Incremental Page Rank Computation on Evolving Graphs : NOTES

Incremental Page Rank Computation on Evolving Graphs
Prasanna Desikan
Dept. of Computer Science
University of Minnesota
Minneapolis, MN 55455
USA
desikan@cs.umn.edu
Nishith Pathak
USA
npathak@cs.umn.edu
Jaideep Srivastava
USA
srivasta@cs.umn.edu
Vipin Kumar
USA
kumar@cs.umn.edu
ABSTRACT
Link Analysis has been a popular and widely used Web mining
technique, especially in the area of Web search. Various ranking
schemes based on link analysis have been proposed, of which the
PageRank metric has gained the most popularity with the success
of Google. Over the last few years, there has been significant
work in improving the relevance model of PageRank to address
issues such as personalization and topic relevance. In addition, a
variety of ideas have been proposed to address the computational
aspects of PageRank, both in terms of efficient I/O computations
and matrix computations involved in computing the PageRank
score. The key challenge has been to perform computation on
very large Web graphs. In this paper, we propose a method to
incrementally compute PageRank for a large graph that is
evolving. We note that although the Web graph evolves over
time, its rate of change is rather slow. When compared to its size.
We exploit the underlying principle of first order markov model
on which PageRank is based, to incrementally compute PageRank
for the evolving Web graph. Our experimental results show
significant speed up in computational cost, the computation
involves only the (small) portion of Web graph that has
undergone change. Our approach is quite general, and can be used
to incrementally compute (on evolving graphs) any metric that
satisfies the first order Markov property.
Keywords
Link Analysis, Web Search, PageRank, Incremental Algorithms
1. INTRODUCTION
The importance of link analysis on the Web graph has gained
significant prominence after the advent of Google [1]. The key
observation is that a hyperlink from a source page to a destination
page serves as an endorsement of the destination page by the
(author of the) source page on some topic. This idea has been
exploited by various researchers and has resulted in a variety of
hyperlink based ranking metrics for ranking of Web Pages.
Kleinberg’s Hubs and Authority [2] and Google’s Pagerank [3]
are the most popular among such metrics. A variety of
modifications and improvements to these approaches have been
developed in recent years[6,7,8,9,10].
Link analysis techniques have adopted different knowledge
models for the measures developed for various applications on the
Web [15]. Kleinberg’s Hubs and Authority is based on the
observation that the Web graph has a number of bipartite cores
[2], while Google’s PageRank is based on the observation that a
user’s browsing of the Web can be approximated as a first order
markov model [3]. Giles, et al [5] have used network flow models
to identify web communities. Thus, a variety of models have been
used to measure different properties of the Web Graph at a given
time instance. Success of Google has signified the importance of
Pagerank as a ranking metric. This has also led to a variety of
modifications and improvisations of the basic PageRank metric.
These have either focused on changing the underlying model or
on reducing the computation cost.
Another important dimension of Web mining is the evolution of
the Web graph [4]. The Web is changing over time, and so is the
users’ interaction on (and with) the Web, suggesting the need to
study and develop models for the evolving Web Content, Web
Structure and Web Usage. The study of such evolution of the
Web would require computing the various existing measures for
the Web graph at different time instances. A straightforward
approach would be to compute these measures for the whole Web
Graph at each time instance. However, given the size of the Web
graph, this is becoming increasingly infeasible. Furthermore,, if
the percent of nodes that change during a typical time interval
when the Web is crawled by search engines is not high, a large
portion of the computation cost may be wasted on re-computing
the scores for the unchanged portion. Hence, there is a need for
computing metrics incrementally, to save on the computation
costs.
Techniques for incremental computations, to study changes in
graph structure over time, would depend on the underlying
knowledge model that defines a metric [15]. For example, the
computation of hub and authority scores is based on mutual
reinforcement of nodes, and hence a change in the indegree or
outdegree of a node may affect its score. Mutual reinforcement
makes hub and authority scores a second order model. However,
for PageRank whose random surfer model is based on the first
order markov property, the change in out degree of the node does
not affect the score of the node. Hence, the level of penetration of
change in scores due to a change in the degree of a node is not as
high in PageRank as in hub and authority scores.
In this paper, we describe an approach to compute PageRank in
an incremental fashion. We exploit the underlying first order
markov model1
property of the metric, to partition the graph
1
The property that the PageRank score of a page depends only on the
PageRank scores of the pages pointing to it.
Copyright is held by the author/owner(s).
WWW 2005, May 10--14, 2005, Chiba, Japan.

