Network analysis methods can be used for sports analytics applications like team and lineup ranking. SportsNetRank ranks teams based on their win-loss network using PageRank centrality. LinNet evaluates lineups based on their matchup network using network embeddings. It learns latent representations of lineups using node2vec and predicts outcomes of new lineup matchups. LinNet outperforms adjusted plus-minus and PageRank in predicting unseen lineup matchups, with probabilities well calibrated and Brier scores around 0.19. Substitution networks also show potential for explaining team performance. Further work could optimize network embeddings and model lineup ability curves.

1. Winning in Sports with Networks
Konstantinos Pelechrinis
University of Pittsburgh
@kpelechrinis

4. Data-Driven Coaches?

5. Data-Driven Front Offices

6. Why now?
• Data analysis & use of statistics is not new in sports!!
• Now we have the technology to collect many more detailed information
about the game
• Detailed/Advanced box score
• Play-by-play data
• Player tracking

7. So…
• If you like data and sports…
• ...and you are too short to play basketball professionally
• You work on sports analytics 

8. In this talk
• Focus on network methods for sports applications
• Team ranking
• Lineup rating
• Player characterization

9. Team sports are won by…teams
• Single player evaluation metrics might not capture the interactions between
players when they appear in the same lineup

10. SportsNetRank

11. Team Rankings
• Team rankings are permanent feature in media content
• Power rankings for NFL are updated after every week’s games
• Typically decided by pundits
• However they are also important in the operations of leagues
• Playoff seeding
• NCAA tournament seeding and bowl games selection

12. Network-based Ranking
• Networks represent connection patterns between objects
• Who-wins-whom networks can utilize vertex ranking metrics to rank teams
Pelechrinis, Konstantinos, Evangelos Papalexakis, and Christos Faloutsos.
"Sportsnetrank: Network-based sports team ranking." ACM SIGKDD
Workshop on Large Scale Sports Analytics. 2016.

13. Network-based Ranking
• PageRank:

14. Network-based Ranking
0.5
0.6
0.7
0.8
1 2 3 4
Season Period
Accuracy
Method
Base
Net

15. Network-based Ranking
• Intransitivity is realized through cycles in the network
• An acyclic network can provide a total ranking
• Compute correlation between minimum arc set and performance differential
between SportsNetRank and baseline
-0.19 (p-value 0.015) -0.21 (p-value 0.039)

16. LinNet

17. Lineup Evaluation
• Summing up individual players’ performance (e.g., PM) is not appropriate
• Why not rely on networks?

18. LinNet: Network-based Lineup Evaluation
wi
wj
wr
node2vec

19. Network Embedding for LinNet
• Consider a network G=(V, E)
• node2vec learns a low dimensional vector representation of the nodes, f: V
ℝd, by optimizing a neighborhood preserving objective
• Maximum likelihood estimation problem
• With NR(u) being the neighborhood identified for node u with random walk strategy R,
f is learned through the following maximization problem
node2vec: Scalable Feature Learning for Networks.
A. Grover, J. Leskovec. ACM SIGKDD 2016.

20. Two standard assumptions
• Conditional independence: Likelihood of observing node ui ∈ NR(u) is
independent of observing any other node in NR(u)
• Feature space symmetry: A source node and a neighborhood node have
symmetric effect over each other in feature space

21. Two standard assumptions
• The objective function simplifies to:

22. Sampling strategies
• For every node u we sample several neighborhoods of size k
• This provides the probability of observing node v in NS(u)
• With ci being the ith node on a walk started at node c0=u:
• πvx is the unnormalized transition probability between nodes v and x

23. Sampling strategies
• In order to allow for a large spectrum of sampling strategies πvx is expressed
as the product of a bias term α and the edge weights w
• 2nd order random walk
• The search bias term α is parametrized by two variables p and q and is a
function of the node t via which the random walker reached current node v
Shortest path distance
between t and x

24. Sampling strategies: Intuition
• p and q control how fast the random walker explores/leaves the vicinity of
the starting node
• High p (>max(q,1))  less likely to sample a previously sampled node
• q > 1  random walker is biased towards nodes close to
node t
node2vec: Scalable Feature Learning for Networks.
A. Grover, J. Leskovec. ACM SIGKDD 2016.
LinNet: p = 0.5 and q = 3
d = 128

25. LinNet: Network-based Lineup Evaluation
wi
wj
wr
node2vec

26. Latent Bradley-Terry Model
• The learned features for the nodes – i.e., lineups – can be used as the input
to a Bradley-Terry logistic regression model
• The difference in the abilities of two lineups can be modeled after lineup-
specific explanatory variables zi
Probability lineup λi
outperforms λj
Conditioned on lineup’s
abilities πi and πj
We use the learned lineup feature from
the network embedding for z

27. Previously unseen lineups
• What if a lineup λi has not played before?
• Not part of the matchup network
• Define the similarity σ(λi,λj) between two lineups λi and λj (of the same team)
as the number of common players between them
• Can we use this similarity value and the latent representation for λj to obtain an
educated approximation of the latent space for λi ?
Yes!

