7 The Genetic Programming Collaboration Network
and its Communities
L. Luthi∗ M. Tomassini∗† M. Giacobini†‡ W. B. Langdon§
Abstract
Useful information about scientific collaboration structures and pat-
terns can be inferred from computer databases of published papers.
The genetic programming bibliography is the most complete reference
of papers on GP. In addition to locating publications, it contains coau-
thor and coeditor relationships from which a more complete picture of
the field emerges. We treat these relationships as undirected small
world graphs whose study reveals the community structure of the GP
collaborative social network. Automatic analysis discovers new com-
munities and highlights new facets of them. The investigation reveals
many similarities between GP and coauthorship networks in other sci-
entific fields but also some subtle differences such as a smaller central
network component and a high clustering.
1 Introduction
The genetic programming (GP) bibliography1, created and maintained by
one of us (WBL) and by S. Gustafson contains most of the GP papers. As
such, it is a rich source of data that implicitly describes many aspects of
the structure of the GP community. Searching the bibliography and looking
at the images2 provides a lot of useful information about the field and the
people working on GP. However, a deeper analysis of the data, that goes
beyond the mere pictorial aspect, provides a much more complete view.
The coauthorship data is a social network since collaborating in a research
∗Information Systems Department, University of Lausanne, Switzerland
†Information Systems Department, University of Lausanne, Switzerland
‡Dpt. of Animal Production Epidemiology and Ecology, University of Torino, Italy
§Department of Computer Science, University of Essex, UK
1http://www.cs.bham.ac.uk/∼wbl/biblio/
2http://www.cs.bham.ac.uk/∼wbl/biblio/gp-coauthors/
http://arxiv.org/abs/0704.0551v1
study usually requires that the coauthors become personally acquainted.
Thus, studying those ties, their structure, and their evolution allows a better
understanding of the factors that shape scientific collaboration.
We present a systematic study of the GP coauthorship data base us-
ing methods and tools pertaining to complex networks and social network
analysis. Social network analysis (see [?] for a survey), although it is an
old discipline, has recently received new impetus and tools from the field of
complex networks (see [?] for an excellent review). This is mainly due to
the relatively recent availability of large machine-readable databases such as
the GP bibliography. Social acquaintances involve psychological and other
human aspects that are difficult to quantify. However, as it has been done
in other fields [?, ?, ?, ?], we use objective data such as joint published work
to stand for social bonds. Since this must ignore subtler aspects of a col-
laboration relationship, it is obviously far from perfect as a social indicator,
yet it is still a good “proxy” for the network of social relationships and can
reveal several interesting facts and trends.
A preliminary investigation of the GP coauthorship network appears in
[?]. In the first part of this article we update this initial study using the
most recent data and adding the study of the influence of excluding co-
edited proceedings and books. In the second part we offer a new analysis of
the finer community structure of the collaboration network. Similar studies
have been performed in the last few years on several other collaboration
networks in disciplines such as physics, mathematics, medicine, biology, and
computer science [?, ?, ?, ?]. A related investigation concerning the EC
collaboration network [?] has appeared recently in popular form, but it does
not take into account, for example the community structure of the network.
[?] deals with some of the same statistical features for the EC community
at large as we describe in detail here for GP. The values reported by [?] are
in line with those found here for the GP field. Given that the intersection
between the GP researchers and general EC is likely to be rather large, it
would be interesting to study how they are related to each other.
2 The GP Collaboration Network
We treat the genetic programming social network as a graph where each
node is a GP researcher, i.e. someone who has at least one entry in the bib-
liography. There is a connection between two people if they have coauthored
at least one paper, or if they have coedited one or more book or proceedings.
As of the start of 2007, there is a total of N = 2809 connected nodes, i.e.
authors that have at least one GP collaborator, and a total of 5853 edges
(collaborations) in the GP coauthorship network. There are 367 isolated
vertices, which represent authors who have not collaborated with others to
the extent of coauthoring a paper. Isolated vertices are ignored in our graph
statistics. We have also excluded a single paper with 108 coauthors in a
nuclear physics journal. This is because we consider it to be an anomalous
entry that is not representative of typical collaborations in our discipline.
Due to the youth of GP, the graph is relatively small compared to some
studied collaboration networks [?, ?, ?]. (Although some published studies
have covered much smaller and more specialised networks, e.g. of only 50
people [?].) The main disadvantage of studying a relatively small database
is that, like any statistical study, more data allows deeper and more mean-
ingful inferences to be drawn. In particular, studies of the form of the
distributions (such as whether they follow exponential or power laws) re-
quire a large amount of data. The advantages include that the graph almost
fully represents the state of the whole GP community. This allows reliable
characterisation of collaboration in the community. Also, the problems of
multiple authors with the same name (e.g. A. Smith), outliers and different
name spelling that plague the larger data sets, are unlikely and easy to spot
in our data.
Although in many cases in our field co-editing a book or proceedings vol-
ume does reflect personal acquaintance, there are some large coeditorships
which are not representative and so may give a slanted view. Therefore in
the following figures we present two kinds of statistics: those that include
all joint publications and those in which co-edited conference proceedings
and co-edited books are excluded (but not their contents, of course). Next
we present and discuss some basic measures that characterise the GP col-
laboration network.
2.1 Number of Papers per Author
The average number of papers per author is 3.16 with co-edited books and
proceedings and it is 3.14 without. The five most prolific authors are, in de-
creasing order: J. Koza, R. Poli, W. B. Langdon, W. Banzhaf and C. Ryan.
If we exclude proceedings’ co-editors the ranking remains unchanged. Nat-
urally the distribution of the number of papers per author, P (k), has some
scatter, particularly in the tail of the distribution. Thus, we present in Fig-
ure 1 the graph of the cumulative distribution P (k ≥ n) which is smoother
and allows the same inferences to be made. The curves are rather well fitted
by a straight line, and thus the distributions follow a power-law P (k) ∝ k−γ
10000
1 10 100 1000
number of entries
with coeditors
without coeditors
power-law fit
Figure 1: Cumulative distribution of the number of entries per author. Log-
log scale. The straight line is the best mean-square fit and shows the number
of authors is ∝ k−2.5.
with a calculated exponent γ of 2.5 for both of them. A power-law distribu-
tion with similar exponents has been observed for analogous collaboration
networks, e.g. 2.86 for a biological publication database (Medline), 3.41 for
a computer science database (NCSTRL), 2.4 for mathematics, and 2.1 for a
neuroscience papers database [?, ?]. A smaller exponent (in absolute value)
means that the tail of the distribution is more stretched towards high values
of degree.
