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OCR for page 84
Colloquium
The simultaneous evolution of author and
paper networks
Katy Bornert$, Jeegar T. Maru§, and Robert L. Gold stone
tSchool of Library and Information Science and Departments of §Computer Science and ~Psychology, Indiana University, Bloomington, IN 47405
There has been a long history of research into the structure and
evolution of mankind's scientific endeavor. However, recent
progress in applying the tools of science to understand science
itself has been unprecedented because only recently has there
been access to high-volume and high-quality data sets of scientific
output (e.g., publications, patents, grants) and computers and
algorithms capable of handling this enormous stream of data. This
article reviews major work on models that aim to capture and
recreate the structure and dynamics of scientific evolution. We
then introduce a general process model that simultaneously grows
coauthor and paper citation networks. The statistical and dynamic
properties of the networks generated by this model are validated
against a 20-year data set of articles published in PNAS. Systematic
deviations from a power law distribution of citations to papers are
well fit by a model that incorporates a partitioning of authors and
papers into topics, a bias for authors to cite recent papers, and a
tendency for authors to cite papers cited by papers that they have
read. In this TARL model (for topics, aging, and recursive linking),
the number of topics is linearly related to the clustering coefficient
of the simulated paper citation network.
M odels capturing the structure and evolution of mankind's
sc~enUf~c endeavor are expected to provide insights into
the inner workings of science. They are developed to provide
objective guidance to augment decisions concerning resource
allocation (identification of research frontiers, determining
award amount, many small vs. a few large grants), optimum
interdisciplinary collaboration (too little collaboration might
lead to duplication, too much may lead to rather shallow
science), the influence of publishing mechanisms (books vs. fast
ejournals), and so on.
Two kinds of models are commonly distinguished: descriptive
models that aim to describe the major features of a (typically
static) data set and process models that model the mechanisms
and temporal dynamics by which real-world networks (e.g.,
coauthor or paper citation networks) are created. Most research
in bibliometrics (1), scientometrics (2), or knowledge domain
visualizations (A has focused on descriptive models. For exam-
ple, research has studied the statistical patterns of coauthorship
networks, paper citation networks, individual differences in
citation practice, the composition of knowledge domains, and
the identification of research fronts as indicated by new but
highly cited papers. Recent work in statistical physics and
sociology aims to design process models. Of particular interest is
the identification of elementary mechanisms that lead to the
emergence of small-world (4, 5) and scale-free network struc-
tures (6, 7~.
The model proposed in this article is unique in that it simulates
the simultaneous growth of more than one network structure,
here authors and papers. The core assumption is that the twin
networks of scientific researchers and scholarly articles mutually
support one another. Researchers connect articles to one an-
other in cocitation networks, and articles link researchers to one
another in coauthorship networks.
5266-5273 1 PNAS 1 April 6, 2004 1 vol. 101 1 suppl. 1
The model provides a grounded mechanism for modeling the
"rich-get-richer" phenomenon for paper citation networks as an
emergent property of the elementary networking activity of
authors reading and citing articles and also the references listed
in read articles. The generalized rich-get-richer phenomenon is
also known as the Mathew effect (8), cumulative advantage (9),
or preferential attachment (104.
The growth of scientific publications and citations is governed
by two underlying processes: growth and aging (114. Growth
seems to be important for the development of scale-free net-
works. Aging is an antagonistic force to preferential attachment.
Even highly connected nodes typically stop receiving links after
time has passed. The bias to cite newer papers frequently
prevents a scale-free distribution of connectivity (12~. In the
proposed model, an aging bias offsets the rich-get-richer phe-
nomenon for paper citation networks.
A 20-year data set of articles published in PNAS is used to
validate the model in terms of major network properties of the
interlinked coauthor and paper citation networks.
The subsequent sections review related research on descrip-
tive and process models of coauthor and paper citation networks,
discuss desirable features and basic assumptions of the process
model, validate the model by comparing simulated data to a
20-year ISIII PNAS data set, and discuss the influence of model
parameters such as the aging of papers in terms of their power
to attract citations, the number of topics, and the length of the
chain of references that authors consider when making citations.
The article concludes with a discussion and outlook.
Related Work
There is a long history in bibliometrics (1) and scientometrics (2)
of describing the structure and evolution of science (3~. As early
as 1964, Garfield and his colleagues (13) proposed using citation
data to study and write the history of science. Citation data has
also been used to identify the associations between authors,
publications, patents, grants, data, and more recently genes,
proteins, diseases, etc. Associations have been discovered over
time, space, and fields to identify changing frontiers of science
(14), measure science (15), or recognize research fronts (16~.
