

{geraldine.del_mondo, pierre-marie.stevenin}@insa-rouen.fr

**Keywords: Graph modelization, metrics, spatio-temporal graph, sports**

## 1 Graph-based model for team sports

Nowadays, the study of complex systems as graph-based models is gaining popularity in the computer scientific community in order to analyze dynamic interactions, involving spatial and temporal considerations, between entities. Among others, the domain of team sport uses such modelization to analyse team coordination patterns such as passing networks [1]. In a previous work, we proposed a new spatio-temporal graph-based model, briefly presented below. In the present work, we address the issue of identifying relevant metrics for the output model.

The spatio-temporal graph-based model contains spatial, temporal and semantic information. Specifically, the output of the model is the following: given the record of a team sport match, the model produces a set of directed labeled paths. A path, representing an action of a team, contains spatial and temporal information on each vertex, and semantic information on each arc. Notice that paths do not necessarily have the same length. An example is given in Figure 1 (in the case of rugby with players).

Note that, in our experimental protocol, we aim at detecting the effect of pedagogy on team coordination patterns, thus the initial position of the game is always the same. Next, we propose some measures to analyse different outputs of the model.

<figure><img src="image_3.png" /><figcaption>FIG. 1: Example of an directed labeled path (in red) on a static graph representing all possible transitions (in grey). Vertices are identified by spatial information (representing relative/absolute positions) with additional temporal labels (in blue, representing a time interval), and the arcs are labeled with semantics (in teal, representing the nature of the evolution).</figcaption></figure>

## 2 Different measures

The output of the model is a set of directed labeled paths with quantitative and qualitative information, both on vertices and arcs. The analysis of such outputs is composed of
two dimensional aspects: the *scale* of the measure (either *local* scale, at the level of a path, or *global* scale, at the level of the subgraph representing the union of paths); and the *nature* of the measure (either *similarity* representing pair-wise comparison, or *features* representing absolute indicators). In this work, we investigate local similarity measures. Mathematically, it amounts to define a function $\Delta : P(G, v) \times P(G, v') \to \mathbb{R}^{+}$, where $P(G, v)$ is the set of directed labeled paths in graph $G$ starting with vertex $v$, quantifying *how much* two paths are *alike*. Ideally, $\Delta$ is defined as a metric (or a quasi-metric). The difficulty in this case is multiple. We want to define a (quasi-)metric $\Delta$, thus with strict properties, which compares paths of different lengths, all starting at the same vertex, with many qualitative and quantitative information. In addition, $\Delta$ must express realistic likeness between the corresponding actions for sport experts.

We consider different measures, at the core of each of them lies the distance between two arcs (extreme vertices included) represented in Figure 2. This distance is itself composed of three types of distances: distance between spatial information $d_{sp}$ and temporal information $d_{temp}$ on vertices, and distance between semantic information $d_{sem}$ on arcs. Note that each of the latter distances also require specific definitions. Given these three distances, we are able to define the arc distance as the sum of five values for each pair of arcs (see Figure 2).

<figure><img src="image_4.png" /><figcaption>FIG. 2: Distances involved when comparing two arcs (arc *a* in blue and *a'* in red).</figcaption></figure>

Once the arc distance is fixed, we still have to define the distance between two paths. We define and characterize several quasi-metrics, for instance with a *sliding window* (comparing at each step two paths of same length, see Figure 3) or a *fuzzy matching* (comparing a limited and *close* (in terms of ranks) set of arcs of path 2 for a given arc of path 1, see Figure 4).

<figure><img src="image_5.png" /><figcaption>FIG. 3: Distance between two paths ($p_1$ in blue, $p_2$ in red) with a sliding window, requiring 3 computations of distances $d_{eqpath}$ between subpaths of same length.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 4: Example of distance between two paths ($p_1$ in blue, $p_2$ in red) with a fuzzy matching. Each arc of $p_1$ is compared with arcs of $p_2$ of close rank (here, $\pm 1$).</figcaption></figure>

Eventually, we test these measures on existing experimental protocols for rugby and basketball, requiring to adjust the various choices of parameters and configurations of these measures to meet all the requirements mentioned above. Preliminary results show that our model detects coordination patterns related to coach directives in some cases.

## References

[1] Q. Bourgeais, R. Charrier, E. Sanlaville, and L. Seifert. A temporal graph model to study the dynamics of collective behavior and performance in team sports: an application to basketball. *Social Network Analysis and Mining*, 14(1):94, 2024.