2.2 Number of Collaborators per Author
The average number of collaborators per author, i.e. the mean degree 〈k〉 of
the coauthorship graph, is 4.17 with proceedings and 3.62 without. This is
close to the values reported by studies of computer science, physics (exclud-
ing high energy physics) and Mathematics, suggesting GP follows similar
collaboration patterns to those disciplines. However it is much less than
found in high energy physics and medicine. See Table 1. In order and
including co-edited volumes, the five authors that have the largest num-
ber of collaborators are: W. Banzhaf, J.A. Foster, P. Nordin, W.B. Lang-
don, U.-M. O’Reilly. Without co-edited books the ranking is: P. Nordin,
W. Banzhaf, J. Daida, C. Ryan and R. Goodacre. The five “pairs” that
have the highest number of coauthored papers are, in decreasing order both
with or without co-edited proceedings: J. Koza–M. A. Keane, R. Poli–W.B.
Langdon, J. Koza–D. Andre, J. Koza–F. Bennet and F. Bennet–M.A. Keane.
This shows that J. Koza’s group has been tightly collaborating for a long
time, a conclusion that is confirmed in the community study of section 4.
It is also evident that the W.B. Langdon–R. Poli association has been an
extremely productive one.
10000
1 10 100
number of collaborators
with coeditors
without coeditors
Figure 2: Cumulative distribution of the number of authors with a given
number of collaborators. Logarithmic scale on both axes.
Figure 2 shows the cumulative distributions of the number of collabora-
tors. One sees that the distributions are not pure power-laws, otherwise the
points would approximately lie on a straight line. Rather, the distributions
shows a power-law regime in the first part followed by an exponential decay
in the tail. That is, the whole network cannot be fitted by a power-law.
This is quite common. In fact, several measured social networks do not
follow a power-law degree distribution [?, ?] and are best fitted either by
an exponential degree distribution P (k) ≈ e−k/〈k〉 or by an exponentially
truncated power-law of the type P (k) ≈ k−γe−k/kc , where kc represents a
critical connectivity and 〈k〉 is the average degree.
2.3 Number of Authors per Paper
Figure 3 shows the cumulative distribution of the number of papers written
by a given number of coauthors. Here the distribution also has a tail that is
longer than that of a Gaussian or exponential distribution, however it does
not follow a power-law. The average number of authors per paper is 2.25
(2.22 without co-editors). From Table 1 we can see that these figures are
close to the equivalent ones for computer science (NCSTRL) and physics,
while Mathematics has a lower number of co-authors per paper. On the
other hand, nuclear physics stands out with an unusually high number of
coauthors per paper.
10000
1 10 100
number of coauthors
with coeditors
without coeditors
Figure 3: Cumulative distribution of the number of papers with a given
number of coauthors on log-log scales.
From Figures 2 and 3 one can see that the tails of the distribution with
co-editors are longer than without them. Thus, taking co-editorship into
account seems to rather artificially inflate the number of publications with
many co-authors and, by consequence, the number of collaborators that a
person has.
2.4 Connected Components
In the theory of Poisson random graphs there is a critical value of average de-
gree 〈k〉 = 1 above which there is a sudden appearance of a giant component.
This is so-called since most vertices belong to it. The other components are
smaller and have an exponentially decreasing size distribution [?]. Although
collaboration graphs are not random, a similar phenomenon appears. In-
cluding coeditors there are 1025 GP authors in the giant component. This
is 36.5% of the total graph. If we exclude coediting proceedings etc. the
size is 743, representing the 26.9% of the total. In the giant component the
average number of collaborators per author is 5.83 with co-editors and 4.39
without them.
The cumulative size distribution of the connected components with and
without co-editors are depicted in Figure 4. Figure 4 shows that the proba-
bility density functions are well approximated by a power law with exponent
of 2.9 (excluding co-editors) and 2.6 (total). Since the other authors did not
provide the analogous data for their databases, we do not know how our
figures would compare with those for other coauthorship databases.
The existence of a big connected component has a social meaning. It
suggests 36.5% of GP researchers are members of a single community, since
those researchers are either directly connected via a collaboration or they
are close to each other in a way that will be made clear in section 3. The
size of the giant component is notably smaller in the GP graph with respect
to other measured coauthorship networks (see Table 1). This may be due
to the comprehensive nature of the GP bibliography. It captures work done
by smaller groups which does not get into major journals, whereas, perhaps,
the other databases concentrate upon higher impact outlets where work is
heavily cited but at the expense of ignoring less regarded authors. This
may artificially inflate the fraction of authors within their giant component.
Alternatively it may be due to the youth of the GP field, with many semi-
isolated individuals and groups starting research independently.
One should also consider that all collaboration networks are in a non-
equilibrium state as they are continuously evolving [?]. Accordingly, as
time goes by, one should observe small components progressively connecting
themselves to the large one. For example, in less than one year the size of
the giant component including co-editors has grown from 942 to 1025 nodes.
This is due in part to a number of newcomers collaborating with people
already belonging to the giant component. The other part comes from the
absorption of a few disconnected small components into the giant one thanks
to one or more new collaborations. This suggests that the size of the giant
component has not yet reached its “steady-state” value and it will continue
to grow in relative size. Since we possess all the time-stamped data, it is
possible to study the evolution of this component, as well as several other
indicators from the beginning and up to the present days. This investigation
is currently under way.
