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The authors have declared that no competing interests exist.

Conceived and designed the experiments: JHF DA. Performed the experiments: JHF JI AP. Analyzed the data: JHF JI AP DA JSW. Contributed reagents/materials/analysis tools: JHF DA. Wrote the paper: JHF DA JI JSW.

We asked how team dynamics can be captured in relation to function by considering games in the first round of the NBA 2010 play-offs as networks. Defining players as nodes and ball movements as links, we analyzed the network properties of degree centrality, clustering, entropy and flow centrality across teams and positions, to characterize the game from a network perspective and to determine whether we can assess differences in team offensive strategy by their network properties. The compiled network structure across teams reflected a fundamental attribute of basketball strategy. They primarily showed a centralized ball distribution pattern with the point guard in a leadership role. However, individual play-off teams showed variation in their relative involvement of other players/positions in ball distribution, reflected quantitatively by differences in clustering and degree centrality. We also characterized two potential alternate offensive strategies by associated variation in network structure: (1) whether teams consistently moved the ball towards their shooting specialists, measured as “uphill/downhill” flux, and (2) whether they distributed the ball in a way that reduced predictability, measured as team entropy. These network metrics quantified different aspects of team strategy, with no single metric wholly predictive of success. However, in the context of the 2010 play-offs, the values of clustering (connectedness across players) and network entropy (unpredictability of ball movement) had the most consistent association with team advancement. Our analyses demonstrate the utility of network approaches in quantifying team strategy and show that testable hypotheses can be evaluated using this approach. These analyses also highlight the richness of basketball networks as a dataset for exploring the relationships between network structure and dynamics with team organization and effectiveness.

Capturing the interactions among individuals within a group is a central goal of network analyses. Useful depictions of network structure should provide information about the networks purpose and functionality. But how do network attributes relate to functional outcomes at the group and/or individual levels? A useful context to ask this question is within small team networks. Teams occur everywhere across the broad array of biological societies, from cooperatively hunting carnivores to social insects retrieving prey

How can we capture the relevance of these interactions to team function? Because teams are dynamic systems, it makes sense to use network analyses to approach this problem. The game of basketball is based on a series of interactions, involving a tension between specialization and flexibility; players must work together to move the ball into the basket while anticipating and responding to the opposing team. Thus, plays that begin as set strategies evolve quickly into dynamic interactions

The dynamic between within-group cooperation and conflict, and group versus individual success, is an inherent feature of both human and biological social systems. This tension, exemplified in the distribution of shooting opportunities in a game across players, or by salary dispersion inequities in a team or organization, is a fundamental issue across cooperative systems

To evaluate basketball teams as networks, we examined the offensive ball sequences by National Basketball Association (NBA) teams during the first round of the 2010 playoffs. We graphed player positions and inbound/outcomes as nodes, and ball movement among nodes (including shots to the basket) as edges. From the iterated offensive 24 second clocks, we recorded sequences of ball movement of each of the 16 play-off teams across two games. We used the compiled data to first ask whether we can capture the game of basketball through a transition network representing the mean flow of the ball through these sequences of play (a stochastic matrix), and secondly whether individual teams have specific network signatures. We then examined how different network metrics may be associated with variation in actual play strategy. We asked whether teams vary strategically in centrality of ball distribution, such that some teams rely more heavily on a key player, such as the point guard, to make decisions on ball movement. We used degree centrality to compare teams using this strategy with those in which the ball is distributed more evenly. We similarly used clustering analyses to examine relative connectedness among players within teams and to ask whether teams differentially engaged players across multiple positions. We also asked whether ball movement rate, measured as path length and path flow rate, could capture the perceived dichotomy of teams using dominant large players, usually centers, versus small ball teams that move the ball quickly across multiple players

We were interested in whether network metrics can usefully quantify team decisions about how to most effectively coordinate players. We examined two network metrics that we hypothesized might capture different offensive strategies. One is to move the ball in a way that is unpredictable and thus less defensible. To measure network unpredictability we calculated team entropy, applying Shannons entropy to the transition networks as a proxy for the unpredictability of individual passing behavior among team players. Another, not mutually exclusive, strategy is to capitalize on individual expertise by moving the ball towards players with high probability of shooting success. In a sense, this strategy reflects a coordinated division of labor between ball distributors early in the play, transitioning to shooting specialists. We looked for evidence of this strategy using a metric of uphill/downhill flux, which estimates the average change in potential shooting percentage as the ball moves between players in relation to their differential percent shooting success. Uphill/downhill and team entropy both recognize the need for coordination within a team, but they emphasize different aspects of network dynamics; one capitalizes on individual specialization while the other emphasizes team cohesion.