into two portions such one of them is unchanged since the last
computation, and it has only outgoing edges to the other partition.
Since there are no coming edges from the other partition, the
distribution of PageRank values of the nodes in this partition will
not be affected by the nodes in the other partition. The other
partition is the rest of the graph, which has undergone changes
since the last time the metric was computed. Figure 1 gives an
overview of our approach and explains the difference between
related work and the work in this paper. This paper is organized
as follows. In Section 2, we give an introduction to the basic
PageRank metric and the various issues involved in its
computation. Section 3 gives an overview of our approach to
incrementally compute PageRank for evolving Web graphs. We
describe the Incremental PageRank Algorithm in Section 4 and
present our experimental results in Section 6. Section 7 discusses
the related work and places our work in context. Finally, in
Section 8 we conclude the and provide directions for future work.
Figure 1. Overview of the Proposed Approach.
2. PAGERANK
PageRank is a metric for ranking hypertext documents that
determines their quality. It was originally developed by Page et
al. [3] for the popular search engine, Google [1]. The key idea is
that a page has high rank if it is pointed to by many highly ranked
pages. Thus, the rank of a page depends upon the ranks of the
pages pointing to it. The rank of a page p can thus be written as:
Here, n is the number of nodes in the graph and OutDegree(q) is
the number of hyperlinks on page q. Intuitively, the approach can
be viewed as a stochastic analysis of a random walk on the Web
graph. The first term in the right hand side of the equation
corresponds to the probability that a random Web surfer arrives at
a page p from somewhere, i.e. (s)he could arrive at the page by
typing the URL or from a bookmark, or may have a particular
page as his/her homepage. d would then be the probability that a
random surfer chooses a URL directly – i.e. typing it, using the
bookmark list, or by default – rather than traversing a link.
Finally, 1/n is the uniform probability that a person chooses page
p from the complete set of n pages on the Web. The second term
in the right hand side of the equation corresponds to a factor
contributed by arriving at a page by traversing a link. 1- d is the
probability that a person arrives at the page p by traversing a link.
The summation corresponds to the sum of the rank contributions
made by all the pages that point to the page p. The rank
contribution is the PageRank of the page multiplied by the
probability that a particular link on the page is traversed. So for
any page q pointing to page p, the probability that the link
pointing to page p is traversed would be 1/OutDegree(q),
assuming all links on the page is chosen with uniform probability.
Figure 2 illustrates an example of computing PageRank of a page
P from the pages, P1, P2, P3 pointing to it.
There are other computational challenges that arise in PageRank.
Apart from the issue of scalability, the other important
computational issues are the convergence of PageRank iteration
and the handling of dangling nodes. The convergence of
PageRank is guaranteed only if the Web graph is strongly
connected and is aperiodic. To ensure the condition of strong
connectedness, the dampening factor is introduced, which assigns
a uniform probability to jumping to any page. In a graph theoretic
sense it is equivalent of adding an edge between every pair of
vertices with a transition probability of d/n. The aperiodic
property is also guaranteed for the Web graph.
Another important issue in computation of PageRank is the
handling of dangling nodes. Dangling nodes are nodes with no
outgoing edge. These nodes tend to act as rank sink, as there is no
way for rank to be distributed among the other nodes. The
suggestion made initially to address this problem, was to
iteratively remove all the nodes that have an outdegree of zero,
−
+
=
G
p
q
q
PR
p
)
,
(
)
(
)
)
1 ∑
∈
⋅
q
OutDegree
d
n
d
PR (
(
)
( (1)
G1
Link Analysis
PageRank
G2
Link Analysis
PageRank
Diff(G1,G2)
G2
’
G2
’’
Scale
Merge
Partition(G2)
Incremental Approach (this paper)
Link Analysis