2.5 Social GP Clusters
The clustering coefficient of a node in a graph is the proportion of its neigh-
bouring nodes which are also neighbours of each other. The average clus-
tering coefficient 〈C〉 is calculated across all nodes in the graph [?]. In other
words, 〈C〉 is a simple statistical measure of the amount of local structure
1 10 100 1000
component size
with coeditors
without coeditors
power-law fit
power-law fit
Figure 4: Cumulative distributions of the number of connected components
in the collaboration graph by number of people. Log-log scale.
that is present in a graph. Most real-world networks, e.g. the world wide
web, roads, electrical power transmission and including the social networks
that have been studied to date, have a much larger clustering coefficient
than would be expected of a random graph with the same number of ver-
tices and edges. Social networks are particularly clustered. For example,
the average clustering coefficient is 〈C〉 = 0.665 for the GP collaboration
graph including book co-editors, and it is 0.660 without. (We would expect
0.0015 and 0.0013 for the corresponding random graphs). In terms of scien-
tific collaborations, a high clustering coefficient means that people tend to
collaborate in groups of three or more. This agrees with what we know of
the GP field. It may mean that two researchers that collaborate indepen-
dently with a third one may, in time, become acquainted and so collaborate
themselves. Alternatively it might be due to collaborators coming from the
same institution. In all cases, a high value of 〈C〉 for a social network is
an indication that collaborations are not made at random at all, and that
social forces and processes are at work in the network structure formation.
Table 1 summarises the results of this section and compares them with
those for some other collaboration networks. Some of the entries in the table
will be discussed in the following section. Most GP statistics are similar to
those of the larger databases. However one notable difference, as we have
already remarked, is the relative smallness of the largest component. The
clustering is rather high, which shows that GP researchers know each other
quite well within the large component, and the community is rather homoge-
Table 1: Basic statistics for some scientific collaboration networks. GP1 is
the GP bibliography at the start of 2007, including coedited books and pro-
ceedings. GP2 is the same but without coeditors. SPIRES is a data set of
papers in high-energy physics. Medline is a database of articles on biomed-
ical research. Mathematics comprises articles from Mathematical Reviews.
NCSTRL is a database of preprints in computer science. Physics has been
assembled from papers posted on the Physics E-print Archive. Details about
these databases can be found in [?, ?, ?].
GP1 GP2 SPIRES Medline Mathematics NCSTRL Physics
Total number of papers 4564 4504 66652 2163923 1600000 13169 98502
Total number of authors 2809 2765 56627 1520251 253339 11994 52909
Average papers per author 3.16 3.14 11.6 6.4 7 2.55 5.1
Average authors per paper 2.25 2.22 8.96 3.754 1.5 2.22 2.53
Average collaborators per author 4.17 3.62 173 18.1 2.94 3.59 9.7
Size of the giant component (%) 36.5 26.9 88.7 92.6 82.0 57.2 85.0
Clustering coefficient 0.665 0.660 0.726 0.066 0.15 0.496 0.43
Average path length 4.74 5.2 4.0 4.6 7.73 9.7 5.9
neous. In contrast, in biology and medicine or mathematics, where scientist
from different sub-disciplines seldom collaborate, the clustering coefficient is
lower. Note also the high number of authors per paper, and especially the
strikingly high number of collaborators per author in the nuclear physics
community (SPIRES). Clearly, nobody can maintain an average of 173 sci-
entific partners on a first-hand acquaintance basis and thus this figure does
not seem to be socially meaningful.
3 Distances and Centrality
A social network can be characterised by a number of measures that give an
idea of “how far” people are from each other, or how “central” they are with
respect to the whole community. These measures are well known in social
network analysis. Here we shall concentrate on average path length and on
betweenness centrality.
3.1 Average Path Length
The average path length L of a graph is the average value of the shortest
paths between all of its pairs of vertices. In random graphs and many real
networks, such as the Internet, the World Wide Web and social networks,
the average path distance scales as a logarithmic function O(logN) of the
number of vertices N . Such networks, if they also have a high clustering
coefficient, are known as small worlds networks [?]. Since, even for very large
graphs, any two nodes in a small world network are only a few steps apart.
In contrast in regular lattices, two nodes are O(N
D ) apart. (Where D is the
lattice’s dimensionality. For example, for a square lattice L ≤ 2
2 ). The
average path length of the giant component of the GP collaboration graph
including coeditors is 4.74 (it is 5.2 without coeditors). The longest among
all the shortest paths (known as the diameter) is 12 (14 without coeditors).
Thus, unsurprisingly, the GP community, as far as its “core” component
is concerned, is indeed a small world and is characterised by values that
are typical of these kinds of network (see Table 1). Being a small world
means that information may circulate quickly and collaborations are easier
to set up. These are clearly advantageous for a research community. The
connected components following the largest one are themselves small worlds.
We expect over time some of them will merge with the largest component.
(For this to happen, only a single new collaboration between two scientists
each belonging to one of the components is needed.)
3.2 Betweenness
The betweenness b(v) of a vertex v is the total number of shortest paths be-
tween all possible pairs of vertices that pass through this vertex. Nodes that
have a high betweenness potentially have more influence, i.e. they are more
central in the network, in that there is more “traffic” that goes through
them. The first five authors in terms of betweenness in the network (in-
cluding co-editors and in decreasing order) are: W. Banzhaf, H. Iba, U.-M.
O’Reilly, H. de Garis and W. B. Langdon. W. Banzhaf is also the researcher
that has the highest number of different collaborators. Without co-editors
the ranking is: W. B. Langdon, U.-M. O’Reilly, W. Banzhaf, M. Tomassini
and P. Nordin. People who have a large value of betweenness play the role
of intermediaries or “brokers” in a social sense.
3.3 Non-random collaborations between
directly connected authors
Most technological and biological networks are disassortive in that they have
negative correlation, meaning that high-degree vertices are preferentially
connected to low-degree vertices. However most measured social networks
are assortative, meaning highly connected nodes tend to be connected with
other highly connected nodes [?]. The GP collaboration network confirms
this general observation with a correlation coefficient of +0.15 for the gi-
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Lanz Pizz
Figure 5: One of the communities belonging to the main network compo-
nent. The thickness of the links gives an indication of the number of co-
authored papers. The largest thickness indicates more than 16 coauthored
works. The thinnest link (light gray) stands for a single collaboration. The
different symbols and colours represent sub-communities of the illustrated
community.
ant component, and +0.30 for the whole graph (including coeditors and
excluding the single physicist’s paper). These are close to the coefficients
observed for other social networks (specifically 0.127 for Medline and 0.120
for Mathematics [?]).
