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Topic-specific PageRank

Thus far we have discussed the PageRank computation with a teleport operation in which the surfer jumps to a random web page chosen uniformly at random. We now consider teleporting to a random web page chosen non-uniformly. In doing so, we are able to derive PageRank values tailored to particular interests. For instance, a sports aficionado might wish that pages on sports be ranked higher than non-sports pages. Suppose that web pages on sports are ``near'' one another in the web graph. Then, a random surfer who frequently finds himself on random sports pages is likely (in the course of the random walk) to spend most of his time at sports pages, so that the steady-state distribution of sports pages is boosted.

Suppose our random surfer, endowed with a teleport operation as before, teleports to a random web page on the topic of sports instead of teleporting to a uniformly chosen random web page. We will not focus on how we collect all web pages on the topic of sports; in fact, we only need a non-zero subset $S$ of sports-related web pages, so that the teleport operation is feasible. This may be obtained, for instance, from a manually built directory of sports pages such as the open directory project ( or that of Yahoo.

Provided the set $S$ of sports-related pages is non-empty, it follows that there is a non-empty set of web pages $Y \supseteq S$ over which the random walk has a steady-state distribution; let us denote this sports PageRank distribution by $\vec{\pi}_s$. For web pages not in $Y$, we set the PageRank values to zero. We call $\vec{\pi}_s$ the topic-specific PageRank for sports.

\includegraphics[width=13cm]{topics.eps} Topic-specific PageRank.In this example we consider a user whose interests are 60% sports and 40% politics. If the teleportation probability is 10%, this user is modeled as teleporting 6% to sports pages and 4% to politics pages.

We do not demand that teleporting takes the random surfer to a uniformly chosen sports page; the distribution over teleporting targets $S$ could in fact be arbitrary.

In like manner we can envision topic-specific PageRank distributions for each of several topics such as science, religion, politics and so on. Each of these distributions assigns to each web page a PageRank value in the interval $[0,1
)$. For a user interested in only a single topic from among these topics, we may invoke the corresponding PageRank distribution when scoring and ranking search results. This gives us the potential of considering settings in which the search engine knows what topic a user is interested in. This may happen because users either explicitly register their interests, or because the system learns by observing each user's behavior over time.

But what if a user is known to have a mixture of interests from multiple topics? For instance, a user may have an interest mixture (or profile) that is 60% sports and 40% politics; can we compute a personalized PageRank for this user? At first glance, this appears daunting: how could we possibly compute a different PageRank distribution for each user profile (with, potentially, infinitely many possible profiles)? We can in fact address this provided we assume that an individual's interests can be well-approximated as a linear combination of a small number of topic page distributions. A user with this mixture of interests could teleport as follows: determine first whether to teleport to the set $S$ of known sports pages, or to the set of known politics pages. This choice is made at random, choosing sports pages 60% of the time and politics pages 40% of the time. Once we choose that a particular teleport step is to (say) a random sports page, we choose a web page in $S$ uniformly at random to teleport to. This in turn leads to an ergodic Markov chain with a steady-state distribution that is personalized to this user's preferences over topics (see Exercise 21.2.3 ).

While this idea has intuitive appeal, its implementation appears cumbersome: it seems to demand that for each user, we compute a transition probability matrix and compute its steady-state distribution. We are rescued by the fact that the evolution of the probability distribution over the states of a Markov chain can be viewed as a linear system. In Exercise 21.2.3 we will show that it is not necessary to compute a PageRank vector for every distinct combination of user interests over topics; the personalized PageRank vector for any user can be expressed as a linear combination of the underlying topic-specific PageRanks. For instance, the personalized PageRank vector for the user whose interests are 60% sports and 40% politics can be computed as

0.6\vec{\pi}_s + 0.4\vec{\pi}_p,
\end{displaymath} (261)

where $\vec{\pi}_s$ and $\vec{\pi}_p$ are the topic-specific PageRank vectors for sports and for politics, respectively.


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Next: Hubs and Authorities Up: PageRank Previous: The PageRank computation   Contents   Index
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