and compute the PageRank on the remaining nodes [3]. The
reasoning here was that dangling nodes do not affect the
PageRank of other nodes. Another suggested approach was to
remove the dangling nodes while computation initially and add
them back during the final iterations of the computation [7]. Other
popular approaches to handling dangling nodes, is to add self
loops to dangling nodes[11,20] and to add links to all nodes in the
graph, G from each of the dangling node to distribute the
PageRank of the dangling node uniformly among all nodes[3].
Figure 2. Illustrative example of PageRank.
3. PROPOSED APPROACH
In the proposed approach, we exploit the underlying first order
Markov Model on which the computation of PageRank is based.
It should be noted that PageRank of a page depends only on the
pages that point to it and is independent of the outdegree of the
page. The principle idea of our approach is to find a partition such
that there are no incoming links from a partition, Q (includes all
changed nodes) to a partition, P. In such a case the PageRank of
the partition, Q is computed separately and later scaled and
merged with the rest of the graph to get the actual PageRanks of
vertices in Q. The scaling is done with respect to the number of
vertices in partition, P-`n(P) to the total number of nodes in the
whole graph, G –n(P UQ)=V. The PageRank of the partition Q is
computed, taking the border vertices that belong to the partition P
and have edges pointing to the vertices in partition Q. The
PageRank values of partition P are obtained by simple scaling.
This basic idea of partitioning the Web graph, and computing the
PageRanks for individual partitions and merging works extremely
well when incrementally computing PageRank for a Web graph
that has evolved over time. Given, the Web graphs at two
consecutive time instances, we first determine the portion of the
graph that has changed. A vertex is declared to be changed when
a new edge added or deleted between the vertex and any other
vertex belonging to the graph or if the weight of a node or an edge
weight adjoined to that node has changed. Once the changed
portion is defined, for each page we determine iteratively all the
pages that will be affected by its PageRank. In this process, we
include pages that remain unchanged but whose; PageRank gets
affected due to the pages that have changed. in partition Q. The
rest of the unchanged graph is in partition P.
The whole concept is illustrated in Figure 3. Let the graph at the
new time be G(V,E), and
=
b
v Vertex on the border of the left partition from which there
are only outgoing edges to the right partition.
unchanged
remains
which
partition
left
on the
vertex
=
ul
v
The set of unchanged vertices can be represented as,
{ }
V
u
v
u
v
u
V ∈
∀
= , where
u
v is a vertex which has not
changed.
=
ur
v Vertex on the right partition which remains unchanged,
but whose PageRank is affected by vertices in the changed
component.
=
cr
v Vertex on the right partition which has changed, or has
been a new addition.
Therefore, the set of changed vertices can be represented as,
{ }
V
cr
v
cr
v
c
V ∈
∀
= ,
In order to compute PageRank incrementally, for every vertex
in
c
V , which is a set of changed vertices, perform a BFS to find
out all vertices reachable from this set. The PageRank of these
vertices will be affected by vertices in
cr
V . These set of vertices
can be denoted by the set,
{ }
V
ur
v
ur
v
ur
V ∈
∀
= ,
Similarly, the set of vertices
b
v can be denoted as,
{ }
V
b
v
b
v
b
V ∈
∀
= ,
Hence the set of vertices whose PageRank has to be computed in
the incremental approach corresponds to the partition Q described
above, and can be denoted as,
b
V
ur
V
c
V
Q
V U
U
=
Let an edge set,
Q
E , be defined as set of edges,
{ }
Q
V
y
x
y
x
e
Q
E ∈
= ,
|
,
, where
y
x
e
,
represents a directed
edge from vertex x to vertex y.
The set of partitioning edges can be defined as,
{ }
Q
V
y
P
V
x
y
x
e
Part
E ∈
∈
= ,
|
,
,
P1
P2
P3
P
)
1
(
1
P
OutDeg
)
2
(
1
P
OutDeg
)
3
(
1
P
OutDeg
N
d
P1
P2
P3
P
)
1
(
1
P
OutDeg
)
2
(
1
P
OutDeg
)
3
(
1
P
OutDeg
N
d
d/N








+
+
−
+
=
)
3
(
)
3
(
)
2
(
)
2
(
)
1
(
)
1
(
)
1
(
)
(
P
OutDeg
P
PR
P
OutDeg
P
PR
P
OutDeg
P
PR
d
N
d
P
PR

Figure 3. Incremental Computation of PageRank.
The vertices in partition P can be defined as,
b
V
Q
V
V
P
V +
−
=
And the edges that correspond to this partition can be defined as,
{ }
P
V
y
x
y
x
e
P
E ∈
= ,
|
,
, where
y
x
e
,
represents a directed
edge from vertex x to vertex y.
Thus, the given graph ( )
E
V
G , can be partitioned into two
graphs namely, ( )
P
E
P
V
P
G , and 





Q
E
Q
V
Q
G , .
Now, since we know that the graph ( )
P
E
P
V
P
G , has remained
unchanged from the previous time instance and the PageRank of
vertices in this partition is not affected by the partition,






