1. Introduction and Motivation
General web search is performed predominantly through text queries to search
engines. Because of the enormous size of the web, text alone is usually not
selective enough to limit the number of query results to a manageable size.
The PageRank algorithm [11], among others
[9], has been proposed (and implemented in
Google [1]) to exploit the linkage structure of the web to
compute global ``importance'' scores that can be used to influence the
ranking of search results. To encompass different notions of importance for
different users and queries, the basic PageRank algorithm can be modified to
create ``personalized views'' of the web, redefining importance according to
user preference. For example, a user may wish to specify his bookmarks as a
set of preferred pages, so that any query results that are important with
respect to his bookmarked pages would be ranked higher. While
experimentation with the use of personalized PageRank has shown its utility
and promise [5,11], the size of the web makes its practical
realization extremely difficult. To see why, let us review the intuition
behind the PageRank algorithm and its extension for personalization.
The fundamental motivation underlying PageRank is the recursive notion that important pages are those linkedto by many important pages. A page with only two inlinks, for example, may seem unlikely to be an important page, but it may be important if the two referencing pages are Yahoo! and Netscape, which themselves are important pages because they have numerous inlinks. One way to formalize this recursive notion is to use the ``random surfer'' model introduced in [11]. Imagine that trillions of random surfers are browsing the web: if at a certain time step a surfer is looking at page , at the next time step he looks at a random outneighbor of . As time goes on, the expected percentage of surfers at each page converges (under certain conditions) to a limit that is independent of the distribution of starting points. Intuitively, this limit is the PageRank of , and is taken to be an importance score for , since it reflects the number of people expected to be looking at at any one time.
The PageRank score reflects a ``democratic'' importance that has no preference for any particular pages. In reality, a user may have a set of preferred pages (such as his bookmarks) which he considers more interesting. We can account for preferred pages in the random surfer model by introducing a ``teleportation'' probability : at each step, a surfer jumps back to a random page in with probability , and with probability continues forth along a hyperlink. The limit distribution of surfers in this model would favor pages in , pages linkedto by , pages linkedto in turn, etc. We represent this distribution as a personalized PageRank vector (PPV) personalized on the set . Informally, a PPV is a personalized view of the importance of pages on the web. Rankings of a user's textbased query results can be biased according to a PPV instead of the global importance distribution.
Each PPV is of length , where is the number of pages on the web. Computing a PPV naively using a fixedpoint iteration requires multiple scans of the web graph [11], which makes it impossible to carry out online in response to a user query. On the other hand, PPV's for all preference sets, of which there are , is far too large to compute and store offline. We present a method for encoding PPV's as partiallycomputed, shared vectors that are practical to compute and store offline, and from which PPV's can be computed quickly at query time.
In our approach we restrict preference sets to subsets of a set of hub pages , selected as those of greater interest for personalization. In practice, we expect to be a set of pages with high PageRank (``important pages''), pages in a humanconstructed directory such as Yahoo! or Open Directory [2], or pages important to a particular enterprise or application. The size of can be thought of as the available degree of personalization. We present algorithms that, unlike previous work [5,11], scale well with the size of . Moreover, the same techniques we introduce can yield approximations on the much broader set of all PPV's, allowing at least some level of personalization on arbitrary preference sets.
The main contributions of this paper are as follows.
 A method, based on new graphtheoretical results (listed next), of encoding PPV's as partial quantities, enabling an efficient, scalable computation that can be divided between precomputation time and query time, in a customized fashion according to available resources and application requirements.
 Three main theorems: The Linearity Theorem allows every PPV to be represented as a linear combination of basis vectors, yielding a natural way to construct PPV's from shared components. The Hubs Theorem allows basis vectors to be encoded as partial vectors and a hubs skeleton, enabling basis vectors themselves to be constructed from common components. The Decomposition Theorem establishes a linear relationship among basis vectors, which is exploited to minimize redundant computation.
