Graph theory
The Geometry of Teams
Turn eleven players into a weighted graph — nodes at average pass origins, edges counting completed passes — and a team’s identity becomes computable. Across eight iconic matches, the two lowest-tempo networks (1.98 and 2.05 passes/min) both won, yet the winner carried the higher algebraic connectivity in six of the eight, including all four finals. Pick a match, hover a player, read the shape.
The network explorer
Each network is built from kickoff to that team’s own first substitution, so the eleven nodes are the same people throughout. Node size scales with touches (passes made + received); edge width with completed passes between the pair (pairs with at least 3 passes drawn). Hover or tab to a player to isolate their connections; hover an edge for the pair count; hover a metric for its definition and rank across all sixteen networks.
Pass network
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Full player centrality table
1 Windows differ by team because they end at that team’s own first change: France’s 2022 final network stops at the famous 41′ double substitution (40.5 min of play), Spain’s Euro 2024 final window ends with Rodri’s half-time injury exit (45.0 min), and Barcelona’s Clásico window is just 27.0 min after Mascherano’s early injury. Rate metrics (passes/min, λ1/min, λ2/min) are normalized by window length, so networks stay comparable.
Same pitch, different games
The eight winners, drawn to one scale and ordered by passing tempo. At one end, Barcelona’s 7.44 passes/min web — 33 strong pairs in a 27-minute window. At the other, Leicester’s 1.98 passes/min spear, and the sparsest network in the set: France won the 2018 World Cup final with only 9 pairs reaching three completed passes. Volume is style, not destiny. Click any panel to load it in the explorer.
Source: StatsBomb open event data · winners’ constant-XI networks, node size = touches, edge width = completed passes (≥3) · ordered by passes/min.
Sixteen networks, side by side
Every metric for every team, sortable. Three patterns worth clicking for:
Tempo doesn’t decide matches. Sort by passes/min: the two slowest networks in the set both won — Leicester at 1.98/min away at Manchester City (who passed at 6.44/min and lost 1–3), and 2018 France at 2.05/min in a World Cup final against Croatia’s 4.66/min.
Connectivity does better. The winner had the higher algebraic connectivity (λ2/min — resistance to being split in two) in 6 of 8 matches, including all four finals. Pre-collapse France 2022 also owns the set’s lowest clustering coefficient (0.685 against a set mean of 0.86): the triangle structure had dissolved before Deschamps made his 41′ double change, already 0–2 down.
Beware the single hub. In the 0–4 Clásico, Real Madrid’s play funneled through Kroos — eigenvector 1.00 and betweenness 0.32, team centralization 0.500 — while Barcelona spread load at 0.327. Croatia’s Brozović (betweenness 0.39, the most concentrated ball-path share of any losing team here) was the same story in the 2018 final: the textbook pressing target.
Passing tempo by match — winner marked
completed passes per minute of each team’s constant-XI window
- Home side
- Away side
- Match winner
| Match | Team | Res | GD | Window (min) | Passes | Passes/min | Centralization | Entropy | λ1/min | λ2/min | Clustering | Hub | Bottleneck | Max btw |
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2 Rates are per minute of each team’s own window; raw counts (passes, window length) are shown so no rate hides its denominator. Entropy is Shannon entropy of the pass-pair distribution normalized by log 110, the maximum for 11 players.
What the metrics mean
Network metrics are seductive; only some carry evidence. The cleanest results in the literature: passing intensity raises goals scored while centralization lowers them (Grund 2012, fixed-effects on 283,259 passes from 760 Premier League matches); algebraic connectivity drops in the windows before a team concedes (Buldú et al.); and evenly spread passing moderately predicts a shot in the following minute (Lopes et al. 2020).
Intensity — passes/min
Completed passes per minute of the constant-XI window. The volume knob of a passing identity: peak Barcelona hit 7.44, Leicester won at 1.98.
+ goals scored · Grund 2012, within-team fixed effects
Centralization
Freeman-style concentration of passing strength in [0,1]: high means play funnels through one hub — a single point of failure for pressing to attack.
− goals scored · Grund 2012
Entropy
Shannon entropy of the pass-pair distribution, normalized by log 110. High entropy = unpredictable, evenly spread passing; low = a few rehearsed channels.
Homogeneous passing predicts a shot next minute (~45%) · Lopes et al. 2020
λ1 — network eigenvalue
Largest eigenvalue of the per-minute pass matrix. Grows with both volume and evenness — a one-number summary of “how much network” a team generates.
Guardiola-Barcelona statistical outlier · Buldú et al. 2019
λ2 — algebraic connectivity
Fiedler value of the symmetrized network: resistance to being split into two disconnected blocks (defense severed from attack). The best story metric in the field.
Drops in the windows before conceding · Buldú et al. 2018/2019
Clustering coefficient
Mean Barrat weighted clustering: the density of passing triangles around each player. Triangle play, literally — Barcelona’s signature in the canon.
Historic-team marker · Buldú et al. 2019
Eigenvector centrality
Importance by association: high if you exchange passes with other important players. The Perron eigenvector of the pass matrix, max-normalized to 1.00.
Xavi dominated the 2010 World Cup by this measure · Peña & Touchette 2012
Betweenness
Share of shortest ball paths (under distance 1/w) that pass through a player. Teams want it low and evenly spread — one high value is a pressing target.
Brozović 2018: 0.39, highest of any losing team here
Closeness
How quickly the ball can reach every teammate from this player, given how strong each link is. High closeness players are where moves restart.
Peña & Touchette 2012, player rankings
Weighted degree
Passes made + received — raw involvement. Jorginho’s 101 in 54 minutes of the Euro 2020 final is the highest in this set.
Volume feeds intensity, the + goals metric · Grund 2012