\section{Methodology}\label{metodologia}
\subsection{The Sport: Mixed Martial Arts}\label{esporte}
\indent
We have chosen Mixed Martial Arts (MMA) as the sport of interest for the initial tests of our proposed temporally-aware ranking strategy for a number of reasons. First, this is a sport that has gained a lot of popularity lately. Second, and more importantly, there are several sites available on the Web with information about events, fights, etc., as well as with rankings of athletes that can serve as a possible gold standard, thus making our work feasible.

Mixed Martial Arts (MMA) is a full contact combat sport that allows several types of fighting techniques. Matches usually take places in events in which fighters try to defeat each other. Although interesting for our study for the reasons mentioned above, there are also some challenges in exploring this particular sport. First, there are several organizations related to the sport that promote events all around the world, with slightly different rules. The current main organization is UFC, although there are several other organizations such as Pride Fighting Championships, DREAM, and WEC. The second challenge relates to the existence of several fighter categories, based on the body weight. Examples include Welterweight and Heavyweight. An important aspect is that each organization adopts its own rules to define the categories. For instance, UFC and Strikeforce consider that the Welterweight category covers body weights between 71--77 kg while in Pride FC the same category covers fighters who weight up to 83 kg. Besides that, another important aspect is that a fighter may compete in different categories in different points in time. For instance,  in a given moment a fighter may belong to category Welterweight while in a different moment in his carrier he fighted under the Middleweight category.

For these reasons we made the following design choices:
\begin{itemize}
\item We have not considered existing differences among organizations and events. We considered all matches from all events equivalent, since the fighters were competing under the same conditions.
\item The analysis of results are perfomed per category, i.e.,  we consider all fights within a given category to produce a rank.
\end{itemize}
\subsection{The Dataset}\label{dataset}
\indent

As far as we know, there is no publicly available dataset with results of MMA fights. To obtain structured data of fight results we \rem{had to find sites} \rev{looked for websites} that made \rem{some of this data} \rev{this kind of data} available \rem{on the Web} and created scrappers ~\cite{_web_scraping} to extract \rem{the data} \rev{it}. The main and most complete site related to MMA \rem{in the world} in terms of match results is Sherdog~\cite{_sherdog_site}. However, Sherdog does not make available information about the category of the matches, only the \rem{category of the fighters} \rev{fighters main categories}. As fighters may belong to different categories at different points in time and \rem{our results are organized the match categories} \rev{we wanted to compare rankings obtained from matches of each category}, we could not use this data.

Instead we used data available in the FightMetrics~\cite{_fightmetric_site} site which also makes available the results of the most important matches in the main MMA events around the world. Although it covers less matches than Sherdog, it has detailed information of each match, including the category of each one. Moreover, FightMetrics has also a cleaner and more structured interface which facilitates the creation of the scrapper for data extraction.

Table \ref{data_summary} summarizes the results obtained from FightMetrics with our scrapper:
\begin{table}[h]
\centering
\begin{tabular}{|l|c|}
\hline Number of fighters & 1838 \\
\hline Number of matches & 3648 \\
\hline Number of organizations & 59 \\
\hline
\end{tabular}
\caption{Summary of the extracted data.}
\label{data_summary}
\end{table}


\subsection{The Network Representation}\label{rede}

\indent
The results of the matches can be better visualized through a network of matches represented as a weighted directed graph. Each node corresponds to a fighter and there is an edge connecting nodes $i$ and $j$ if the fighter represented by node $i$ was defeated in some match by the fighter represented by node $j$.
The weight of an edge represents the weight associated with one or more matches between two fighters. This will be further discussed in Section \ref{temporal}. As an example, we can observe in Figure \ref{fig:rede_full} the network constructed with the results of all matches in all categories of our collected dataset.
%{\vspace{-0.1cm}}
% EXAMPLE OF HOW TO INCLUDE eps FIGURES
\begin{figure*}[tb] % FIGURES SHOULD BE AT THE TOP [t] OR BOTTOM [b} OF PAGES
\centering % FIGURES SHOULD BE CENTERED
\includegraphics[width=0.7\textwidth]{figuras/rede_full.eps}
\caption{Network of macthes among figthers of all categories.} % ALWAYS INCLUDE CAPTIONS IN YOUR FIGURES
\label{fig:rede_full} 
\end{figure*}

It is important to notice that we organized the data in order to generate network of matches according to various criteria, such as matches in the same category, that happened in the same year, etc. for future developments.


\subsection{The Ranking Method}\label{ranqueamento}
We want to obtain a ranking of fighters based on the results of the matches up to a given point in time. By analyzing the network of matches, we can, intuitively, make the conclusion that nodes with high indegree represented those with more wins, and, consequently, with higher chance of being highly ranked. This intuition, however, takes into account only the number of wins, ignoring the ``quality'' of these wins. For instance, using only this criteria a fighter that has defeated several other not so``prestigious'' fighters would be better ranked than another fighter that has obtained fewer victories but over stronger opponents. A way around this problem is to apply the well known PageRank algorithm ~\cite{page_pagerank_1999} in the network of matches. The intuition here is analogous to that of Web page rank. With each fighter is associated a certain ``prestige'' that defines its importance and that flows through the network according to the direction of the edges. Thus, fighters who defeat other fighters with high prestige are more benefited.

