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Abstract

We present a method to analyze lineup usage patterns in the NBA, based on topological data analysis. Traditional network analysis is based on which pairs of players are on the court together. Instead, we construct a simplicial complex using complete information about the lineups – all five players at once. We weight the lineups by possessions played and compute the persistent homology to detect ‘holes’ in the complex, reflecting groups of players who do not play together at the same time. We demonstrate the method on twenty years of NBA lineup data, give examples of how the results can be used and interpreted, and discuss further applications.

Keywords

Basketball lineup analysis topological data analysis persistent homology simplicial complex

Introduction

Different NBA teams employ different lineup usage patterns. One group of players might not play together often because they play the same position, another because they are collectively weak on defense, or another because the team prefers to keep the starters and reserves in separate units, for example. These lineup usage patterns vary from team to team, and from year to year. We have data on NBA lineups from the website cleaningtheglass.com (Cle, 2023) – for each season, for each team, how many possessions each lineup played together – and we want to understand how these patterns vary and what it means for the teams.

The traditional way to analyze lineup data is to put it into a graph, or network: each vertex is a player, and two vertices are connected if the two players were ever on the court together. This approach has been very fruitful for lineup and other types of data in sports; see, for example, Ahmadalinezhad and Makrehchi (2020); Bai and Bai (2022); Brave et al. (2019); Clemente and Martins (2017); Clemente et al. 2016; Fewell et al. (2012); Hambrick 2019; Korte and Lames (2018); Korte et al. (2019); Mora-Cantallops and Sicilia (2019); Wäsche et al. (2017). However, the network approach considers only information about pairs of players, and thus we lose information about the other players in the lineup with them.

Another approach is to include information about which groups of players have been in a lineup together, in the form of a simplicial complex, specifically a collaboration simplicial complex (Moore et al., 2012; Ramanathan et al., 2011). A simplicial complex encodes all of the pairs, trios, etc., who have played together (Section ‘Simplicial complexes’). (See Devlin and Uminsky, 2020 for another approach to analyzing lineups and subsets of lineups.) Similar complexes have been used to analyze interactions in academic collaborations, scientific ideas, neural networks, actors, wireless networks, and social contagion, among others (Ebli et al., 2020; Iacopini et al., 2019; Pal et al., 2017; Patania et al., 2017; Salnikov et al., 2018).

Example 1

Team 1 has two wings, $A$ and $B$ , and a center, $C$ (along with other players, of course). $A$ , $B$ , and $C$ are sometimes all in the lineup together. Then part of the lineup graph would look like Figure 1(a).

Now suppose that Team 2 has wings $A^{'}$ and $B^{'}$ and a center $C^{'}$ . $A^{'}$ and $B^{'}$ are not good shooters, so for spacing reasons the team never plays both of them with $C^{'}$ . But they can play together if $C^{'}$ is not on the court, and $C^{'}$ can play with each of them individually. Then part of Team 2’s lineup graph would look like Figure 1(b). Notice that we can’t tell from the lineup graph whether or not $A^{'}$ , $B^{'}$ , and $C^{'}$ are ever all on the court together, since it tells us only about pairs of players. The graphs for Team 1 and Team 2 are identical.

Figure 1.

Lineup graphs – pairwise information. (a) $A B C$ played together. (b) $A^{'} B^{'}$ played together, $A^{'} C^{'}$ played together, and $B^{'} C^{'}$ played together, but $A^{'} B^{'} C^{'}$ never played all together.

Since five players play together at a time, we should also consider information about which groups (not just pairs, but trios, quartets, and quintets) played together. We can do that with a simplicial complex (Moore et al., 2012; Ramanathan et al., 2011).

Example 2

Consider again the teams from the previous example. We can build the simplicial complex as follows. Again the vertices correspond to players, and there’s an edge between two players if they were ever on the court together. But now we add information about trios: if three players were ever all on the court together, then we fill in the triangle that they form. So part of the simplicial complex for Team 1 looks like Figure 2(a), while Team 2’s looks like Figure 2(b). The hollow triangle in Figure 2(b) shows that the three players, who all played together pairwise, never played together as a trio. We will want to count holes like this. (Since the boundary of the hole consists of one-dimensional objects (line segments), we use $H_{1}$ to denote the group of such holes, and $β_{1}$ to count the number of them. See Section ‘Simplicial complexes’.)