 Several algorithms for computing basis vectors, specializations of these algorithms for computing partial vectors and the hubs skeleton, and an algorithm for constructing PPV's from partial vectors using the hubs skeleton.
 Experimental results on real web data demonstrating the effectiveness and scalability of our techniques.
Due to space constraints, this paper omits proofs of the theorems and algorithms presented. These proofs are included as appendices in the full version of this paper [7].
2. Preliminaries
Let denote the web graph, where is the set of all web
pages and contains a directed edge
iff page
links to page . For a page , we denote by and the set of
inneighbors and outneighbors of , respectively. Individual inneighbors
are denoted as (
), and individual
outneighbors are denoted analogously. For convenience, pages are numbered
from to , and we refer to a page and its associated number
interchangeably. For a vector , denotes entry , the
th component of . We always typeset vectors in boldface and
scalars (e.g., ) in normal font. All vectors in this paper are
dimensional and have nonnegative entries. They should be thought of as
distributions rather than arrows. The magnitude of a vector
is defined to be
and is written
.
In this paper, vector magnitudes are always in . In an implemention,
a vector may be represented as a list of its nonzero entries, so another
useful measure is the size of , the number of nonzero entries
in .
We generalize the preference set discussed in Section 1 to a preference vector , where and denotes the amount of preference for page . For example, a user who wants to personalize on his bookmarked pages uniformly would have a where if , and if . We formalize personalized PageRank scoring using matrixvector equations. Let be the matrix corresponding to the web graph , where if page links to page , and otherwise. For simplicity of presentation, we assume that every page has at least one outneighbor, as can be enforced by adding selflinks to pages without outlinks. The resulting scores can be adjusted to account for the (minor) effects of this modification, as specified in the appendices of the full version of this paper [7].
For a given , the personalized PageRank equation can be written as
For the reader's convenience, Table 1 lists terminology
that will be used extensively in the coming sections.

3. Basis Vectors
We present the first step towards encoding PPV's as shared components. The
motivation behind the encoding is a simple observation about the
linearity[footnote 2] of PPV's, formalized by the following theorem.
Let be the unit vectors in each dimension, so that for each , has value at entry and everywhere else. Let be the PPV corresponding to . Each basis vector gives the distribution of random surfers under the model that at each step, surfers teleport back to page with probability . It can be thought of as representing page 's view of the web, where entry of is 's importance in 's view. Note that the global PageRank vector is , the average of every page's view.
An arbitrary personalization vector can be written as a weighted
sum of the unit vectors :
(3) 
(4) 
Recall from Section 1 that preference sets (now preference vectors) are restricted to subsets of a set of hub pages . If a basis hub vector (or hereafter hub vector) for each were computed and stored, then any PPV corresponding to a preference set of size (a preference vector with nonzero entries) can be computed by adding up the corresponding hub vectors with the appropriate weights .
Each hub vector can be computed naively using the fixedpoint computation in [11]. However, each fixedpoint computation is expensive, requiring multiple scans of the web graph, and the computation time (as well as storage cost) grows linearly with the number of hub vectors . In the next section, we enable a more scalable computation by constructing hub vectors from shared components.
4. Decomposition of Basis Vectors
In Section 3 we represented PPV's as a linear combination
of hub vectors , one for each . Any PPV based on
hub pages can be constructed quickly from the set of precomputed hub
vectors, but computing and storing all hub vectors is impractical. To
compute a large number of hub vectors efficiently, we further decompose them
into partial vectors and the hubs skeleton, components from
which hub vectors can be constructed quickly at query time. The
representation of hub vectors as partial vectors and the hubs skeleton
saves both computation time and storage due to sharing of components among
hub vectors. Note, however, that depending on available resources and
application requirements, hub vectors can be constructed offline as well.
Thus ``query time'' can be thought of more generally as ``construction
time''.
