Abstract
Modern bibliographic databases provide the basis for scientific research and its evaluation. While their content and structure differ substantially, there exist only informal notions on their reliability. Here we compare the topological consistency of citation networks extracted from six popular bibliographic databases including Web of Science, CiteSeer and arXiv.org. The networks are assessed through a rich set of local and global graph statistics. We first reveal statistically significant inconsistencies between some of the databases with respect to individual statistics. For example, the introduced field bowtie decomposition of DBLP Computer Science Bibliography substantially differs from the rest due to the coverage of the database, while the citation information within arXiv.org is the most exhaustive. Finally, we compare the databases over multiple graph statistics using the critical difference diagram. The citation topology of DBLP Computer Science Bibliography is the least consistent with the rest, while, not surprisingly, Web of Science is significantly more reliable from the perspective of consistency. This work can serve either as a reference for scholars in bibliometrics and scientometrics or a scientific evaluation guideline for governments and research agencies.
Introduction
Bibliographic databases range from expensive handcurated professional solutions like Web of Science and Scopus to preprint repositories^{1}, public servers^{2} and automated services that collect freely accessible manuscripts from the Web^{3,4}. These provide the basis for scientific research, where new knowledge is derived from the existing, while also the main source of its evaluation. Undoubtedly, the number of citations a paper receives is still considered to be the main indicator of its importance or relevance^{5,6}. However, the probability distribution of scientific citations has been shown to follow a wide range of different forms including powerlaw^{7}, shifted powerlaw^{8}, stretched exponential^{9}, lognormal^{10}, Tsallis^{11} and modified Bessel^{12}, to name just a few. Although some methods used in these studies might be questionable, more importantly, they are based on different bibliographic data. In fact, the content and structure of modern bibliographic databases differ substantially, while there exist only informal notions on their reliability.
One way to assess the databases is simply by the amount of literature they cover. Web of Science spans over 100 years and includes several dozens of millions of publication records^{13,14}, an extent similar to that of Scopus, which, however, came into existence only some ten years ago. On the other hand, the preprint repository arXiv.org^{1} and the digital library DBLP Computer Science Bibliography^{2} both date back to 1990s and include only millions of publications or publication records. The coverage of different bibliographic databases has else been investigated by various scholars^{14,15,16,17}, while others have analyzed also their temporal evolution^{1,18}, available features^{15,19}, data acquisition and maintenance methodology^{14,20} and the use within a typical scientific workflow^{21}.
Yet, despite some notable differences, the reliability of bibliographic databases is primarily seen as the accuracy of its citation information. While citations are input by hand in the case of professional databases, services like CiteSeer and Google Scholar use information retrieval and machine learning techniques to automatically parse citations from publication manuscripts^{3,4}. Expectedly, this greatly impacts bibliometric analyses^{20} and standard metrics of scientific evaluation like citation counts and hindex^{17,22}. Although networks of citations between scientific papers have been studied since the 1950s^{7,13} and are also commonly used in the modern network analysis literature^{23,24}, there exists no statistical comparison of citation topology of different bibliographic databases.
In this study, we compare the topological consistency of citation networks extracted from six popular bibliographic databases (see Methods). The networks are assessed through local and global graph statistics by a methodology borrowed from the machine learning literature^{25}. We first reveal statistically significant inconsistencies between some of the databases with respect to individual graph statistics. For example, the introduced field bowtie decomposition of DBLP Computer Science Bibliography substantially differs from the rest due to the coverage of the database or the sampling procedure, while the citation information within arXiv.org is proven to be the most exhaustive. Finally, we compare the consistency of databases over multiple graph statistics. The citation topology of DBLP Computer Science Bibliography is the least consistent with the rest, while, not surprisingly, Web of Science is significantly more reliable from this perspective. Note that the reliability is here seen as a deviation from the majority (see Discussion). Differences between other databases are not statistically significant. This work can serve either as a reference for scholars in bibliometrics and scientometrics or a scientific evaluation guideline for governments and research agencies.