Research on process models seeks to simulate, statistically
describe, or formally reproduce statistical characteristics of
interest. Of particular interest are models that "conform to the
measured data not only on the level where the discovery was
This paper results from the Arthur M. Sackler Colloquium of the National Academy of
Sciences, "Mapping Knowledge Domains," held May 9-11, 2003, at the Arnold and Mabel
Beckman Center of the National Academies of Sciences and Engineering in Irvine, CA.
$To whom correspondence should be addressed. E-mail: katy~?indiana.edu.
These data are extracted from Science Citation Index Expanded [Institute for Scientific
Information, Inc. (ISI), Philadelphia, PA; Copyright ISI]. All rights reserved. No portion of
these data may be reproduced or transmitted in any form or by any means without the
prior written permission of ISI.
2004 by The National Academy of Sciences of the USA
www.pnas.org/cgi/doi/10. 1 073/pnas.03076251 00
OCR for page 85
originally made but also at the level where the more elementary
mechanisms are observable and verifiable" (17~.
Recent work in statistical physics aims to design models and
tools to analyze the statistical mechanics of topology and dy-
namics of diverse physical, biological, and social networks. A
major goal is to identify elementary mechanisms that lead to the
emergence of small-world (4, 5) and scale-free network struc-
tures (6, 7) commonly observed in the real world.
Small-world networks have a short average path length among
nodes but a high local clustering coefficient compared to random
networks (18~. Important small-world graph properties in-
clude the number of vertices (n), the average degree (k>, the
characteristic path length (1), and the clustering coefficient (C).
The degree of a vertex is the total number of its connections. The
characteristic path length describes how far apart any two nodes
in the graph network are. It is computed by determining the
shortest path I (i, j) between any two nodes i andj in the network
and calculating the average of all 1. The clustering coefficient is
a more local measure of how "cliquish" a graph is or how tightly
nodes in the graph are connected to each other. If a node has K
edges that connect it to its neighbors, then the node's clustering
coefficient is given by C = N/(K*(K - 1~/2), where N is the
number of edges connecting neighbors of the node to each other.
The strength of a connection (e.g., the number of times two
authors wrote papers together) is not considered during the
computation of I or C.
In scale-free networks, the frequency f of the degree of
connectivity k of a vertex is a power function of k, f ~ key.
Examples of real-world data sets that are well approximated by
power law relationships include actor collaborations, power
grids, and the worldwide web (10~. For these data sets, the power
law applies over many orders of magnitude, and hence these
networks are known as "scale-free." With very few highly
interlinked nodes and many weakly interlinked nodes, scale-free
networks are surprisingly robust against random deletion of
edges, e.g., network failures (19~.
The Watts-Strogatz model was the first model to generate
graphs with small-world properties (20~. Their process model
starts with a regular lattice network configuration. Each edge is
redirected with a probability p to another randomly selected
node. This process model is of limited direct utility for coauthor
and paper citation networks because in those networks the links
are fixed; no rewiring takes place. That is, once a paper has cited
another paper, or two authors have collaborated on a paper,
these associations are forever part of the permanent historical
record.
The Barabasi and Albert (BA) model has been a highly
influential and successful attempt to simulate networks that
show scale-free properties (10~. It starts with a small number
(No) of nodes, and at every time step, a new node is added as well
as a set of m new edges that link the new node to the nodes
already present in the system. The probabilityp that a new node
will be connected to node i is proportional to the degree ki Of
node i. Hence a new node is preferentially attached to an already
highly connected node. After t time steps the network has n =
t + No nodes and m*t edges. This network evolves into a
stationary scale-free state with the probability that a node has k
edges following a power law with an exponent CUBA = 3.
Gradually adding nodes to the network over time appears to be
critical in obtaining scale-free distributions (21~.
Copying behavior was introduced as an alternative to explain
the power law degree distribution for the worldwide web (22~.
Recent workit models the probability distribution of citations by
copying references used in other papers. The resulting network
tiSimkin, M.V. & Roychowdhury, V.P. (2003) Condens. Matter, cond-mat/0305150 (abstr.).
t$Vazquez, A. (2001) Condens. Matter, cond-mat/0006132 (abstr.).
Borner et a/.
quantitatively matches the citation distribution observed in real
citation networks. Vazquez¢t even suggested that authors do a
recursive bibliographic search. In his model a new node is
connected to a randomly selected node as well as nodes linked
from (referenced by) this node. Although these models attempt
to capture preferential attachment, less is known about their
small-world properties. Numerous other attempts to model
small-world and scale-free networks are reviewed in refs. 4 and 7.
