The Missing Link - A Probabilistic Model of 
Document Content and Hypertext Connectivity 
David Cohn 
Burning Glass Technologies 
201 South Craig St, Suite 2W 
Pittsburgh, PA 15213 
david. cohn @ burn ing-glass. corn 
Thomas Hofmann 
Department of Computer Science 
Brown University 
Providence, RI 02192 
th@cs. brown. edu 
Abstract 
We describe a joint probabilistic model for modeling the contents and 
inter-connectivity of document collections such as sets of web pages or 
research paper archives. The model is based on a probabilistic factor 
decomposition and allows identifying principal topics of the collection 
as well as authoritative documents within those topics. Furthermore, the 
relationships between topics is mapped out in order to build a predictive 
model of link content. Among the many applications of this approach are 
information retrieval and search, topic identification, query disambigua- 
tion, focused web crawling, web authoring, and bibliometric analysis. 
1 Introduction 
No text, no paper, no book can be isolated from the all-embracing corpus of documents 
it is embedded in. Ideas, thoughts, and work described in a document inevitably relate 
to and build upon previously published material. 1 Traditionally, this interdependency has 
been represented by citations, which allow authors to explicitly make references to related 
documents. More recently, a vast number of documents have been "published" electron- 
ically on the world wide web; here, interdependencies between documents take the form 
of hyperlinks, and allow instant access to the referenced material. We would like to have 
some way of modeling these interdependencies, to understand the structure implicit in the 
contents and connections of a given document base without resorting to manual clustering, 
classification and ranking of documents. 
The main goal of this paper is to present a joint probabilistic model of document con- 
tent and connectivity, i.e., a parameterized stochastic process which mimics the generation 
of documents as part of a larger collection, and which could make accurate predictions 
about the existence of hyperlinks and citations. More precisely, we present an extension of 
our work on Probabilistic Latent Semantic Analysis (PLSA) [4, 7] and Probabilistic HITS 
(PHITS) [3, 8] and propose a mixture model to perform a simultaneous decomposition of 
the contingency tables associated with word occurrences and citations/links into "topic" 
factors. Such a model can be extremely useful in many applications, a few of which are: 
 Identifying topics and common subjects covered by documents. Representing 
Although the weakness of our memory might make us forget this at times. 
documents in a low-dimensional space can help understanding of relations be- 
tween documents and the topics they cover. Combining evidence from terms and 
links yields potentially more meaningful and stable factors and better predictions. 
 Identifying authoritative documents on a given topic. The authority of a document 
is correlated with how frequently it is cited, and by whom. Identifying topic- 
specific authorities is a key problems for search engines [2]. 
 Predictive navigation. By predicting what content might be found "behind" a link, 
a content/connectivity model directly supports navigation in a document collec- 
tion, either through interaction with human users or for intelligent spidering. 
 Web authoring support. Predictions about links based on document contents can 
support authoring and maintenance of hypertext documents, e.g., by (semi-) auto- 
matically improving and updating link structures. 
These applications address facets of one of the most pressing challenges of the "informa- 
tion age": how to locate useful information in a semi-structured environment like the world 
wide web. Much of this difficulty, which has led to the emergence of an entire new indus- 
try, is due to the impoverished explicit structure of the web as a whole. Manually created 
hypefiinks and citations are limited in scope - the annotator can only add links and pointers 
to other document they are aware of and have access to. Moreover, these links are static; 
once the annotator creates a link between documents, it is unchanging. If a different, more 
relevant document appears (or if the cited document disappears), the link may not get up- 
dated appropriately. These and other deficiencies make the web inherently "noisy" - links 
between relevant documents may not exist and existing links might sometimes be more or 
less arbitrary. Our model is a step towards a technology that will allow us to dynamically 
infer more reliable inter-document structure from the impoverished structure we observe. 
In the following section, we first review PLSA and PHITS. In Section 3, we show how 
these two models can be combined into a joint probabilistic term-citation model. Section 4 
describes some of the applications of this model, along with preliminary experiments in 
several areas. In Section 5 we consider future directions and related research. 
2 PLSA and PHITS 
PLSA [7] is a statistical variant of Latent Semantic Analysis (LSA) [4] that builds a fac- 
tored multinomial model based on the assumption of an underlying document generation 
process. The starting point of (P)LSA is the term-document matrix N of word counts, i.e., 
Nij denotes how often a term (single word or phrase) i occurs in document dj. In LSA, N 
is decomposed by a SVD and factors are identified with the left/right principal eigenvectors. 
