# -*- coding: utf-8 -*- """ ICM 2020 D — The Wisdom of Teams（团队协作的智慧） 真实赛题：COMAP 国际交叉学科建模竞赛（ICM）D 题——足球传球网络与团队协作分析。 T1 为球员间传球建立网络（节点=球员，边=传球），识别二元/三元结构、阵型。 T2 跨尺度/跨时间指标（微观成对→宏观全体；每分钟→整场/赛季）。 T3 确定反映成功团队合作的绩效指标（打法多样性、协调性、贡献分配；适应性/节奏/流程）。 T4 建模型捕捉团队合作的结构-配置-动态；给 Huskies 教练结构性策略建议。 T5 推广：如何设计更有效的（通用）团队，通用团队绩效模型还需哪些维度。 真源模型（纯标准库，确定性；SEED=2020 锚定随机流，便于逐次复现）： 采用「确定性合成传球网络」方法，参数取自真实赛题数据量级 （366 球员 / 23,429 传球 / 59,271 事件；本模型聚焦 Huskies 11 人首发网络， 赛季 38 场、场均 618 次传球，与真实量级一致）： · 11 名首发按 4-3-3 阵型布置于 105×68 球场坐标，角色 GK/DEF/MID/FWD。 · 传球生成：每场按角色权重选传球者，按「角色亲和矩阵 × 空间邻近 (exp(-d/σ))」选接球者，累计成加权有向邻接 W[i][j]（i→j 传球量）。 · 网络指标：出/入度、强度、密度、互惠性、加权聚类系数(Fagiolo)、 加权介数(Brandes/Dijkstra)、特征向量中心性、直径/平均路径长。 · 二元结构：最强传球对（dyad）；三元结构：三元普查（transitive/cyclic/ divergent/convergent 等 16 类归并）。 · 阵型识别：按 x 坐标带聚类 → 1-4-3-3。 · 团队绩效指标(TPI)：归一化加权合成 div=打法多样性(出边分布香农熵) / coord=协调性(0.5密度+0.5互惠) / balance=贡献均衡(1-Gini) / adapt=适应性(上下半场 TPI 稳定度) / tempo=节奏(每分钟传球)。权重 [0.25,0.20,0.25,0.15,0.15]。 · 对手对照：8 支对手队（参数扰动）各算 TPI，取均值对照 Huskies。 · 教练建议：敏感度 what-if（跨线传球 +15% → TPI 增量）。 · 通用团队设计：由网络科学提炼原则（密度最优区间、模块性、桥接角色、 中心化–分布式均衡）。 输出：assets/problems/data/icm2020d.csv（9 行队伍面板：Huskies + 8 对手 + TPI 与分项） 作者说明：本模型为方法论演示用确定性合成网络（种子 2020），几何/传球参数取自公开赛事量级， 非 Huskies 真实逐球数据；真实参赛应以官方逐球事件数据(csv)标定，本模型聚焦 「传球网络-结构指标-团队绩效-策略建议-通用推广」的可复现链路。 """ import os import math import csv import random SEED = 2020 HERE = os.path.dirname(os.path.abspath(__file__)) # --------------------------------------------------------------------------- # 0. 模型常数（显式声明，便于敏感性分析） # --------------------------------------------------------------------------- PITCH_W = 105.0 PITCH_H = 68.0 N_MATCHES = 38 PASS_PER_MATCH = 618 # 场均传球；38×618 = 23484 ≈ 真实 23429 量级 SIGMA = 40.0 # 空间邻近尺度（球场单位） TH = 150.0 # 显著边阈值（赛季累计传球；用于二值结构指标） ROLES = ["GK", "DEF", "MID", "FWD"] # Huskies 首发 11 人（4-3-3，攻击方向 +x；坐标为确定性合成） _SQUAD = [ (0, "GK1", "GK", 8, 34), (1, "DF1", "DEF", 22, 12), (2, "DF2", "DEF", 22, 27), (3, "DF3", "DEF", 22, 41), (4, "DF4", "DEF", 22, 56), (5, "MF1", "MID", 48, 20), (6, "MF2", "MID", 48, 34), (7, "MF3", "MID", 48, 48), (8, "FW1", "FWD", 78, 22), (9, "FW2", "FWD", 78, 34), (10, "FW3", "FWD", 78, 46), ] # 传球者角色权重（Huskies） ROLE_W_HUSK = {"GK": 0.04, "DEF": 0.30, "MID": 0.42, "FWD": 0.24} # 角色亲和矩阵 AFF[from][to]：接球者角色偏好 AFF_HUSK = { "GK": {"GK": 0.00, "DEF": 0.85, "MID": 0.15, "FWD": 0.00}, "DEF": {"GK": 0.05, "DEF": 0.20, "MID": 0.65, "FWD": 0.10}, "MID": {"GK": 0.02, "DEF": 0.18, "MID": 0.45, "FWD": 0.35}, "FWD": {"GK": 0.00, "DEF": 0.05, "MID": 0.55, "FWD": 0.40}, } # TPI 权重 W_DIV, W_COORD, W_BAL, W_ADAPT, W_TEMPO = 0.25, 0.20, 0.25, 0.15, 0.15 # --------------------------------------------------------------------------- # 1. 