We compute one partial vector for each hub page , which essentially encodes the part of the hub vector unique to , so that components shared among hub vectors are not computed and stored redundantly. The complement to the partial vectors is the hubs skeleton, which succinctly captures the interrelationships among hub vectors. It is the ``blueprint'' by which partial vectors are assembled to form a hub vector, as we will see in Section 4.3.
The mathematical tools used in the formalization of this decomposition are presented next.[footnote 3]
4.1 Inverse Pdistance
To formalize the relationship among hub vectors, we relate the personalized
PageRank scores represented by PPV's to inverse Pdistances in the
web graph, a concept based on expected distances as introduced in
[8].
Let . We define the inverse Pdistance
from to as
Let be some nonempty set of pages. For , we
define as a restriction of that considers only tours
which pass through some page in equation (5). That
is, a page must occur on somewhere other than the endpoints.
Precisely, is written as
4.2 Partial Vectors
Intuitively, , defined in equation (6), is the
influence of on through . In particular, if all paths from to
pass through a page in , then separates and , and
. For wellchosen sets (discussed in Section
4.4.2), it will be true that
for many pages . Our strategy is to take advantage of this property by
breaking into two components:
and
, using the equation
We first precompute and store the partial vector instead of the full hub vector . Partial vectors are cheaper to compute and store than full hub vectors, assuming they are represented as a list of their nonzero entries. Moreover, the size of each partial vector decreases as increases, making this approach particularly scalable. We then add back at query time to compute the full hub vector. However, computing and storing explicitly could be as expensive as itself. In the next section we show how to encode so it can be computed and stored efficiently.
4.3 Hubs Skeleton
Let us briefly review where we are: In Section 3 we
represented PPV's as linear combinations of hub vectors , one for
each , so that we can construct PPV's quickly at query time if we
have precomputed the hub vectors, a relatively small subset of PPV's. To
encode hub vectors efficiently, in Section 4.2
we said that instead of full hub vectors , we first compute and
store only partial vectors
, which intuitively
account only for paths that do not pass through a page of (i.e., the
distribution is ``blocked'' by ). Computing and storing the difference
vector efficiently is the topic of this section.
It turns out that the vector can be be expressed in terms of the partial vectors , for , as shown by the following theorem. Recall from Section 3 that has value at and everywhere else.
In terms of inverse Pdistances (Section 4.1), the Hubs Theorem says roughly that the distance from page to any page through is the distance from to each times the distance from to , correcting for the paths among hubs by . The terms and deal with the special cases when or is itself in . The proof, which is quite involved, is in the full version [7].
The quantity
appearing on the righthand
side of (8) is exactly the partial vectors discussed in
Section 4.2. Suppose we have computed
for a hub page . Substituting the Hubs
Theorem into equation 7, we have the following Hubs
Equation for constructing the hub vector from partial vectors:
The set has size at most , much smaller than the full hub vector , which can have up to nonzero entries. Furthermore, the contribution of each entry to the sum is no greater than (and usually much smaller), so that small values of can be omitted with minimal loss of precision (Section 6). The set forms the hubs skeleton, giving the interrelationships among partial vectors.
An intuitive view of the encoding and construction suggested by the Hubs Equation (9) is shown in Figure 1.
4.4 Discussion
4.4.1 Summary
In summary, hub vectors are building blocks for PPV's corresponding to
preference vectors based on hub pages. Partial vectors, together with the
hubs skeleton, are building blocks for hub vectors. Transitively, partial
vectors and the hubs skeleton are building blocks for PPV's: they can be
used to construct PPV's without first materializing hub vectors as an
intermediate step (Section 5.4). Note that for
preference vectors based on multiple hub pages, constructing the
corresponding PPV from partial vectors directly can result in significant
savings versus constructing from hub vectors, since partial vectors are
shared across multiple hub vectors.