In contrast, PLSA performs a probabilistic decomposition which is closely related to the 
non-negative matrix decomposition presented in [9]. Each factor is identified with a state zk 
(1 _< k _< K) of a latent variable with associated relative frequency estimates ?(i Izk) for 
each term in the corpus. A document di is then represented as a convex combination of fac- 
tors with mixing weights ? (zld ), i.e., the predictive probabilities for terms in a particular 
document are constrained to be of the functional form ?(ild) = y ? (i Iz) 19 (z I dj), 
with non-negative probabilities and two sets of normalization constraints Yi ?(i I zk) = 1 
for all k and y ?(zld): 1 for all j. 
Both the factors and the document-specific mixing weights are learned by maximizing the 
likelihood of the observed term frequencies. More formally, PLSA aims at maximizing 
 = Yi,j Nii logy Since factors z can be interpreted as states 
of a latent mixing variable associated with each observation (i.e., word occurrence), the 
Expectation-Maximization algorithm can be applied to find a local maximum of . 
PLSA has been demonstrated to be effective for ad hoc information retrieval, language 
modeling and clustering. Empiric ally, different factors usually capture distinct "topics" 
of a document collection; by clustering documents according to their dominant factors, 
useful topic-specific document clusters often emerge (using the Gaussian factors of LSA, 
this approach is known as "spectral clustering"). 
It is important to distinguish the factored model used here from standard probabilistic mix- 
ture models. In a mixture model, each object (such as a document) is usually assumed to 
come from one of a set of latent sources (e.g. a document is either from z or z2). Credit for 
the object may be distributed among several sources because of ambiguity, but the model 
insists that only one of the candidate sources is the true origin of the object. In contrast, a 
factored model assumes that each object comes from a mixture of sources -- without ambi- 
guity, it can assert that a document is half z and half z2. This is because the latent variables 
are associated with each observation and not with each document (set of observations). 
PHITS [3] performs a probabilistic factoring of document citations used for bibliometric 
analysis. Bibliometrics attempts to identify topics in a document collection, as well as in- 
fluential authors and papers on those topics, based on patterns in citation frequency. This 
analysis has traditionally been applied to references in printed literature, but the same tech- 
niques have proven successful in analyzing hyperlink structure on the word wide web [8]. 
In traditional bibliometrics, one begins with a matrix A of document-citation pairs. Entry 
Aij is nonzero if and only if document di is cited by document dj or, equivalently, if dj 
contains a hyperlink to di.2 The principal eigenvectors of AA  are then extracted, with each 
eigenvector corresponding to a "community" of roughly similar citation patterns. The co- 
efficient of a document in one of these eigenvectors is interpreted as the "authority" of that 
document within the community -- how likely it is to by cited within that community. A 
document's coefficient in the principal eigenvectors of AA is interpreted as its "hub" value 
in the community -- how many authoritative documents it cites within the community. 
In PHITS, a probabilistic model replaces the eigenvector analysis, yielding a model that 
has clear statistical interpretations. PHITS is mathematically identical to PLSA, with one 
distinction: instead of modeling the citations contained within a document (corresponding 
to PLSA's modeling of terms in a document), PHITS models "inlinks," the citations to a 
document. It substitutes a citation-source probability estimate P(c Izk) for PLSA's term 
probability estimate. As with PLSA and spectral clustering, the principal factors of the 
model are interpreted as indicating the principal citation communities (and by inference, 
the principal topics). For a given factor/topic zk, the probability that a document is cited, 
P(dj Iz), is interpreted as the document's authority with respect to that topic. 
3 A Joint Probabilistic Model for Content and Connectivity 
Linked and hyperlinked documents are generally composed of terms and citations; as such, 
both term-based PLSA and citation-based PHITS analyses are applicable. Rather than 
applying each separately, it is reasonable to merge the two analyses into a joint probabilistic 
model, explaining terms and citations in terms of a common set of underlying factors. 
Since both PLSA and PHITS are based on a similar decomposition, one can define the 
following joint model for predicting citations/links and terms in documents: 
(1) 
Notice that both decompositions share the same document-specific mixing proportions 
?(z I d j). This couples the conditional probabilities for terms and citations: each "topic" 
2In fact, since multiple citations/links may exist, we treat Aij as a count variable. 
has some probability P(cz Izk) of linking to document dz as well as some probability 
P(i I zk) of containing an occurrence of term i. The advantage of this joint modeling ap- 
proach is that it integrates content- and link-information in a principled manner. Since the 
mixing proportions are shared, the learned decomposition must be consistent with content 
and link statistics. In particular, this coupling allows the model to take evidence about link 
structure into account when making predictions about document content and vice versa. 