A number of mathematical models of network evolution have
been developed in sociology. Several models (25) assume a fixed
number of edges. Snijders (26) proposed a class of statistical
models for longitudinal network data that assumes a directed
graph with a fixed set of actors. However, neither the number of
nodes nor edges is fixed for evolving coauthor or paper citation
networks. The model by Gilbert (27) aims to simulate the
structure of academic science. It assumes that papers generate
future papers, giving authors a rather incidental role. The model
was validated based on the number and distribution of citation
counts. The small-world and scale-free properties of the result-
ing networks are unknown.
To our knowledge no algorithmic approach exists that simul-
taneously models the evolution of different networks such as
coauthor and paper citation networks within an ecology of
multiple interacting networks. Here, we argue that to fully
understand the structure, evolution, and utilization of networks,
coauthor and paper citation networks need to be considered
simultaneously. For example, to understand how knowledge
diffuses across authors via their papers at the same time that new
authors and papers are accumulated, it is essential to model the
coupled growth of both network structures.
Process Model for Author-Paper Networks
This section motivates the features and simplifying assumptions
of a process model for the simultaneous growth of coauthor and
paper citation networks as seen in citation databases like PNAS.
Given the importance of the interplay of topics, aging, and
recursive follow-up of links (here citation references), it was
named the TARL (topics, aging, and recursive linking) model.
The TARL model attempts to capture the roles of authors and
papers in the production, storage, and dissemination of knowl-
edge. Information diffusion is assumed to occur directly via
coauthorships and indirectly via the consumption of other
authors' papers. It assumes the existence of a set of authors and
papers. Each author and paper is assigned a single topic. Ideally,
several levels of topics would be organized hierarchically in
terms of specificity. The same paper may belong to the coarse
topic of immunology, the more specific topic of HIV infection,
and the still more specific topic of hemolytic anemia in HIV
patients with G6-PD deficiency. The current modeling uses the
simplifying assumption that there is a single level of relatively
specific topics. In contrast to the ephemeral lifespan of authors,
papers, once written, exist forever.
The set of authors is interlinked via undirected coauthorship
relations. Papers are interconnected via directed "provides input
to" links. Authors and papers are interlinked via directed
"consumed" links denoting the flow of information from papers
to authors as well as directed "produced" links representing the
act of paper generation by authors. Note that the decision to
direct links according to the flow of information reverses the
direction of the commonly used "cited by" link. The in-degree
of a paper node refers to its number of references and the
out-degree to its number of received citations.
Coauthor, citation, consumed, and produced links, once cre-
ated, are permanent. Coauthorship links may become stronger
as more and more papers are coauthored together. The number
of provides input to links representing received citations may
grow over time. Note that citation links can be created to any
existing paper. However, coauthorship links can only be made to
PNAS 1 Aprii 6, 2004 1 vof. 101 1 suppl. ~ 1 5267
OCR for page 86
}
II Initialization
generate #_papers papers and assign a random topic to each paper;
generate #_authors authors and assign a random topic to each author;
randomly assign #_co-authors+1 authors to papers of the same topic;
II Simulation
for each year do {
add #_new_authors new authors, deactivate authors older than #_author_age;
for each topic do {
randomly partition set of authors into author_groups of size #_co-authors+1;
for each author_group do {
for each new_paper to be produced, do {
generate new_paper;
randomly select #_read_ papers from existing papers;
_
get all references of read_ papers up to #_reference_path_length;
for each new_paper_reference do {
select a time_slice from (start year to curr_year-1) with probability given in aging_function;
randomly select a paper published or cited in this time_slice; as a new_paper_reference;
add the new_paper_reference to new_paper;
}
add all new papers to the set of existing papers;
add new links to author and paper information;
Fig. 1. Process model in pseudo code. If notopics are considered then the number of topics is one, i.e., all papers and authors havethe sametopic. If no coauthors
are considered then each paper has exactly one author. If the reference path length is O then no references are considered for citation. If no aging function is
given then all papers have the same probability of getting selected.
currently active authors. For simplicity, each paper has a fixed
number of authors and a fixed number of references.
Topics are randomly assigned to authors at initialization time.
Papers inherit the topic of their authoress. In the current
instantiation of the model, each author and each paper has
exactly one topic. The initial number of used topics is typically
lower than the total number of available topics. Eventually all of
the topics are covered as new authors with randomly assigned
topics are added gradually. Consumed-produced relationships
among papers and authors are restricted to authors and papers
within the same topic. Although this is a rather unrealistic
assumption, it parsimoniously models the fact that authors from
one knowledge domain typically do not frequently read journals,
attend conferences, coauthor, or interact from/with other
domains.