传球网络模拟 # --------------------------------------------------------------------------- def _squad(): return [(i, nm, role, x, y) for (i, nm, role, x, y) in _SQUAD] def _simulate(seed, role_w, aff, ppm, sigma, n_matches=N_MATCHES): rng = random.Random(seed) squad = _squad() n = len(squad) W = {} W1 = {} # 上半场(≤45min) W2 = {} # 下半场(>45min) per_min = [0] * 91 for _m in range(n_matches): for _ in range(ppm): # 选传球者：按角色权重 × 同角色内均匀 pw = [role_w[squad[i][2]] / 1.0 for i in range(n)] tot = sum(pw) pw = [p / tot for p in pw] i = rng.choices(range(n), weights=pw, k=1)[0] ri = squad[i][2] xi, yi = squad[i][3], squad[i][4] # 选接球者：AFF × 邻近 cands = [] wts = [] for j in range(n): if j == i: continue rj = squad[j][2] xj, yj = squad[j][3], squad[j][4] d = math.hypot(xi - xj, yi - yj) w = aff[ri][rj] * math.exp(-d / sigma) cands.append(j) wts.append(w) s = sum(wts) if s <= 0: continue wts = [w / s for w in wts] j = rng.choices(cands, weights=wts, k=1)[0] W[(i, j)] = W.get((i, j), 0.0) + 1.0 mn = int(min(90, max(1, rng.gauss(45, 22)))) per_min[mn] += 1 if mn <= 45: W1[(i, j)] = W1.get((i, j), 0.0) + 1.0 else: W2[(i, j)] = W2.get((i, j), 0.0) + 1.0 return W, W1, W2, per_min # --------------------------------------------------------------------------- # 2. 网络指标 # --------------------------------------------------------------------------- def _adj_bin(W, n): A = [[0] * n for _ in range(n)] for (i, j), v in W.items(): if v > 0: A[i][j] = 1 return A def _entropy_norm(W, n): """每球员出边分布香农熵，归一化到 0-1；团队多样性=均值。""" hs = [] for i in range(n): s = sum(W.get((i, j), 0.0) for j in range(n) if j != i) if s <= 0: hs.append(0.0) continue h = 0.0 for j in range(n): if j == i: continue p = W.get((i, j), 0.0) / s if p > 0: h -= p * math.log(p) hs.append(h / math.log(n - 1) if n > 1 else 0.0) return sum(hs) / n def _gini(vals): xs = sorted(vals) m = len(xs) if m == 0 or sum(xs) == 0: return 0.0 cum = 0.0 for k, x in enumerate(xs, 1): cum += k * x return (2.0 * cum) / (m * sum(xs)) - (m + 1.0) / m def _clustering(W, n): """加权聚类系数(Fagiolo)，权重先归一化到 [0,1]，取均值。""" maxw = max(W.values()) if W else 1.0 maxw = maxw or 1.0 cs = [] for i in range(n): si = (sum(W.get((i, j), 0.0) for j in range(n)) + sum(W.get((j, i), 0.0) for j in range(n))) / maxw if si == 0: cs.append(0.0) continue num = 0.0 for j in range(n): if j == i: continue for k in range(n): if k == i or k == j: continue wij = W.get((i, j), 0.0) / maxw wik = W.get((i, k), 0.0) / maxw wjk = W.get((j, k), 0.0) / maxw if wij > 0 and wik > 0 and wjk > 0: num += (wij * wik * wjk) ** (1.0 / 3.0) den = si * (si - 1.0) if si > 1.0 else 0.0 cv = num / den if den > 0 else 0.0 cs.append(min(1.0, max(0.0, cv))) return sum(cs) / n def _brandes(W, n): """加权介数中心性（距离=1/(1+w)）。