4.4.2 Choice of
So far we have made no assumptions about the set of hub pages . Not
surprisingly, the choice of hub pages can have a significant impact on
performance, depending on the location of hub pages within the overall graph
structure. In particular, the size of partial vectors is smaller when pages
in have higher PageRank, since highPageRank pages are on average close
to other pages in terms of inverse Pdistance (Section
4.1), and the size of the partial vectors is related to the
inverse Pdistance between hub pages and other pages according to the Hubs
Theorem. Our intuition is that highPageRank pages are generally more
interesting for personalization anyway, but in cases where the intended hub
pages do not have high PageRank, it may be beneficial to include some
highPageRank pages in to improve performance. We ran experiments
confirming that the size of partial vectors is much smaller using
highPageRank pages as hubs than using random pages.
4.4.3 Web Skeleton
The techniques used in the construction of hub vectors can be extended to
enable at least approximate personalization on arbitrary preference vectors
that are not necessarily based on . Suppose we want to personalize on a
page . The Hubs Equation can be used to construct
from partial vectors, given that we have computed . As discussed in
Section 4.3, the cost of computing and storing
is orders of magnitude less than . Though is
only an approximation to , it may still capture significant
personalization information for a properlychosen hub set , as
can be thought of as a ``projection'' of onto .
For example, if contains pages from Open Directory,
can capture information about the broad topic of . Exploring the
utility of the web skeleton
is an area
of future work.
5. Computation
In Section 4 we presented a way to construct hub
vectors from partial vectors
, for , and
the hubs skeleton
. We also discussed the web
skeleton
. Computing these partial
quantities naively using a fixedpoint iteration [11] for each
would scale poorly with the number of hub pages. Here we present scalable
algorithms that compute these quantities efficiently by using dynamic
programming to leverage the interrelationships among them. We also show how
PPV's can be constructed from partial vectors and the hubs skeleton at query
time. All of our algorithms have the property that they can be stopped at
any time (e.g., when resources are depleted), so that the current ``best
results'' can be used as an approximation, or the computation can be resumed
later for increased precision if resources permit.
We begin in Section 5.1 by presenting a theorem underlying all of the algorithms presented (as well as the connection between PageRank and inverse Pdistance, as shown in the full version [7]). In Section 5.2, we present three algorithms, based on this theorem, for computing general basis vectors. The algorithms in Section 5.2 are not meant to be deployed, but are used as foundations for the algorithms in Section 5.3 for computing partial quantities. Section 5.4 discusses the construction of PPV's from partial vectors and the hubs skeleton.
5.1 Decomposition Theorem
Recall the random surfer model of Section 1, instantiated
for preference vector
(for page 's view of the web).
At each step, a surfer teleports to page with some probability .
If is at , then at the next step, with probability will be
at a random outneighbor of . That is, a fraction
of the time, surfer will be at any given outneighbor of one step
after teleporting to . This behavior is strikingly similar to the model
instantiated for preference vector
, where surfers teleport directly to each
with equal probability
. The similarity is
formalized by the following theorem.
(10) 
The Decomposition Theorem gives another way to think about PPV's. It says that 's view of the web () is the average of the views of its outneighbors, but with extra importance given to itself. That is, pages important in 's view are either itself, or pages important in the view of 's outneighbors, which are themselves ``endorsed'' by . In fact, this recursive intuition yields an equivalent way of formalizing personalized PageRank scoring: basis vectors can be defined as vectors satisfying the Decomposition Theorem.
While the Decomposition Theorem identifies relationships among basis vectors, a division of the computation of a basis vector into related subproblems for dynamic programming is not inherent in the relationships. For example, it is possible to compute some basis vectors first and then to compute the rest using the former as solved subproblems. However, the presence of cycles in the graph makes this approach ineffective. Instead, our approach is to consider as a subproblem the computation of a vector to less precision. For example, having computed to a certain precision, we can use the Decomposition Theorem to combine the 's to compute to greater precision. This approach has the advantage that precision needs not be fixed in advance: the process can be stopped at any time for the current best answer.