In the general model, the number of papers produced by an
author would be a random variable. However, for the current
instantiation of the model, a single fixed number of papers per
author per year is assumed. We are aware that this is unrealistic
and will not result in a Gaussian or power law distribution for the
number of coauthors nor the number of papers published per
author. Hence, only the properties of the paper citation network
will be validated against the PNAS data set.
During model initialization, a set of authors and a set of papers
with randomly assigned topics are generated (see pseudocode in
Fig. 1~. Subsequently, a predefined number of coauthors sharing
the same topic is randomly selected and assigned to each paper
via produced by links. All papers have authors but there may be
authors that have not yet produced papers. There are initially no
coauthor or paper citation links, making it advantageous to start
the model at least 1 year earlier than the time period of interest.
At each time step (year), a specified number of new authors
A' is created with a specific time stamp and added to the set of
existing authors Al = A' USA_. A number of authors can be
deactivated as well and subtracted from the set of authors.
Subsequently, each author in set Al randomly identifies a set of
coauthors, reads a specified number of randomly selected papers
from within his/her topic, and produces a specified number of
new papers. Each new paper will cite a fixed number of existing
5268 1 www.pnas.org/cgi/doi/10.1073/pnas.0307625100
papers, a number frequently stipulated by publishers and con-
strained by the number of pages available per journal. To select
the papers cited, authors consume (read) a rather small set of
papers because of finite cognitive and time constraints. To model
the local networking activity of authors, the number of levels of
paper references that are followed up by an author is modeled
explicitly. If it is zero, then only the papers that an author reads
in a year, the set PO, can be cited. If it is two, then an author can
cite any paper that they have read, any paper Pi that is cited in
one of the PO papers, or any paper that is cited in any of the
papers in set Pi. With each deeper level of references, the set of
papers, P', is added to Pr = P' UPr_1. The set of papers available
as citation references for a given year t and depth of reference
level r is denoted by Pro.
A parameterized model was implemented in Java to simulate
the simultaneous growth of interlinked coauthor networks and
paper citation networks as described above. The model takes as
input parameters that specify the number of authors and papers
created in the initial year as well as the number of topics, the
number of authors to be deleted per year, the number of papers
an author produces per year, the number of papers cited by a new
paper, the number of coauthors, the number of levels references
are followed up, and the parameters of the aging function.
The model can be started with or without topics, coauthor-
ships, following up of references, aging of papers, or any
combination of these variables. A small author-paper network
for topics only and for coauthors only is shown in Fig. 2 a and
b, respectively. Fig. 2a shows three unconnected topic-based
author and paper networks. During the simulation, authors read,
cite, and produce papers from their topic area exclusively. Given
that authors exclusively coauthor with authors within their own
topic, each paper has exactly one topic. Author ad was assigned
a topic for which two papers existed after the initialization and
hence both papers are assigned to the author. Later on the
author generates one paper each year that cites the author's own
work on the topic. Without any new authors or new topic areas
for existing authors, all subsequently produced papers will
belong to one of these three topic areas. With authors reading
papers only of their own topic area, there will never be any links
Borner et a/.
OCR for page 87
a ,~44\
t
l MA ~ 5
10
~~.C
b
..^
\ 4
~7
11
,..
~ /
1
\ ^w ,~ ~19
i2 >
/.Y'''~
8
I.
Or
.,
i..
-
3,`~
\ \
_~a4~
~0
\ . `~ ,,j",,,~r. ,y,i~ \
4', ~ ~
\ ,j,,( ~~:-~-
~=== 4~
~ ~3~
v'.1~1 ~
; ~-7
= i..
,,,
Fig. 2. Author-paper network generated by using the model with topics only (a) and coauthors only (b). The model was started with five topics, authors, and
papers and run for 2 years. In each year, each author produces one paper, which cites two earlier papers. No authors were added or deactivated. The resulting
networks has five authors (labeled a1-a5, blue circles) and 15 or 9 papers (labeled O. 2, 3 . . ., red triangles). Papers are linked via red directed provided input
to links. Authors are connected by blue coauthorships links. Light green indicates directed links denoting the flow of information from papers to authors and
from authors to new papers via consumed and produced relations.
between the three topical clusters. If authors do not coauthor
then there are no coauthor links.
If coauthorship is simulated, then each paper is authored by
a predefined number of authors. At each time step, each author
will select a number of coauthors to produce papers, and each
produced paper has multiple authors. The model stops when the
number of specified time steps is reached. In the network shown
in Fig. 2b each newly generated paper has exactly two authors.
