python中colley排名算法实现的无效结果

Question

我在这篇文章之后为体育比赛实现了 colley 排名算法： https://towardsdatascience.com/generate-sports-rankings-with-data-science-4dd1979571da

但是我得到了无效的结果。 r 结果应该是获胜概率，所以它必须在 0 和 1 之间。但是当使用大量输入时，我得到了 1.4 和负结果。

这是Colley算法的一个已知问题吗？ 是否有可以正确处理大量数据的修复或替代算法？

我的代码：

import json
import numpy as np

c=None
b=None

def iter_game(t1, t2, r1, r2):

    global c, b

    # Updating vecotr b based on result of each game
    if r1 > r2:
        b[(t1 - 1)] += 1
        b[(t2 - 1)] -= 1
    elif r1 < r2:
        b[(t1 - 1)] -= 1
        b[(t2 - 1)] += 1
    else: return

    c[(t1 - 1)][(t1 - 1)] += + 1  # Updating diagonal element
    c[(t2 - 1)][(t2 - 1)] += + 1  # Updating diagonal element
    c[(t1 - 1)][(t2 - 1)] -= 1  # Updating off - diagonal element
    c[(t2 - 1)][(t1 - 1)] -= 1  # Updating off - diagonal element

def main():

    global c,b

    num_players = 2537

    # Initializing Colley Matrix 'c'and vector 'b'
    c = np.zeros([num_players, num_players])
    b = np.zeros(num_players)

    with open("colley_games.json") as f:
        games = json.load(f)

    for game in games:
        iter_game(game["home"], game["away"], game["score_home"], game["score_away"])

    # Adding 2 to diagonal elements (total number of games) of Colley matrix
    diag = c.diagonal() + 2
    np.fill_diagonal(c, diag)

    # Dividing by 2 and adding one to vector b
    for i, value in enumerate(b):
        b[i] = b[i] / 2
        b[i] += 1

    # Solving N variable linear equation
    r = np.linalg.solve(c, b)

    # Displaying ranking for top 10 teams
    top_teams = r.argsort()[-10:][::-1]
    for i in top_teams:
        print (str(r[i]) + " " + str(i))

    print ("----------------------------")

    # Displaying ranking for lower 10 teams
    top_teams = r.argsort()[:10][::-1]
    for i in top_teams:
        print (str(r[i]) + " " + str(i))

main()

使用colley_games.json输入执行时的结果：

# python3 rank_colley_ask.py
1.409508465374069 2135
1.1358580974322448 1759
1.134486271801534 2126
1.1314563266569193 1763
1.0930304236523831 2134
1.0809741214865278 1243
1.0633760655215825 2143
1.049467222041803 1748
1.0391031285438894 1470
1.0288821673935697 1453
----------------------------
-0.1304799893162797 1954
-0.15012844703440156 1929
-0.19462901224272772 2121
-0.20745023863077341 1930
-0.21188300405221577 890
-0.24910253479694192 968
-0.25265547797693333 2155
-0.34306068196974493 930
-0.3468485876254179 913
-0.3792348796324475 2151

在这里你有小提琴： https://pyfiddle.io/fiddle/ffc0e2d7-3d47-4ee9-b9f8-9c28e6e3b500/?i=true （我不得不gzip，因为最大上传是1MB）