Once a decomposition is learned, the model may be used to address questions like "What 
words are likely to be found in a document with this link structure?" or "What link structure 
is likely to go with this document?" by simple probabilistic inference. 
The relative importance one assigns to predicting terms and links will depend on the spe- 
cific application. In general, we propose maximizing the following (normalized) log- 
likelihood function with a relative weight c. 
- 
(2) 
The normalization by term/citation counts ensures that each document is given the same 
weight in the decomposition, regardless of the number of observations associated with it. 
Following the EM approach it is straightforward to derive a set of re-estimation equations. 
For the E-step one gets formulae for the posterior probabilities of the latent variables asso- 
ciated with each observation 3 
J'(:k Ii, ' 
(3) 
The class-conditional distributions are recomputed in the M-step according to 
along with the mixing proportions 
Nj 
P(zkldj) (x o  i, Nitj 
AO 
P(zklidj) - (1 - o)  .l Alj 
l 
4 Experiments 
In the introduction, we described many potential applications of the the joint probabilistic 
model. Some, like classification, are simply extensions of the individual PHITS and PLSA 
models, relying on the increased power of the joint model to improve their performance. 
Others, such as intelligent web crawling, are unique to the joint model and require its 
simultaneous modelling of a document's contents and connections. 
In this section, we first describe experiments verifying that the joint model does yield im- 
proved classification compared with the individual models. We then describe a quantity 
called "reference flow" which can be computed from the joint model, and demonstrate its 
use in guiding a web crawler to pages of interest. 
3Our experiments used a tempered version of Equation 3 to minimize overfitting; see [7] for 
details. 
0.38 
0.36 
0.34 
0.32 
0.3 
0.28 
0.26 
0.24 
WebKB data 
std error 
0.5 
0.45 
0.4 
0.35 
0.3 
0.25 
Cora data 
std error 
0 0.2 0.4 0.6 0.8 I 0 0.2 0.4 0.6 0.8 
alpha alpha 
Figure 1: Classification accuracy on the WebKB and Cora data sets for PHITS (ct = 0), 
PLSA (ct = 1) and the joint model (0 < ct < 1). 
We used two data sets in our experiments. The WebKB data set [11], consists of ap- 
proximately 6000 web pages from computer science departments, classified by school and 
category (student, course, faculty, etc.). The Cora data set [10] consists of the abstracts and 
references of approximately 34,000 computer science research papers; of these, we used 
the approximately 2000 papers categorized into one of seven subfields of machine learning. 
4.1 Classification 
Although the joint probabilistic model performs unsupervised learning, there are a number 
of ways it may be used for classification. One way is to associate each document with its 
dominant factor, in a form of spectral clustering. Each factor is then given the label of the 
dominant class among its associated documents. Test documents are judged by whether 
their dominant factor shares their label. 
Another approach to classification (but one that forgoes clustering) is a factored nearest 
neighbor approach. Test documents are judged against the label of their nearest neighbor, 
but the "nearest" neighbor is determined by cosines of their projections in factor space. 
This is the method we used for our experiments. 
For the Cora and WebKB data, we used seven factors and six factors respectively, arbitrarily 
selecting the number to correspond to the number of human-derived classes. We compared 
the power of the joint model with that of the individual models by varying ct from zero to 
one, with the lower and upper extremes corresponding to PHITS and PLSA, respectively. 
For each value of ct, a randomly selected 15% of the data were reserved as a test set. The 
models were tempered (as per [7]) with a lower limit of/ = 0.8, decreasing/ by a factor 
of 0.9 each time the data likelihood stopped increasing. 
Figure 1 illustrates several results. First, the accuracy of the joint model (where ct is neither 
0 nor 1), is greater than that of either model in isolation, indicating that the contents and link 
structure of a document collection do indeed corroborate each other. Second, the increase 
in accuracy is robust across a wide range of mixing proportions. 
4.2 Reference Flow 
The previous subsection demonstrated how the joint model amplifies abilities found in 
the individual models. But the joint model also provides features found in neither of its 
progenitors. 