Blue, undirected lines represent coauthorships. Line thickness
indicates the number of papers that have been coauthored
together, e.g., al and a3 coauthored several times. The total
number of papers produced each year is lower than in Fig. 2a
because two authors produce one paper together. If a topic area
has fewer authors than needed for the collaborative production
of a paper then no papers are produced.
If no references and no aging are considered then references
are randomly selected from the set of papers that a coauthor
team selected for reading. When references in papers are
followed up then authors consider not only the papers they
read as potential reference candidates but also papers linked
to those via citation references up to a path of a certain length.
Thus, a paper that was cited five times has six chances (or
tokens) to get selected. The resulting paper citation network
has some nodes, typically older papers, which are very highly
cited, whereas the majority of papers are rarely, if at all, cited;
see Fig. 3a.
If references as well as aging are considered, then the prob-
ability of papery being cited, P(y), corresponds to the normalized
sum of the aging-dependent probability for each of its tokens,
n
t= 1 i ~ PP{Ai=Y
w(t)
Pa) = ~
n
~ ~ W(t)
t=1 i ~Pr.t
Borner et a/.
where n is the total number of years considered. Hence a paper
that was published in year y and received four citations in year
y + 1 and two citations in year y + 2 has seven tokens that are
weighted by the probability value for each year. The probability
of citing a paper written t years ago can be fit by a Weibull
distribution of the form
~ t a
W(t) = cab to lie lob ~
where c is a scaling factor, a controls the variability of distribu-
tion, and b controls the rightward extension of the curve. As b
increases, the probability of citing older papers increases. For the
present purposes, a small value of b represents a strong aging bias
that favors citing papers that have been published recently. For
small values of b, the function predicts very few citations for
older papers. The introduction of aging offsets the rich-get-
richer effect that favors the citation of older papers that have
already been frequently cited.
The parameters specified in the input script file provide
flexibility to fit the model output to diverse data sets. The model
is used to fit the PNAS data in the next section. Later, we
examine the influence of aging, reference path length, and
number of topics on the structure of interlinked coauthor and
paper citation networks.
Model Validation
To validate the TARL model, a 20-year (1982-2001) data set of
PNAS was used. Subsequently, we describe the data set, select a
set of model parameters, and compare the model output with the
PNAS data set in terms of network properties.
The PNAS Data Set. The PNAS data set contains 45,120 regular
articles. The number of unique authors for those papers is
105,915. Table 1 provides counts of the total number of papers,
authors, references, and citations received by all of the papers for
each of the 20 years, as well as the average number of coauthors
per author. The average number of papers published per author
PNAS 1 April 6, 2004 1 vol. 101 1 suppl. ~ 1 5269
OCR for page 88
a ~12
A: t
b
r ~ '~:~ 79
of
I..
Irk /
i,1 4
6
~ 1 ~
~ I
13
\
\ ~
~ \
10 ~
\
11
Fig. 3. Paper network without aging (a) and with aging (b). Without aging, older papers are more likely to provide input to younger papers; i.e., they are
attracting most of the citation links. For example, in a, paper no. O generated at initialization time and paper no. 5 generated in year 1 provide input to four
and six papers, respectively.
and the average number of references and citations per paper
can be easily derived from Table 1. Note that the citation counts,
particularly for younger papers, are artificially low because they
have not existed in the literature long enough to garner many
citations. Table 1 also provides information on the number of
citations received from papers within the PNAS data set in the
right-most column. Only those intra-PNAS citation links will be
modeled. The paper most highly cited by papers within the set
received 285 citations.
Fig. 4 visualizes the limited coverage of the data set. It neither
contains all work by many authors for the 20-year time span as
they may have published in other venues as well, nor does it
provide information about citations received from PNAS papers
Table 1. PNAS statistics in terms of total number of papers (#p),
unique authors (#a), references (#r), citations received per paper
(#c), number of coauthors per paper (a#ca), and the number of
citations (#=win) within the PNAS data set for each year
Year #p
1982 1,669
1983 1,611
1984 1,695
1985 1,846
1986 2,042
1987 1,924
1988 2,035
1989 2,088
1990 2,066
1991 2,382
1992 2,500
1993 2,413
1994 2,600
1995 2,476
1996 2,765
1997 2,618
1998 2,711
1999 2,603
2000 2,501
2001 2,575
Total 45,120
#a #r #c a#ca
5,201 46,665
5,142 46,685
5,583 49,834
6,325 55,662
7,209 64,379
7,061 59,110
7,471 63,116
7,959 65,883
8,031 66,019
9,559 77,740
9,812 80,949
9,770 79,848
10,656 86,176
10,429 82,021
11,803 99,061
11,255 96,788
12,328 100,973
12,182 97,018
12,201 94,181
13,038 97,450
1,509,558
56,690
161,437
174,161
191,750
218,229
207,729
215,227
215,437
207,138
223,102
211,238
193,867
187,353
151,249
148,622
122,908
107,764
76,080
44,131
16,357
3,230,469
5270 1 www.pnas.org/cgi/doi/10.1073/pnas.0307625100
3.92
3.98
4.22
4.38
4.76
4.88
4.8
5.01
5.15
5.25
5.29
5.55
5.56
5.66
5.96
6.12
6.48
6.69
7.6
8.4
published past 2001 or non-PNAS papers. References to papers
outside the 20-year data set will be ignored.