5.2 Algorithms for Computing Basis Vectors
We present three algorithms in the general context of computing full basis
vectors. These algorithms are presented primarily to develop our algorithms
for computing partial quantities, presented in Section
5.3. All three algorithms are iterative
fixedpoint computations that maintain a set of intermediate results
. For each , is a
lowerapproximation of on iteration , i.e.,
for all . We build solutions (
) that are successively better approximations to , and
simultaneously compute the error components , where
is the ``projection'' of the vector
onto the (actual) basis vectors. That is, we maintain the
invariant that for all and all ,
Thus, is a lowerapproximation of with error
We begin with and , so that logically, the approximation is initially and the error is . To store and efficiently, we can represent them in an implementation as a list of their nonzero entries. While all three algorithms have in common the use of these intermediate results, they differ in how they use the Decomposition Theorem to refine intermediate results on successive iterations.
It is important to note that the algorithms presented in this section and their derivatives in Section 5.3 compute vectors to arbitrary precision; they are not approximations. In practice, the precision desired may vary depending on the application. Our focus is on algorithms that are efficient and scalable with the number of hub vectors, regardless of the precision to which vectors are computed.
5.2.1 Basic Dynamic Programming Algorithm
In the basic dynamic programming algorithm, a new basis vector for
each page is computed on each iteration using the vectors computed for
's outneighbors on the previous iteration, via the Decomposition
Theorem. On iteration , we derive
from
using the equations:
A proof of the algorithm's correctness is given in the full version [7], where the error is shown to be reduced by a factor of on each iteration.
Note that although the values help us to see the correctness of the algorithm, they are not used here in the computation of and can be omitted in an implementation (although they will be used to compute partial quantities in Section 5.3). The sizes of and grow with the number of iterations, and in the limit they can be up to the size of , which is the number of pages reachable from . Intermediate scores will likely be much larger than available main memory, and in an implementation could be read off disk and written to disk on each iteration. When the data for one iteration has been computed, data from the previous iteration may be deleted. Specific details of our implementation are discussed in Section 6.
5.2.2 Selective Expansion Algorithm
The selective expansion algorithm is essentially a version of the
naive algorithm that can readily be modified to compute partial vectors, as
we will see in Section 5.3.1.
We derive
by ``distributing'' the error at
each page (that is, ) to its outneighbors via the
Decomposition Theorem. Precisely, we compute results on iteration using
the equations:
for a subset . If for all , then the error is reduced by a factor of on each iteration, as in the basic dynamic programming algorithm. However, it is often useful to choose a selected subset of as . For example, if contains the pages for which the error is highest, then this top scheme limits the number of expansions and delays the growth in size of the intermediate results while still reducing much of the error. In Section 5.3.1, we will compute the hub vectors by choosing . The correctness of selective expansion is proven in the full version [7].
5.2.3 Repeated Squaring Algorithm
The repeated squaring algorithm is similar to the selective expansion
algorithm, except that instead of extending
one step using equations (14) and
(15), we compute what are essentially
iteration results using the equations
where . For now we can assume that for all ; we will set to compute the hubs skeleton in Section 5.3.2. The correctness of these equations is proven in the full version [7], where it is shown that repeated squaring reduces the error much faster than the basic dynamic programming or selective expansion algorithms. If , the error is squared on each iteration, as equation (17) reduces to:
As an alternative to taking , we can also use the top scheme of Section 5.2.2.
Note that while all three algorithms presented can be used to compute the set of all basis vectors, they differ in their requirements on the computation of other vectors when computing : the basic dynamic programming algorithm requires the vectors of outneighbors of to be computed as well, repeated squaring requires results to be computed for such that , and selective expansion computes independently.
5.3 Computing Partial Quantities
In Section 5.2 we presented iterative algorithms for
computing full basis vectors to arbitrary precision. Here we present
modifications to these algorithms to compute the partial quantities:
 Partial vectors , .
 The hubs skeleton (which can be computed more efficiently by itself than as part of the entire web skeleton).