28. Previously unseen lineups
Weak but significant (linear) correlation

29. Datasets
• Five year of NBA lineup matchup data
• 2007-08 to 2011-12
• basketballvalue.com
• ({Players of lineup λi},{Players of lineup λj}, Total point differential, Total time of
matchup)

30. Evaluation
• Out-of-sample accuracy: using the first 70% of the lineup matchups how
does LinNet perform over the rest 30%
• Probability calibration: how well calibrated the probabilities output are
• Brier score
• Validation curves
• Four baselines for comparison

31. Plus-Minus (+/-)
• Boxplot  limited view of players contributions
Points scored: s1
Points allowed: a1
Points scored: s2
Points allowed: a2
Points scored: sn
Points allowed: an
.
.
.
𝑖=1
𝑛
(𝑠𝑖 − 𝑎𝑖)

32. Plus-Minus (+/-)
.
.
.

33. Adjusted Plus-Minus
• Controls for the presence of other players on the court
• Both offense and defense
• Each stint is a data point
• DV: PM/possession
• IVs: Dummy variables for all players
• 1 for home team players in the stint, -1 for visiting team players in the stint and 0 for the rest

34. Adjusted Plus-Minus
• Pass all the stints through a linear regression
• The coefficient for each player αi is the adjusted plus-minus of the player
• Using the adjusted plus-minus of each player in a lineup we can consider the
ability of a lineup to be the average adjusted plus-minus
𝑦 = 𝑎1 𝑥1 + 𝑎1 𝑥1+…+𝑎 𝑟 𝑥 𝑟 + 𝜀

35. Page Rank
• We can use network centrality metrics to rank nodes – i.e., lineups - directly
from the matchup network
• Our previous study has shown that this is a better predictor than win-loss%
• We calculate the Page Rank of each lineup based on the matchup network
and use this as the performance predictor for future lineup matchups
Pelechrinis, Konstantinos, Evangelos Papalexakis, and Christos Faloutsos.
"Sportsnetrank: Network-based sports team ranking." ACM SIGKDD
Workshop on Large Scale Sports Analytics. 2016.

36. 0.00
0.25
0.50
0.75
1.00
2007-08 2008-09 2009-10 2010-11 2011-12
Season
Accuracy
Method
APM
LinNet
PageRank

37. 0.00
0.05
0.10
0.15
0.20
0.25
2007-08 2008-09 2009-10 2010-11 2011-12
NBA Season
BrierScore
Method
APM
LinNet
PageRank
Predicted probability
Actual binary outcome

38. Lineup-based Plus-Minus
• We also compared the performance of lineups plus-minus with LinNet
• Raw lineup +/-
• Adjusted lineup +/-
• Using a regression similar to the players adjusted +/
Raw +/- Adjusted +/- LinNet
Mean Accuracy 54.5% 59.1% 67%
Mean Brier score 0.24 0.23 0.19

39. LinNet validation curves
0.00
0.25
0.50
0.75
1.00
0.00 0.25 0.50 0.75 1.00
Predicted Lineup Matchup Probability
FractionofCorrectPredictions
Count
2000
4000
6000
Linear fit R2 = 96%
Intercept: 0.1, [0.02, 0.14]
Slope: 0.85, [0.79, 0.98]

40. Season Performance – LinNet Score
• How well can lineup ratings obtained from LinNet explain the win-loss
record for a team?
• With LT being all the lineups of team T and γλ being the time that lineup λ
was on the field, we can define the LinNet rating of team T as:

41. Season Performance – LinNet Score
• Linear fit has significant slope
(p<0.001)
• Correlation coefficient: 0.53
• R2 = 27%
• Teams do not use their best
lineups (?)
• Time on the court might be
important for a lineup’s
performance

42. Embedding dimensionality
0.00
0.25
0.50
0.75
1.00
0 100 200 300 400 500
Embedding Dimensionality
Accuracy

43. Discussion
• How crucial is the time that a lineup stays on the court?
• Is there a lineup ability curve similar to Oliver’s player skill curve?
• Is there a better way to predict outcomes for previously unseen lineups?
• Clearly there is not strong linearity in the latent space
• Can a task-specific network embedding perform better?
• How can we further optimize node2vec (e.g., choice of p and q)?

44. Substitution Networks
(work in progress)

45. Substitute Networks
Who-subs-whom
K. Pelechrinis, “Can NBA Substitution Networks Explain
a Team’s Performance?”, the Athlytics Blog (2017)

46. Substitute Networks
• Simple network metrics of the sub
network can explain 33% of the variation
of winning percentage
• This is not small!
• Sub networks say nothing about the quality of
the subs
• Network embedding can also reveal latent
relationships between starters and bench

47. Conclusions & Discussion
• Networks offer a strong tool for sports-related problems
• Many interesting problems and extensions of the work presented here

48. Conclusions & Discussion
• How crucial is the time that a lineup stays on the court?
• Is there a lineup ability curve similar to Oliver’s player skill curve?
• Is there a better way to predict outcomes for previously unseen lineups?
• Clearly there is not strong linearity in the latent space
• Can a task-specific network embedding perform better?
• How can we further optimize node2vec (e.g., choice of p and q)?