Table 2 lists small-world properties and power law exponents
for diverse coauthor and paper citation networks. The values for
the PNAS data set under examination and the simulated paper
citation network are also given.
Note that for undirected coauthor networks, the in-degree of
a node equals its out-degree and hence the exponents for both
distributions are identical. For directed paper citation networks,
the number of references is rather small and constant. As typical,
only the in-degree distribution (received citations) are consid-
ered (7) and the By reported values for paper citation networks
characterize the in-degree distribution. For paper citation net-
works, we do not report the value for the characteristic path
length as it reflects the time duration of the sample but little
about the structure of the network.
Based on these values, the PNAS data set can be classified as
a medium-sized data set that has a similar average node degree
sky, path length 1, cluster coefficient C, and power law exponent
By to the networks previously examined. The (k) value of the
paper citation network is rather low. The total number of links
6,749 within the PNAS citation network is 114,003. On average, each
7,188
6,928
7,425
7,985
7,340
7,547
7,386
7,089
7,511
6,932
5,979
5,910
4,922
5,013
4,290
3,580
2,453
1,354
422
114,003
~ # papers
papers citing
papers papers in X
in X
papers cited by
1~< -—~
.
Other
Publications
PNAS
~ 1982 200 1
time
Fig. 4. Coverage of the PNAS data set in terms of time span, total papers, and
complete authors) work.
Borner et a/.
OCR for page 89
Table 2. Properties of coauthor and paper citation networks
comprising number of nodes n, average node degree (k), path
length 1, cluster coefficient C, and power law exponent by
Network n (k) I C
Coauthorship networks
LANL
MEDLINE
SPI RES
NCSTRL
Mathematics
Neuroscience
PNAS
Paper citation networks
151
PhysRev
PNAS
SIM
52,909 9.7
1,520,251 18.1
56,627 1.73
11,994 3.59
70,975 3.9
209,293 11.5
105,915 8.97
783,339 8.57
24,296 14.5
45,120 3.53
37,114 2.13
5.9 0.43
4.6 0.066
4.0 0.726
9.7 0.496
9.5 0.59
6 0.76
5.89 0.399
0.081
0.074
Values for the first four coauthorship networks are taken from refs. 28-30.
Math and neuroscience network were analyzed (21); Redner (31) reported the
paper citation network values for ISI and PhysRev. Values for PNAS and
simulated network data were acquired by the authors.
paper receives about three citations from another paper in this
data set. The coauthor network has 472,552 links.
The power law exponent for the PNAS coauthor network is
2.54 and seems to match the values reported for other networks
well. It accounts for 91% of the variance in the relation between
number of coauthors and frequency of occurrence. The number
of authors with very few coauthors is less than predicted by a
power law relation, and the number of authors with a moderate
number of coauthors is more than predicted. The best-fitting
power law exponent for the paper citation network is 2.29, and
the power law accounts for 87% of the variance. The systematic
deviations from a power law are that most cited papers are cited
less often than predicted by a power law, and the less cited papers
are cited more often than predicted. A plausible account for
these deviations are that networks in which aging occurs, e.g.,
actor networks, friendship networks, but also paper citation
networks, show a connectivity distribution that has a power law
regime followed by an exponential or Gaussian decay or have an
exponential or Gaussian connectivity distribution (12~. Newman
(30) showed that connectivity distributions of coauthor networks
from astrophysics, condensed matter, high energy, and computer
science databases can be fitted by a power law form with an
a 1 4000
1 2000-
8000
6000
4000
2000
O -
Borner et al.
exponential cutoff. Following this lead, we fit a power law with
exponential cutoff of the form fix) = A)c-BeX/C. This function
provided an excellent fit to the PNAS paper citation network
with values of A = 13,652, B = 0.49, and C = 4.21 (R2 = 1.00~.