Results
Citation networks representing bibliographic databases are compared through 21 graph statistics described in Methods. In the following, we discuss the values of statistics in the context of complex network theory. Next, we reveal some statistically significant differences in individual statistics using Student ttest^{26}. We then select ten statistics whose independence is confirmed by Fisher ztest^{27} and show that the databases display significant inconsistencies in the selected statistics using Friedman rank test^{28,29}. Last, the databases with no significant inconsistencies are revealed by Nemenyi posthoc test^{30} and the critical difference diagram^{25}. Finally, we also compare the bibliographic databases with the selected online databases to verify the predictive power of the employed statistical methodology. See Methods for further details on statistical comparison.
Graph statistics of citation networks
Table 1 shows descriptive statistics of citation networks. The networks range from thousands of nodes to millions of links, while the largest weakly connected components contain almost all the nodes. This is consistent with the occurrence of a giant connected component in random graphs^{31}. Directed networks are often assessed also according to their bowtie structure^{32}. However, due to the acyclic nature of citation networks where papers can only cite papers from the past, the decomposition proves meaningless. We introduce the field bowtie decomposition into the infield component, which consists of papers citing no other paper, the outfield component, which consists of papers not cited by any other paper and the field core. The outfield component thus includes the research front^{7} and the infield and core components include the knowledge or intellectual base^{33}. Table 1 shows the percentage of nodes in each of the field components, while a visual representation is given in Fig. 1. Notice that, in most cases, the majority of papers is included in the core and outfield components of the citation networks. Nevertheless, the main mass of the papers shifts towards the incomponent in HistCite and DBLP databases (Figure 1, panels D and E). Since the former consists of papers from merely major journals and conferences and the latter is based on the bibliography of a single author, many of the papers in the databases cite no other. Hence, reducing a bibliographic database to only a subset of publications or authors gives notably different citation structure and also influences many common graph statistics.
Table 2 shows degree statistics of citation networks. Observe that the mean degree 〈k〉 is around 8.8 in all cases except arXiv database, which, somewhat surprisingly, coincides with the common density of realworld networks^{34}. Note, however, that since 〈k〉/2 = 〈k_{in}〉 = 〈k_{out}〉 for any network, the papers cite and are cited by only four other papers on average. This number becomes meaningful when one considers that far more citations come from outside the field^{18,35}, whereas all databases are subsets of their respective fields in some sense. Considerably higher 〈k〉 in arXiv database is most likely due to several reasons. In contrast to other databases, arXiv.org stores journal and conference papers, technical reports, draft manuscripts that never came to print etc. Next, the citation network studied has been released within the KDD Cup 2003 (http://www.cs.cornell.edu/projects/kddcup) and has thus presumably been cleansed appropriately. Also, the subset of arXiv.org considered consists of physics publications, while other databases consist of computer science publications. Regardless of the true reason, the citation information within arXiv database is notably more exhaustive, which clearly reflects in its graph structure (see field bowtie in Fig. 1, panel F).
Figure 1 plots degree distributions of citation networks, while the corresponding scalefree^{36} exponents γ, γ_{in} and γ_{out} are given in Table 2. We stress that not all distributions, especially outdegree distributions, are a valid fit to a powerlaw form^{37}. Nevertheless, the degree distributions further confirm the inconsistencies observed above. A larger number of nonciting papers results in a less steep outdegree distribution, whereas γ_{out} ≈ 2.6 for HistCite and DBLP databases, while γ_{out} ≈ 3.8 otherwise. On the contrary, the indegree distribution of HistCite database is much steeper with γ_{in} = 3.5, while γ_{in} ≈ 2.5 for the rest. In fact, γ_{in} > γ_{out} for HistCite database, whereas γ_{in} < γ_{out} for all others. Finally, the lack of lowciting papers in arXiv database prolongs the degree distributions towards the righthand side of the scale (see Fig. 1, panel F).
Degree mixing^{38} in Table 2 reveals no particularly strong correlations. Still, the indegree and outdegree mixing coefficients r_{(in,in)} and r_{(out,out)} show positive correlation, while the undirected degree mixing r is negative. For comparison, in social networks and for Internet and the Web^{38,39}. Again, HistCite and DBLP databases deviate from common behaviour due to the reasons given above. For example, the directed degree mixing coefficient r_{(out,in)} is substantially lower for HistCite database, while all directed coefficients are relatively low for DBLP database. Figure 1 plots also neighbour connectivity profiles of citation networks. Notice dichotomous degree mixing^{40} that is positive for smaller outdegrees and negative for larger indegrees, represented by increasing or decreasing trend, respectively (see, e.g., Fig. 1, panels A and B). Similar observations were recently made also in software^{41} and undirected biological^{40} networks. Consistent with the above, these trends are not present in HistCite and DBLP databases (see Fig. 1, panels D and E).