A document d may be thought of as occupying 
a point 2: {P(ld),...,P(kld)} in the joint 
model's space of factor mixtures. The terms in d 
act as "signposts" describing 2, and the links act as 
directed connections between that point and others. 
Together, they provide a reference flow, indicating 
a referential connection between one topic and an- 
other. This reference flow exists between arbitrary 
points in the factor space, even in the absence of 
documents that map directly to those points. 
Consider a reference from document di to document 
dj, and two points in factor space 2, and 2n, not 
particularly associated with di or dj. Our model 
allows us to compute and ?(dale'n), the 
probability that the combination of factors at 2, 
and 2,are responsible for di and di respectively. 
Their product is then the prob- 
student/ 
department 
'ct 
)t./ 
course 
Figure 2: Principal reference 
flow between the primary topics 
identified in the examined subset 
of the WebKB archive. 
ability that the observed link represents a reference between those two points in fac- 
tor space. By integrating over all links in the corpus we can compute, f, = 
i,:,4jo P(dilz-)?(dJ Iz), an unnormalized "reference flow" between , and . 
Figure 2 shows the principal reference flow between several topics in the WebKB archive. 
4.3 Intelligent Web Crawling with Reference Flow 
Let us suppose that we want to find new web pages on a certain topic, described by a set 
of words composed into a target pseudodocument dr. We can project dt into our model to 
identify the point gt in factor space that represents that topic. Now, when we explore web 
pages, we want to follow links that will lead us to new documents that also project to 
To do so, we can use reference flow. 
Consider a web page ds (or section of 
300 
a web page4). Although we don't know 
where its links point, we do know what 
words it contains. We can project them 
as a peudodocument to find s the point  200 
in factor space the page/section occupies, 
'- 150 
prior to any information about its links. 
We can then use our model to compute lOO 
the reference flow fst indicating the (un- 
normalized) probability that a document 
at 2s would contain a link to one at 2t. 
o 
As a greedy solution, we could simply 
follow links in documents or sections 
that have the highest reference flow to- 
ward the target topic. Or if computation 
is no barrier, we could (in theory) use ref- 
erence flow as state transition probabili- 
ties and find an optimal link to follow by 
treating the system as a continuous-state 
true source 
'placebo' 
10 20 30 40 50 60 70 80 90 
rank 
I 
O0 
Figure 3: When ranked according to mag- 
nitude of reference flow to a designated tar- 
get, a "true source" scores much higher than 
a placebo source document drawn at random. 
Markov decision process. 
4Though not described here, we have had success using our model for document segmentation, 
following an approach similar to that of [6]. By projecting successive n-sentence windows of a 
document into the factored model, we can observe its trajectory through "topic space." A large jump 
in the factor mixture between successive windows indicates a probable topic boundary in document. 
To test our model's utility in intelligent web crawling, we conducted experiments on the 
WebKB data set using the greedy solution. On each trial, a "target page" dt was selected 
at random from the corpus. One "source page" ds containing a link to the target was 
identified, and the reference flow fst computed. The larger the reference flow, the stronger 
our model's expectation that there is a directed link from the source to the target. 
We ranked this flow against the reference flow to the target from 100 randomly chosen "dis- 
tractor" pages d, d2 ..., doo. As seen in Figure 3, reference flow provides significant 
predictive power. Based on 2400 runs, the median rank for the "true source" was 27/100, 
versus a median rank of 50/100 for a "placebo" distractor chosen at random. Note that the 
distractors were not screened to ensure that they did not also contain links to the target; as 
such, some of the high-ranking distractors may also have been valid sources for the target 
in question. 
5 Discussion and Related Work 
There have been many attempts to combine link and term information on web pages, though 
most approaches are ad hoc and have been aimed at increasing the retrieval of authoritative 
documents relevant to a given query. Bharat and Henzinger [1] provide a good overview 
of research in that area, as well as an algorithm that computes bibliometric authority after 
weighting links based on the relevance of the neighboring terms. The machine learning 
community has also recently taken an interest in the sort of relational models studied by 
Bibliometrics. Getoor et al. [5] describe a general framework for learning probabilistic 
relational models from a database, and present experiments in a variety of domains. 
In this paper, we have described a specific probabilistic model which attempts to explain 
both the contents and connections of documents in an unstructured document base. While 
we have demonstrated preliminary results in several application areas, this paper only 
scratches the surface of potential applications of a joint probabilistic document model. 
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