We present equivalent formalizations that show CoSimRank’s close relationship to Personalized PageRank and SimRank and also show how we can take advantage of fast matrix multiplication algorithms to compute CoSimRank.

We first first give an intuitive introduction of CoSimRank as a Personalized PageRank (PPR) derivative.

3.1 Personalized PageRank

Haveliwala (2002) introduced Personalized PageRank — or topic-sensitive PageRank — based on the idea that the uniform damping vector 19(0) can be replaced by a personalized vector, which depends on node i.

The use of weighted edges was first proposed in the PageRank patent.

These algorithms are often based on PageRank (Erin and Page, 1998) and other centrality measures (e.g., (Erkan and Radev, 2004)).

This paper introduces CoSimRank,1 a new graph-theoretic algorithm for computing node similarity that combines features of SimRank and PageRank .

Another important similarity measure is cosine similarity of Personalized PageRank (PPR) vectors.

LexRank (Erkan and Radev, 2004) is similar to PPR+cos in that it combines PageRank and cosine; it initializes the sentence similarity matrix of a document using cosine and then applies PageRank to compute lexical centrality.

These approaches use at least one of cosine similarity, PageRank and SimRank.

Besides, the PageRank algorithm (Page et al., 1998) is adopted to optimize the graph model.

Finally, the weights of word nodes are calculated using the PageRank algorithm as follows:

where d is the damping factor as in the PageRank algorithm.

Besides, the standard PageRank algorithm is employed to optimize the graph model.

After creating the graph, PageRank is run to rank sentences.

Finally, instead of PageRank , we used SimRank (Haveliwala, 2002) to identify the nodes most similar to the query node and not only the central sentences in the graph.

After the graph is generated, the PageRank algorithm (Page et al., 1999) is used to determine the most central linguistic units in the graph.

PageRank spreads the query similarity of a vertex to its close neighbors, so that we rank higher sentences that are similar to other sentences which are similar to the query.

We set the damping factor ,u to 0.85, following the standard PageRank paradigm.

The standard PageRank algorithm starts from an arbitrary node and randomly selects to either follow a random outgoing edge (considering the weighted transition matrix) or to jump to a random node (treating all nodes with equal probability).

where 1 is a vector with all elements equaling to l and the size is correspondent to the size of V0 or VT. ,u is the damping factor usually set to 0.85, as in the PageRank algorithm.