\section{Introduction}

Network Science has emerged with the goal of understanding properties of dynamic and connected systems, providing several 
models and tools to characterize their behavior~\cite{barabasi2009}. Facebook, the Web, protein interaction and computer networks are all 
examples of networks that have been widely studied in the literature. Moreover, the last decade has evidenced a growing 
interest in the study of networks, partially due to the availability of large amounts of empirical data and the increase in 
computational power \cite{Newman2010}.

Social Networks are among the most studied kind of network in part due to the surge of Online Social Networks and Social Media, 
such as Twitter, Facebook, YouTube and Google Plus. Among these, real sport social networks have also received attention recently. 
In such networks, nodes are athletes or teams and edges indicate some sort of interaction among them, such as direct matches.  
Soccer ~\cite{cotta_fifa_2011,onody_complex_2004}, basketball~\cite{vaz_de_melo_can_2008} and tennis~\cite{radicchi_who_2011}, for 
instance, are sport modalities that have been studied considering player-level interactions. 

An important problem related to network study is vertex ranking, where the goal is to determine an ordering of network nodes to 
reflect some relative aspect of their importance. For example, ranking in social networks usually attempts to order individuals by 
their importance~\cite{pires2011,newman2004}. Ranking has also been applied to sport social networks to identify the better teams or athletes in a 
particular sport~\cite{radicchi_who_2011}. More specifically, ~\citeN{radicchi_who_2011} have recently considered the problem of identifying the 
best tennis players of all times. They consider a network where nodes correspond to 
athletes and directed edges correspond to match results. There is an edge from athlete $i$ to athlete $j$ if $j$ has defeated 
$i$ in at least one official tennis match. Figure \ref{exemplo_rede} illustrates an example of such a network. The authors apply 
a variation of the well-known PageRank algorithm~\cite{page_pagerank_1999} to such a network created using thousands of official 
matches in order to obtain a ranking of the best players in history. The intuition behind this idea is analogous to that of Web 
page ranking. Each player has an associated ``prestige'' that defines her importance and flows through the network according to the direction of the edges. In such a configuration, players that defeat other players with high prestige are highly benefited. 

\begin{figure}[t!]
  \centering
  \subfigure {
    \begin{minipage}[c]{0.2\textwidth}
      \centering
      \begin{scriptsize}
        \begin{tabular}{lcc}
          \toprule
          \multirow{2}{*}{}\textbf{Time}    & \multicolumn{2}{c}{\textbf{Match}} \tabularnewline
          \midrule
          \cmidrule(r){2-3} & \textbf{Winner}   & \textbf{Loser}    \tabularnewline  
            2007            & A                 & D                  \tabularnewline
            2008            & A                 & C                  \tabularnewline
            2009            & A                 & B                  \tabularnewline
            2010            & C                 & B                  \tabularnewline      
            2011            & C                 & D                  \tabularnewline
            2012            & C                 & A                  \tabularnewline            
         \bottomrule
        \end{tabular}
      \end{scriptsize}
    \end{minipage}
  }
  \subfigure {
    \begin{minipage}[c]{0.23\textwidth}
      \centering
      \epsfig{file=figuras/exemplo_rede.eps,width=.55\textwidth} \\
      \phantom{a}
    \end{minipage}
  }
  \caption{Example of a network of matches.}
  \label{exemplo_rede} % USE THE CONTENT WITHIN label FOR REFERENCING IT (USE \ref WITHIN THE TEXT)
\end{figure}

Although good results have been obtained when compared to some official rankings, this strategy does not consider the temporal factor inherent to 
the events that make up the network. For instance, a match that took place recently is more important than one that happened far in the 
past, with respect to establishing a current ranking. Intuitively, sport social networks are dynamic with respect to athletes and matches 
and this time factor clearly influences official rankings. For example, an athlete that has won more matches than any other athlete 
in the past is not necessarily at the top of the current ranking. The importance of her victories diminishes over time when considering the 
ranking problem. This is also illustrated in Figure~\ref{exemplo_rede}. Looking only at the network no clear winner emerges, but based on the match results it is clear that in the last three years athlete C is the dominant one.

In this article we address this limitation by explicitly considering temporal aspects when constructing a sport social network. Our approach 
is to define a time-varying weight for the edges of the network to allow for more accurate rankings. Intuitively, an edge weight decreases with time and increases when a match is won by an athlete. This approach can be used to produce a ranking for any point in time, taking 
into consideration any particular period of time (e.g., the ranking of the 80s). We apply our approach to data obtained from two popular sports, Mixed Martial Arts (MMA) and tennis. Experimental results indicate that the rankings produced with our approach are 
superior to those provided by a classic approach (where edge weights do not decay) when compared to gold standards derived from credible known rankings. In some cases, the rankings 
produced by our approach are very similar to the gold standard ones.

The rest of this article is organized as follows. Section 2 addresses related work. Section 3 describes the methodological
aspects of our work. Section 4 details our experiments. Finally, Section 5 presents our conclusions and comments on future work.


