Rating and ranking

Rating and ranking in sport have a flourishing tradition. Each sport competition has its own official rating, from which a ranking of players and teams can be compiled.

The challenge of many sports' fans and bettors is to beat the official rating method: to develop an alternative rating algorithm that is better than the official one in the task of predicting future results. As a consequence, many sport rating methods have been developed. Amy N. Langville and Carl D. Meyer even wrote a (compelling) book about (general) rating and ranking methods entitled Who's #1?.

These methods are in fact valid as a centrality measures even outside the sport world. Immagine a weighted directed network in which nodes have an advantage in having weighty edges exiting from the node, while they have a disadvantage in having weighty edges entering to the node. For instance, this holds when nodes are countries and edges corresponds to valued transfer of goods between countries: an edge weighted $k$ from A to B means that country A exports to country B for a value of $k$ (or B imports from A for a value o $k$). As another example, imagine if nodes are financial actors (like banks) and edges are money that have been loaned between banks: an edge weighted $k$ from A to B means that bank A loaned to bank B a value of $k$ (or B borrowed from A a value o $k$).

We can view such a weighted directed network as a competition between nodes of the network as follows:

1. for any pair of nodes $i$ and $j$ for which there is an edge $(i,j)$ with weight $k_1$ as well as the reciprocal edge $(j,i)$ with weight $k_2$, we have a match between $i$ and $j$ in which $i$ scores $k_1$ points and $j$ scores $k_2$ points;
2. for any pair of nodes $i$ and $j$ for which there is an edge $(i,j)$ with weight $k_1$ but there is no reciprocal edge $(j,i)$, we have a match between $i$ and $j$ in which $i$ scores $k_1$ points and $j$ scores $0$ points;
3. for any pair of nodes $i$ and $j$ that are not connected by an edge (in any direction), we have no match between $i$ and $j$.

Having defined the matches of the competition, we can apply any sport rating method to estimate the importance of nodes in the network. For instance, in the case of Massey's method and the banking scenario illustrated above, we are looking for ratings of banks such that $r_i - r_j = y_k$, where $y_k$ is the financial balance (credit minus debit) between banks $i$ and $j$. Hence if $j$ is highly indebted with $i$, then the difference in rankings between $i$ and $j$ is large (in favor of $i$).