Network Status and Centrality

Who are the key actors in the football network? What makes them the key actors

Isha Mahajan, Ankit Kumar (UMass Amherst)
2022-05-09

Assignment Description

Calculate closeness, betweenness and eigenvector centrality measures for your network data, and bonachic-power if possible. Compare these measures to basic degree centrality measures. Try to interpret the results. Are there any interesting patterns in the distribution of measures or correlations between them that provide insight into the measures?

You may also want to identify whether the same node(s) are more/less central using the different measures, and see if you can find any patterns in the results based on which nodes are high/low on each measures. Discuss (with any related evidence) whether or not the node(s) behavior is in line with or violates expectations based on the degree centrality measure, comparing across those measures using a data frame similar to that constructed in the syntax. Be sure to share your assignment with group members for comments.

Eigenvector Centrality

The eigenvector centrality determines how important the neighborhood of the nodes is in the network. The higher the eigenvalue centrality, the more important the neighbors of that node are with respect to other nodes in the network. (Jackson 08) The eigenvalues also take into account the proximity of the node to many other important nodes in the network.

AS Roma has the highest eigenvalue in the network. This can be an additional support to our previous speculation on how AS Roma might influential in the influx and outflow of football talent as they are well connected to the other influential nodes in our football transfer network.

temp <- igraph::centr_eigen(ig,directed=T)
transfer.nodes$eigen <- temp$vector
#arrange descending and return top 5 nodes
h <- arrange(transfer.nodes, desc(eigen))
kable(head(h))
name degree indegree outdegree eigen
AS Roma AS Roma 78 39 39 1.0000000
Genoa CFC Genoa CFC 73 46 27 0.9612356
Inter Milan Inter Milan 68 32 36 0.8787351
ACF Fiorentina ACF Fiorentina 66 38 28 0.7988873
US Sassuolo US Sassuolo 70 35 35 0.7789739
AC Milan AC Milan 60 38 22 0.7611737

Derived and Reflected Centrality

We compute the derived and reflected Eigenvector centrality to better understand whether the influence is more observed in the neighborhood of the nodes or is it more observed on the strength of the node itself.

matrix <- as.matrix(as_adjacency_matrix(ig, attr = "weight"))
#square the adjacency matrix
matsq<-t(matrix) %*% matrix
#Calculate the proportion of reflected centrality.
transfer.nodes$rc<-diag(matsq)/rowSums(matsq)
#replace missing values with 0
transfer.nodes$rc<-ifelse(is.nan(transfer.nodes$rc),0,transfer.nodes$rc)
#Calculate received eigenvalue centrality
transfer.nodes$eigen.rc<-transfer.nodes$eigen*transfer.nodes$rc
#Calculate the proportion of derived centrality.
transfer.nodes$dc<-1-diag(matsq)/rowSums(matsq)
#replace missing values with 0
transfer.nodes$dc<-ifelse(is.nan(transfer.nodes$dc),1,transfer.nodes$dc)
#Calculate received eigenvalue centrality
transfer.nodes$eigen.dc<-transfer.nodes$eigen*transfer.nodes$dc

filter(transfer.nodes,name%in%c("Chelsea FC","FC Barcelona"))

                     name degree indegree outdegree     eigen
FC Barcelona FC Barcelona     59       21        38 0.2704215
Chelsea FC     Chelsea FC     48       12        36 0.2449357
                    rc   eigen.rc        dc  eigen.dc
FC Barcelona 0.2218696 0.05999830 0.7781304 0.2104232
Chelsea FC   0.2352494 0.05762098 0.7647506 0.1873148
##cen <- eigen_centrality(ig, directed = TRUE)
#cen$vector

library(ggthemes)
transfer.nodes%>%
  select(-name) %>% 
  gather() %>% 
  ggplot(aes(value)) +
    geom_histogram() +
    facet_wrap(~key, scales = "free")+
    theme_minimal()


#build on the dataframe
temp<-transfer.nodes %>% 
  select(degree,indegree,outdegree,eigen,eigen.rc,eigen.dc)%>%
  correlate() %>%
  rearrange()

fashion(temp)

       term outdegree degree indegree eigen.rc eigen.dc eigen
1 outdegree              .94      .79      .67      .67   .69
2    degree       .94             .95      .73      .79   .80
3  indegree       .79    .95               .70      .82   .82
4  eigen.rc       .67    .73      .70               .82   .86
5  eigen.dc       .67    .79      .82      .82           1.00
6     eigen       .69    .80      .82      .86     1.00      
rplot(temp)

Betweeness Centrality

Betweeness centrality quantifies the number of times a node acts as a bridge along the shortest path between two other nodes.

transfer.nodes$betweeness <- igraph::betweenness(ig, directed=TRUE)
transfer.nodes%>%
arrange(desc(betweeness))%>%
slice(1:5)

