Example: Football (soccer) players

Our network example describes the 32 soccer teams which participated in the World Championship in Paris, 1998. Players of the national team often have contracts in other countries. This constitutes a players market where national teams export players to other countries. Members of the 32 teams had contracts in altogether 35 countries. Counting which team exports how many players to which country can be described with a valued, asymmetric graph. The graph is highly a symmetric: some countries only export players, some countries are only importers (data collected by Lothar Krempel).

Hubs and authorities

In directed networks we can usually identify two types of important vertices: hubs and authorities. A vertex is a good hub, if it points to many good authorities, and it is a good authority, if it is pointed to by many good hubs. Hubs and authorities are related to eigenvector centrality.
See:
KLEINBERG, Jon M.:
Authoritative Sources in a Hyperlinked Environment. Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms, edited by Howard Karloff (SIAM/ACM-SIGACT, 1998).
SVG layout - export

Network Analysis

                                                         Valued     Valued
          Input       Output                             Input      Output
          Degree      Degree  Hubs         Authorities   Degree     Degree
----------------------------------------------------------------------------
 ARG      2           2       0.2764013    0.0073580     4          13
 AUT      3           5       0.1691211    0.0289530     3          11
 BEL      2           4       0.0871006    0.0295895     3           8
 BGR      0           4       0.1087431    0.0000000     0          10
 BRA      2           5       0.2103657    0.0322629    12          12
 CHE      2           0       0.0000000    0.0189380     2           0
 CHL      0           3       0.0204822    0.0000000     0           3
 CMR      0           9       0.1803360    0.0000000     0          20
 COL      0           5       0.0496744    0.0000000     0           9
 DEU     16           3       0.0713116    0.3097197    39           4
 DNK      0           7       0.2454492    0.0000000     0          16
 ESP     18           0       0.0000000    0.5789962    54           0
 FRA     13           0       0.0000000    0.1634971    24           0
 GBR     12           0       0.0000000    0.5355642    50           0
 GRE      3           0       0.0000000    0.0635664     5           0
 HRV      0           6       0.2602221    0.0000000     0          14
 IRN      0           1       0.0403306    0.0000000     0           3
 ITA     16           3       0.1038528    0.4502114    42           5
 JAM      0           1       0.1627250    0.0000000     0           7
 JPN      4           0       0.0000000    0.0637349     9           0
 KOR      0           2       0.0181625    0.0000000     0           5
 MAR      0           6       0.1746280    0.0000000     0          13
 MEX      1           0       0.0000000    0.0027950     1           0
 NGA      0          10       0.2098329    0.0000000     0          21
 NLD      6           4       0.2711707    0.0799073    10          12
 NOR      0           6       0.3800853    0.0000000     0          22
 PRT      4           0       0.0000000    0.0522481     7           0
 PRY      0           3       0.0643884    0.0000000     0          13
 ROM      0           7       0.2620101    0.0000000     0          18
 SCO      2           2       0.1698217    0.0814487     6           8
 TUN      1           2       0.0481771    0.0075801     1           5
 TUR      7           0       0.0000000    0.1587900    18           0
 USA      3           3       0.0768484    0.0143101     4           5
 YUG      0           6       0.4122957    0.0000000     0          21
 ZAF      1           9       0.2264670    0.0091084     1          17
  • Degree Centrality
    • There exist countries that are only exporters:
      Yugoslavia, Romania, Norway, Nigeria...
    • There exist countries that are only importers:
      Spain, France, Turkey, Greece, GBR...
    • Spain imports players from the highest number of different countries (18); following by Italy and Germany importing from 16 countries.
    • Nigeria exports players into the highest number of different countries (10).
  • Hubs and Authorities
    Authorities are countries having good leagues, hubs are countries having good players.
    • Hubs: Yugoslavia, Norway, Argentina, Romania, Croatia, Denmark...
      Yugoslavia (0.41) and Norway (0.38) are much more important hubs than the others (Argentina - 0.28).
    • Hubs and authorities: The Netherlands, Scotland, Brazil, Austria...
    • Authorities: Spain (0.58), GBR (0.54), Italy (0.45), Germany (0.31), France (0.16), Turkey (0.16)...
    Countries are written in descending order according to hubs/authorities weights.
    SVG layout - hubs/authorities
Comparing degree centrality to hubs and authorities two questions arise:
  1. Why is Nigeria not an important hub, although it exports to the highest number of different countries?
    There are two reasons:
    • Degree centrality used counts only different countries where players are exported and not how many players, while this information is used when computing hubs and authorities. If we count how many players are exported (valued degree) we got:
      Ranking according to the number of imported players:
      Spain (54), GBR (50), Italy (42), Germany (39),...
      Ranking according to the number of exported players:
      Norway (22), Yugoslavia (21), Nigeria (21), Cameroon (20),...
      We see that Nigeria is not in the first place any more.
    • Even more important reason is that Nigeria exports much smaller proportion of its players to the best leagues (authorities) than Yugoslavia and Norway, for example: Into the best leagues (Spain, Italy, Germany, France and GBR)
      • Nigeria exports only 10 of its 21 exported players (48%);
      • Yugoslavia exports 19 of its 20 exported players (90%);
      • Norway exports 17 of its 22 exported players (77%).
  2. Why is Argentina among hubs although it does not export many players (only 13)?
    Argentina exports this 13 players only to two countries, but these two countries have the best leagues (are authorities): it exports 4 players to Spain and 9 players to Italy.