Q
E
Q
V
Q
G , . Now a change in a node induces a change in
the distribution of PageRank values for all its children and since
all the nodes that are influenced by changes are already separated
in the partition Q. The distribution of PageRank values for the
nodes in partition GP is going to be the same as it was for the
corresponding nodes in the previous time instance G’. Thus the
PageRank of the vertices in partition P could be calculated by
simply scaling the scores from the previous time instance. And
the scaling factor will be
( )
( )
G
n
G
n ′
, where G′ is the graph at
the previous time instance. And the PageRank for the partition,






Q
E
Q
V
Q
G , can be computed using the regular PageRank
Algorithm and scaled for the size of the graph, G. Since the
percent change in the structure of the Web is not high, the
computation of the changed portion will be a smaller graph
compared to the whole Web. And the existence of such partitions
is also suggested by the bow-tie model of the Web [12], where
about 27% of Web contributes to the influx. It should also be
noted that while computing PageRanks for the changed portion, in
order to maintain the stochastic property of the incremental
matrix, we have to scale the PageRanks of nodes in Vb such that
they correspond to the number of nodes for which the PageRank
is actually computed. Also taken into account is the outdegree of
these border nodes that have edges in partition P, since the way
they distribute their PageRanks to nodes in partition Q, will
depend on their outdegree.
4. INCREMENTAL ALORITHM
In this section, we will describe the incremental algorithm to
compute PageRank. The initial step is to read the graph at a new
instance and determine the vertices that have changed. This does
not require additional time as it can be computed as we read the
new graph. Thus, after reading the graph, we can assume that we
are given two sets of vertices – one containing the vertices which
ul
v
b
v
ur
v
cr
v
Vertex on the border of left
partition from which there are
outgoing edges to the right
partition
Vertex on left partition, which
remains unchanged
Vertex on right partition, which
remains unchanged but whose
PageRank gets affected by
vertices in changed partition
Vertex on right partition, which
has changed or has been added.
Partition
The distribution of
PageRanks of the vertices in
the left remains unchanged,
since their structure has not
changed and there are no
incoming links from the right
partition.
The distribution of
PageRanks of the vertices
changes and also depends
on the PageRanks of the
border nodes of the left
partition.
Vertices whose edges have
not changed
Vertices whose edges have
not changed