We recorded and analyzed transition networks for the 16 teams in televised games of the 2010 NBA first round play-offs. The sequential ball movement for each teams offensive plays was recorded across two games for each pair; games were picked haphazardly a priori, not based on outcome (analyzed games and outcomes in

Matchup | Games | Game Winner | Series Winner |

Bobcats vs. Magic | Game 1 | Magic | Magic |

Bobcats vs. Magic | Game 2 | Magic | Magic |

Cavaliers vs. Bulls | Game 2 | Cavaliers | Cavaliers |

Cavaliers vs. Bulls | Game 4 | Cavaliers | Cavaliers |

Hawks vs. Bucks | Game 3 | Bucks | Hawks |

Hawks vs. Bucks | Game 4 | Bucks | Hawks |

Celtics vs. Heat | Game 1 | Celtics | Celtics |

Celtics vs. Heat | Game 3 | Celtics | Celtics |

Lakers vs. Thunder | Game 1 | Lakers | Lakers |

Lakers vs. Thunder | Game 2 | Lakers | Lakers |

Jazz vs. Nuggets | Game 1 | Nuggets | Jazz |

Jazz vs. Nuggets | Game 4 | Jazz | Jazz |

Mavericks vs. Spurs | Game 5 | Mavericks | Spurs |

Mavericks vs. Spurs | Game 6 | Spurs | Spurs |

Suns vs. Blazers | Game 1 | Blazers | Suns |

Suns vs. Blazers | Game 6 | Suns | Suns |

For initial analyses, all possible start-of-play (inbounds, rebounds and steals) and outcomes (successful/failed two point or three point shots, fouls, shooting fouls with different success outcomes, steals and turnovers) were recorded as nodes. Data per offensive play generated a sequential pathway

Although we chose games haphazardly, the differential in total points in analyzed games generally reflected outcomes for the play-off round (

We generated weighted graphs from the cumulative transition probabilities. When all data were analyzed, almost all nodes became connected, making it difficult to differentiate across graphs. Therefore, we generated a series of weighted graphs at increasing cut-off weights from the 30th to 70th percentiles (with the 30th percentile graphs highlighting only the most frequently seen transitions). This allowed us to analyze changes in network structure as we move from the most likely links between players to those that were least frequent. We used the entire matrix of transitions for each team to perform structural network analyses

Path length and path flow rate compared the number of passes and the speed of ball movement involved in team play. Path length simply included the number of passes between players per play, ignoring inbound and outcome nodes. Paths included all between-player edges, such that a given player could be involved twice or more across the path. Path flow rate was calculated as the number of edges per unit time from inbound to shot clock time at the end of the play. To calculate degree centrality we used the weighted graphs from iterated offensive plays across the two games. However, we aggregated outcome data into two categories of shoot and other, to reduce weighting bias from multiple outcome nodes. Degree was first calculated per position as the weighted sum of total out-edges per player. The relative distributions of player degrees were then calculated across the graph, such that a homogeneous graph (connectivity distributed most equally across all players) has zero degree centrality. For a weighted graph

To calculate team entropy, we first determined individual player entropy. For this metric we excluded inbound passes because of the strong weight of the inbound edge. We included outcome, because the possibility of shooting the ball represents a decision point contributing to uncertainty of ball movement. As with centrality, outcomes were collapsed into two node categories of shooting or not shooting. We used Shannons entropy

We then combined player entropies to determine entropy of the whole team. There are multiple ways to calculate network entropy. One possibility is to use a simple averaging of player entropies. A second is Markov chain entropy, which incorporates the conditional probability of any given player moving the ball to any other player, conditioned on the probability that the given player has the ball. However, from the opposing teams perspective, the real uncertainty of team play is the multiplicity of options across all ball movements rather than just across players. We thus calculated a whole-network or Team Entropy from the transition matrix describing ball movement probabilities across the five players and the two outcome options.