Figure 2.

Lineup simplicial complexes. (a) $A B C$ played together – the triangle is filled in. (b) $A^{'} B^{'}$ played together, $A^{'} C^{'}$ played together, and $B^{'} C^{'}$ played together, but $A^{'} B^{'} C^{'}$ never played all together – the triangle is hollow.

Example 3

We add more information about Team 2’s lineups. Say that there is a lineup including players $B^{'} C^{'} D^{'}$ , another with $A^{'} B^{'} E^{'}$ , and a third with $A^{'} C^{'} F^{'}$ . Then part of the simplicial complex with those lineups would look like Figure 3. Again, observe the hollow triangle $A^{'} B^{'} C^{'}$ , reflecting the fact that $A^{'} B^{'} C^{'}$ never played together. Likewise, since there is no filled-in triangle with vertices $A^{'}$ , $B^{'}$ , and $D^{'}$ , those three players have also not all played together; in fact, since there is no segment joining $A^{'}$ and $D^{'}$ , those two have not played together.

Figure 3.

Lineup simplicial complex for Example 3.

Example 4

We go back to Team 1 and add more information. Say that there is another player, $D$ , and that the trios $A B C$ , $A B D$ , $A C D$ , and $B C D$ all played together, but the quartet $A B C D$ never played together. Then we get a tetrahedron for the simplicial complex (Figure 4). Since the four of them never played together, the tetrahedron is hollow. This gives another kind of hole. Since the boundary of the hole is formed by two-dimensional objects (the triangles), we denote the group of such holes by $H_{2}$ and the number by $β_{2}$ . (If all four had played together, we would fill in the tetrahedron to make it solid.)

Figure 4.

$A B C$ , $A B D$ , $A C D$ , and $B C D$ all played together, but the quartet $A B C D$ never played together, so the tetrahedron is hollow.

As we add more players and lineups to the simplicial complex, we will get objects (and holes) in higher and higher dimensions. Since a lineup has five players, the complete lineup simplicial complex will reside in five dimensions. We can no longer visualize it, but fortunately our intuition from two and three dimensions is still essentially correct. We can use topology, a kind of generalization of geometry, to count the holes of different dimensions in the simplicial complex, and we can perform the actual calculations regardless of dimension.

We also want to add information about the number of possessions played together by each lineup. We can analyze this using persistent homology.

Example 5

We go back to Team 2, and now add information about how many possessions each lineup played together. Say that the lineup including

B^{'} C^{'} D^{'}

played 1000 possessions together, the lineup including

A^{'} C^{'} F^{'}

played 800 possessions together, and the lineup including

A^{'} B^{'} E^{'}

played 500 possessions together. In addition, let’s say that there was a lineup including

E^{'} F^{'} G^{'}

that played 300 possessions, and a lineup including

A^{'} E^{'} F^{'}

that played only 10 possessions together. We summarize this information in Table 1.

Table 1.

Possessions played by different lineups, Team 2.

lineup	possessions
$B^{'} C^{'} D^{'} x x$	1000
$A^{'} C^{'} F^{'} x x$	800
$A^{'} B^{'} E^{'} x x$	500
$E^{'} F^{'} G^{'} x x$	300
$A^{'} E^{'} F^{'} x x$	10

Now, when we build our simplicial complex out of the lineups, we can filter it based on the number of possessions played. If we include only lineups playing 1000 or more possessions, we get Figure 5(a). Similarly, if we include only lineups playing 800 or more possessions, we get Figure 5(b). A cutoff of 500 possessions gives Figure 5(c), a cutoff of 300 possessions gives Figure 5(d), and a cutoff of 10 possessions gives Figure 5(e). (Note that the lengths and angles in the complex don’t matter, just the ways in which the vertices and edges are joined.) We see that a hole appears at possessions threshold 500, and another hole appears at 300 and disappears at 10.