                                 name degree indegree outdegree
UC Sampdoria             UC Sampdoria     57       32        25
Olympiacos Piraeus Olympiacos Piraeus     41       19        22
Genoa CFC                   Genoa CFC     73       46        27
ACF Fiorentina         ACF Fiorentina     66       38        28
Hellas Verona           Hellas Verona     39       27        12
                       eigen         rc   eigen.rc        dc
UC Sampdoria       0.6156139 0.06250452 0.03847865 0.9374955
Olympiacos Piraeus 0.1548258 0.02184055 0.00338148 0.9781595
Genoa CFC          0.9612356 0.08798435 0.08457368 0.9120157
ACF Fiorentina     0.7988873 0.11743415 0.09381665 0.8825658
Hellas Verona      0.6603528 0.02847622 0.01880435 0.9715238
                    eigen.dc betweeness
UC Sampdoria       0.5771352   45262.42
Olympiacos Piraeus 0.1514443   43409.28
Genoa CFC          0.8766619   34603.48
ACF Fiorentina     0.7050706   31710.00
Hellas Verona      0.6415485   31489.50

Network Constraint

transfer.nodes$constraint<-constraint(ig)

Authority

Authorities are those actors which get many incoming links from hubs, presumably because of their high quality relevant information. (Klien 16) In this case Juventus, Chelsea, Barcelona,PSG, Manchester United and Athleteco have the highest authority scores. This means that these clubs are attracting talent from the hubs like Real Madrid, which we will see includes Eden Hazard’s transfer in 2019

transfer.nodes$authority <- authority_score(ig)$vector

a <- transfer.nodes %>% 
  arrange(desc(authority))
kable(head(a))
name degree indegree outdegree eigen rc eigen.rc dc eigen.dc betweeness constraint authority
Juventus FC Juventus FC 68 27 41 0.6770212 0.2150981 0.1456259 0.7849019 0.5313953 20394.667 0.1013719 1.0000000
Chelsea FC Chelsea FC 48 12 36 0.2449357 0.2352494 0.0576210 0.7647506 0.1873148 470.000 0.1043159 0.7752539
FC Barcelona FC Barcelona 59 21 38 0.2704215 0.2218696 0.0599983 0.7781304 0.2104232 4600.083 0.0962553 0.7566895
Paris Saint-Germain Paris Saint-Germain 37 17 20 0.3905048 0.2504831 0.0978149 0.7495169 0.2926900 1106.000 0.1181023 0.6491154
Manchester United Manchester United 29 15 14 0.1991738 0.2315679 0.0461223 0.7684321 0.1530515 1222.500 0.0935063 0.5523492
Atlético de Madrid Atlético de Madrid 48 26 22 0.3488429 0.1835874 0.0640432 0.8164126 0.2847998 1422.662 0.0980257 0.5507481

Hubs

Many Ties to Peripheral Actors

transfer.nodes$hub_score <- hub_score(ig)$vector
kable(head(transfer.nodes))
name degree indegree outdegree eigen rc eigen.rc dc eigen.dc betweeness constraint authority hub_score
Aston Villa Aston Villa 35 27 8 0.2139009 0.1837553 0.0393054 0.8162447 0.1745954 788.000 0.1133398 0.1109352 0.2347654
Inter Milan Inter Milan 68 32 36 0.8787351 0.2193920 0.1927874 0.7806080 0.6859477 2799.590 0.0962341 0.2636786 0.7339420
Borussia Dortmund Borussia Dortmund 45 18 27 0.2370659 0.1480322 0.0350934 0.8519678 0.2019725 2073.000 0.0640196 0.1477605 0.5191494
ACF Fiorentina ACF Fiorentina 66 38 28 0.7988873 0.1174342 0.0938166 0.8825658 0.7050706 31709.998 0.1108554 0.0614752 0.4296561
Real Madrid Real Madrid 46 17 29 0.2296944 0.2975292 0.0683408 0.7024708 0.1613536 4185.762 0.1078723 0.3925985 1.0000000
RB Leipzig RB Leipzig 44 26 18 0.3082591 0.1717242 0.0529356 0.8282758 0.2553235 2936.417 0.0909443 0.1002923 0.3521672

Consider Real Madrid a hub given that their hub_score is 1. We can confirm this by looking at higher eigenvector derived centrality and lower reflected centrality.

Bridges

A few ties to central actors =

bridges(ig)

+ 238/4386 edges from f1949b0 (vertex names):
 [1] Maccabi Haifa      ->Levski Sofia           
 [2] SC Farense         ->Vancouver Whitecaps FC 
 [3] Moreirense FC      ->Al-Batin FC            
 [4] Grenoble Foot 38   ->FC Famalicão           
 [5] CA Fénix           ->FC Famalicão           
 [6] Maccabi Netanya    ->Hapoel Beer Sheva      
 [7] Ironi Kiryat Shmona->Hapoel Beer Sheva      
 [8] Hapoel Haifa       ->Hapoel Beer Sheva      
 [9] FC Basel 1893 U18  ->Juventus Primavera     
[10] Amiens SC U19      ->Manchester United Youth
+ ... omitted several edges
bridges <- transfer.nodes %>% 
  filter(name == "Maccabi Haifa")
kable(head(bridges))
name degree indegree outdegree eigen rc eigen.rc dc eigen.dc betweeness constraint authority hub_score
Maccabi Haifa Maccabi Haifa 1 0 1 0 0 0 1 0 0 1 0 0

We test the understanding whether these teams actually serve as bridges in the network. By looking at the derived and reflected centrality scores, we see that the the reflected centrality for Maccabi Haifa is low while the derived centrality is high, thereby making it a bridge.