 The web skeleton .
5.3.1 Partial Vectors
Partial vectors can be computed using a simple specialization of the
selective expansion algorithm (Section
5.2.2): we take and
for , for all . That is, we never ``expand'' hub
pages after the first step, so tours passing through a hub page are
never considered. Under this choice of ,
converges to
for all . Of
course, only the intermediate results
for should be computed. A proof is presented in the full version
[7].
This algorithm makes it clear why using highPageRank pages as hub pages improves performance: from a page we expect to reach a highPageRank page sooner than a random page, so the expansion from will stop sooner and result in a shorter partial vector.
5.3.2 Hubs Skeleton
While the hubs skeleton is a subset of the complete web skeleton and can be
computed as such using the technique to be presented in Section
5.3.3, it can be computed much faster by itself if we are
not interested in the entire web skeleton, or if higher precision is desired
for the hubs skeleton than can be computed for the entire web skeleton.
We use a specialization of the repeated squaring algorithm (Section 5.2.3) to compute the hubs skeleton, using the intermediate results from the computation of partial vectors. Suppose , for , have been computed by the algorithm of Section 5.3.1, so that , for some error . We apply the repeated squaring algorithm on these results using for all successive iterations. As shown in the full version [7], after iterations of repeated squaring, the total error is bounded by . Thus, by varying and , can be computed to arbitrary precision.
Notice that only the intermediate results for are ever needed to update scores for , and of the former, only the entries , for , are used to compute . Since we are only interested in the hub scores , we can simply drop all nonhub entries from the intermediate results. The running time and storage would then depend only on the size of and not on the length of the entire hub vectors . If the restricted intermediate results fit in main memory, it is possible to defer the computation of the hubs skeleton to query time.
5.3.3 Web Skeleton
To compute the entire web skeleton, we modify the basic dynamic programming
algorithm (Section 5.2.1) to compute only the hub scores
, with corresponding savings in time and memory usage. We restrict
the computation by eliminating entries from the intermediate
results
, similar to the technique used in
computing the hubs skeleton.
The justification for this modification is that the hub score is affected only by the hub scores of the previous iteration, so that in the modified algorithm is equal to that in the basic algorithm. Since is likely to be orders of magnitude less than , the size of the intermediate results is reduced significantly.
5.4 Construction of PPV's
Finally, let us see how a PPV for preference vector can be
constructed directly from partial vectors and the hubs skeleton using the
Hubs Equation. (Construction of a single hub vector is a specialization of
the algorithm outlined here.) Let
be a preference vector, where for
. Let
, and let
which can be computed from the hubs skeleton. Then the PPV for can be constructed as
Both the terms and are partial vectors, which we assume have been precomputed. The term represents a simple subtraction from . If , then represents a full construction of . However, for some applications, it may suffice to use only parts of the hubs skeleton to compute to less precision. For example, we can take to be the hubs for which is highest. Experimentation with this scheme is discussed in Section 6.3. Alternatively, the result can be improved incrementally (e.g., as time permits) by using a small subset each time and accumulating the results.
6. Experiments
We performed experiments using real web data from Stanford's WebBase
[6], a crawl of the web containing 120 million pages. Since the
iterative computation of PageRank is unaffected by leaf pages (i.e.,
those with no outneighbors), they can be removed from the graph and added
back in after the computation [11]. After removing leaf pages, the
graph consisted of 80 million pages
Both the web graph and the intermediate results were too large to fit in main memory, and a partitioning strategy, based on that presented in [4], was used to divide the computation into portions that can be carried out in memory. Specifically, the set of pages was partitioned into arbitrary sets of equal size ( in our experiments). The web graph, represented as an edgelist , is partitioned into chunks ( ), where contains all edges for which . Intermediate results and were represented together as a list where and , for . Only pages for which either or were included. The set of intermediate results was partitioned into chunks , so that contains triples of for which and . In each of the algorithms for computing partial quantities, only a single column was kept in memory at any one time, and part of the nextiteration results were computed by successively reading in individual blocks of the graph or intermediate results as appropriate. Each iteration requires only one linear scan of the intermediate results and web graph, except for repeated squaring, which does not use the web graph explicitly.