Figure 5.

Lineup simplicial complexes for varying possessions thresholds. (a) Threshold 1000 (b) Threshold 800 (c) Threshold 500 (d) Threshold 300 (e) Threshold 10.

For large data sets, it will be difficult to keep track of all the simplicial complexes this way (and impossible to draw them in higher dimensions), so we summarize the information about the holes, the homology (Section ‘Homology’), in the form of a barcode. Since in this example the only holes have a boundary formed by one-dimensional objects, we only need to record $H_{1}$ in the barcode. See Figure 6(a). Since $β_{1}$ counts the elements of $H_{1}$ , we have that $β_{1} = {\begin{matrix} 0 & if threshold > 500 \\ 1 & if 500 \geq threshold > 300 \\ 2 & if 300 \geq threshold > 10 \\ 1 & if threshold \leq 10. \end{matrix}$

Figure 6.

Homology barcodes. (a) Barcode for Example 5 (b) Barcode for Example 6.

We can also put that information into a chart, in Table 2, where all other entries are understood to be 0:

Example 6

As another example, consider Team 1. If the lineup data are as shown in Table 3, then the

H_{2}

barcode would be as shown in Figure 6(b); the hole appears at possessions threshold 600.

Table 2.

Barcode summary for Example 5.

possessions	$β_{1}$
500	1
300	2
10	1

Table 3.

Possessions played by different lineups, Team 1.

lineup	possessions
$A B C x x$	900
$A B D x x$	800
$A C D x x$	700
$B C D x x$	600

So, to summarize our method of analysis of the lineups data:

We combine all the information about which players and groups of players were on the court at the same time into a simplicial complex. This is high-dimensional and can be very complicated, so:

We compute the homology of the simplicial complex to detect holes, roughly corresponding to groups that don’t play together.

Persistent homology adds information about how many possessions each lineup played together.

Simplicial complexes

We will give the bare minimum of background information necessary to understand the paper. There are many references for simplicial complexes and homology; see, for example, Edelsbrunner and Harer (2010); Fugacci et al. (2016, 2024); Nanda (2024). For a more intuitive and less formal explanation of homology, see Houston-Edwards (2021).

An abstract simplicial complex consists of a set $V$ of vertices and a non-empty set $S$ of subsets (simplices) of $V$ , with the property that if the subset $σ$ is in $S$ , then any subset $τ \subset σ$ is also in $S$ . The dimension of a simplex is the number of vertices it contains minus one.

Every abstract simplicial complex can be realized geometrically as a simplicial complex in $n$ -dimensional Euclidean space $R^{n}$ , for large enough $n$ . The vertices are points in the space, and a simplex $σ \in S$ corresponds to the convex hull of the vertices in $σ$ . Intuitively, we can think of a simplicial complex as a collection of lines (1-simplices), triangles (2-simplices), tetrahedra (3-simplices), and their higher-dimensional analogues, glued together.

The simplicial complex determined by an NBA team’s lineups has a particularly simple form. Since every lineup has exactly five players, the simplicial complex is generated by the 4-simplices, each representing a single lineup of five players. So, for example, the simplicial complex contains a given 3-simplex if and only if those four players played together in at least one five-player lineup.

Homology and persistent homology

Homology

We can put an algebraic structure on a simplicial complex by giving each simplex an orientation. Intuitively, an orientation for a line (1-simplex) corresponds to a choice of direction, an orientation for a triangle (2-simplex) corresponds to a choice of top and bottom, and analogously in higher dimensions. Each simplex has two possible orientations, and we can begin to do algebra by saying that a simplex with a given orientation is equal to minus the same simplex with the opposite orientation; then their algebraic sum would be zero. (An orientation for a point (0-simplex) is just a choice of positive or negative.)