Model Initialization. The statistical properties of the PNAS data
set were used to select the initialization values for the model. The
model was run with topics, coauthors, references, and aging for
2 21 years covering 1981-2001. The year 1981 was used for
— initialization purposes. In 1981, 4,809 authors and 1,624 papers
2.5 covering 1,000 topics were generated (see discussion of the linear
2.1 relation between cluster coefficient and topics in the next
2 54 section). In accordance with the PNAS data, the number of
active authors was increased by 430 each year. Note that this
3 increase in authors is caused mostly by external factors such as
3 funding, which are not modeled in the current simulation and
2 29 hence have to be supplied by hand. Even though 20 years is a
2 05 rather large time span, the simplifying assumption was made that
all authors remained alive/active. Although the number of
coauthors increases continuously over time we decided to use the
average value of 4. Hence the number of authors per paper is
five. One paper is produced by each author per year. The average
number of references per paper to papers within PNAS was set
to 3 as determined by the actual data. One level of references was
considered and the Weibull aging function was used, with a
parameter value of b = 3, providing a 12-year time window in
Statistics. Simulated data have been compared to the PNAS
article data set in terms of total number of papers, unique
authors, and citations received per paper for each year given in
Table 1, as well as in terms of their small-world properties.
Interestingly, the total number of papers in the simulation is
slightly lower than the actual PNAS data. This is because authors
who do not manage to find a sufficient number of coauthors in
their topic area will not produce any paper in this particular year.
Similarly, papers that are produced in a topic area with very few
papers will not be able to reference the called for number of
three papers. Hence the average degree (k) is slightly lower than
the value observed for the PNAS paper network.
As shown in Fig. Sa the number of papers published in PNAS
increases slowly but steadily over the 20-year time period. The
number of authors publishing in PNAS increases more rapidly
than the number of papers because the average number of
coauthors per paper increases from 3.49 in 1982 to 5.42 in 2001.
The simulation assumes a linear increase in authors over time,
1 981 1 986
12000
1 0000
8000 -
- 7T~ ~ .~r~
~ #a SIM O 6000 -
_ map PNAS
#pSIM
4000 -
1 991 1 996 2001
Year
r+#c_win PNAS|
~ l l ~
~ 0 tic win SIM
\
I ~~\
\\
, , , ~
1981 1986 1991 1996 2001
Year
Fig. 5. Total number of actual and simulated papers (#p) and authors (#a) (a) and received citations (#c win) (b).
PNAS 1 April 6, 2004 1 vol. 101 1 suppl. 1 1 5271
OCR for page 90
a 0.016
~ 0.014
._
.~ 0.01 2
8
~ 0.008
._
~ 0.006
In
`, 0.004
0.01
0.002
o
`~ .....
0 10
b 0.016
~ 0.014
._
.o 0.012
0 0.01
~ 0.008-
·~
0.Q06-
0.004
0.002 -
o
20 30 40 50 0 5 10 15 20 25
Parameter in Aging Function
Reference Path Length
Fig. 6. Cluster coefficient as a function of the aging function (a) and the reference path length (b).
but the increase in the number of caners Produced naturally
comes out of this increase in authors.
The average number of received citations for each year is
displayed in Fig. Sb. The model closely tracks the number of
actual citations for all years after 1984. The fit for the first 2 years
is poor because the model has no initial citation links nor record
of papers before 1981. Given that no papers before 1981 are
available as references, papers published in early years of the
simulation receive a disproportionately large number of cita-
tions. This effect fades away in 1985 as the aging function selects
mostly papers published in the last 12 years and papers published
in the last 7 years have a particularly high probability of being
cited. The total number of citations received by papers within the
PNAS data set is 111,341. The artifacts during the initial phase
of the model run could be eliminated by starting the model 10
years earlier and analyzing only the final 20 years. However, we
believe the graph in Fig. Sb nicely illustrates the influence of
aging and the model in action. Both actual and modeled data sets
reflect the fact that younger papers have shorter periods of time
in which to draw citations.
rear--- r-
Network Properties. This section discusses the fit of the simulated
paper citation network to the PNAS data in terms of small-world
properties as well as the power law exponent By for the paper
connectivity graph. Results for the best-fitting model parameters
are reported in the last row of Table 2.
The cluster coefficient for simulations in which all authors and
papers have the same topic is rather low. The cluster coefficient
increases considerably if topics are considered; see also the
discussion on topics in the next section.
The simulation with 1,000 topics and an aging parameter of
b = 3 provides a good fit to the PNAS data set in terms of the
distribution of citations. The model data R2 was 0.996, which is
substantially better than the best-fitting power law to the PNAS
data (R2 = 0.87), and almost as good as the best-fitting power law
with exponential tail (R2 = 1.00~. As with the PNAS data, the
simulated data were fit much better with a power law with
exponential tail (R2 = o.g99) than simple power law (0.987~.