4 Communities in the Giant Component
All the researchers belonging to the largest component of the network can be
said to form part of the GP community at large, in the sense that they are all
only a few steps away from any other member of the community. However,
we know from direct experience that some groups of GPers are more closely
connected between themselves than with other people. In other words, they
belong to what one might call a group or a tighter community within the
global one. It is not easy to give a rigorous quantitative definition of a
community within a network. For our purposes a community can be seen
as a set of highly connected vertices having few connections with vertices
belonging to other communities. In the analysis of social networks, several
algorithms that attempt to split a network into communities have been
proposed. We used Newman’s method [?], which is based on a measure
of the fraction of edges that fall within communities minus the expected
value of the same quantity if edges fall at random without regard for the
community structure.
Since the GP bibliography contains the number of papers that any two
collaborators have published together, it is possible to go a step further
than just saying that two people have coauthored at least a paper, and
give a measure of the intensity of the collaboration. We use the number of
papers that two given authors have in common as a measure of the strength
of their collaboration. Newman [?] has proposed a more refined measure
which takes into account the actual number of coauthors of each paper.
However this is more complicated than we need, instead we ignore the total
number of coauthors for each paper. Our measure of collaboration strength
is used in our communities algorithm to highlight groups of researchers that
have collaborated strongly with the aim of uncovering the stability of the
scientific relationship. We have also excluded coedited proceedings, books,
etc., as we have already seen that these might sometimes represent spurious
collaboration relationships.
The results of running the algorithm on the subgraph represented by the
largest connected component are qualitatively surprisingly close to what one
would expect, given our knowledge of the GP field. The advantage is that the
analysis makes them explicit and uncovers a number of other relationships
that would be difficult to infer without an explicit study of the raw data.
As an example of the about 25 communities that the algorithm discovers,
Figure 5 shows the structure of the groups around one of us (“Toma”). If we
now consider this community as an isolated subgraph and apply again New-
man’s algorithm to it, we obtain the groups highlighted by different symbols
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Chio BuxtHoll
Figure 6: Another community belonging to the main network component.
The thickness of the links gives an indication of the number of co-authored
papers. The largest thickness indicates more than 16 coauthored works.
The thinnest link (light gray) stands for a single collaboration. The different
symbols and colours represent sub-communities.
and colours in the figure. Thus, the groups correspond to sub-communities
within the main community. The thickness of the links represents the inten-
sity of the relationship. It is easy to recognise a “hard core” of collaborating
researchers strongly connected to “Toma” forming triads and higher poly-
gons of order four and five. The strong triangle (“Foli”, “Pizz”, “Spez”) is
relatively loosely connected to the rest, showing that these researchers be-
long to the community but often collaborate between themselves. It is also
possible to discern institutional and geographical components in the com-
munity. For example, most of the upper right part of the figure through the
node “Chop” comprises researchers essentially belonging to the University
of Geneva, which is close to the University of Lausanne, to which “Toma”
belongs. However, geographical closeness is not the key factor in the other
groups which belong to Universities in France, Italy, Spain, and the US.
We might conjecture that many collaborations start locally at the same or
at close institutions and then they spread through people being introduced
to others via a common acquaintance, or through people physically moving
or visiting other institutions. This is the case in the figure, where “Vann”,
”Chop”, and “Vega” among others have played the role of “bridges” between
different institutions and across countries.
As a second illustration, let us look at Figure 6 which is the community
that revolves around one of us (“Lang”) and “Poli”. In contrast to the pre-
vious case, one can see that the graph structure is more “star-like”, with two
large directly connected big hubs (“Lang” and “Poli”) who have about 70 co-
authored papers, and three other highly connected nodes (“Buxt”,“McPh”,
“Rowe”) which are strongly connected to one of the main hubs but not to
both. It is interesting to observe the role of “McPh” who, like “Vega” in the
previous community (cf. Figure 5), plays a bridging role, this time between
some UK and some American researchers. We can also recognise a strong
”theory-oriented” group, which is almost a clique in the graph, formed by
(“McPh”,“Poli”,“Rowe”, “Steph”, “Wrig”). There is also another bridge
formed by “Cagn” from UK to Italy, again due to a long-standing collabo-
ration and friendship. The small cliques or almost cliques at the periphery
of the figure essentially represent people that have worked at the same in-
stitution in either Italy or the US.
The discussion above, motivated by our belonging to the mentioned com-
munities, and thus by our direct human knowledge about them, should be
enough to get an impression of the many useful observations that one can
make on the communities that interlock in the main network component.
There are of course several other large well known and interesting commu-
nities in the network but unfortunately we cannot describe them here for
reasons of space.
5 Conclusions
In sections 2 and 3 we characterised the genetic programming (GP) coau-
thorship graph using a number of local and global statistics. We extended
and updated the findings presented in [?] by studying the influence of
coedited volumes and by using the latest data available. Section 3 showed
the GP field to be highly clustering and that the GP coauthorship network
has a small mean path length. Together these suggest that, at least for the
core, GP is indeed a “small world”. We also found, compared with other
published collaboration networks, that the fraction of GP authors connected
by coauthorship is a relatively small fraction of all GP authors.
Section 4 is a study of the community structure of GP. It uses a more pre-
cise definition of collaboration, which takes into account the intensity of the
relationship. This uncovers many groups of tightly interacting researchers.
From the detailed study of two of the communities we have drawn inferences
about the pivotal role of some researchers or groups of researchers in pro-
moting collaborations within and between academic institutions. Adding
our human knowledge about geographical location and personal acquain-
tance, allows some conjectures to be drawn about the way in which different
continents and countries collaborate on research projects.
It should be obvious that the present data driven approach to social
network analysis can only provide some answers but not all of them. Algo-
rithms and data cannot take into account human aspects such as friendship
in scientific collaboration. While these may be buried in the sea of numbers
they will never appear explicitly from such analyses. Nevertheless, we feel
that our results are interesting and useful in the way that they characterise
our community.
There is another aspect of the collaboration graph that would be re-
vealing: the analysis of its development over the years. Indeed, since each
paper has a date of publication, we possess all the data that are needed
for such an investigation. This would allow the detailed study of how the
network has grown to its present size and structure from the beginning and
might give hints as to its future progress. This extension is currently under
investigation.