An orientation for a $n$ -simplex $σ$ automatically induces an orientation for the $n$ $(n - 1)$ -simplices in its boundary $\partial (σ)$ . For example, if we take the 2-simplex (solid triangle) $A B C$ , oriented so that we are looking at it from the top in Figure 7(a), then the orientations for its boundary $\partial (A B C)$ (the 1-simplices (lines) $A B$ , $B C$ , and $C A$ ) are as shown; we denote the orientations as $\vec{A B}$ , $\vec{B C}$ , $\vec{C A}$ . We can keep going: for example, $\partial (A B)$ , the boundary of the 1-simplex $A B$ , consists of the points $A$ and $B$ . Since $A B$ is oriented from $A$ to $B$ , we give $A$ the negative orientation ( $- A$ ) and $B$ the positive orientation ( $B$ ).

Figure 7.

Lineup simplicial complexes. (a) A boundary (b) A cycle that is not a boundary.

In this example, we observe that $\partial (\partial (A B C)) = \partial (\vec{A B} +$ $\vec{B C} + \vec{C A}) = (- A + B) + (- B + C) + (- C + A) = 0$ . This is not a coincidence. It is not hard to prove that the boundary of a boundary is always zero algebraically. We call an algebraic sum of simplices a cycle if its boundary is zero; thus every boundary is a cycle. However, not every cycle is a boundary. For example, in Figure 7(b), $\vec{D E} + \vec{E F} + \vec{F D}$ is not the boundary of any region, because the triangle is not filled in (the 2-simplex $D E F$ is not present). But it is a cycle, since $\partial (\vec{D E} + \vec{E F} + \vec{F D}) = (- D + E) + (- E + F) +$ $(- F + D) = 0$ .

Intuitively, the cycles that are not boundaries correspond to ‘holes’ in the simplicial complex. More formally, we take the algebraic set of cycles in each dimension, and divide out the set of boundaries. What remains are the homology groups of the simplicial complex, $H_{1}$ , $H_{2}$ , $\dots$ . We are often interested only in the size, or rank, of these groups; we call these the Betti numbers $β_{1}$ , $β_{2}$ , $\dots$ . For example, in Figure 7(b), $\vec{D E} + \vec{E F} + \vec{F D}$ is a 1-dimensional simplicial complex that is not a boundary, so it contributes a 1 to $β_{1}$ . It is the only one, so in fact $β_{1} = 1$ . In Figure 7(a), there are no cycles that are not boundaries, so $β_{1} = 0$ .

Similarly, a hollow tetrahedron would contribute a 1 to $β_{2}$ . Roughly, we can think of $β_{n}$ as corresponding to the number of holes whose boundary consists of $n$ -dimensional simplices. Since we are counting algebraically, different complexes can correspond to the same geometric hole. For example, in Figure 8, $\vec{A B} + \vec{B C} + \vec{C A}$ and $\vec{D E} + \vec{E F} + \vec{F D}$ are equal in the homology group $H_{1}$ , so they contribute only 1 to $β_{1}$ , not 2. This feature can make the homological results somewhat difficult to interpret geometrically; we will see another example when we look at the NBA lineups data.

Figure 8.

A hollow cylinder with no top or bottom.

Persistent homology

Again, we give only a superficial overview of persistent homology; for details see, for example, Edelsbrunner and Harer (2010); Fugacci et al. (2016, 2024); Nanda (2024). The basic idea is straightforward. We add an appropriately defined “weight” function to the simplices, then filter the simplicial complex to include only simplices at or above a given weight.¹ As we decrease the threshold, the simplicial complex grows, and the homology can change, as in Examples 5 and 6 above. We record the changes in a barcode, which shows when each hole is created and when it disappears, as in Figure 6.

NBA lineups results

Using the data from cleaningtheglass.com, for each season we create simplicial complexes for each team in Python using pandas, and then use the software package Perseus (Nanda, 2023) to compute the persistent homology. We used a minimum of 100 possessions played, to eliminate lineups that were not part of the real rotation and instead played only a few possessions at the end of blow-outs, for example. The results are available online (Wiseman, 2024). As an example, we show the results for the 2022-23 San Antonio Spurs and 2018-19 Minnesota Timberwolves in Figure 9, and the corresponding barcodes in Figure 10. For the Spurs, we see that a one-dimensional hole appears at possessions threshold 105. For the Timberwolves, we see that two one-dimensional holes appear at threshold 197, of which one disappears at 166 and the other at 162, and a two-dimensional hole appears at 158.

