Scientometrics

, Volume 110, Issue 2, pp 765–777 | Cite as

Ruling out static latent homophily in citation networks

  • Peter Wittek
  • Sándor Darányi
  • Gustaf Nelhans
Open Access
Article

Abstract

Citation and coauthor networks offer an insight into the dynamics of scientific progress. We can also view them as representations of a causal structure, a logical process captured in a graph. From a causal perspective, we can ask questions such as whether authors form groups primarily due to their prior shared interest, or if their favourite topics are ‘contagious’ and spread through co-authorship. Such networks have been widely studied by the artificial intelligence community, and recently a connection has been made to nonlocal correlations produced by entangled particles in quantum physics—the impact of latent hidden variables can be analyzed by the same algebraic geometric methodology that relies on a sequence of semidefinite programming (SDP) relaxations. Following this trail, we treat our sample coauthor network as a causal graph and, using SDP relaxations, rule out latent homophily as a manifestation of prior shared interest only, leading to the observed patternedness. By introducing algebraic geometry to citation studies, we add a new tool to existing methods for the analysis of content-related social influences.

Keywords

Citation network Causal network Semidefinite programming Hidden variables Sum-of-squares decomposition 

Mathematics Subject Classification

90C22 

Introduction

Clarifying a line of argumentation by references, citations as a legacy mapping and orientation tool have been in use by knowledge organization for a long time. Their respective importance has led to the birth of new fields of study like scientometrics and altmetrics (Borgman and Furner 2005; Zahedi et al. 2014; Cronin and Sugimoto 2014), permeating funding decisions and ranking efforts (Vanclay 2012; Hicks 2012). At the same time, citations embody scholarly courtesy as well as a form of social behavior, maintaining or violating norms (Cronin and Overfelt 1994; Kaplan 1965; Mitroff 1974; Gilbert 1977; Ziman 2000; Sandstrom 2001; Börner et al. 2006). Due to this, as is often the case when individual and social patterns of action are contrasted, one can suspect that factors not revealed to the observer of a single individual may point at underlying group norms when communities of individuals are scrutinized. To understand our own behavior as a species, it is important to detect any such influence.

Lately, the idea that multiple versions of probabilities do exist brought new ideas to the foreground (Mugur-Schächter 2014; Khrennikov 2010). Eventually the testing of a second probability alternative has made it clear that by its use, rules that were known to apply to the subatomic world of quantum mechanics only start making sense in the atomic world too. Examples include decision theory and cognition (Busemeyer and Bruza 2012), economy (Haven 2015), biology (Asano et al. 2012; Wittek et al. 2013), and language (Bruza and Woods 2008; Darányi and Wittek 2012; Cohen et al. 2010).

With the above unexpected development in the history of science, and departing from earlier work in social network research (Aral et al. 2009; Ver Steeg and Galstyan 2011), we turned to citation studies to find supporting evidence for signs of quantum-likeness in co-author behaviour, captured by longitudinal datasets. Our working hypothesis was that in citation patterns, a more fundamental layer would correspond to research based on shared interest between the author and her/his predecessors called latent homophily, whereas a more ephemeral second layer would link in current trends in science. Due to this, e.g. for a funding agency to find citation patterns going back to latent homophily as a single source would amount to better founded decisions, with such a pattern playing the role of a knowledge nugget. Consequently, ruling out latent homophily would correspond to a sieve filtering out cases where correlations in the data go back to more than latent homophily, one important step in an anticipated workflow to dig for such nuggets by stratification in citations.

Related research and conceptual clarifications

The notion of the citation network was famously developed by de Solla Price (1965) and since then it has evolved in many different directions. Incidentally, Garfield et al. (1964) had already proposed the use of “Network Charts” of papers for the study of the history of science, but see also Garfield et al. (2003) and Garfield (2009) for a newfound interest in algorithmic historiography. Although fruitful for analysis at a less aggregated level, these maps provide the possibility to visualize the network structure of single citing/cited papers of up to, say, the lower hundreds of papers before becoming too complex to overview. To remedy this, aggregated forms of citation networks have been developed, most notably bibliographic coupling (Kessler 1963), ‘co-mentions’ of literary authors (Rosengren 1968), and the more established concept of ‘co-citation’ of papers (Small 1973). Eventually, over time these aggregated forms of measurement were extended to analyse network structures of authors (McCain 1986; White and Griffith 1981). By today, possibilities include the coverage of source titles and, for bibliographic coupling to reveal the networks based on address data such as department, institution and country, are limited only to the kind of structured data available in the database used for sampling (van Eck and Waltman 2010, 2014). Common for many of these efforts is that the network structure is used to map or represent bibliometric data for descriptive purposes in visualization, while attempts at analyzing the relationships dynamically in more causal ways have not been considered to the same extent. A notable exception is Bar-Ilan (2008) for an overview of a third mode of aggregated co-studies, namely co-authorship studies that incorporate complex systems research and Social Network Analysis.

To address a different subject area, graphical models capture the qualitative structure of the relationships among a set of random variables. The conditional independence implied by the graph allows a sparse description of the probability distribution (Pearl 2009). Therefore by combining co-authorship and citation data we propose to view co-author and citation graphs as examples of such graphical models.