	Introduction
	The GP Collaboration Network
	Number of Papers per Author
	Number of Collaborators per Author
	Number of Authors per Paper
	Connected Components
	Social GP Clusters
	Distances and Centrality
	Average Path Length
	Betweenness
	Non-random collaborations between directly connected authors
	Communities in the Giant Component
	Conclusions

  Useful information about scientific collaboration structures and patterns can
be inferred from computer databases of published papers. The genetic
programming bibliography is the most complete reference of papers on GP\@. In
addition to locating publications, it contains coauthor and coeditor
relationships from which a more complete picture of the field emerges. We treat
these relationships as undirected small world graphs whose study reveals the
community structure of the GP collaborative social network. Automatic analysis
discovers new communities and highlights new facets of them. The investigation
reveals many similarities between GP and coauthorship networks in other
scientific fields but also some subtle differences such as a smaller central
network component and a high clustering.

Introduction
The genetic programming (GP) bibliography1, created and maintained by
one of us (WBL) and by S. Gustafson contains most of the GP papers. As
such, it is a rich source of data that implicitly describes many aspects of
the structure of the GP community. Searching the bibliography and looking
at the images2 provides a lot of useful information about the field and the
people working on GP. However, a deeper analysis of the data, that goes
beyond the mere pictorial aspect, provides a much more complete view.
The coauthorship data is a social network since collaborating in a research
∗Information Systems Department, University of Lausanne, Switzerland
†Information Systems Department, University of Lausanne, Switzerland
‡Dpt. of Animal Production Epidemiology and Ecology, University of Torino, Italy
§Department of Computer Science, University of Essex, UK
1http://www.cs.bham.ac.uk/∼wbl/biblio/
2http://www.cs.bham.ac.uk/∼wbl/biblio/gp-coauthors/
http://arxiv.org/abs/0704.0551v1
study usually requires that the coauthors become personally acquainted.
Thus, studying those ties, their structure, and their evolution allows a better
understanding of the factors that shape scientific collaboration.
We present a systematic study of the GP coauthorship data base us-
ing methods and tools pertaining to complex networks and social network
analysis. Social network analysis (see [?] for a survey), although it is an
old discipline, has recently received new impetus and tools from the field of
complex networks (see [?] for an excellent review). This is mainly due to
the relatively recent availability of large machine-readable databases such as
the GP bibliography. Social acquaintances involve psychological and other
human aspects that are difficult to quantify. However, as it has been done
in other fields [?, ?, ?, ?], we use objective data such as joint published work
to stand for social bonds. Since this must ignore subtler aspects of a col-
laboration relationship, it is obviously far from perfect as a social indicator,
yet it is still a good “proxy” for the network of social relationships and can
reveal several interesting facts and trends.
A preliminary investigation of the GP coauthorship network appears in
[?]. In the first part of this article we update this initial study using the
most recent data and adding the study of the influence of excluding co-
edited proceedings and books. In the second part we offer a new analysis of
the finer community structure of the collaboration network. Similar studies
have been performed in the last few years on several other collaboration
networks in disciplines such as physics, mathematics, medicine, biology, and
computer science [?, ?, ?, ?]. A related investigation concerning the EC
collaboration network [?] has appeared recently in popular form, but it does
not take into account, for example the community structure of the network.
[?] deals with some of the same statistical features for the EC community
at large as we describe in detail here for GP. The values reported by [?] are
in line with those found here for the GP field. Given that the intersection
between the GP researchers and general EC is likely to be rather large, it
would be interesting to study how they are related to each other.
2 The GP Collaboration Network
We treat the genetic programming social network as a graph where each
node is a GP researcher, i.e. someone who has at least one entry in the bib-
liography. There is a connection between two people if they have coauthored
at least one paper, or if they have coedited one or more book or proceedings.
As of the start of 2007, there is a total of N = 2809 connected nodes, i.e.
authors that have at least one GP collaborator, and a total of 5853 edges
(collaborations) in the GP coauthorship network. There are 367 isolated
vertices, which represent authors who have not collaborated with others to
the extent of coauthoring a paper. Isolated vertices are ignored in our graph
statistics. We have also excluded a single paper with 108 coauthors in a
nuclear physics journal. This is because we consider it to be an anomalous
entry that is not representative of typical collaborations in our discipline.
Due to the youth of GP, the graph is relatively small compared to some
studied collaboration networks [?, ?, ?]. (Although some published studies
have covered much smaller and more specialised networks, e.g. of only 50
people [?].) The main disadvantage of studying a relatively small database
is that, like any statistical study, more data allows deeper and more mean-
ingful inferences to be drawn. In particular, studies of the form of the
distributions (such as whether they follow exponential or power laws) re-
quire a large amount of data. The advantages include that the graph almost
fully represents the state of the whole GP community. This allows reliable
characterisation of collaboration in the community. Also, the problems of
multiple authors with the same name (e.g. A. Smith), outliers and different
name spelling that plague the larger data sets, are unlikely and easy to spot
in our data.
Although in many cases in our field co-editing a book or proceedings vol-
ume does reflect personal acquaintance, there are some large coeditorships
which are not representative and so may give a slanted view. Therefore in
the following figures we present two kinds of statistics: those that include
all joint publications and those in which co-edited conference proceedings
and co-edited books are excluded (but not their contents, of course). Next
we present and discuss some basic measures that characterise the GP col-
laboration network.
2.1 Number of Papers per Author
The average number of papers per author is 3.16 with co-edited books and
proceedings and it is 3.14 without. The five most prolific authors are, in de-
creasing order: J. Koza, R. Poli, W. B. Langdon, W. Banzhaf and C. Ryan.
If we exclude proceedings’ co-editors the ranking remains unchanged. Nat-
urally the distribution of the number of papers per author, P (k), has some
scatter, particularly in the tail of the distribution. Thus, we present in Fig-
ure 1 the graph of the cumulative distribution P (k ≥ n) which is smoother
and allows the same inferences to be made. The curves are rather well fitted
by a straight line, and thus the distributions follow a power-law P (k) ∝ k−γ
10000
1 10 100 1000
number of entries
with coeditors
without coeditors
power-law fit
Figure 1: Cumulative distribution of the number of entries per author. Log-
log scale. The straight line is the best mean-square fit and shows the number
of authors is ∝ k−2.5.
with a calculated exponent γ of 2.5 for both of them. A power-law distribu-
tion with similar exponents has been observed for analogous collaboration
networks, e.g. 2.86 for a biological publication database (Medline), 3.41 for
a computer science database (NCSTRL), 2.4 for mathematics, and 2.1 for a
neuroscience papers database [?, ?]. A smaller exponent (in absolute value)
means that the tail of the distribution is more stretched towards high values
of degree.