We used individual flow centrality to characterize player/position importance within the ball distribution network

To capture a teams ability to move the ball towards their better shooters, we developed a metric we call uphill/downhill flux, defined as the average change in potential shooting percentage per pass. A team that has a high positive uphill/downhill flux moves the ball consistently to their better shooters; a team that with a negative value moves the ball on average to the weaker shooters. The latter can happen if the ball distributor (e.g. the Point Guard) is also the best shooter on the team. Letting

Finally, we wanted to compare teams in terms of relative player involvement, such that we can differentiate those teams for which most players are interconnected from those that rely consistently on a defined subset for offensive plays. One way to do so is to look for the occurrence of triangles, or connected 3-node subgraphs within the network. Teams with higher connectedness will contain more cases in which sets of 3 players have a link to each other; the maximum number of these triangles in a group of 5 players is 10. The clustering coefficient measures the number of triangles in a network as a percentage of all possible triangles. However, a single evaluation of this metric is again problematic. If we use all ball movement data, all nodes become connected to all other nodes, and the clustering coefficient is uniformly high. Additionally, it is important to remember that the triangles in these networks are association links and not necessarily sequences of plays. Hence we decided that the most meaningful measure to characterize the association structure of the ball movements was to calculate the clustering coefficients for undirected unweighted graphs across the different cutoffs of the cumulative weight, beginning with the 30 percentile when triangles first appear. This allowed us to compare teams with consistently high clustering to those that showed triangles only when less frequent links were included.

The first question posed by this study was how well a network approach can capture the game of basketball from a team-level perspective. We constructed transition networks (i.e. stochastic matrices) as first-order characterization of team play style for each team individually and for the pooled set of all observed transitions across all teams. Because even a single game generates a rich dataset, we imposed thresholds to clarify the dominant transitions, highlighting from most to least frequent the minimal set of transitions representing a particular percentile of all ball movements. At the 60th percentile, players in all but one network were connected to at least one other player (the San Antonio Spurs Center was disconnected) and all teams had an edge to at least one outcome, generally success. This matched the expectation that these are elite and cohesive teams and gave us a starting point for comparative analyses (weighted graphs for all teams across the 30th to 70th percentile thresholds shown in Supplemental

To look at the NBA as a whole, we combined the transition data across all teams in a compiled network (

Edge width is proportional to probability of transition between nodes. Red edges represent transition probabilities summing to the 60^{th} percentile.

The importance of the Point Guard in distributing the ball identifies this as the primary leadership position in the team network. If we define leadership as the relative importance of any player or position in the network, we can capture this quantitatively using individual flow centrality, or the proportion of paths (offensive plays) involving a particular node

Dark bars represent flow centrality calculated across all player possessions in a sequence, and light bars represent flow centrality calculated across the last 3 player possessions in successful sequences.

How do individual teams vary around this centralized model? The star pattern was most exemplified by the Bulls (

Red edges represent transition probabilities summing to the 60^{th} percentile. Player nodes are sorted by decreasing degree clockwise from the left.

Deviations from the Point-Guard centered star pattern confirmed known team playing styles (

Networks are ordered according to the average clustering coefficient across all cutoffs.

The network concept of triangles as a fully connected subgroups translates well to the Lakers highly discussed triangle offense. Jackson and Winter

An important question is whether differences in the weighted team graphs can be captured more quantitatively by network metrics. As discussed above, a primary visual distinction in our weighted graphs was between teams using a central player to distribute the ball, and those moving the ball across multiple players. Our calculated degree centralities in general matched our visual networks (