We see that the Timberwolves’ lineup data are more homologically complicated than the Spurs’, in two different ways. First, Minnesota has more holes, in more dimensions. Second, Minnesota’s holes persist longer, reflecting a more robust structure. The single hole for the Spurs, in $H_{1}$ , persists only from 105 possessions to the cutoff of 100 possessions, giving a length of only 5. For Minnesota, the elements of $H_{1}$ have lengths of 31 and 35, and the element of $H_{2}$ has length 58.

Figure 9.

Betti numbers. (a) 2022-23 San Antonio Spurs (b) 2018-19 Minnesota Timberwolves.

Figure 10.

Barcodes. (a) 2022-23 San Antonio Spurs (b) 2018-19 Minnesota Timberwolves.

In this setting, the ‘holes’ detected by homology tell us about the set of lineups. For example, if $β_{1}$ is not zero, then there is some collection of players $P_{1}, P_{2}, \dots, P_{n}$ such that $\vec{P_{1} P_{2}} + \vec{P_{2} P_{3}} + \dots + \vec{P_{n - 1} P_{n}} + \vec{P_{n} P_{1}}$ is a cycle but not a boundary. In basketball terms, this implies that $P_{1}$ and $P_{2}$ were in a lineup together, $P_{2}$ and $P_{3}$ were in a lineup together, and so on up to $P_{n - 1}$ and $P_{n}$ , as well as $P_{n}$ and $P_{1}$ , but no three of them were ever in a lineup together.²

A nonzero $β_{2}$ is similar, but harder to visualize. If $β_{2}$ is not zero, then there is a set of players where each member is in at least one playing trio with other members, forming a cycle, but no four of whom were ever in a lineup together. The lineup interpretations for $β_{3}$ and $β_{4}$ are analogous.

The manner in which the homology algorithm works makes it difficult to determine exactly which lineups are creating the holes. This is not a real problem, since we are interested in the overall pattern rather than individual lineups, but we can get the information by analyzing the data by hand if necessary. For example, for the 2022-23 Spurs, we see that the $H_{1}$ entry appears at a possessions cutoff of 105. The lineup that played 105 possessions consisted of Tre Jones, Devin Vassell, Keldon Johnson, Keita Bates-Diop, and Zach Collins. Looking at the lineups that played more than 105 possessions, we find that in addition to Vassell and Bates-Diop playing together, Bates-Diop and Jeremy Sochan played together, as did Vassell and Sochan, but Bates-Diop, Sochan, and Vassell never played all together. This creates a hole, explaining the entry in $H_{1}$ ; see Figure 11.

Figure 11.

Each pair played together, but not the trio, creating a one-dimensional hole.

(Direct examination of the data also shows that Bates-Diop, Vassell, and Jakob Poetl played together pairwise, but not as a trio, so that hole can also explain the $H_{1}$ entry. As we saw in Section Homology, because homology counts holes algebraically, different holes can contribute to the same element in $H_{1}$ . This is one reason that interpreting the results of homology calculations can be challenging.)

The persistent homology tells us how the number of lineup ‘holes’ changes as we decrease the possessions threshold. In some cases, we can gain homology, as the newly added lineups create new cycles. In other cases, we can lose homology, as the existing holes are filled in by simplices created by the new lineups.

We have hundreds of individual barcodes, one for each team for each season going back to 2003-04. To analyze them collectively, we need to condense the data. There are many ways to do this. Here, as an example, we will examine trends over time by taking the maximum, over all thresholds, for each $β_{i}$ for each team, then taking the average over all teams for each season. For example, the maximum for the 2022-23 Spurs for $β_{1}$ is 1; the maximum for the 2018-19 Timberwolves for $β_{1}$ is 2 and for $β_{2}$ is 1.

The complete results are displayed in Figure 12.³ The most interesting cases, $H_{1}$ and $H_{2}$ , are also broken out individually in Figure 13.