2.2 Number of Collaborators per Author
The average number of collaborators per author, i.e. the mean degree 〈k〉 of
the coauthorship graph, is 4.17 with proceedings and 3.62 without. This is
close to the values reported by studies of computer science, physics (exclud-
ing high energy physics) and Mathematics, suggesting GP follows similar
collaboration patterns to those disciplines. However it is much less than
found in high energy physics and medicine. See Table 1. In order and
including co-edited volumes, the five authors that have the largest num-
ber of collaborators are: W. Banzhaf, J.A. Foster, P. Nordin, W.B. Lang-
don, U.-M. O’Reilly. Without co-edited books the ranking is: P. Nordin,
W. Banzhaf, J. Daida, C. Ryan and R. Goodacre. The five “pairs” that
have the highest number of coauthored papers are, in decreasing order both
with or without co-edited proceedings: J. Koza–M. A. Keane, R. Poli–W.B.
Langdon, J. Koza–D. Andre, J. Koza–F. Bennet and F. Bennet–M.A. Keane.
This shows that J. Koza’s group has been tightly collaborating for a long
time, a conclusion that is confirmed in the community study of section 4.
It is also evident that the W.B. Langdon–R. Poli association has been an
extremely productive one.
10000
1 10 100
number of collaborators
with coeditors
without coeditors
Figure 2: Cumulative distribution of the number of authors with a given
number of collaborators. Logarithmic scale on both axes.
Figure 2 shows the cumulative distributions of the number of collabora-
tors. One sees that the distributions are not pure power-laws, otherwise the
points would approximately lie on a straight line. Rather, the distributions
shows a power-law regime in the first part followed by an exponential decay
in the tail. That is, the whole network cannot be fitted by a power-law.
This is quite common. In fact, several measured social networks do not
follow a power-law degree distribution [?, ?] and are best fitted either by
an exponential degree distribution P (k) ≈ e−k/〈k〉 or by an exponentially
truncated power-law of the type P (k) ≈ k−γe−k/kc , where kc represents a
critical connectivity and 〈k〉 is the average degree.
2.3 Number of Authors per Paper
Figure 3 shows the cumulative distribution of the number of papers written
by a given number of coauthors. Here the distribution also has a tail that is
longer than that of a Gaussian or exponential distribution, however it does
not follow a power-law. The average number of authors per paper is 2.25
(2.22 without co-editors). From Table 1 we can see that these figures are
close to the equivalent ones for computer science (NCSTRL) and physics,
while Mathematics has a lower number of co-authors per paper. On the
other hand, nuclear physics stands out with an unusually high number of
coauthors per paper.
10000
1 10 100
number of coauthors
with coeditors
without coeditors
Figure 3: Cumulative distribution of the number of papers with a given
number of coauthors on log-log scales.
From Figures 2 and 3 one can see that the tails of the distribution with
co-editors are longer than without them. Thus, taking co-editorship into
account seems to rather artificially inflate the number of publications with
many co-authors and, by consequence, the number of collaborators that a
person has.
2.4 Connected Components
In the theory of Poisson random graphs there is a critical value of average de-
gree 〈k〉 = 1 above which there is a sudden appearance of a giant component.
This is so-called since most vertices belong to it. The other components are
smaller and have an exponentially decreasing size distribution [?]. Although
collaboration graphs are not random, a similar phenomenon appears. In-
cluding coeditors there are 1025 GP authors in the giant component. This
is 36.5% of the total graph. If we exclude coediting proceedings etc. the
size is 743, representing the 26.9% of the total. In the giant component the
average number of collaborators per author is 5.83 with co-editors and 4.39
without them.
The cumulative size distribution of the connected components with and
without co-editors are depicted in Figure 4. Figure 4 shows that the proba-
bility density functions are well approximated by a power law with exponent
of 2.9 (excluding co-editors) and 2.6 (total). Since the other authors did not
provide the analogous data for their databases, we do not know how our
figures would compare with those for other coauthorship databases.
The existence of a big connected component has a social meaning. It
suggests 36.5% of GP researchers are members of a single community, since
those researchers are either directly connected via a collaboration or they
are close to each other in a way that will be made clear in section 3. The
size of the giant component is notably smaller in the GP graph with respect
to other measured coauthorship networks (see Table 1). This may be due
to the comprehensive nature of the GP bibliography. It captures work done
by smaller groups which does not get into major journals, whereas, perhaps,
the other databases concentrate upon higher impact outlets where work is
heavily cited but at the expense of ignoring less regarded authors. This
may artificially inflate the fraction of authors within their giant component.
Alternatively it may be due to the youth of the GP field, with many semi-
isolated individuals and groups starting research independently.
One should also consider that all collaboration networks are in a non-
equilibrium state as they are continuously evolving [?]. Accordingly, as
time goes by, one should observe small components progressively connecting
themselves to the large one. For example, in less than one year the size of
the giant component including co-editors has grown from 942 to 1025 nodes.
This is due in part to a number of newcomers collaborating with people
already belonging to the giant component. The other part comes from the
absorption of a few disconnected small components into the giant one thanks
to one or more new collaborations. This suggests that the size of the giant
component has not yet reached its “steady-state” value and it will continue
to grow in relative size. Since we possess all the time-stamped data, it is
possible to study the evolution of this component, as well as several other
indicators from the beginning and up to the present days. This investigation
is currently under way.