返回列表。""" import heapq bet = [0.0] * n for s in range(n): dist = [float("inf")] * n dist[s] = 0.0 prev = [[] for _ in range(n)] pq = [(0.0, s)] while pq: du, u = heapq.heappop(pq) if du > dist[u]: continue for (i, j), w in W.items(): if i != u: continue v = j nd = du + 1.0 / (1.0 + w) if nd < dist[v]: dist[v] = nd prev[v] = [u] heapq.heappush(pq, (nd, v)) elif abs(nd - dist[v]) < 1e-9: prev[v].append(u) # accumulation delta = [0.0] * n stk = [v for v in range(n) if v != s] # topological by distance stk.sort(key=lambda x: -dist[x]) for w in stk: for p in prev[w]: if dist[w] < float("inf"): c = (1.0 + delta[w]) / 1.0 delta[p] += c if w != s: bet[w] += delta[w] / 2.0 return bet def _eigenvector(W, n, iters=60): v = [1.0] * n for _ in range(iters): nv = [0.0] * n for (i, j), w in W.items(): nv[j] += w * v[i] s = math.sqrt(sum(x * x for x in nv)) or 1.0 v = [x / s for x in nv] return v def _triad_class(A, i, j, k): e = lambda a, b: 1 if A[a][b] > 0 else 0 m = sum(1 for (a, b) in [(i, j), (i, k), (j, k)] if e(a, b) and e(b, a)) a = sum(1 for (x, y) in [(i, j), (i, k), (j, i), (j, k), (k, i), (k, j)] if (e(x, y) or e(y, x)) and not (e(x, y) and e(y, x))) t = m * 2 + a if t == 0: return "003" if t == 1: return "012" if t == 2: if m == 1: return "102" outs = [e(i, j) + e(i, k), e(j, i) + e(j, k), e(k, i) + e(k, j)] return "021D" if max(outs) == 2 else "021U" if t == 3: if m == 1: return "120" outs = [e(i, j) + e(i, k), e(j, i) + e(j, k), e(k, i) + e(k, j)] ins = [e(j, i) + e(k, i), e(i, j) + e(k, j), e(i, k) + e(j, k)] if all(o == 1 and ins[idx] == 1 for idx, o in enumerate(outs)): return "030C" return "030T" if t == 4: return "201" if m == 2 else "120" if t == 5: return "210" return "300" def _triadic(W, n, th=TH): from itertools import combinations A = [[0] * n for _ in range(n)] for i in range(n): for j in range(n): if i != j and W.get((i, j), 0.0) >= th: A[i][j] = 1 cnt = {} for (i, j, k) in combinations(range(n), 3): c = _triad_class(A, i, j, k) cnt[c] = cnt.get(c, 0) + 1 # 归并展示类 show = { "divergent(021D)": cnt.get("021D", 0), "convergent(021U)": cnt.get("021U", 0), "transitive(030T)": cnt.get("030T", 0), "cyclic(030C)": cnt.get("030C", 0), "dense(120/201/210/300)": cnt.get("120", 0) + cnt.get("201", 0) + cnt.get("210", 0) + cnt.get("300", 0), "open(003/012/102)": cnt.get("003", 0) + cnt.get("012", 0) + cnt.get("102", 0), } return show, cnt def _metrics(W, squad, th=TH): n = len(squad) # 强度（加权，全网络） strength = [sum(W.get((i, j), 0.0) for j in range(n)) + sum(W.get((j, i), 0.0) for j in range(n)) for i in range(n)] total = sum(W.values()) # 显著边（二值，用于结构指标）：赛季累计传球 ≥ TH A = [[0] * n for _ in range(n)] sig = 0 mutual = 0 for i in range(n): for j in range(n): if i == j: continue if W.get((i, j), 0.0) >= th: A[i][j] = 1 sig += 1 if W.get((j, i), 0.0) >= th: mutual += 1 dens = sig / (n * (n - 1)) if n > 1 else 0.0 recip = (mutual / sig) if sig else 0.0 # 多样性（出边分布熵，加权全网络） div = _entropy_norm(W, n) # 贡献均衡 gini = _gini(strength) balance = 1.0 - gini # 聚类（权重归一化） clust = _clustering(W, n) # 介数 / 特征向量（加权全网络） bet = _brandes(W, n) eig = _eigenvector(W, n) # 直径 / 平均路径（跳数，基于显著边二值图） Dhop = [[float("inf")] * n for _ in range(n)] for i in range(n): Dhop[i][i] = 0 for i in range(n): for j in range(n): if A[i][j]: Dhop[i][j] = 1 for k in range(n): for i in range(n): for j in range(n): if Dhop[i][k] + Dhop[k][j] < Dhop[i][j]: Dhop[i][j] = Dhop[i][k] + Dhop[k][j] ds = [Dhop[i][j] for i in range(n) for j in range(n) if i != j and Dhop[i][j] < float("inf")] diam = max(ds) if ds else 0.0 avgpath = (sum(ds) / len(ds)) if ds else 0.0 # 阵型（按 x 坐标带聚类） bands = {"GK": 0, "DEF": 0, "MID": 0, "FWD": 0} for (i, nm, role, x, y) in squad: if x < 15: bands["GK"] += 1 elif x < 35: bands["DEF"] += 1 elif x < 60: bands["MID"] += 1 else: bands["FWD"] += 1 formation = "%d-%d-%d-%d" % (bands["GK"], bands["DEF"], bands["MID"], bands["FWD"]) # 最强 dyad（互传总量，全网络） dyads = [] for i in range(n): for j in range(i + 1, n): wij = W.get((i, j), 0.0) + W.get((j, i), 0.0) if wij > 0: dyads.append((squad[i][1], squad[j][1], wij)) dyads.sort(key=lambda t: -t[2]) # 三元结构（显著边二值图） triadic, _ = _triadic(W, n, th) return dict( n=n, strength=strength, total_passes=total, density=dens, reciprocity=recip, diversity=div, gini=gini, balance=balance, clustering=clust, betweenness=bet, eigenvector=eig, diameter=diam, avgpath=avgpath, formation=formation, top_dyads=dyads[:6], triadic=triadic, ) def _tpi_components(m, adaptability=None, tempo=None): div = m["diversity"] coord = 0.5 * m["density"] + 0.5 * m["reciprocity"] balance = m["balance"] adapt = adaptability if adaptability is not None else 0.8 t = tempo if tempo is not None else 7.0 tempo_n = min(t / 9.0, 1.0) tpi = (W_DIV * div + W_COORD * coord + W_BAL * balance + W_ADAPT * adapt + W_TEMPO * tempo_n) return dict(div=div, coord=coord, balance=balance, adapt=adapt, tempo=tempo_n, tpi=tpi) # --------------------------------------------------------------------------- # 3. 主入口 # --------------------------------------------------------------------------- def gen_icm2020d(seed=SEED, out_csv="icm2020d.csv"): squad = _squad() n = len(squad) W, W1, W2, per_min = _simulate(seed, ROLE_W_HUSK, AFF_HUSK, PASS_PER_MATCH, SIGMA) m = _metrics(W, squad) # 节奏（每分钟平均传球） tempo = sum(per_min) / float(N_MATCHES) / 90.0 * 90.0 / N_MATCHES # 实际：场均传球 / 90 分钟 tempo = m["total_passes"] / float(N_MATCHES) / 90.0 # 适应性：上下半场结构与贡献稳定度（与阈值无关，用加权全网络指标） m1 = _metrics(W1, squad) m2 = _metrics(W2, squad) d1, b1 = m1["diversity"], m1["balance"] d2, b2 = m2["diversity"], m2["balance"] adaptability = (max(0.0, 1.0 - 0.5 * (abs(d2 - d1) / d1 + abs(b2 - b1) / b1)) if (d1 > 0 and b1 > 0) else 0.0) c1 = _tpi_components(m1, adaptability, tempo) c2 = _tpi_components(m2, adaptability, tempo) tpi1, tpi2 = c1["tpi"], c2["tpi"] comp = _tpi_components(m, adaptability, tempo) TPI = comp["tpi"] # 对手对照：8 队（参数扰动） opp_defs = [ ("Opp-A 控球型", {"GK": 0.03, "DEF": 0.25, "MID": 0.48, "FWD": 0.24}, 660, 36), ("Opp-B 防守型", {"GK": 0.06, "DEF": 0.40, "MID": 0.34, "FWD": 0.20}, 560, 44), ("Opp-C 边路型", {"GK": 0.04, "DEF": 0.28, "MID": 0.40, "FWD": 0.28}, 600, 38), ("Opp-D 长传型", {"GK": 0.05, "DEF": 0.35, "MID": 0.35, "FWD": 0.25}, 540, 55), ("Opp-E 均衡型", {"GK": 0.04, "DEF": 0.30, "MID": 0.42, "FWD": 0.24}, 618, 40), ("Opp-F 高位逼抢", {"GK": 0.03, "DEF": 0.22, "MID": 0.45, "FWD": 0.30}, 650, 34), ("Opp-G 快速反击", {"GK": 0.05, "DEF": 0.33, "MID": 0.37, "FWD": 0.25}, 580, 50), ("Opp-H 传控渗透", {"GK": 0.03, "DEF": 0.26, "MID": 0.46, "FWD": 0.25}, 640, 37), ] opponents = [] for idx, (nm, rw, ppm, sg) in enumerate(opp_defs): Wo, _, _, _ = _simulate(seed + 10 + idx, rw, AFF_HUSK, ppm, sg) mo = _metrics(Wo, squad) to = mo["total_passes"] / float(N_MATCHES) / 90.0 co = _tpi_components(mo, 0.80, to) opponents.append(dict(name=nm, tpi=co["tpi"], div=co["div"], coord=co["coord"], balance=co["balance"], adapt=co["adapt"], tempo=co["tempo"], density=mo["density"], reciprocity=mo["reciprocity"], gini=mo["gini"], formation=mo["formation"], clustering=mo["clustering"], total_passes=mo["total_passes"])) opp_mean = sum(o["tpi"] for o in opponents) / len(opponents) # what-if：传球分布向均值均衡化 15% → 贡献更均衡、打法更多样（正增益） mean_w = m["total_passes"] / float(n * (n - 1)) Ww = {} for (i, j), v in W.items(): Ww[(i, j)] = v * 0.85 + mean_w * 0.15 mw = _metrics(Ww, squad) cw = _tpi_components(mw, adaptability, tempo) whatif_delta = cw["tpi"] - TPI # 敏感性：TPI 对各参数扰动 sens = [] # 多样性权重 ±0.05 base_div_w = W_DIV for dw, label in [(+0.05, "多样性权重+0.05"), (-0.05, "多样性权重-0.05")]: tpi_s = (base_div_w + dw) * m["diversity"] + W_COORD * comp["coord"] + W_BAL * comp["balance"] + W_ADAPT * comp["adapt"] + W_TEMPO * comp["tempo"] sens.append((label, round((tpi_s - TPI) * 100, 2))) # 传球量 ±20% for mult, label in [(1.20, "传球量+20%"), (0.80, "传球量-20%")]: Ws = {(i, j): v * mult for (i, j), v in W.items()} ms = _metrics(Ws, squad) cs = _tpi_components(ms, adaptability, tempo * mult) sens.append((label, round((cs["tpi"] - TPI) * 100, 2))) # σ 邻近 ±20% for mult, label in [(1.20, "邻近尺度σ+20%"), (0.80, "邻近尺度σ-20%")]: Ws, _, _, _ = _simulate(seed, ROLE_W_HUSK, AFF_HUSK, PASS_PER_MATCH, SIGMA * mult) ms = _metrics(Ws, squad) cs = _tpi_components(ms, adaptability, tempo) sens.append((label, round((cs["tpi"] - TPI) * 100, 2))) # 时序（每分钟平均传球，按场均） temporal = [per_min[t] / float(N_MATCHES) for t in range(1, 91)] # ---- 写 CSV（9 行队伍面板）---- OUT_DIR = os.path.normpath(os.path.join(HERE, "..", "assets", "problems", "data")) os.makedirs(OUT_DIR, exist_ok=True) csv_path = os.path.join(OUT_DIR, out_csv) cols = ["team", "formation", "total_passes", "density", "reciprocity", "diversity", "gini", "balance", "clustering", "tempo_ppm", "TPI"] rows = [("Huskies(主队)", m["formation"], round(m["total_passes"], 0), round(m["density"], 3), round(m["reciprocity"], 3), round(m["diversity"], 3), round(m["gini"], 3), round(m["balance"], 3), round(m["clustering"], 3), round(tempo, 2), round(TPI, 3))] for o in opponents: rows.append((o["name"], o["formation"], round(o["total_passes"], 0), round(o["density"], 3), round(o["reciprocity"], 3), round(o["div"], 3), round(o["gini"], 3), round(o["balance"], 3), round(o["clustering"], 3), round(o["tempo"] * 9.0, 2), round(o["tpi"], 3))) with open(csv_path, "w", newline="", encoding="utf-8-sig") as f: w = csv.writer(f) w.writerow(cols) for r in rows: w.writerow(r) D = dict( SEED=SEED, squad=squad, n=n, metrics=m, comp=comp, TPI=TPI, tempo=tempo, adaptability=adaptability, tpi_h1=tpi1, tpi_h2=tpi2, opponents=opponents, opp_mean_tpi=opp_mean, whatif_delta=whatif_delta, whatif_tpi=cw["tpi"], sensitivity=sens, temporal=temporal, csv_path=csv_path, summary=dict( total_passes=round(m["total_passes"], 0), formation=m["formation"], density=round(m["density"], 3), reciprocity=round(m["reciprocity"], 3), diversity=round(m["diversity"], 3), gini=round(m["gini"], 3), balance=round(m["balance"], 3), clustering=round(m["clustering"], 3), diameter=round(m["diameter"], 3), avgpath=round(m["avgpath"], 3), tpi=round(TPI, 3), opp_mean_tpi=round(opp_mean, 3), whatif_delta=round(whatif_delta, 3), adapt=round(adaptability, 3), top_dyad=m["top_dyads"][0], triadic=m["triadic"], ), ) return D if __name__ == "__main__": D = gen_icm2020d() s = D["summary"] m = D["metrics"] print("=== Huskies 传球网络（4-3-3，赛季 %d 场）===" % N_MATCHES) print("总传球量 = %d（场均 %.0f，≈真实 23429 量级）" % (int(s["total_passes"]), s["total_passes"] / N_MATCHES)) print("识别阵型 = %s" % s["formation"]) print("密度 = %.3f 互惠性 = %.3f" % (s["density"], s["reciprocity"])) print("打法多样性(熵归一) = %.3f 贡献 Gini = %.3f → 均衡 = %.3f" % (s["diversity"], s["gini"], s["balance"])) print("加权聚类系数 = %.3f 直径 = %.3f 平均路径 = %.3f" % (s["clustering"], s["diameter"], s["avgpath"])) print("最强传球对 = %s ↔ %s（%d 次）" % (s["top_dyad"][0], s["top_dyad"][1], int(s["top_dyad"][2]))) print("三元结构：", s["triadic"]) c = D["comp"] print("\nTPI 分项：多样性=%.3f 协调性=%.3f 均衡=%.3f 适应性=%.3f 节奏=%.3f" % (c["div"], c["coord"], c["balance"], c["adapt"], c["tempo"])) print("Huskies TPI = %.3f 对手均值 TPI = %.3f" % (s["tpi"], s["opp_mean_tpi"])) print("适应性（上下半场稳定度）= %.3f" % s["adapt"]) print("What-if 传球分布均衡化15%% → TPI 增量 = %+.3f（→ %.3f）" % (s["whatif_delta"], D["whatif_tpi"])) print("\n敏感性（TPI 变化百分点）：") for lbl, d in D["sensitivity"]: print(" %-18s %+.2f" % (lbl, d)) print("\n对手 TPI：") for o in D["opponents"]: print(" %-14s TPI=%.3f 阵型=%s" % (o["name"], o["tpi"], o["formation"])) print("\nCSV ->", D["csv_path"])