have changed from a previous time instance and the other
containing vertices that have remain unchanged. Hence, the input
to the algorithm is the graph G, and the two lists Vc and Vu.. The
outline of the algorithm is shown in Figure 3. We will describe
each step briefly:
Step 1 - Initialize a list VQ
Step 2 - A change in a vertex induces a change in the PageRank
distribution of all its children. All such changed vertices are in the
queue Vc. In this step, the list of “changed vertices” is extended to
a partition to include all descendents of the initial list of “changed
vertices”. All these vertices are pushed into the list Q2.
Step 3 – For the remaining vertices are there is no change in their
PageRank distribution. The New PageRank is simply obtained by
scaling the previous PageRank scores. The scaling factor is
simply:
( )
( )
G
n
G
n ′
=Order of graph at previous time instance/Order of
the graph at the present time instance.
Also all those vertices from this set of unchanged vertices that
point to a changed vertex, will influence the PageRank value of
that changed vertex, hence these too must be included in the list
VQ as their PageRank scores will be required for computing the
PageRank scores for the changed vertices.
Step 4 – Now original PageRank computation algorithm along
with steps taken to ensure stochastic property of transition matrix
is performed on the nodes that are in Q2 and colored violet (i.e.
nodes which have changed) to get the new PageRank values for
these changed nodes.
Thus, we end up localizing the changed partition to a certain sub-
graph of the web which consists of all changed nodes and then
basic PageRank algorithm is performed only on this changed sub-
graph. The PageRank value for the rest of the nodes is simply a
matter of scaling the previous values.
Figure 4. Incremental PageRank Algorithm.
Step 2 has a cost of E’, where Part
Q E
E
E −
=
′ , is the number
of edges in the partition Q. Now the PageRank values for the
partition P are obtained by scaling the PageRank values with
respect to ranks in the previous time instance. This step requires a
cost of V’’, where V’’ is number of vertices in partition P. Now
using these scaled values and the naïve approach PageRank for
the vertices in partition Q is calculated. This step (including that
required to scale the border nodes) requires a cost of nE + E’ +Vb,
where n is number of iterations required for PageRank values to
converge and E’ is again number of edges in partition Q. Thus,
the total cost for incremental PageRank can be summed up to be
O(2E’+V’’+nE+Vb).
5. EXPERIMENTAL RESULTS
To test our theoretical approach on real datasets, we needed
graphs at two different time instances to compute the incremental
version. We performed the experiments on two different web
sites- the Computer Science website and the Institute of
Technology website at the University of Minnesota. We
performed the experiments at different time intervals to study the
change and effect of the incremental computation. For the
Computer Science website our analysis was done at a time
interval of two days, eight days and ten days. We also performed
the analysis for a time interval of two days for the Institute of
technology web site.
In our experiments we also simulated the focused crawling, by not
considering the Web pages that have very low PageRank into our
graph construction and PageRank Computation. This was to
emulate the real world scenario where not all pages are crawled.
We wanted to analyze, how the incremental approach performs
when pages with low PageRank are not crawled.
We used the following approximate measure to compare the
computational costs of our method versus the naïve method.
Number of Times Faster = Num of Iterations(PR)/(1 + (fraction
of changed portion)*Number of iterations(IPR))
The intuition behind the measure was how fast the convergence
threshold will be reached computing PageRank incrementally
versus computing PageRank in a naïve method for the whole
graph. The convergence threshold that was chosen on our
experiments was 1x10-8
The experimental results are presented in Figure 5. These results
are from actual experiments conducted on the Computer Science
and Institute of Technology websites. For the Computer Science
website, in the first time interval of eight days, there seemed to be
a significant change in the structure of the Website – about 60%
of the pages had changed their link structure. We found out such a
sea change occurred because a whole subgraph that contained the
documentation for Matlab help was removed. The incremental
approach still however, performed 1.86 as much faster as the
naïve PageRank. Similarly, for a period of ten days the
incremental approach performed around 1.75 times faster. For a
period of two days the improvement was 8.65 times faster. These
results are for the case of an unfocussed crawl. The results for
focused crawl for the CS Website were better. In the first case,
when the time interval was eight days, the improvement was 1.9
times and when the time interval was 10 days, the improvement
was 1.76 times. For a period of two days the improvement with
IPR(G,Vu,Vc) :-
Step 1 – Initialize the list VQ
Step 2 – Pop a Vertex N from Vc
2.1 For all the children of N
if children of N list Vu
remove them from Vu
push them in Vc
2.2. Push N in VQ and repeat step 2 till
queue Vc is empty
Step 3 – For each element in list Vu
3.1 Take the element and scale the
previous pagerank value to get new
pagerank value.
3.2 Look up whether any of the children, of the
element of Vu belong to VQ, if so remove this
element of Vu, copy it in Vb .
Step 4 – Scale Border Nodes in Vb for stochastic
property
Perform Original PageRank(VQ Vb)
U
∈

focused crawling was 9.88 times. Thus, it suggests that focused
crawling can also improve the computational costs of the
incremental algorithm.
Computer Science Website
Focussed Crawl
July19 vs July 27th
percentage of change = 53.1429% L1 -norm : 4.38609e-05 NumTimes faster= 1.900538
10 iteration(s) for inc_pagerank
12 iteration(s) for actual pagerank
July 27th vs July 29th
percentage of change = 5.25071% L1-norm : 1.60988e-07 NumTimes faster= 9.885481
July19th vs 29th
percentage of change = 58.3493% L1-norm : 4.38692e-05 NumTimes faster= 1.755669
Unfocussed Crawl
July19 vs July 27th
percentage of change = 60.2997% norm : 1.70552e-07 NumTimes faster= 1.867123
Institute of Technology Website
Unfocussed/Focussed Crawl
July 30th vs Aug 1st
percentage of change = 0% norm : 8.15708e-07 NumTimes faster= 11
Figure 5. Comparison of results for Incremental PageRank Algorithm versus Naïve PageRank Algorithm for the following
departments at the University: (a) Computer Science Website, (b) Institute of Technology Website.
The Institute of technology website typically represented a
website that doesn’t change too often. The change over a period
of two days in the Web Structure was none. Since there was no
change detected, there was no necessity to compute the PageRank
for the graph at the new time instance. And by our measure, it was
11 times faster. Since, there was no change in the graph structure,
the improvements for the case of focused crawling and
unfocussed crawling remain the same.
6. RELATED WORK
Determining the quality of a page has been the primary focus of
Web mining research community and various measures and
metrics have been developed for the same for different
applications. PageRank [3] was developed by Google founders,
for ranking hypertext documents. The overall idea is described in
detail in Section 2. The other popular metric based on link
analysis is hub and authority scores. From a graph theoretic point
(a)
(b)