2.5 Social GP Clusters
The clustering coefficient of a node in a graph is the proportion of its neigh-
bouring nodes which are also neighbours of each other. The average clus-
tering coefficient 〈C〉 is calculated across all nodes in the graph [?]. In other
words, 〈C〉 is a simple statistical measure of the amount of local structure
1 10 100 1000
component size
with coeditors
without coeditors
power-law fit
power-law fit
Figure 4: Cumulative distributions of the number of connected components
in the collaboration graph by number of people. Log-log scale.
that is present in a graph. Most real-world networks, e.g. the world wide
web, roads, electrical power transmission and including the social networks
that have been studied to date, have a much larger clustering coefficient
than would be expected of a random graph with the same number of ver-
tices and edges. Social networks are particularly clustered. For example,
the average clustering coefficient is 〈C〉 = 0.665 for the GP collaboration
graph including book co-editors, and it is 0.660 without. (We would expect
0.0015 and 0.0013 for the corresponding random graphs). In terms of scien-
tific collaborations, a high clustering coefficient means that people tend to
collaborate in groups of three or more. This agrees with what we know of
the GP field. It may mean that two researchers that collaborate indepen-
dently with a third one may, in time, become acquainted and so collaborate
themselves. Alternatively it might be due to collaborators coming from the
same institution. In all cases, a high value of 〈C〉 for a social network is
an indication that collaborations are not made at random at all, and that
social forces and processes are at work in the network structure formation.
Table 1 summarises the results of this section and compares them with
those for some other collaboration networks. Some of the entries in the table
will be discussed in the following section. Most GP statistics are similar to
those of the larger databases. However one notable difference, as we have
already remarked, is the relative smallness of the largest component. The
clustering is rather high, which shows that GP researchers know each other
quite well within the large component, and the community is rather homoge-
Table 1: Basic statistics for some scientific collaboration networks. GP1 is
the GP bibliography at the start of 2007, including coedited books and pro-
ceedings. GP2 is the same but without coeditors. SPIRES is a data set of
papers in high-energy physics. Medline is a database of articles on biomed-
ical research. Mathematics comprises articles from Mathematical Reviews.
NCSTRL is a database of preprints in computer science. Physics has been
assembled from papers posted on the Physics E-print Archive. Details about
these databases can be found in [?, ?, ?].
GP1 GP2 SPIRES Medline Mathematics NCSTRL Physics
Total number of papers 4564 4504 66652 2163923 1600000 13169 98502
Total number of authors 2809 2765 56627 1520251 253339 11994 52909
Average papers per author 3.16 3.14 11.6 6.4 7 2.55 5.1
Average authors per paper 2.25 2.22 8.96 3.754 1.5 2.22 2.53
Average collaborators per author 4.17 3.62 173 18.1 2.94 3.59 9.7
Size of the giant component (%) 36.5 26.9 88.7 92.6 82.0 57.2 85.0
Clustering coefficient 0.665 0.660 0.726 0.066 0.15 0.496 0.43
Average path length 4.74 5.2 4.0 4.6 7.73 9.7 5.9
neous. In contrast, in biology and medicine or mathematics, where scientist
from different sub-disciplines seldom collaborate, the clustering coefficient is
lower. Note also the high number of authors per paper, and especially the
strikingly high number of collaborators per author in the nuclear physics
community (SPIRES). Clearly, nobody can maintain an average of 173 sci-
entific partners on a first-hand acquaintance basis and thus this figure does
not seem to be socially meaningful.
3 Distances and Centrality
A social network can be characterised by a number of measures that give an
idea of “how far” people are from each other, or how “central” they are with
respect to the whole community. These measures are well known in social
network analysis. Here we shall concentrate on average path length and on
betweenness centrality.
3.1 Average Path Length
The average path length L of a graph is the average value of the shortest
paths between all of its pairs of vertices. In random graphs and many real
networks, such as the Internet, the World Wide Web and social networks,
the average path distance scales as a logarithmic function O(logN) of the
number of vertices N . Such networks, if they also have a high clustering
coefficient, are known as small worlds networks [?]. Since, even for very large
graphs, any two nodes in a small world network are only a few steps apart.
In contrast in regular lattices, two nodes are O(N
D ) apart. (Where D is the
lattice’s dimensionality. For example, for a square lattice L ≤ 2
2 ). The
average path length of the giant component of the GP collaboration graph
including coeditors is 4.74 (it is 5.2 without coeditors). The longest among
all the shortest paths (known as the diameter) is 12 (14 without coeditors).
Thus, unsurprisingly, the GP community, as far as its “core” component
is concerned, is indeed a small world and is characterised by values that
are typical of these kinds of network (see Table 1). Being a small world
means that information may circulate quickly and collaborations are easier
to set up. These are clearly advantageous for a research community. The
connected components following the largest one are themselves small worlds.
We expect over time some of them will merge with the largest component.
(For this to happen, only a single new collaboration between two scientists
each belonging to one of the components is needed.)
3.2 Betweenness
The betweenness b(v) of a vertex v is the total number of shortest paths be-
tween all possible pairs of vertices that pass through this vertex. Nodes that
have a high betweenness potentially have more influence, i.e. they are more
central in the network, in that there is more “traffic” that goes through
them. The first five authors in terms of betweenness in the network (in-
cluding co-editors and in decreasing order) are: W. Banzhaf, H. Iba, U.-M.
O’Reilly, H. de Garis and W. B. Langdon. W. Banzhaf is also the researcher
that has the highest number of different collaborators. Without co-editors
the ranking is: W. B. Langdon, U.-M. O’Reilly, W. Banzhaf, M. Tomassini
and P. Nordin. People who have a large value of betweenness play the role
of intermediaries or “brokers” in a social sense.
3.3 Non-random collaborations between
directly connected authors
Most technological and biological networks are disassortive in that they have
negative correlation, meaning that high-degree vertices are preferentially
connected to low-degree vertices. However most measured social networks
are assortative, meaning highly connected nodes tend to be connected with
other highly connected nodes [?]. The GP collaboration network confirms
this general observation with a correlation coefficient of +0.15 for the gi-
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Lanz Pizz
Figure 5: One of the communities belonging to the main network compo-
nent. The thickness of the links gives an indication of the number of co-
authored papers. The largest thickness indicates more than 16 coauthored
works. The thinnest link (light gray) stands for a single collaboration. The
different symbols and colours represent sub-communities of the illustrated
community.
ant component, and +0.30 for the whole graph (including coeditors and
excluding the single physicist’s paper). These are close to the coefficients
observed for other social networks (specifically 0.127 for Medline and 0.120
for Mathematics [?]).