Table 3 shows clustering^{42} statistics of citation networks. The mean clustering coefficients 〈c〉, 〈b〉 and 〈d〉 greatly vary across the databases, whereas 〈c〉 ≈ 0.15 for WoS, CiteSeer and DBLP databases and 〈c〉 ≈ 0.3 in the case of Cora, HistCite and arXiv databases. This may be an artefact of the coverage or the sampling procedure used for citation extraction, while clustering can also reflect the amount of citations copied from other papers^{43,44} known as indirect citation^{45}. Unbiased clustering mixing coefficients r_{b} and r_{d} in Table 3 reveal strong positive correlations, similar to other realworld networks^{41}. However, as before, r_{d} = 0.26 for DBLP database, while r_{d} ≈ 0.4 for all others. Figure 1 plots clustering profiles of citation networks. Due to degree mixing biases^{46}, C(k) ~ k^{−α} for α ≈ 1^{47}, while this behaviour is absent from corrected profiles B(k) and D(k).
Table 3 shows also diameter statistics of citation networks. Undirected effective diameter is somewhat consistent across the databases, in contrast to the directed variant δ_{90}, where δ_{90} ≈ 8.5 for WoS, HistCite and DBLP databases, while δ_{90} > 20 for other databases. Low value of δ_{90} for HistCite and DBLP databases is due to the limited coverage discussed above, whereas the respective networks are also much smaller (see Table 1). On the other hand, low δ_{90} for WoS database is due to a rather nonintuitive phenomena that realworld networks shrink as they grow^{23}. WoS database includes 50 years of literature, while the time span of, e.g., arXiv database is merely 10 years. The databases are thus not directly comparable in δ_{90} and neither is indeed inconsistent with the rest. Described can be more clearly observed in hop plots shown in Fig. 1 (see, e.g., panels A and B).
Comparison of databases by individual statistics
The above discussion was in many cases just qualitative. In the following, we reveal also statistically significant differences between some of the databases with respect to individual graph statistics. Since their values of a true citation network are, obviously, not known, we compute externally studentized residuals that measure the consistency of each database with the rest (Figure 2, panels A–F). Statistically significant inconsistencies in individual statistics are revealed by independent twotailed Student ttests (see Methods).
WoS, CiteSeer and Cora databases show no significant differences at Pvalue = 0.05. On the contrary, the scalefree indegree exponent γ_{in} in HistCite database is significantly higher than in other databases, while the directed degree mixing coefficient r_{(out,in)} is significantly lower (Pvalue = 0.019 and Pvalue = 0.033, respectively; see Table 2 and Fig. 2, panel D). This is a direct consequence of the limited coverage already noted above. For example, since the database is derived from a bibliography of a single author, highly cited papers are likely missing, which results in a much steeper citation distribution P(k_{in}) and thus higher γ_{in}. Next, the unbiased clustering mixing coefficient r_{d} is significantly lower in DBLP database (Pvalue = 0.017; see Table 3 and Fig. 2, panel E). Apparently, reducing the bibliographic database to only selected publications gives a rather heterogeneous citation structure, which does not share high clustering assortativity^{41}, , of other citation networks. Note that the differences in the field bowtie decomposition of DBLP database become statistically significant at Pvalue = 0.052 (see below). Finally, as thoroughly discussed above, the citation information within arXiv database is significantly more exhaustive with much higher mean degree 〈k〉 (Pvalue = 0.009; see Table 1 and Fig. 2, panel F). Notice that statistically significant inconsistencies between the databases are, expectedly, merely a subset of the differences exposed through the expert analysis above. Still, in summary, the results reveal that bibliographic databases with substantially different coverage have significantly different citation topology.