\subsection{Time-Varying Edge Weights}\label{temporal}

The time-dependent interactions among athletes clearly has a fundamental impact on their rankings. For example, a series of recent matches is 
likely to be more important in determining the current ranking of set of athetes than a series of matches among the same set of athletes 
that occured far in the past. Intuitively, the importance of match towards ranking of athetes decays over time: the older a match is, the 
less important is its result. Thus, a good ranking for today's athletes should give more importance to recent matches. 

The importance of relationships is usually captured by assiging weights to network edges. Thus, based on our 
intuition, we propose time-varying edge weights to relfect the fact that importance of match results decay with time. 
In particular, we will consider an exponentially decaying weight, controlled by a parameter that determines how fast 
importance decays with time. Consider two atheltes $i$ and $j$ and let $t_{i,j}^k$ denote the time instant of their 
$k$-th match, for $k = 1, 2, \ldots$. Moreover let $w_{i,j}(t)$ denote the edge weight of the directed edge 
from $i$ to $j$ at time $t \geq 0$. We define the edge weight $w_{i,j}(t)$ recursively as follows:

\begin{equation}
w_{i,j}(t) = \left\{
\begin{array}{l l}
 0                                                      & \text{if $t = 0$} \\
 w_{i,j}(t-\epsilon) + I                                & \text{if $t = t_{i,j}^k$ and if $j$ defeated $i$ in match $t_{i,j}^k$} \\
 w_{i,j}(t_{i,j}^k) \times \exp^{\alpha (t-t_{i,j}^k)}  & \mbox{if $t_{i,j}^k \leq t < t_{i,j}^{k+1}$} \\
\end{array} \right.
\label{eq:weight}
\end{equation}

Note that the weight of edge $w_{i,j}(0)$ is zero. At the time of a match, namely at the instants $t_{i,j}^k$ for $k=1, \ldots$, 
the weight of the edge increases by $I$, if $j$ defeats $i$. Note that $I$ is a constant that indicates how much weight is 
added to the edge when an athlete defeats another. Note that this value could depend on the importance of the match (e.g., a final), 
but in this paper we assume $I=1$ for all matches. Finally, in between matches the weight decreases exponentially with 
parameter $\alpha$ according to the amount of time elapsed since the last match. Note that if $j$ never again defeats $i$, 
then the weight $w_{i,j}$ will go to zero. 

A key parameter of the formulation above is $\alpha$ which denotes how fast the weight of an edge decreases. If $\alpha$ is too 
small, then the edge weight will have a long memory. In particular, if $\alpha = 0$ then edge weights do not decrease with time. 
On the other hand if $\alpha$ is too large, then the weights have very short memory, quickly going to zero. Intuitively, $\alpha$ 
should be set according to the timescale of the sport to reflect how fast real rankings change. Moreover, it should also be 
related to the number of matches per unit of time (e.g., year) of the sport. In what follows, we investigate the impact of various 
values for $\alpha$.

Figures \ref{fig:fd1} and \ref{fig:fd1} illustrate the function $w_{i,j}(t)$ for two athletes in our dataset. Note that there was a match 
at times $0$, $7$, $15$, $18$, $19$ and $20$ when $j$ defeated $i$, thus indicating the increments ($I=1$). In between matches, the weight 
decreases exponentially. Figures \ref{fig:fd1} and \ref{fig:fd1} illustrate $\alpha = 0.05$ and $\alpha = 0.20$, respectively. Note that 
for $\alpha = 0.05$ the edge accumulates more weight over time, indicating its longer memory.

\begin{figure}[tb] % FIGURES SHOULD BE AT THE TOP [t] OR BOTTOM [b} OF PAGES
\begin{minipage}[b]{0.5\linewidth}
\centering % FIGURES SHOULD BE CENTERED
\includegraphics[width=1\textwidth]{figuras/funcao_decaimento_1_alpha_005.eps} 
\caption{Evolution of edge weight $w_{i,j}(t)$ over time ($\alpha = 0.05$).} % ALWAYS INCLUDE CAPTIONS IN YOUR FIGURES
\label{fig:fd1} % USE THE CONTENT WITHIN label FOR REFERENCING IT (USE \ref WITHIN THE TEXT)
\end{minipage}
\hspace{0.2cm}
\begin{minipage}[b]{0.5\linewidth}
\centering % FIGURES SHOULD BE CENTERED
\includegraphics[width=1\textwidth]{figuras/funcao_decaimento_2_alpha_020.eps} 
\caption{Evolution of edge weight $w_{i,j}(t)$ over time ($\alpha = 0.20$).} % ALWAYS INCLUDE CAPTIONS IN YOUR FIGURES
\label{fig:fd2} % USE THE CONTENT WITHIN label FOR REFERENCING IT (USE \ref WITHIN THE TEXT)
\end{minipage}
\end{figure}