4 Communities in the Giant Component
All the researchers belonging to the largest component of the network can be
said to form part of the GP community at large, in the sense that they are all
only a few steps away from any other member of the community. However,
we know from direct experience that some groups of GPers are more closely
connected between themselves than with other people. In other words, they
belong to what one might call a group or a tighter community within the
global one. It is not easy to give a rigorous quantitative definition of a
community within a network. For our purposes a community can be seen
as a set of highly connected vertices having few connections with vertices
belonging to other communities. In the analysis of social networks, several
algorithms that attempt to split a network into communities have been
proposed. We used Newman’s method [?], which is based on a measure
of the fraction of edges that fall within communities minus the expected
value of the same quantity if edges fall at random without regard for the
community structure.
Since the GP bibliography contains the number of papers that any two
collaborators have published together, it is possible to go a step further
than just saying that two people have coauthored at least a paper, and
give a measure of the intensity of the collaboration. We use the number of
papers that two given authors have in common as a measure of the strength
of their collaboration. Newman [?] has proposed a more refined measure
which takes into account the actual number of coauthors of each paper.
However this is more complicated than we need, instead we ignore the total
number of coauthors for each paper. Our measure of collaboration strength
is used in our communities algorithm to highlight groups of researchers that
have collaborated strongly with the aim of uncovering the stability of the
scientific relationship. We have also excluded coedited proceedings, books,
etc., as we have already seen that these might sometimes represent spurious
collaboration relationships.
The results of running the algorithm on the subgraph represented by the
largest connected component are qualitatively surprisingly close to what one
would expect, given our knowledge of the GP field. The advantage is that the
analysis makes them explicit and uncovers a number of other relationships
that would be difficult to infer without an explicit study of the raw data.
As an example of the about 25 communities that the algorithm discovers,
Figure 5 shows the structure of the groups around one of us (“Toma”). If we
now consider this community as an isolated subgraph and apply again New-
man’s algorithm to it, we obtain the groups highlighted by different symbols
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Chio BuxtHoll
Figure 6: Another community belonging to the main network component.
The thickness of the links gives an indication of the number of co-authored
papers. The largest thickness indicates more than 16 coauthored works.
The thinnest link (light gray) stands for a single collaboration. The different
symbols and colours represent sub-communities.
and colours in the figure. Thus, the groups correspond to sub-communities
within the main community. The thickness of the links represents the inten-
sity of the relationship. It is easy to recognise a “hard core” of collaborating
researchers strongly connected to “Toma” forming triads and higher poly-
gons of order four and five. The strong triangle (“Foli”, “Pizz”, “Spez”) is
relatively loosely connected to the rest, showing that these researchers be-
long to the community but often collaborate between themselves. It is also
possible to discern institutional and geographical components in the com-
munity. For example, most of the upper right part of the figure through the
node “Chop” comprises researchers essentially belonging to the University
of Geneva, which is close to the University of Lausanne, to which “Toma”
belongs. However, geographical closeness is not the key factor in the other
groups which belong to Universities in France, Italy, Spain, and the US.
We might conjecture that many collaborations start locally at the same or
at close institutions and then they spread through people being introduced
to others via a common acquaintance, or through people physically moving
or visiting other institutions. This is the case in the figure, where “Vann”,
”Chop”, and “Vega” among others have played the role of “bridges” between
different institutions and across countries.
As a second illustration, let us look at Figure 6 which is the community
that revolves around one of us (“Lang”) and “Poli”. In contrast to the pre-
vious case, one can see that the graph structure is more “star-like”, with two
large directly connected big hubs (“Lang” and “Poli”) who have about 70 co-
authored papers, and three other highly connected nodes (“Buxt”,“McPh”,
“Rowe”) which are strongly connected to one of the main hubs but not to
both. It is interesting to observe the role of “McPh” who, like “Vega” in the
previous community (cf. Figure 5), plays a bridging role, this time between
some UK and some American researchers. We can also recognise a strong
”theory-oriented” group, which is almost a clique in the graph, formed by
(“McPh”,“Poli”,“Rowe”, “Steph”, “Wrig”). There is also another bridge
formed by “Cagn” from UK to Italy, again due to a long-standing collabo-
ration and friendship. The small cliques or almost cliques at the periphery
of the figure essentially represent people that have worked at the same in-
stitution in either Italy or the US.
The discussion above, motivated by our belonging to the mentioned com-
munities, and thus by our direct human knowledge about them, should be
enough to get an impression of the many useful observations that one can
make on the communities that interlock in the main network component.
There are of course several other large well known and interesting commu-
nities in the network but unfortunately we cannot describe them here for
reasons of space.
5 Conclusions
In sections 2 and 3 we characterised the genetic programming (GP) coau-
thorship graph using a number of local and global statistics. We extended
and updated the findings presented in [?] by studying the influence of
coedited volumes and by using the latest data available. Section 3 showed
the GP field to be highly clustering and that the GP coauthorship network
has a small mean path length. Together these suggest that, at least for the
core, GP is indeed a “small world”. We also found, compared with other
published collaboration networks, that the fraction of GP authors connected
by coauthorship is a relatively small fraction of all GP authors.
Section 4 is a study of the community structure of GP. It uses a more pre-
cise definition of collaboration, which takes into account the intensity of the
relationship. This uncovers many groups of tightly interacting researchers.
From the detailed study of two of the communities we have drawn inferences
about the pivotal role of some researchers or groups of researchers in pro-
moting collaborations within and between academic institutions. Adding
our human knowledge about geographical location and personal acquain-
tance, allows some conjectures to be drawn about the way in which different
continents and countries collaborate on research projects.
It should be obvious that the present data driven approach to social
network analysis can only provide some answers but not all of them. Algo-
rithms and data cannot take into account human aspects such as friendship
in scientific collaboration. While these may be buried in the sea of numbers
they will never appear explicitly from such analyses. Nevertheless, we feel
that our results are interesting and useful in the way that they characterise
our community.
There is another aspect of the collaboration graph that would be re-
vealing: the analysis of its development over the years. Indeed, since each
paper has a date of publication, we possess all the data that are needed
for such an investigation. This would allow the detailed study of how the
network has grown to its present size and structure from the beginning and
might give hints as to its future progress. This extension is currently under
investigation.
	Introduction
	The GP Collaboration Network
	Number of Papers per Author
	Number of Collaborators per Author
	Number of Authors per Paper
	Connected Components
	Social GP Clusters
	Distances and Centrality
	Average Path Length
	Betweenness
	Non-random collaborations between directly connected authors
	Communities in the Giant Component
	Conclusions
