Chapter 10: Key Takeaways

Quick Reference Summary

Passing networks transform qualitative team play into quantitative, analyzable structures using graph theory.

Core Concepts

Network Fundamentals

Graph G = (V, E)
- V = Players (nodes)
- E = Passes (edges)
- Weighted: Edge weight = number of passes
- Directed: A→B distinct from B→A

Adjacency Matrix

$$A_{ij} = \text{number of passes from player } i \text{ to player } j$$

For 11 players: 11×11 matrix with up to 110 directed edges.

Essential Metrics

Player-Level (Centrality)

Network-Level

Quick Interpretation Guide

Centrality Patterns

Network Properties

Typical Benchmarks

Network Density by Style

• Possession teams: 0.40 - 0.50
• Balanced teams: 0.30 - 0.40
• Direct teams: 0.20 - 0.30

Centralization Thresholds

• Balanced team: < 0.25
• Moderate dependency: 0.25 - 0.35
• Star dependency: > 0.35

Clustering Coefficient

• High combination play: > 0.40
• Average: 0.30 - 0.40
• Limited triangles: < 0.30

Python Quick Reference

Build Network

import networkx as nx

G = nx.DiGraph()
pass_counts = passes.groupby(['player', 'pass_recipient']).size()
for (passer, receiver), count in pass_counts.items():
    G.add_edge(passer, receiver, weight=count)

Calculate Centralities

# Degree
degree = dict(G.degree(weight='weight'))

# Betweenness
betweenness = nx.betweenness_centrality(G, weight='weight')

# PageRank
pagerank = nx.pagerank(G, weight='weight')

# Closeness
closeness = nx.closeness_centrality(G, distance='weight')

Network Metrics

# Density
density = nx.density(G)

# Clustering (undirected)
clustering = nx.average_clustering(G.to_undirected(), weight='weight')

Basic Visualization

from mplsoccer import Pitch

pitch = Pitch()
fig, ax = pitch.draw()

for u, v, d in G.edges(data=True):
    ax.annotate('', xy=pos[v], xytext=pos[u],
               arrowprops=dict(arrowstyle='->', lw=d['weight']/5))

for player, (x, y) in pos.items():
    ax.scatter(x, y, s=degree[player]*10)

Common Mistakes to Avoid

1. Ignoring direction: Always use directed graphs for passing networks
2. Neglecting weights: Unweighted analysis misses pass frequency information
3. Single-match conclusions: Network properties vary match-to-match
4. Context blindness: Score, opponent, and game state affect networks
5. Over-interpretation: High centrality isn't always "good"
6. Formation effects: Different formations naturally produce different structures

Practical Workflow

1. Data Preparation

Load events → Filter successful passes → Extract player pairs

2. Network Construction

Create DiGraph → Add weighted edges → Calculate positions

3. Analysis

Player centralities → Network metrics → Top combinations

4. Visualization

Network plot → Matrix heatmap → Comparative charts

5. Interpretation

Combine metrics → Consider context → Draw tactical insights

Key Formulas Reference Card

Density

$$D = \frac{|E|}{|V|(|V|-1)}$$

Degree Centrality

$$C_D(v) = \frac{k_v}{n-1}$$

Betweenness Centrality

$$C_B(v) = \sum_{s \neq v \neq t} \frac{\sigma_{st}(v)}{\sigma_{st}}$$

Closeness Centrality

$$C_C(v) = \frac{n-1}{\sum_{u \neq v} d(v, u)}$$

Clustering Coefficient

$$C_i = \frac{2T_i}{k_i(k_i-1)}$$

Network Entropy

$$H = -\sum_{i,j} p_{ij} \log(p_{ij})$$

Centralization

$$C = \frac{\sum_{i=1}^{n}(C_{max} - C_i)}{max\sum_{i=1}^{n}(C_{max} - C_i)}$$

Applications Summary

Next Chapter Preview

Chapter 11: Possession and Territorial Control extends network concepts to understand spatial dominance, ball control zones, and territorial metrics that capture how teams control areas of the pitch beyond individual passing actions.