of view, hubs and authorities can be interpreted as ‘fans’ and
‘centers’ in a bipartite core of a Web graph. The hub and
authority scores computed for each Web page indicate the extent
to which the Web page serves as a ‘hub’ pointing to good
‘authority’ pages or the extent to which the Web page serves as an
‘authority’ on a topic pointed to by good hubs. The hub and
authority scores for a page are not based on a formula for a single
page, but are computed for a set of pages related to a topic using
an iterative procedure, namely HITS algorithm [2]. A detail study
of link analysis techniques can be found in [13, 14, 15, 21, 22].
There have been a number of extensions of the basic PageRank
that have been proposed, such as including the topic information
of page to determine the topic relevance. One approach [17] was
to precalculate different PageRank vectors for a given number of
terms, focusing on the subset of pages that contain the term of
interest. The search results for a query would be ranked according
to scores that were precalculated for the collection of terms that
contain the query words. Another approach for introducing topic
relevance was addressed by Haveliwala et al [9]. In the approach,
PageRank is calculated for all pages according to each category of
the Open directory project. The pages that belong to a particular
category have higher scores for the PageRank values computed
for that category. Ranking of results of a search query is done
according to scores of the category in which the query terms
belong to. Oztekin et al [16], proposed Usage Aware PageRank.
Their modified PageRank metric incorporates usage information.
Weights are assigned to a link based on number of traversals on
that link, and thus modifying the probability that a user traverses a
particular link. Also the probability to arrive at a page directly is
computed using the usage statistics.
There has been a variety of work on improving on the PageRank
computation. I/O efficient techniques for computing PageRank
has been addressed by Haveliwala et al [8] and Yen Yu Chen[10].
The basic of their approach is to partition the link file of the
whole graph into partitions such that destination vector of each
partition fits into the main memory. Kamvar et al in [6] have
suggested quadratic extrapolation techniques to accelerate the
convergence of PageRank. In a different paper [7], they have also
suggested a way of exploiting the block-structure of the Web to
compute Block Ranks for different domains and compute local
PageRanks. Chein et al [19] have also exploited the idea of
evolving graph to compute PageRank. However, their idea is to
collapse the unchanged portion to a single node and compute the
PageRank for the new graph. This leads to approximate PageRank
values.
In out paper, we provide an approach to incrementally compute
PageRanks. We do so by exploiting the underlying first order
Markov model on which PageRank is based and partition the
graph in such a manner so that we compute the exact PageRank
values for a graph at a new time instance. Incremental
computations are very useful to study the evolution of graphs. The
significance of to study the temporal behavior of graph is
addressed in our earlier paper [4].
7. CONCLUSIONS AND FUTURE
DIRECTIONS
In this paper we have provided an approach to compute PageRank
incrementally for evolving graphs. The key observation is that
evolution of the Web graph is slow, with large parts of it
remaining unchanged. By carefully delineating the changed and
unchanged portions, and the dependence across them, it is
possible to develop efficient algorithms for computing the
PageRank metric incrementally. We follow a generic approach
that can be applied to any algorithm that has been developed for
efficient computation of the PageRank metric. Experimental
results show significant speed up in computation of PageRank
using our approach as compared to naive approach. Also, in the
incremental approach, if the partitioned sub-graph that has
changed is small, the whole PageRank computation might perhaps
be performed in main memory.
Many issues remain to be investigated. In this paper we have
proposed an incremental approach that applies to graph metrics
based on first order Markov model, such as PageRank. An area to
explore is for similar incremental approaches for other link based
metrics. We have provided a method for an efficient incremental
computation of relevance metric for a single node level. However,
to study graph evolution, we would need measures and metrics
defined at the level of a subgraph and a whole graph, and efficient
methods to incrementally compute them
8. ACKNOWLEDGMENTS
We would like to thank Data Mining Research group at the
Department of Computer Science for providing valuable
suggestions. This work was been partially supported by the
ARDA Agency under contract F30602-03-C-0243 and Army
High Performance Computing Research Center contract number
DAAD19-01-2-0014. The content of the work does not
necessarily reflect the position or policy of the government and no
official endorsement should be inferred. Access to computing
facilities was provided by the AHPCRC and the Minnesota
Supercomputing Institute.
9. REFERENCES
[1] http://www.google.com
[2] J.M. Kleinberg, “Authoritative Sources in Hyperlinked
Environment”, 9th
Annual ACM-SIAM Symposium on
Discrete Algorithms, pages 668-667, 1998
[3] L. Page, S. Brin, R. Motwani and T. Winograd “The
PageRank Citation Ranking: Bringing Order to the
Web” Stanford Digital Library Technologies, January
1998.
[4] P. Desikan and J. Srivastava, "Mining Temporally
Evolving Graphs", WebKDD Workshop, Seattle
(2004).
[5] Gary William Flake, Steve Lawrence, C. Lee Giles .
Efficient Identification of Web Communities. Sixth
ACM SIGKDD International Conference on
Knowledge Discovery and Data Mining. August 2000,
pp. 150-160.
[6] Sepandar D. Kamvar, Taher H. Haveliwala,
Christopher D. Manning, and Gene H. Golub,
"Extrapolation Methods for Accelerating PageRank
Computations." In Proceedings of the Twelfth