At Pvalue = 0.1, several other inconsistencies become statistically significant. For CiteSeer database, the largest weakly connected component is significantly smaller than in other databases (Pvalue = 0.059; see Table 1 and Fig. 2, panel B); for HistCite database, the clustering mixing coefficient r_{c} is lower (Pvalue = 0.066; see Table 3 and Fig. 2, panel D); for DBLP database, the infield component is larger (Pvalue = 0.052; see Table 1 and Fig. 2, panel E), while the field core and the directed degree mixing coefficient r_{(in,in)} are smaller (Pvalue = 0.090 and Pvalue = 0.095, respectively; see Table 1 and Table 2 and Fig. 2, panel E); and for arXiv database, the undirected degree mixing coefficient r and the corrected clustering coefficient 〈b〉 are higher (Pvalue = 0.081; see Table 2 and Table 3 and Fig. 2, panel F). Note that, due to space limitations, not all inconsistencies at Pvalue = 0.1 are discussed in the analysis above.
Selection of independent graph statistics
Since the adopted graph statistics of citation networks are by no means independent^{42,46}, one cannot simply compare the bibliographic databases over all. For this purpose, we select ten statistics listed in Fig. 2, panel G and verify their statistical independence (see Methods). We compute Fisher transformations of the pairwise Spearman correlations between the statistics, while significant correlations are revealed by independent twotailed ztests (Figure 2, panel H). Notice that no correlation is statistically significant at Pvalue = 0.01.
The selection of independent graph statistics proceeds as follows. We first discard statistics that are sums or aggregates of the others by definition. Namely, the sizes of the largest weakly connected and outfield components (see Table 1), the scalefree degree exponent γ, the undirected degree mixing r and also both mixed directed mixing coefficients r_{(in,out)} and r_{(out,in)} (see Table 2). We next discard statistics whose correlations have been proven in the literature^{46} or are dependent on some intrinsic characteristic of the database like the time span of the publications (see above). Namely, the standard clustering 〈c〉 and the corresponding mixing coefficient r_{c} and the directed effective diameter δ_{90} (see Table 3). Finally, out of the both unbiased clustering coefficients 〈b〉 and 〈d〉, we decide for the latter and its corresponding mixing coefficient r_{d} (see Table 3). We are thus left with ten statistics (Figure 2, panel G). Namely, the sizes of the infield and core components (see Table 2), the mean degree 〈k〉, the directed scalefree exponents γ_{in} and γ_{out} and the directed degree mixing coefficients r_{(in,in)} and r_{(out,out)} (see Table 2), the unbiased clustering 〈d〉 and its corresponding mixing coefficient r_{d} and the undirected effective diameter (see Table 3).
For some further notes on statistics independence see Discussion.
Comparison of databases over multiple statistics
In the following, we compare the bibliographic databases over independent graph statistics selected above. We rank the databases according to the studentized statistics residuals and compute their mean ranks over all statistics (see Methods). The final ranks are 2.2 for WoS database, 3.1 for both CiteSeer and Cora databases, 3.6 for arXiv database, 4.0 for HistCite database and 5.0 for DBLP database. Notice that the ranks indeed reflect the conclusions on database consistency given above. We reject the null hypothesis that the ranks of the databases are statistically equivalent by onetailed Friedman test at Pvalue = 0.05 and thus compare the ranks by twotailed Nemenyi posthoc test (Figure 2, panel I). The databases whose ranks differ by more than a critical distance 2.38 show statistically significant inconsistencies in the selected statistics at Pvalue = 0.05. Hence, the citation topology of WoS database is significantly more reliable than that of DBLP database, which is the least consistent with the rest. On the other hand, the differences between other databases are not statistically significant, whereas concluding that these are consistent with both WoS and DBLP databases would be a statistical nonsense^{25}. At Pvalue = 0.1, the critical distance drops to 2.17, while all conclusions still remain the same. Interestingly, neglecting the requirement for the independence of graph statistics and comparing the bibliographic databases over all 21 statistics, again gives exactly the same conclusions on their consistency. Although, the ranking changes, since arXiv database is ranked in front of Cora database.
For some further notes on database consistency see Discussion.
Comparison of bibliographic and online databases
To assess the power of the employed statistical methodology for quantifying the differences in network topology, we compare citation networks representing different bibliographic databases with two networks extracted from online databases. Namely, a technological network of Gnutella peertopeer file sharing (http://rfcgnutella.sourceforge.net) from August 2002^{23}, where nodes are hosts and links are shares between them; and a social network representing Twitter social circles (http://twitter.com) crawled from public repositories^{48}, where nodes are users and links are follows between them. Both these networks are provided within SNAP (http://snap.stanford.edu), while their basic descriptive statistics are given in Table 1.