6.1 Computing Partial Vectors
For comparison, we computed both (full) hub vectors and partial vectors for
various sizes of , using the selective expansion algorithm with (full hub vectors) and
(partial vectors). As discussed
in Section 4.4.2, we found the partial vectors approach
to be much more effective when contains highPageRank pages rather than
random pages. In our experiments ranged from the top to top
pages with the highest PageRank. The constant was set to
.
To evaluate the performance and scalability of our strategy independently of implementation and platform, we focus on the size of the results rather than computation time, which is linear in the size of the results. Because of the number of trials we had to perform and limitations on resources, we computed results only up to 6 iterations, for up to .
Figure 2 plots the average size of (full) hub vectors and partial vectors (recall that size is the number of nonzero entries), as computed after 6 iterations of the selective expansion algorithm, which for computing full hub vectors is equivalent to the basic dynamic programming algorithm. Note that the xaxis plots in logarithmic scale.Experiments were run using a 1.4 gigahertz CPU on a machine with 3.5 gigabytes of memory. For , the computation of full hub vectors took about seconds per vector, and about seconds for each partial vector. We were unable to compute full hub vectors for due to the time required, although the average vector size is expected not to vary significantly with for full hub vectors. In Figure 2 we see that the reduction in size from using our technique becomes more significant as increases, suggesting that our technique scales well with .
6.2 Computing the Hubs Skeleton
We computed the hubs skeleton for by running the selective
expansion algorithm for iterations using , and then running
the repeated squaring algorithm for iterations (Section
5.3.2), where is chosen to be the top 50 entries
under the top scheme (Section 5.2.2).
The average size of the hubs skeleton is entries. Each iteration of
the repeated squaring algorithm took about an hour, a cost that depends only
on and is constant with respect to the precision to which the partial
vectors are computed.
6.3 Constructing Hub Vectors from Partial Vectors
Next we measured the construction of (full) hub vectors from partial vectors
and the hubs skeleton. Note that in practice we may construct PPV's directly
from partial vectors, as discussed in Section 5.4.
However, performance of the construction would depend heavily on the user's
preference vector. We consider hub vector computation because it better
measures the performance benefits of our partial vectors approach.
As suggested in Section 4.3, the precision of the hub vectors constructed from partial vectors can be varied at query time according to application and performance demands. That is, instead of using the entire set in the construction of , we can use only the highest entries, for .
Figure 3 plots the average size and time required to construct a full hub vector from partial vectors in memory versus , for . Results are averaged over randomlychosen hub vectors. Note that the xaxis is in logarithmic scale.Recall from Section 6.1 that the partial vectors from which the hubs vector is constructed were computed using 6 iterations, limiting the precision. Thus, the error values in Figure 3 are roughly (ranging from for to for ). Nonetheless, this error is much smaller than that of the iteration full hub vectors computed in Section 6.1, which have error . Note, however, that the size of a vector is a better indicator of precision than the magnitude, since we are usually most interested in the number of pages with nonzero entries in the distribution vector. An iteration6 full hub vector (from Section 6.1) for page contains nonzero entries for pages at most 6 links away from , pages on average. In contrast, from Figure 3 we see that a hub vector containing 14 million nonzero entries can be constructed from partial vectors in 6 seconds.
7. Related Work
The use of personalized PageRank to enable personalized web search was first
proposed in [11], where it was suggested as a modification of the
global PageRank algorithm, which computes a universal notion of importance.
The computation of (personalized) PageRank scores was not addressed beyond
the naive algorithm.