Figure 12.

NBA lineups, average maximum Betti numbers, 2003–04 to 2022–23.

Figure 13.

NBA lineups, average maximum $β_{1}$ and $β_{2}$ , 2003–04 to 2022–23. (a) Average maximum $β_{1}$ (b) Average maximum $β_{2}$ .

Interpretation and uses

Missing edges in the network reflect pairs of players who do not play together. Holes in the simplicial complex, as measured by homology, reflect groups of players who do not play together. If there were no positions in basketball (formal or informal), then everyone could play together, and we would not expect to see any lower-dimensional holes. The more restrictions there are on groups playing with each other, the more holes we would expect to see.

An obvious source of these restrictions is shooting ability. Over the last decade, teams have placed enormous importance on spacing. As in Example 1, where Team 2 doesn’t play two poor-shooting wings and the center at the same time, this can lead to certain lineup combinations receiving very few possessions.

Another obvious potential restriction is defensive ability. As shooting has become more important, so has the ability to defend it. For example, a team might be able to play two weak perimeter defenders if a strong defensive center is in the game to protect the rim, but not if a smaller player is in at center. Again, this creates restrictions on playable lineups.

This emphasis on spacing, and the resulting limitations on lineups, could explain the pattern we see in Figures 12 and 13. In both $H_{1}$ and $H_{2}$ , the dimensions with by far the largest values, we see a roughly steady increase until around the 2016–17 season, and then a marked decrease. The increase may reflect the growing importance of spacing, as teams gradually adjusted the kinds of lineups they play. The subsequent decrease may indicate the fact that roster-building was catching up to the lineup trends – teams would no longer employ as many players who couldn’t shoot or couldn’t defend and thus couldn’t be on the floor much. (Some of the change is certainly attributable to the COVID-19 pandemic as well.)

Example 7

For example, Coach Tom Thibodeau has a reputation for playing his starters longer than most other coaches. That could mean using more rigid lineups overall, which we should be able to detect in the data. He has been the head coach for three teams (Chicago Bulls, 2010–2015; Minnesota Timberwolves, 2016–2019; New York Knicks, 2020–present). For each of these teams, we compare the number (maximum Betti number over all possession thresholds) and persistence (length of the longest barcode) of holes in his first year with the team versus the average of the two previous years. The results are in Table 4. We see that in all three cases, both the number and the persistence of the holes increased in Thibodeau’s first year, reflecting the change in coaching styles.

Table 4.

Lineups persistent homology, before and after Thibodeau takes over.

season(s)	team	$β_{1}$	$β_{2}$	persistence
2008–10	CHI	0	0.5	4.5
2010–11	CHI	1	2	132
2014–16	MIN	0.5	1	73
2016–17	MIN	1	2	107
2018–20	NYK	0.5	0	25.5
2020–21	NYK	2	0	70

Example 8

In the playoffs, when rotations are tightened and player weaknesses are more likely to be exploited by the other team, we would expect to see fewer, more rigid lineups and more holes. This is exactly what we do see in the data for the 2022-23 season. We used playoff lineup data from the website pbpstats.com (PBP, 2024), with a minimum of 10 possessions played, instead of the 100 that we used for the regular season, since there are fewer games in the postseason. Table 5 shows the average, over all playoff teams, of their maximum Betti numbers

β_{1}

and

β_{2}

for the regular season and the playoffs; both numbers increased dramatically in the playoffs. This makes sense, since by the end of the season teams are likely to have figured out what works and what doesn’t and thus experiment less with mixing and matching in their lineups. Also, as in the example with Coach Thibodeau, in the playoffs teams are likely to give more minutes to their best players.

Table 5.

Betti numbers for 2022-23 playoff teams in the regular season and playoffs.