Note that online databases reveal knowingly different network topology than reliable bibliographic databases. For example, the majority of nodes in Gnutella database is included in the infield component (see Methods), similarly as in DBLP database (see Table 1). Next, the mean degree 〈k〉 is considerably higher in Twitter database and lower in Gnutella database (see Table 2). Furthermore, the degree distributions of Gnutella database are not a valid fit to a powerlaw form^{37} with higher scalefree degree exponents γs than in other databases (see Table 2). On the contrary, the scalefree outdegree exponent γ_{out} of Twitter database is lower, similarly as in HistCite database. Online databases also reveal notably different clustering regimes than bibliographic databases (see Table 3). The standard and unbiased clustering coefficients 〈c〉 and 〈d〉 are much higher in Twitter database, while much lower in Gnutella database. Finally, Gnutella database shows relatively heterogeneous clustering structure with very low unbiased clustering mixing coefficients r_{b} and r_{d}.
In the following, we reveal statistically significant inconsistencies between some of the databases with respect to individual graph statistics (see Methods). We consider the online databases and four most reliable bibliographic databases so that all critical values remain the same as before. Under this setting, the bibliographic databases show no inconsistencies at Pvalue = 0.05 (Figure 3, panels A–D). On the other hand, five most significant inconsistencies of online databases almost precisely coincide with the differences exposed through the analysis above (Figure 3, panels E and F). For Gnutella database, the infield component is larger (Pvalue = 0.008), the degree and indegree scalefree exponents γ and γ_{in} are higher (Pvalue = 0.011 and Pvalue = 0.008, respectively) and the unbiased clustering mixing coefficients r_{b} and r_{d} are lower (Pvalue = 0.032 and Pvalue = 0.011, respectively); and for Twitter database, the mean degree 〈k〉 is higher (Pvalue = 0.039), the outdegree scalefree exponent γ_{out} and the directed degree mixing coefficient r_{(in,in)} are lower (Pvalue = 0.063 and Pvalue = 0.066, respectively) and the standard and unbiased clustering coefficients 〈c〉 and 〈d〉 are higher (Pvalue = 0.056 and Pvalue = 0.065, respectively).
In the remaining, we also rank the databases over multiple graph statistics as before (see Methods). We select ten statistics listed in Fig. 3, panel G, whose pairwise independence is confirmed at Pvalue = 0.001 (Figure 3, panel H). The overall ranks of the databases are not statistically equivalent at Pvalue = 0.05 and are given in Fig. 3, panel I. Expectedly, the online databases are the least consistent with the rest, whereas the ranks are 4.6 and 4.9 for Gnutella and Twitter databases, respectively and 1.9–3.3 for the bibliographic databases. Yet, merely WoS bibliographic database significantly differs from the online databases at Pvalue = 0.05 (see Fig. 3, panel I).
In summary, the employed statistical testing proves to be rather effective in quantifying the inconsistencies between network databases with respect to individual graph statistics. On the contrary, the comparison over multiple statistics appears to be less powerful and cannot distinguish between the online databases and all bibliographic databases considered above. Nevertheless, the statistically significant inconsistencies between WoS and DBLP bibliographic databases highlighted in the study can thus indeed be regarded as rather substantial.
Discussion
We conduct an extensive statistical analysis of the citation information within six popular bibliographic databases. We extract citation networks and compare their topological consistency through a large number of graph statistics. We expose statistically significant inconsistencies between some of the databases with respect to individual graph statistics and compare the databases over multiple statistics. DBLP Computer Science Bibliography is found to be the least consistent with the rest, while Web of Science is significantly more reliable from this perspective. The result is somewhat surprising, since DBLP Computer Science Bibliography is informally considered as one of the most accurate freely available sources of computer science literature. The analysis further reveals that the coverage of the database and the time span of the literature greatly affect the overall citation topology, although this can be avoided in the case of the latter. This work can serve either as a reference for the analyses of citation networks in bibliometrics and scientometrics literature or a guideline for scientific evaluation based on some particular bibliographic database or literature coverage policy.
We introduce the field bowtie decomposition of a citation network (see Methods), which proves to be one of the most discriminative approaches for comparing the citation topology of bibliographic databases (see Results). We also consider 18 other local and global graph statistics. Nevertheless, we neglect some possible common patterns of nodes like motifs^{49} and graphlets^{50} and the occurrence of larger characteristic groups of nodes like communities^{51} and modules^{52}. Yet, these structures are not well understood for the specific case of citations networks and thus not easily interpretable.