International World Wide Web Conference, May,
2003.
[7] Sepandar D. Kamvar, Taher H. Haveliwala,
Christopher D. Manning, and Gene H. Golub,
"Exploiting the Block Structure of the Web for
Computing PageRank." Preprint, March, 2003
[8] Taher Haveliwala. "Efficient Computation of
PageRank," Stanford University Technical Report,
September 1999.
[9] Taher Haveliwala. "Topic-Sensitive PageRank," In
Proceedings of the Eleventh International World Wide
Web Conference, May 2002
[10]Y. Chen, Q. Gan, and T. Suel. I/O-efficient techniques
for computing pagerank. In Proc. of the 11th
International Conf. on Information and Knowledge
Management, pages 549--557, November 2002.
[11]G. Jeh and J. Widom. Scaling personalized web
search. In 12th Int. World Wide Web Conference,
2003.
[12]R. Kumar, P. Raghavan, S. Rajagopalan, D.
Sivakumar, A. Tomkins, and E. Upfal. The Web as a
graph. In ACM SIGMOD-SIGACT-SIGART
Symposium on Principles of Database Systems, pages
1-10, 2000.
[13]Kemal Efe, Vijay Raghavan, C. Henry Chu, Adrienne
L. Broadwater, Levent Bolelli, Seyda Ertekin (2000),
The Shape of the Web and Its Implications for
Searching the Web, International Conference on
Advances in Infrastructure for Electronic Business,
Science, and Education on the Internet- Proceedings at
http://www.ssgrr.it/en/ssgrr2000/proceedings.htm,
Rome. Italy, Jul.-Aug. 2000.
[14]Monika Henzinger, Link Analysis in Web Information
Retrieval, ICDE Bulletin Sept 2000, Vol 23. No.3.
[15]P. Desikan, J. Srivastava, V. Kumar, P.-N. Tan,
“Hyperlink Analysis – Techniques & Applications”,
Army High Performance Computing Center Technical
Report, 2002.
[16]B.U. Oztekin, L. Ertoz and V. Kumar, “Usage Aware
PageRank”, World Wide Web Conference, 2003.
[17]M. Richardson and P. Domingos. The intelligent
surfer: Probabilistic combination of link and content
information in pagerank. In Advances in Neural
Information Processing Systems, 2002.
[18]J. Srivastava, P.Desikan and V.Kumar. "Web Mining -
Concepts, Applications and Research Directions."
Book Chapter in Data Mining: Next Generation
Challenges and Future Directions, MIT/AAAI 2004.
[19]S. Chien, C. Dwork, S. Kumar, and D. Sivakumar.
Towards exploiting link evolution. Unpublished
manuscript, 2001.
[20]Eiron, N., McCurley, K., Tomlin, J.: Ranking the Web
frontier. In: Proc. 13th conference on World Wide
Web, ACM Press (2004) 309—318
[21]S.Acharyya and J.Ghosh, “A Maximum Entropy
Framework for Link Analysis on Directed Graphs”, in
LinkKDD2003, pp 3-13, Washington DC, USA, 2003
[22]C. Ding, H. Zha, X. He, P. Husbands and H.D. Simon,
“Link Analysis: Hubs and Authorities on the World
Wide Web” May 2001. LBNL Tech Report 47847.

