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\ShortHeadings{LexRank: Graph-based Centrality as Salience 
in Text Summarization}
{Erkan \& Radev}



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{\scriptsize\SetKwInOut{Input}{input}
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\Input{An array $S$\  of $n$\  sentences, cosine threshold $t$}
\Output{An array $C$\  of Centroid scores}
Hash $WordHash$\;
Array $C$\;
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\caption{Computing Centroid scores.}\end{algorithm}%
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\begin{figure}\scriptsize\center
\begin{tabular}{|c c l|}
\hline
SNo & ID & Text\\
\hline
1 &d1s1 & Iraqi Vice President Taha Yassin Ramadan announced today, Sunday, \\
&     & that Iraq refuses to back down from its decision to stop cooperating\\
&&      with disarmament inspectors before its demands are met.\\
2 &d2s1 & Iraqi Vice president Taha Yassin Ramadan announced  today, Thursday,\\
&&      that Iraq rejects cooperating with the United Nations except on the\\
&&      issue of lifting the blockade imposed upon it since the year 1990.\\
3&d2s2 & Ramadan told reporters in Baghdad that "Iraq cannot deal positively\\
&&      with whoever represents the Security Council unless there was a clear\\
&&      stance on the issue of lifting  the blockade off of it.\\
4&d2s3 & Baghdad had decided late last October to completely cease cooperating\\
&&      with the inspectors of the United Nations Special Commission \\
&&      (UNSCOM), in charge of disarming Iraq's weapons, and whose work \\
&&      became very limited since the fifth of August, and announced it will not \\
&&      resume its cooperation with the Commission even if it were subjected \\
&&	to a military operation.\\
5&d3s1 & The Russian Foreign Minister, Igor Ivanov, warned today, Wednesday \\
&&      against using force against Iraq, which will destroy, according to \\
&&      him, seven years of difficult diplomatic work and will complicate \\
&&      the regional situation in the area.\\
6&d3s2 & Ivanov contended that carrying out air strikes against Iraq, who refuses\\
&&      to cooperate with the United Nations inspectors, ``will end  the\\
&&      tremendous work achieved by the international group during the past\\
&&      seven years and will complicate the situation in the region.''\\
7&d3s3 & Nevertheless, Ivanov stressed that Baghdad must resume working \\
&&       with the Special Commission in charge of disarming the Iraqi \\
&&       weapons of mass destruction (UNSCOM).\\
8&d4s1 & The Special Representative of the United Nations Secretary-General \\
&&      in Baghdad, Prakash Shah, announced today, Wednesday, after \\
&&      meeting with the Iraqi Deputy Prime Minister Tariq Aziz, that Iraq \\
&&      refuses to back down from its decision to cut off cooperation with \\
&&  the disarmament inspectors.\\
9&d5s1 & British Prime Minister Tony Blair said today, Sunday, that the crisis\\
&&      between the international community and Iraq ``did not end'' and that \\
&&      Britain is still ``ready, prepared,  and able to strike Iraq.''\\
10&d5s2 & In a gathering with the press held at the Prime Minister's office, \\
&&      Blair contended that the crisis with Iraq ``will not end until Iraq has\\
&&      absolutely and unconditionally respected its commitments'' towards \\
&&      the United Nations. \\
11&d5s3 & A spokesman for Tony Blair had indicated that the British Prime \\
& &     Minister gave permission to British Air Force Tornado planes stationed \\
& &     in Kuwait to join the aerial  bombardment against Iraq. \\
\hline
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&1&2&3&4&5&6&7&8&9&10&11\\
\hline
1 & 1.00 & 0.45 & 0.02 & 0.17 & 0.03 & 0.22 & 0.03 & 0.28 & 0.06 & 0.06 & 0.00 \\
2 & 0.45 & 1.00 & 0.16 & 0.27 & 0.03 & 0.19 & 0.03 & 0.21 & 0.03 & 0.15 & 0.00 \\
3 & 0.02 & 0.16 & 1.00 & 0.03 & 0.00 & 0.01 & 0.03 & 0.04 & 0.00 & 0.01 & 0.00 \\
4 & 0.17 & 0.27 & 0.03 & 1.00 & 0.01 & 0.16 & 0.28 & 0.17 & 0.00 & 0.09 & 0.01 \\
5 & 0.03 & 0.03 & 0.00 & 0.01 & 1.00 & 0.29 & 0.05 & 0.15 & 0.20 & 0.04 & 0.18 \\
6 & 0.22 & 0.19 & 0.01 & 0.16 & 0.29 & 1.00 & 0.05 & 0.29 & 0.04 & 0.20 & 0.03 \\
7 & 0.03 & 0.03 & 0.03 & 0.28 & 0.05 & 0.05 & 1.00 & 0.06 & 0.00 & 0.00 & 0.01 \\
8 & 0.28 & 0.21 & 0.04 & 0.17 & 0.15 & 0.29 & 0.06 & 1.00 & 0.25 & 0.20 & 0.17 \\
9 & 0.06 & 0.03 & 0.00 & 0.00 & 0.20 & 0.04 & 0.00 & 0.25 & 1.00 & 0.26 & 0.38 \\
10 & 0.06 & 0.15 & 0.01 & 0.09 & 0.04 & 0.20 & 0.00 & 0.20 & 0.26 & 1.00 & 0.12 \\
11 & 0.00 & 0.00 & 0.00 & 0.01 & 0.18 & 0.03 & 0.01 & 0.17 & 0.38 & 0.12 & 1.00 
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$i$%
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$gcd\{n:X^n(i,i) > 0\}=1$%
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$\mathbf{B}$%
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$\mathbf{p}$%
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$\mathbf{B}$%
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\begin{displaymath}
p(u) = \frac{d}{N}+{(1-d)}\sum_{v\in{}adj[u]}\frac{p(v)}{deg(v)}
\end{displaymath}%
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\mathbf{p} = [d\mathbf{U}+(1-d)\mathbf{B}]^\mathrm{T}\mathbf{p}
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$\mathbf{B}$%
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\begin{algorithm}
% latex2html id marker 427