Degree Centrality | Team Entropy | Uphill/Downhill | ||||

1 | Lakers* | 0.084 | Lakers* | 3.234 | Mavericks | 0.093 |

2 | Spurs* | 0.087 | Celtics* | 3.229 | Jazz* | 0.044 |

3 | Heat | 0.089 | Bobcats | 3.224 | Nuggets | 0.025 |

4 | Bobcats | 0.093 | Heat | 3.194 | Lakers* | 0.016 |

5 | Celtics* | 0.117 | Nuggets | 3.189 | Bucks | 0.009 |

6 | Blazers | 0.119 | Hawks* | 3.180 | Blazers | 0.007 |

7 | Mavericks | 0.127 | Magic* | 3.178 | Bobcats | 0.005 |

8 | Bucks | 0.135 | Spurs* | 3.171 | Celtics* | 0.001 |

9 | Thunder | 0.148 | Suns* | 3.132 | Cavaliers* | 0.001 |

10 | Suns* | 0.154 | Thunder | 3.119 | Bulls | 0.000 |

11 | Cavaliers* | 0.158 | Blazers | 3.117 | Magic* | −0.001 |

12 | Nuggets | 0.162 | Cavaliers* | 3.112 | Suns* | −0.001 |

13 | Magic* | 0.171 | Bucks | 3.079 | Spurs* | −0.003 |

14 | Hawks* | 0.176 | Bulls | 3.041 | Hawks* | −0.006 |

15 | Jazz* | 0.211 | Mavericks | 2.949 | Heat | −0.014 |

16 | Bulls | 0.219 | Jazz* | 2.934 | Thunder | −0.048 |

Like degree centrality, entropy should be strongly influenced by the extent to which multiple players distribute the ball. Degree centrality and team entropy were negatively correlated (Pearson product moment correlation = −0.6; p<0.003; n = 16), but they captured somewhat different aspects of ball distribution, because team entropy takes into account probabilities outside the network topology. Variation in team entropy was more closely connected to individual team success/failure; winners in 6 of the 8 first round match-ups had higher team entropy, and when entropies were ranked from highest to lowest, 5 of the 8 highest entropies were for winning teams. The play-offs only provide 8 match-ups, too small a sample size to make a statistically meaningful claim (and it would be a simplistic game that allowed a predictive single metric). However, our analyses do suggest that these combined network metrics have value in: (1) capturing variation in team offense, and (2) supporting the hypothesis that complex and unpredictable ball distribution pattern is an important component of team strategy. Indeed, the 2010 Lakers and Celtics teams were arguably built around this principle. The highest entropies overall were achieved by the Lakers and Celtics, and the Lakers simultaneously had the lowest degree centrality. These assertions would be tested by the subsequent play-off seasons, one in which a team known for its dominant forward was successful (2011 Dallas Mavericks) and the next in which the winning team was built around the multi-player model (2012 Miami Heat).

The Dallas Mavericks, who lost in the first round in 2010 but won the title in 2011, are an important counter-point. Their strategy was clear; move the ball consistently to their best shooter. To capture this quantitatively, we developed a new metric that uses flow flux to compare individual player flow centrality with calculated shooting percentage for each player across the two games. Uphill/downhill flux measures the degree to which teams move the ball towards versus away from players relative to their differential shooting success (

Data collected across two games for the (a) Mavericks (highest uphill/downhill), (b) Thunder (lowest uphill/downhill), and (c) Lakers.

Our final team-level metrics were path length and flow rate (speed of ball movement through the path;

Path Length | Flow Rate | |||

Mean | Variance | Mean | Variance | |

Lakers* | 5.81 | 3.67 | 0.60 | 0.28 |

Blazers | 5.52 | 3.53 | 0.53 | 0.22 |

Heat | 5.28 | 3.83 | 0.72 | 0.66 |

Mavericks | 5.24 | 2.89 | 0.50 | 0.16 |

Bobcats | 5.15 | 2.09 | 0.58 | 0.37 |

Spurs* | 5.14 | 1.87 | 0.46 | 0.17 |

Bucks | 4.96 | 1.94 | 0.55 | 0.34 |

Celtics* | 4.93 | 2.75 | 0.68 | 0.52 |

Thunder | 4.88 | 3.15 | 0.65 | 0.35 |

Nuggets | 4.77 | 1.81 | 0.57 | 0.34 |

Cavaliers* | 4.72 | 1.74 | 0.59 | 0.38 |

Jazz* | 4.70 | 1.55 | 0.52 | 0.24 |

Hawks* | 4.69 | 2.22 | 0.71 | 0.74 |

Suns* | 4.68 | 1.88 | 0.53 | 0.22 |

Magic* | 4.65 | 1.91 | 0.55 | 0.28 |

Bulls | 4.48 | 1.62 | 0.69 | 0.53 |

A question in evaluating any organizational network is the relative value of its individual members