Although the simulation does not fit the PNAS data any better
than the power law with exponential tail, it does provide a
process model for why this functional relation applies. Very
highly cited papers are more rare in the PNAS and simulated
data sets than predicted by a power law because of the bias
toward citing recent papers. The tendency for highly cited older
papers to attract still more citations is offset by a counteracting
tendency to cite recent papers.
5272 1 www.pnas.org/cgi/doi/10.1073/pnas.0307625100
The Influence of Model Parameters
This section discusses the influence of different TARL model
parameters on network properties such as cluster coefficient and
power law exponent for the citation distribution.
Interestingly, the number of topics is linearly correlated with
the clustering coefficient of the resulting network: C = 0.000073
* no. topics. Hence our knowledge about the clustering coeffi-
cient in the PNAS network governed the choice of 1,000 topics.
The linear relation entails a desirable property of the simulation;
a simple method exists for creating networks with a specific
degree of clustering.
Topics also influence the power law exponent for the citation
distribution. Increasing the number of topics increases the power
law exponent as authors are now restricted to cite papers in their
own topics area. By dividing science into separate fields, the
global rich-get-richer effect is broken down into many local
rich-get-richer effects, leading to a more egalitarian distribution
of received citations.
Aging refers to the distribution of probabilities for papers
being cited by new papers. The influence of the b value used
to generate different Weibull aging functions is shown in Fig.
6a. The aging distribution observed in the PNAS data was used
to determine the parameter value b = 3, marked with a star in
Fig. 6. By increasing b, and hence increasing the number of
older papers cited as references, the clustering coefficient
decreases. This effect suggests a second kind of clustering that
parallels the strong topic-induced clustering described previ-
ously. Papers are not only clustered by topic, but also in time,
and as a community becomes increasingly nearsighted in terms
of their citation practices, the degree of temporal clustering
Increases.
Last but not least, the length of the chain of paper citation
links that is followed to select references for a new paper also
influences the clustering coefficient. The dependence of the
clustering coefficient on the reference path length is given in Fig.
6b. This result indicates that temporal clustering is ameliorated
by the practice of citing (and hopefully reading!) the papers that
were the earlier inspirations for read papers.
Note that the aging and reference path length examinations
were conducted for 200 topics.
Discussion and Outlook
This article presented results on modeling the simultaneous
evolution and structure of author-paper networks. Although
prior research has described the associations among different
Borner et a/.
OCR for page 91
scientific structure (e.g., authors, publications, topics, web) (23),
to our knowledge, nobody has yet attempted to model the
simultaneous growth and dynamic interactions of multiple net-
works dealing with scientific output.
Models based on preferential attachment assume that new
papers are linked to highly connected (cited) papers and new
authors tend to coauthor with already highly interlinked authors.
However, in today's dynamic scientific world of increasing
specialization, an overview of the connectivity of a scientist or
paper is not available to authors (even experts in a field). Instead,
each author can be seen as a part of a complex network with local
connections. Each author interacts directly only with a rather
limited number of other authors and papers. However, papers
that are cited frequently have a higher probability of being cited
again. Similarly, authors that are highly interconnected with
other authors in social networks are likely to attract more
coauthors if we assume that authors tend to coauthor with
coauthors of their coauthors. The presented model uses the
reading and citing of paper references as a grounded mechanism
to generate paper citation networks that are approximately
scale-free. Moreover, the particular deviations from scale-free
properties are well predicted by a version of the model that
incorporates a bias to cite recent papers and a scientific com-
munity that is subdivided into specialized topics. The model
parameters that governed these two factors were b that reflects
that influence of aging and "number of topics" reflecting the
degree of splintering within science. The values for these pa-
rameters were not freely fit to the citation distribution data.
Instead, the number of topics was selected to approximate
PNAS's clustering coefficient, and b was selected to provide the
optimal Weibull fit to PNAS's distribution of citations as a
function of lag in years. Thus, the highly respectable model-data
fit involving the number of citations and their frequency is
impressive because it involves no true free parameters.
The incorporation of topics and recency bias was instru-
mental in achieving the qualitative violations of a power law
distribution. There are fewer papers that receive a large number
of citations than is predicted by a power law, because the bias
toward citing recent papers offsets the rich-get-richer effect that
generates a power law relation. It is difficult for a well cited paper
to continue to receive additional citations as it ages. The citation
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number of coauthors and the amount and quality of paper output
and hence the likelihood of attracting still more funding.
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code available to determine the small-world properties of networks.
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PNAS 1 April 6, 2004 1 vol. 101 1 suppl. 1 1 5273
Representative terms from entire chapter:
power law