	$β_{1}$	$β_{2}$
regular season	0.125	0.375
playoffs	0.375	0.6875
difference	0.25	0.3125

Example 9

In Figure 14, we plot average player age versus three-point attempts per game for teams from the 2022–23 season. We see that teams with nontrivial persistent homology (meaning those with maximum $β_{1}$ or $β_{2}$ greater than 0) are clustered in the lower left quadrant; they were all relatively young and they all took relatively few three-pointers per game, compared to the rest of the league. Both associations make sense. Young players might not have the experience to play multiple positions or take on multiple roles, so the team would have fewer lineup options. A team that doesn’t take many three-pointers might have fewer shooters, which would lead to less flexibility in constructing lineups in the modern NBA where spacing is important. However, not every team in the lower left quadrant has nontrivial homology, reflecting different team approaches to roster construction, player rotations, and other factors.

Figure 14.

2022–23 average age, three-point attempts per game, and homology, by team.

Example 10

Teams from the 2022-23 season with nontrivial persistent homology (again, meaning those with maximum $β_{1}$ or $β_{2}$ greater than 0) tended to have a better, or at least no worse, winning percentage the following season. We see this in Figure 15, plotting teams’ winning percentages for 2022-23 versus those of 2023-24; almost all of the teams with nontrivial persistent homology show up near or above the diagonal, meaning that their winning percentage either decreased only slightly or increased. (The outlier is the Memphis Grizzlies, who lost their star player to suspension and injury.) One explanation might be that the nontrivial homology, the holes, reflect a rigidity in lineups due to players’ having limited skills and thus limited possible roles. With more experience, in the next season they could be deployed more flexibly in lineups and contribute more to winning. Furthermore, we do not see the same relationship with winning percentage change if we look at average age and three-point attempts instead of homology. More research is called for to better understand the connection.

Figure 15.

Team winning percentages, 2022–23 vs. 2023–24, and homology.

There are many other ways to use the data. On the level of a single team, we could track the evolution of the simplicial complex throughout the season. As the team experiments initially and then begins to abandon lineups that don’t perform well, we would expect to see more holes form. When a player is injured, new lineups would be used, eliminating holes.

We can also compare teams, either different teams in the same season or the same team from year to year. Different substitution patterns (for example, resting most of the starters at once versus staggering their rest) would show up in the simplicial complex. Some teams might replace an injured starter by moving a top reserve into the starting lineup, while others might prefer to start a little-used player in order to keep their second unit intact; this would lead to different patterns in homology. Similarly, we could detect different approaches to load management and resting players.

The techniques could also be applied at the level of individual games. We would expect to see more different lineups in a blowout than in a close game, for example. Games with more fouls called would also be expected to involve more kinds of lineups. Some teams might use different types of lineups in home games versus away games. By including data on when in the games lineups appear, and not just the total possessions played, we could detect usage patterns at different times in the game, and relate that to the game situation at the time.

There are many other ways to generalize this approach. We could include information on wins and losses, or plus/minus, to analyze the relationship with lineup usage. Instead of looking at each team’s lineup separately, we could analyze the ten players on the floor at a time, to detect reactions to the other team’s lineup changes. (For example, a key advantage of having a good shooter in the lineup is that it forces the other team to defend him – how is this reflected in the lineups that are employed against that player?) Or, instead of using individual players as the vertices in the simplicial complex, we could use clusters of similar players – similar heights and weights, or playing the same position, or playing similar roles (as determined by clustering or other methods Muniz and Flamand, 2022). And of course this technique can be applied to other team sports, many of which have very different player usage patterns.

Conclusion

We describe a technique to analyze NBA lineup usage by considering the persistent homology of a simplicial complex, based on information on which groups of players are in a lineup together, rather than pairs of players, as in traditional network analysis. Groups of players who aren’t on the floor together show up as holes in the simplicial complex and can be detected by homology. As is frequently the case in topological data analysis, the interpretation of the homology data can be somewhat challenging. Nonetheless, there are many possible applications in which the technique can offer an improvement over network analysis.

Footnotes

Acknowledgements

I am grateful to Benjamin Markovits,David Richeson,and Thomas Wiseman for their helpful comments and ideas for this paper,and to the anonymous referees for their thoughtful feedback on improving its content and clarity.

Funding

The authors received no financial support for the research,authorship,and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

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