Multiplex Football Simulator

Core Multiplex Knobs

Team Familiarity (Layer 1) 35%

Concept: Represents off-pitch friendship + training chemistry (relational agency). Increasing this adds dashed amber links across the squad without moving player positions. Density ρ = 2L / [N(N − 1)]

Controls Layer 1 (Amber Dashed Ties). Independent of the coach's tactical arrows!

Clique Bias (d) vs. Tactics (1−d) d = 0.80

PageRank-Inspired Choice: Players balance passing to familiar friends (d) vs. following the Coach's Tactical Layer (1−d) to progress toward goal. P(i → j) = d · Fam(i,j) + (1 − d) · Tactics(i,j)

High d = keep ball inside friend clique; Low d = follow blue tactical arrows even to strangers.

Show Clique Bubbles (Modularity Q)

Layer 2: Team Strategy & Squad Size

Coach's Tactical Layer (Blue Arrows)

Distinct Layer 2: The tactical passing structure is set by the coach's game plan—not by who is friends with whom!
• Build-Up: Short progressive passes (DEF → MID → ST).
• Direct Long: Long balls bypassing midfield to Attackers.
• Pass to Star: Heliocentric tactics funnelling balls to #7 / Messi.

Squad Size

Network & Multiplex Statistics

Modularity (Q) — Cliquishness 0.41

High Q (>0.30) = fractured into separate cliques

Newman Modularity (Q): Measures whether familiarity links stay trapped inside sub-communities rather than bridging the whole team. Q = (1 / 2m) Σ [ A_ij − (k_i k_j / 2m) ] δ(c_i, c_j)

Multiplex QAP Overlap (r) 0.48**

Correlation between Familiarity (L1) & Tactics (L2)

Quadratic Assignment Procedure (QAP): Compares Layer 1 (Familiarity matrix F) against Layer 2 (Tactical Passing matrix T). High QAP means your friendship pipes align with the tactical passes required to score! r_QAP = Corr( vec(F), vec(T) )

Familiarity Density (ρ) 0.33

Proportion of active familiarity ties across squad

HCT vs. SNT Proof Experiments

Pure SNT Mode (Equalise HCT)

Set all players to 75 skill so only networks matter

Bench / Injure Star Player

Remove #7 Star / Messi to test network resilience

Star Player Tactical Centrality: 0.32

Pass Completion %

Ball Flow Along Pipes

76%

Expected Goals (xG)

Chance Quality Created

0.68

Simulated Win Rate

Avg Goals: 0.7 / match

29%