Correlations for Centrality Measures

head(transfer.nodes)

                               name degree indegree outdegree
Aston Villa             Aston Villa     35       27         8
Inter Milan             Inter Milan     68       32        36
Borussia Dortmund Borussia Dortmund     45       18        27
ACF Fiorentina       ACF Fiorentina     66       38        28
Real Madrid             Real Madrid     46       17        29
RB Leipzig               RB Leipzig     44       26        18
                      eigen        rc   eigen.rc        dc  eigen.dc
Aston Villa       0.2139009 0.1837553 0.03930541 0.8162447 0.1745954
Inter Milan       0.8787351 0.2193920 0.19278741 0.7806080 0.6859477
Borussia Dortmund 0.2370659 0.1480322 0.03509339 0.8519678 0.2019725
ACF Fiorentina    0.7988873 0.1174342 0.09381665 0.8825658 0.7050706
Real Madrid       0.2296944 0.2975292 0.06834078 0.7024708 0.1613536
RB Leipzig        0.3082591 0.1717242 0.05293556 0.8282758 0.2553235
                  betweeness constraint  authority hub_score
Aston Villa          788.000 0.11333984 0.11093521 0.2347654
Inter Milan         2799.590 0.09623411 0.26367856 0.7339420
Borussia Dortmund   2073.000 0.06401961 0.14776045 0.5191494
ACF Fiorentina     31709.998 0.11085545 0.06147522 0.4296561
Real Madrid         4185.762 0.10787227 0.39259852 1.0000000
RB Leipzig          2936.417 0.09094430 0.10029228 0.3521672
cor<-transfer.nodes %>% 
  select(degree,eigen,betweeness, authority, hub_score)%>%
  correlate() %>%
  rearrange()
rplot(cor)

Brokerage

tempo <-as.data.frame(brokerage(network, cl  = get.vertex.attribute(network,"league"))$raw.nli)
head(tempo)

                  w_I w_O b_IO b_OI b_O   t
Aston Villa       209   0    0    0   0 209
Inter Milan       925   0    0    0   0 925
Borussia Dortmund 416   0    0    0   0 416
ACF Fiorentina    897   0    0    0   0 897
Real Madrid       420   0    0    0   0 420
RB Leipzig        410   0    0    0   0 410
transfer.nodes<-transfer.nodes %>%
mutate(broker.tot = tempo$t, # AS ROMA, AS Monaco, US Sassuolo, Genoa CFC, Atlanta BC
       broker.coord = tempo$w_I,
       broker.itin = tempo$w_O, # Aston Villa, Inter Milan, Borussia Dortmund, ACF florentina
       broker.rep = tempo$b_IO, # # Aston Villa, Inter Milan, Borussia Dortmund, ACF florentina
       broker.gate = tempo$b_OI,
       broker.lia = tempo$b_O)
broker <- transfer.nodes %>% 
  arrange(desc(broker.tot)) %>% 
  slice(1:10)
kable(head(broker))
name degree indegree outdegree eigen rc eigen.rc dc eigen.dc betweeness constraint authority hub_score broker.tot broker.coord broker.itin broker.rep broker.gate broker.lia
AS Roma AS Roma 78 39 39 1.0000000 0.1130986 0.1130986 0.8869014 0.8869014 20233.50 0.0830009 0.3603443 0.2527186 1286 1286 0 0 0 0
AS Monaco AS Monaco 72 39 33 0.3837432 0.1410820 0.0541392 0.8589180 0.3296040 920.00 0.0822568 0.2386720 0.6582977 1157 1157 0 0 0 0
US Sassuolo US Sassuolo 70 35 35 0.7789739 0.0487223 0.0379534 0.9512777 0.7410205 19581.79 0.1146313 0.1001784 0.1704004 1026 1026 0 0 0 0
Genoa CFC Genoa CFC 73 46 27 0.9612356 0.0879843 0.0845737 0.9120157 0.8766619 34603.48 0.1803853 0.1525313 0.4153337 1015 1015 0 0 0 0
Atalanta BC Atalanta BC 68 35 33 0.7380989 0.0801672 0.0591713 0.9198328 0.6789275 17953.23 0.0997573 0.1513361 0.3371393 966 966 0 0 0 0
Inter Milan Inter Milan 68 32 36 0.8787351 0.2193920 0.1927874 0.7806080 0.6859477 2799.59 0.0962341 0.2636786 0.7339420 925 925 0 0 0 0

AS Roma plays a role of coordinator in the network which means that many ties go to and from AS Roma. This is in alignment with the degrees of the network as we saw earlier and the Eigenvector centrality which was also the highest for AS Roma in the network.

Aston Villa is an iterant broker in the network.