In the following, we provide some further notes on the representativeness and reliability of the bibliographic databases and the independence of the databases and adopted graph statistics.
As discussed in Methods, citation networks extracted from bibliographic databases are not necessarily representative due to citation retrieval procedure, data preprocessing techniques, size or other. It should, however, be noted that this work has been done after realizing that citation networks available from the Web provide a rather inconsistent view on the structure of bibliographic information. We have therefore collected and compared all such networks, while including also a citation network extracted from Web of Science. In that sense, the adopted networks are representative of the data readily available for the analyses and thus also commonly used in the literature^{23,24}. Still, other citation networks could give different conclusions on the reliability of bibliographic databases. In particular because the reliability is measured through consistency of the databases. The concepts are of course not equivalent, yet the study reveals that, in most cases, only a single database deviates from a common behaviour for some particular graph statistic (see Results). Hence, the reliability can indeed be seen as a deviation from the majority to a rather good approximation.
Independence between bibliographic databases is obtained trivially, since these are either based on independent bibliographic sources or cover different literature (see Methods). On the other hand, adopted graph statistics of citation networks are by no means independent^{42,46}. As this is required by several statistical tests, we reduce the statistics to a subset whose pairwise independence could be proven. Nevertheless, we only show that the statistics are not clearly dependent and we do not ensure their mutual independence. Although the conclusions of the study are exactly the same regardless of whether it is based on all or merely independent statistics (see Results), further reducing the subset of statistics would discard relevant information and no statistically significant conclusions could be made. We also stress that all results have been verified by an independent expert analysis. An alternative solution would be to transform the statistics into uncorrelated representatives using matrix factorization techniques like principal component analysis^{53}. However, interpreting inconsistencies in, e.g., 0.9γ_{in} − 1.4r_{c} + 0.3δ_{90} would most likely be far from trivial.
Methods
Bibliographic sources
In this study, we conduct a networkbased comparison of citation topology of six bibliographic databases. These have been extracted from publicly available and commercial bibliographic sources, services, software and a preprint repository with particular focus on computer science publications. For bibliographic sources based on a similar methodology^{14,15} (e.g., Web of Science and Scopus, CiteSeer and Google Scholar), a single exemplar has been selected. We have extracted a citation network from each of the selected databases. Publications neither citing nor cited by any other are discarded and any selfcitations that occur due to errors in the databases are removed prior to the analysis (see below and Table 1 for details). Although the databases contain fair portions of the respective bibliographic sources, we stress that they are not all necessarily representative. Still, in most cases, these are the only examples of citation networks readily available online (due to our knowledge) and thus also often used in the network analysis literature^{23,24}.
WoS database
Web of Science (WoS) is informally considered as the most accurate bibliographic source in the world. It is handmaintained by professional staff at Thomson Reuters (http://thomsonreuters.com), previously Institute for Scientific Information. It dates back to the 1950s^{7,13} and contains over 45 million records of publications from all fields of science^{14}. For this study, we consider all journal papers in WoS category Computer Science, Artificial Intelligence as of October 2013. The extracted database spans 50 years and contains 179,510 papers from 877 journals and 639,126 citations between them. Note that 39,148 papers neither cite nor are cited by any other, while the database includes 16 selfcitations.
CiteSeer database
CiteSeer or CiteSeer^{x} (CiteSeer) is constructed by automatically crawling the Web for freely accessible manuscripts of publications and then analyzing the latter for potential citations to other publications^{3} (http://citeseer.ist.psu.edu). It became publicly available in 1998 and is maintained by Pennsylvania State University. It contains over 32 million publication records from computer and information science^{14}. For this study, we consider a snapshot of the database provided within KONECT (http://konect.unikoblenz.de) that contains 723,131 publications and 1,751,492 citations between them. Note that 338,718 publications neither cite nor are cited by any other, while the database includes 6,873 selfcitations.
Cora database
Computer Science Research Paper Search Engine (Cora) is a service for automatic retrieval of publication manuscripts from the Web using machine learning techniques^{4} (http://people.cs.umass.edu/~mccallum). It contains over 50,000 publication records collected from the websites of computer science departments at major universities in August 1998. For this study, we consider a subset of the database that contains 23,166 publications and 91,500 citations between them^{54} (http://lovro.lpt.fri.unilj.si). Note that all papers either cite or are cited by some other, while the database includes no selfcitations.