{\scriptsize\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\Input{A stochastic, irreducible and aperiodic matrix $\mathbf{M}$}
\Input{matrix size $N$, error tolerance $\epsilon$}
\Output{eigenvector $\mathbf{p}$}
$\mathbf{p}_0$=$\frac{1}{N}\mathbf{1}$\;
t=0\;
\Repeat{$\delta<\epsilon$}{
	t=t+1\;
	$\mathbf{p}_t$\  = $\mathbf{M}^\mathrm{T}\mathbf{p}_{t-1}$\;
	$\delta$\  = $||\mathbf{p}_t-\mathbf{p}_{t-1}||$\;
}
\Return{$\mathbf{p}_t$}\;
}
\caption{Power Method for computing the stationary distribution of a Markov chain.}\end{algorithm}%
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$Degree$%
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$0.85$%
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$1$%
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\begin{algorithm}
% latex2html id marker 457

{\scriptsize\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\SetKwFunction{Insert}{Insert}
\SetKwFunction{idfmodifiedcosine}{idf-modified-cosine}
\SetKwFunction{PowerMethod}{PowerMethod}
MInput{An array $S$\  of $n$\  sentences, cosine threshold $t$}
\Output{An array $L$\  of LexRank scores}
Array $CosineMatrix[n][n]$\;
Array $Degree[n]$\;
Array $L[n]$\;
\For{$i\leftarrow 1$\  to n}{
\For{$j\leftarrow 1$\  to n}{
	$CosineMatrix[i][j]$\  $=$\  \idfmodifiedcosine{$S[i]$,$S[j]$}\;
	\If{$CosineMatrix[i][j] > t$}{
		$CosineMatrix[i][j] = 1$\;
		$Degree[i]++$\;
	}
	\Else{$CosineMatrix[i][j] = 0$\;}
}}
\For{$i\leftarrow 1$\  to n}{
\For{$j\leftarrow 1$\  to n}{
$CosineMatrix[i][j]$\  = $CosineMatrix[i][j] / Degree[i]$\;
}}
$L$\  $=$\  \PowerMethod{$CosineMatrix$,$n$,$\epsilon$}\;
\Return{$L$}\;
}
\caption{Computing LexRank scores.}\end{algorithm}%
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p(u) = \frac{d}{N}+{(1-d)}\sum_{v\in{}adj[u]}\frac{\textrm{idf-modified-cosine}(u,v)}{\sum_{z\in{}adj[v]}\textrm{idf-modified-cosine}(z,v)}p(v)
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$d$%
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$0.2$%
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$0.5$%
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{\newpage\clearpage
\lthtmlfigureA{figure520}%
\begin{figure}\center
\scriptsize\begin{boxedverbatim}

feature LexRank LexRank.pl 0.2
Centroid 1 Position 1 LengthCutoff 9 LexRank 1
mmr-reranker-word.pl 0.5 MEAD-cosine enidf\end{boxedverbatim}
\end{figure}%
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$1$%
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\end{document}