Team | Position | ||||

PG | SG | SF | PF | CN | |

Bobcats | 0.94 | 0.87 | 1.17 | 0.92 | 1.42 |

Bucks | 0.94 | 0.65 | 1.25 | 0.87 | 1.54 |

Bulls | 0.72 | 0.55 | 0.95 | 0.78 | 1.36 |

Cavaliers | 0.76 | 1.24 | 0.81 | 1.51 | 0.87 |

Celtics | 1.01 | 0.88 | 1.44 | 1.43 | 0.96 |

Hawks | 0.95 | 1.00 | 0.54 | 0.76 | 0.82 |

Heat | 0.54 | 1.63 | 0.97 | 0.78 | 0.48 |

Magic | 1.07 | 0.55 | 0.94 | 0.91 | 1.70 |

Blazers | 0.99 | 0.77 | 1.24 | 0.77 | 1.86 |

Jazz | 0.95 | 1.13 | 0.86 | 1.09 | 0.80 |

Lakers | 0.67 | 0.85 | 0.63 | 1.31 | 1.94 |

Mavericks | 1.04 | 0.87 | 1.13 | 1.51 | 0.60 |

Nuggets | 0.99 | 1.06 | 1.06 | 0.62 | 1.33 |

Spurs | 1.04 | 1.00 | 1.83 | 0.58 | 1.70 |

Suns | 0.96 | 0.76 | 1.20 | 1.29 | 0.34 |

Thunder | 0.89 | 0.50 | 0.83 | 0.46 | 1.52 |

We found an interesting positional bias in the data, with the Center often having the highest success/failure ratio. In contrast, Point Guards tended to have success/failure ratios at or below 1.0. Although the ratio measure should statistically control for frequency effects, we suggest this metric might be biased mechanistically by relative player involvement. The low flow centrality of the most highly utilized position reflects the argument that high frequency player contributions become negatively affected by exposure. The nonlinear relationship between player involvement and success in our metrics may thus illustrate the price of anarchy

We have presented a network structure analysis of basketball teams in the context of team coordination and strategy. As a starting point, we applied network-level metrics to quantitatively measure fundamental components of team offensive strategy, moving currently available individual player metrics (examples at NBA.com). The study involved more than a thousand ball movements and typically more than one hundred sequences or paths for each team. This dataset allowed us to capture the game of basketball as a network. Because our team comparisons were limited to the pairs in the first round of the play-offs, correlations between game outcome and specific aspects of network structure could not definitively test the specific hypotheses suggested. Answering the question of how network dynamics contribute to successful team strategy will be more complex than a single network variable can capture. We also expect intransitivity across games and opponents, such that the success of emphasizing any given strategy is dependent on the behavior of the opposing team. However our data do suggest that certain metric combinations, particularly entropy, centrality, and clustering, are relevant components of team strategy.

One of the advantages of this beautiful game is the wealth of available data. We encourage the expansion of both the network toolbox and the datasets analyzed. Analyses across a season will help determine whether network structures for a given team are stable or whether they respond flexibly to different defense strategies. Dissecting network shifts within games (e.g. the final quarter or as point differentials change) could help explore game dynamics. Analyses across multiple seasons could track the development of team cohesion. It would also be extremely useful to connect network with spatial and temporal models; this may not be practical with current data acquisition methods, but recent publications

Beyond basketball, this approach may act as a template for evaluating other small team collaborations. Although the specific network metrics will vary across the disparate contexts in which teams occur, the general approach of analyzing network interactions and function is robust

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We thank Alex Gutierrez and Mark Goldfarb for their help in data collection and analysis, and Jon Harrison and an anonymous reviewer for comments on the manuscript.