HistCite database
Algorithmic Historiography (HistCite) is a software package for analysis and visualization of bibliographic databases owned by Thomson Reuters (http://www.histcite.com). It was developed in the 2000s for extracting publication records from WoS database^{55}. For this study, we consider a complete bibliography of Nobel laureate Joshua Lederberg produced by HistCite in February 2008. The database contains 8,843 publications and 41,609 citations between them (http://vlado.fmf.unilj.si). Note that 4,519 publications neither cite nor are cited by any other, while the database includes 14 selfcitations.
DBLP database
DBLP Computer Science Bibliography (DBLP) indexes major journals and proceedings from all fields of computer science^{2} (http://dblp.unitrier.de). It is freely available since 1993 and handmaintained by University of Trier. It contains more than 2.3 million records of publications, while the citation information is extremely scarce compared to WoS and CiteSeer databases^{14}. For this study, we consider a snapshot of the database provided within KONECT (http://konect.unikoblenz.de) that contains 12,591 journal and conference papers and 49,759 citations between them. Note that all papers either cite or are cited by some other, while the database includes 15 selfcitations.
arXiv database
arXiv.org (arXiv) is a public preprint repository of publication drafts uploaded by the authors prior to an actual journal or conference submission (http://arxiv.org). It began in 1991^{1} and is hosted at Cornell University. It currently contains almost one million publications from physics, mathematics, computer science and other fields. For this study, we consider all publications in arXiv category High Energy Physics Phenomenology as of April 2003^{23} provided within SNAP (http://snap.stanford.edu). The database spans over 10 years and contains 34,546 publications and 421,578 citations between them. Note that all publications either cite or are cited by some other, while the database includes 44 selfcitations.
Citation topology
Citation networks extracted from bibliographic databases are represented with directed graphs, where papers are nodes of the graph and citations are directed links between the nodes. The topology of citation networks is assessed through a rich set of local and global graph statistics.
Descriptive and field statistics
The citation network is a simple directed graph G(V, L), where V is the set of nodes, n = V and L is the set of links, m = L. Weakly connected component (WCC) is a subset of nodes reachable from one another not considering the directions of the links. Field bowtie is a decomposition of the largest WCC of a citation network into the infield component, which consists of nodes with no outgoing links, the outfield component, which consists of nodes with no incoming links and the field core.
Degree distributions and mixing
The indegree k_{in} or outdegree k_{out} of a node is the number of incoming and outgoing links, respectively. k is the degree of a node, k = k_{in} + k_{out} and 〈k〉 denotes the mean degree. γ is the scalefree exponent of a powerlaw degree distribution P(k) ~ k^{−γ} and γ_{in} and γ_{out} are the scalefree exponents of P(k_{in}) and P(k_{out})^{36}. Powerlaws are fitted to the tails of the distributions by maximumlikelihood estimation, for k_{min} ∈ {10, 25}. Neighbour connectivity plots show the mean neighbour degree N(k.) of nodes with degree k.^{56}. The degree mixing r_{(α,β)} is the Pearson correlation coefficient of αdegrees or βdegrees at links' source and target nodes, respectively^{57}:
where 〈k.〉 and σ_{k.} are the means and standard deviations, α, β ∈ {in, out}. r is the mixing of degrees k^{39}.
Clustering distributions and mixing
Node clustering coefficient c is the density of its neighbourhood^{42}:
where t is the number of linked neighbours and k(k − 1)/2 is the maximum possible number, c = 0 for k ≤ 1. The mean 〈c〉 is denoted network clustering coefficient^{42}, while the clustering mixing r_{c} is defined as before. Clustering profile shows the mean clustering C(k) of nodes with degree k^{58}. Note that the denominator in equation (2) introduces biases^{46}, particularly when r < 0. Thus, deltacorrected clustering coefficient b is defined as c · k/Δ^{59}, where Δ is the maximal degree k and b = 0 for k ≤ 1. Also, degreecorrected clustering coefficient d is defined as t/ω^{46}, where ω is the maximum number of linked neighbours with respect to their degrees k and d = 0 for k ≤ 1. By definition, b ≤ c ≤ d.