In [5], personalized PageRank scores were used to enable ``topicsensitive'' web search. Specifically, precomputed hub vectors corresponding to broad categories in Open Directory were used to bias importance scores, where the vectors and weights were selected according to the text query. Experiments in [5] concluded that the use of personalized PageRank scores can improve web search, but the number of hub vectors used was limited to 16 due to the computational requirements, which were not addressed in that work. Scaling the number of hub pages beyond 16 for finergrained personalization is a direct application of our work.
Another technique for computing webpage importance, HITS, was presented in [9]. In HITS, an iterative computation similar in spirit to PageRank is applied at query time on a subgraph consisting of pages matching a text query and those ``nearby''. Personalizing based on userspecified web pages (and their linkage structure in the web graph) is not addressed by HITS. Moreover, the number of pages in the subgraphs used by HITS (order of thousands) is much smaller than that we consider in this paper (order of millions), and the computation from scratch at query time makes the HITS approach difficult to scale.
Another algorithm that uses querydependent importance scores to improve upon a global version of importance was presented in [12]. Like HITS, it first restricts the computation to a subgraph derived from text matching. (Personalizing based on userspecified web pages is not addressed.) Unlike HITS, [12] suggested that importance scores be precomputed offline for every possible text query, but the enormous number of possibilities makes this approach difficult to scale.
The concept of using ``hub nodes'' in a graph to enable partial computation of solutions to the shortestpath problem was used in [3] in the context of database search. That work deals with searches within databases, and on a scale far smaller than that of the web.
Some system aspects of (global) PageRank computation were addressed in [4]. The diskbased datapartitioning strategy used in the implementation of our algorithm is adopted from that presented therein.
Finally, the concept of inverse Pdistance used in this paper is based on the concept of expected distance introduced in [8], where it was presented as an intuitive model for a similarity measure in graph structures.
8. Summary
We have addressed the problem of scaling personalized web search:
 We started by identifying a linear relationship that allows personalized PageRank vectors to be expressed as a linear combination of basis vectors. Personalized vectors corresponding to arbitrary preference sets drawn from a hub set can be constructed quickly from the set of precomputed basis hub vectors, one for each hub .
 We laid the mathematical foundations for constructing hub vectors efficiently by relating personalized PageRank scores to inverse Pdistances, an intuitive notion of distance in arbitrary directed graphs. We used this notion of distance to identify interrelationships among basis vectors.
 We presented a method of encoding hub vectors as partial vectors and the hubs skeleton. Redundancy is minimized under this representation: each partial vector for a hub page represents the part of 's hub vector unique to itself, while the skeleton specifies how partial vectors are assembled into full vectors.
 We presented algorithms for computing basis vectors, and showed how they can be modified to compute partial vectors and the hubs skeleton efficiently.
 We ran experiments on real web data showing the effectiveness of our approach. Results showed that our strategy results in significant resource reduction over full vectors, and scales well with , the degree of personalization.
9. Acknowledgment
The authors thank Taher Haveliwala for many useful discussions and extensive help with implementation.
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About this paper
This work was supported by the National Science Foundation under grant IIS9817799. This is an abbreviated version of the full paper that omits appendices. The full version is available on the web at http://dbpubs.stanford.edu/pub/200212.Copyright is held by the author/owner(s).
WWW2003, May 2024, 2003, Budapest, Hungary.
ACM 1581134495/02/0005
Footnotes
 [footnote 1]
 Specifically, corresponds to the steadystate distribution of an ergodic, aperiodic Markov chain.
 [footnote 2]
 More precisely, the transformation from personalization vectors to their corresponding solution vectors is linear.
 [footnote 3]
 Note that while the mathematics and computation strategies in this paper are presented in the specific context of the web graph, they are general graphtheoretical results that may be applicable in other scenarios involving stochastic processes, of which PageRank is one example.
 [footnote 4]
 The definition here of inverse Pdistance differs slightly from the concept of expected distance in [8], where tours are not allowed to visit multiple times. Note that general expected distances have the form ; in our definition, .
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