10. APPENDIX: PROOF OF ALGORITHM’S
SCALABILITY
Consider a Graph )
,
( E
V
G
Let order of graph G be n
Let weight of a node i
v be i
w , V
vi ∈
∀
Also, W
w
n
i
i =
∑
=1
, sum of the weights of all nodes.
PageRank score calculation is analogous to the convergence of a
first order Markov chain.
For calculation of PageRank we perform the operation,
1
+
= i
i M
M
T
over a number of iterations. Here, T is the transition matrix and
i
M is the PageRank score vector at the end of ith
iteration.
We also have initial PageRank score vector,


























=
W
w
W
w
W
w
M
n
.
.
.
.
.
2
1
0
For a node s we have,
∑
=
−
+
=
1
1
1 1
1
)
(
deg
)
(
)
1
(
)
(
)
(
k
i i
i
f
s
f x
ree
Out
x
PR
d
W
w
d
s
PR
where, f
d is the dampening factor.
1
,...,
2
,
1
1
1 k
i
xi =
∀ are all those nodes that have at
least one outgoing edge to s
For the sake of representation let us assume that all those nodes
that have outgoing edge(s) to a node im
x are represented as
)
1
( +
m
i
x .
Now, if l iterations are required for convergence towards the
PageRank score vector then we have,
∑
=
−
+
=
1
1
1 1
1
)
(
deg
)
(
)
1
(
)
(
)
(
k
i i
i
f
s
f x
ree
Out
x
PR
d
W
w
d
s
PR
∑
∑ ∑
=
= =
⋅
−
+
−
+
=
1
1
1 1
2
1
2 1
2
)
(
deg
)}
(
/
)..}
)
(
deg
(
{..
)
1
(
)
(
{
)
1
(
)
(
)
(
1
k
i i
k
i
kl
il
i
il
x
f
x
f
f
s
f
x
ree
Out
x
Outdergee
x
ree
Out
W
w
d
W
w
d
d
W
w
d
s
PR
il
i
Notice that the term W can be taken out as common. Thus, we
have,
]
)
(
deg
)}
(
deg
/
)..}
)
(
deg
(
{..
)
1
(
{
)
1
(
[
1
)
(
1
1
1 1
2
1
2 1
2
1
∑
∑ ∑
=
= =
−
+
−
+
=
k
i i
k
i
kl
il
i
il
x
f
x
f
f
f
x
ree
Out
x
ree
Out
x
ree
Out
w
d
w
d
d
d
W
s
PR
il
i
W
s
PR
α
=
∴ )
(
where,
=
α
]
)
(
deg
)}
(
deg
/
)..}
)
(
deg
(
{..
)
1
(
{
)
1
(
[
1
1
1 1
2
1
2 1
2
1
∑
∑ ∑
=
= =
−
+
−
+
k
i i
k
i
kl
il
i
il
x
f
x
f
f
f
x
ree
Out
x
ree
Out
x
ree
Out
w
d
w
d
d
d
il
i
Now, suppose graph )
,
( E
V
G changes to )
,
( '
'
'
E
V
G
However the changes that occur are such that there is no change
in the structure and weight of the of the node s as well as no
change in the in the structure and weight of all the ancestors of the
node s ……………… [condition (1)].
Now following along the lines of the previous graph we have, for
the new graph
'
G
'
'
'
)
(
W
s
PR
α
=
The terms α and
'
α depend mainly only on the structure of the
graph and the weights of, node s and its ancestors, from
condition(1) we can see that both of them are identical in the
graphs G and
'
G for the node s , assuming that the same
dampening factor is used for both the pagerank calculations.
Thus,
'
α
α =
)
(
)
( '
'
s
PR
W
s
PR
W ⋅
=
⋅
∴
)
(
)
(
)
( '
'
s
PR
W
W
s
PR =
⇒ ………….result(1)
Thus, the new pagerank score of any node x in
'
G is simply
obtained by scaling the pagerank of score of the same node x in
G by a factor of )
( '
W
W provided that condition(1) holds
for the node x and the same dampening factors are used for both
pagerank calculations.
In most cases all nodes of a graph are equally likely here we have,
1
....
3
2
1 =
=
=
=
= n
w
w
w
w
)
(G
n
W =
∴ ,
where )
(G
n is the order of the graph G .
When this graph changes to a graph
'
G we have result(1); in
this case as,
)
(
)
(
)
(
)
( '
'
x
PR
G
n
G
n
x
PR =
From the above result it is also trivial to deduce that a change in a
node influences the pagerank values of all its descendents.

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