Diameter statistics
Hop plot shows the percentage of reachable pairs of nodes H(δ) within δ hops^{23}. The diameter is the minimal number of hops δ for which H(δ) = 1, while the effective diameter δ_{90} is defined as the number of hops at which 90% of such pairs of nodes are reachable^{23}, H(δ_{90}) = 0.9. δ′ denotes the respective number of hops in a corresponding undirected graph. Hop plots are estimated over 100 realizations of the approximate neighbourhood function with 32 trials^{60}.
Statistical comparison
Citation networks representing bibliographic databases are compared through 21 graph statistics introduced above. These are by no means independent^{42,46}, neither are their values of a true citation network known. We thus compute externally studentized residuals of graph statistics that measure the consistency of each bibliographic database with the rest. Statistically significant inconsistencies in individual graph statistics are revealed by Student ttest^{26}. We select ten graph statistics whose pairwise independence is verified using Fisher ztransformation^{27}. Friedman rank test^{28} confirms that bibliographic databases display significant inconsistencies in the selected statistics, while the databases with no significant differences are revealed by Nemenyi test^{25,30}.
Studentized statistics residuals
Denote x_{ij} to be the value of jth graph statistic of ith bibliographic database, where N is the number of databases, N = 6. Corresponding externally studentized residual is:
where and are the sample mean and corrected standard deviation excluding the considered ith database, and . Assuming that the errors in x are independent and normally distributed, the residuals have Student tdistribution with N − 2 degrees of freedom. Significant differences in individual statistics x are revealed by independent twotailed Student ttests^{26} at Pvalue = 0.05, rejecting the null hypothesis H_{0} that x are consistent across the databases, . Notice that the absolute values of individual residuals imply a ranking R over the databases, where the database with the lowest has rank one, the second one has rank two and the one with the largest has rank N.
Pairwise statistics independence
Denote r_{ij} to be the Pearson productmoment correlation coefficient of the residuals for ith and jth graph statistics over all bibliographic databases. Spearman rank correlation coefficient ρ_{ij} is defined as the Pearson coefficient of the ranks R for ith and jth statistics. Under the null hypothesis of statistical independence of ith and jth statistics, H_{0}:ρ_{ij} = 0, adjusted Fisher transformation^{27}:
approximately follows a standard normal distribution. Pairwise independence of the selected graph statistics is thus confirmed by independent twotailed ztests at Pvalue = 0.01.
Comparison of bibliographic databases
Significant inconsistencies between bibliographic databases are exposed using the methodology introduced for comparing classification algorithms over multiple data sets^{25}. Denote R_{i} to be the mean rank of ith database over the selected graph statistics, , where K is the number of statistics, K = 10. Onetailed Friedman rank test^{28,29} first verifies the null hypothesis that the databases are statistically equivalent and thus their ranks R_{i} should equal, H_{0}:R_{i} = R_{j}. Under the assumption that the selected statistics are indeed independent, the Friedman testing statistic^{28}:
has χ^{2}distribution with N − 1 degrees of freedom. By rejecting the hypothesis at Pvalue = 0.05, we proceed with the Nemenyi posthoc test that reveals databases whose ranks R_{i} differ more than the critical difference^{30}:
where q is the critical value based on the studentized range statistic^{25}, q = 2.85 at Pvalue = 0.05. A critical difference diagram plots the databases with no statistically significant inconsistencies in the selected statistics^{25}.
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Acknowledgements
Authors thank J. Demšar, V. Batagelj, M. Žitnik and Z. Levnajić for comments and discussions and Thomson Reuters for providing the access to bibliographic data. This work has been supported in part by the Slovenian Research Agency Program No. P20359, by the Slovenian Ministry of Education, Science and Sport Grant No. 430168/2013/91, by the European Union, European Social Fund and by the European Regional Development Fund Grant No. CZ.1.05/1.1.00/02.0090.
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L.Š. designed and performed the experiments. L.Š., D.F. and M.B. wrote the main manuscript text. All authors reviewed the manuscript. The authors have no competing financial interests.
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Šubelj, L., Fiala, D. & Bajec, M. Networkbased statistical comparison of citation topology of bibliographic databases. Sci Rep 4, 6496 (2014). https://doi.org/10.1038/srep06496
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