However, not all random variables can always be observed in a graphical model: there can be hidden variables. Ruling these out is a major challenge. Take, for instance, obesity, which was claimed to be socially contagious (Christakis and Fowler 2007). Is it not possible that a latent variable was at play that caused both effects: becoming friends and obesity? The above assumption of latent homophily, Ver Steeg and Galstyan (2011) asks whether there is a limit to the amount of correlation between friends, at the same time being separable from other sources different from friendship. Or, do some smokers become connected because they had always smoked, or because copying an example may bring social rewards? To cite a methodological parallel, in quantum physics, the study of nonlocal correlations also focuses on classes of entanglement that cannot be explained by local hidden variable models—these are known as Bell scenarios, initially stated as a paradox by Einstein, Podolsky and Rosen in their so-called EPR paper (Einstein et al. 1935).

As is well known, the EPR paper proposed a thought experiment which presented then newborn quantum theory with a choice: either supraluminal speed for signaling is part of nature but not part of physics, or quantum mechanics is incomplete. Thirty years later, in a modified version of the same thought experiment (Bell 1964), Bell’s Theorem suggested that two hypothetical observers, now commonly referred to as Alice and Bob, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. Once Alice measures spin in one direction, Bob’s measurement in that direction is determined with certainty, as being the opposite outcome to that of Alice, whereas immediately before Alice’s measurement Bob’s outcome was only statistically determined (i.e., was only a probability, not a certainty). This is an unusually strong correlation that classical models with an arbitrary predetermined strategy (that is, a local hidden variable) cannot replicate.

Recently, algebraic geometry offered a new path to rule out local hidden variable models following from Bell’s Theorem (Ver Steeg and Galstyan 2011; Ma et al. 2015; Ver Steeg 2015). By describing probabilistic models as multivariate polynomials, we can generate a sequence of semidefinite programming relaxations which give an increasingly tight bound on the global solution of the polynomial optimization problem (Lasserre 2001). Depending on the solution, one might be able to reject a latent variable model with a high degree of confidence. In our case, Alice and Bob decide about references to be picked in complete isolation, yet their decisions, in spite of being independent from each other’s, may be still correlated. If we identify the source of the shared state preceding their decisions as they make their choices, we can observe correlations between author pairs, and conclude that their patterns of citing behaviour cannot be explained alone by the fact that they have always liked each other. In other words, experimental findings may rule out latent homophily as a single source of correlations in certain scenarios. In a Bell scenario, this means that Alice and Bob can agree on a strategy beforehand (latent hidden variable), but at the end of the day, their observed correlations are so strong that they could only be caused by shared entanglement.

Due to these conceptual overlaps, we believe there is value in introducing this algebraic geometric framework to citation analysis for the following reasons:
  • It can indicate the presence of peer influence (e.g. intellectual fashion, social pressures etc.) interfering with scientific conviction. Also, following Aral et al. (2009) and offering a different angle on it, this would correspond to correlations that cannot be explained by latent homophily alone. Singling out such cases could be a methodological step forward for citation studies;

  • In our model, latent homophily corresponds to what we call a latent hidden variable model in Bell scenarios in quantum information theory. Rejecting such a model indicates entanglement in quantum mechanics, promising a next stepping stone for methodological progress in the study of citation patterns;

  • Given that entanglement in QM goes back to non-classical correlations, it would be a valuable finding that given such outcome, classical and non-classical correlations both contribute to patternedness in citation data. This provides a new research alliance prospect between citation studies and quantum theory based approaches, e.g. new trends in computational linguistics (Widdows and Cohen 2009; Blacoe et al. 2013) or decision theory (Bruza et al. 2009; Khrennikov 2010; Busemeyer and Bruza 2012; Wittek et al. 2013).

Citation networks and latent homophily

To translate the above to experiment design, we must discuss how latent homophily manifests in citation networks and why we want to restrict our attention to static models. We shall be interested in citation patterns of individual authors who have co-authored papers previously. Social ‘contagion’ means that authors will cite similar papers later on if they previously co-authored a paper. On the other hand, latent homophily means that some external factor—such as shared scientific interest—can explain the observed correlations on its own.

Given an influence model in which a pair of authors make subsequent decisions, if we allow the probability of transition to change in between time steps, then arbitrary correlations can emerge. Static latent homophily means that the impact of the hidden variable is constant over time, that is, the transition probabilities do not change from one time step to the other. We restrict our attention to such models, this being a necessary technical assumption for the algebraic geometric framework. In practice, this means that an author does not get more or less inclined over time to cite a particular paper.

A straightforward way to analyze correlations is to look at citation patterns between authors. Departing from a set of authors in an initial period, we can study whether the references an author makes influence the subsequent references of her or his coauthors as defined in the initial period. In this sense, we define a graph where each node is an author-reference. Two nodes are connected if the authors have co-authored a paper at some initial time step. A node is assigned a binary state \(\pm 1\), reflecting whether that author-reference pair is actually present. The influence model is outlined in Fig. 1.
Fig. 1

Outline of the influence model. The latent variables \(R_A\) and \(R_B\) cause the edges in the co-author network and are also the sole influence in changes whether an author-reference pair changes in subsequent time steps

We cannot, however, look at all the references that an author made until the end of some time period. If we assign +1 to the condition that an author-reference pair exists, i.e. the author cited the paper until the end of the specified period, this node state will never flip back to \(-1\). In other words, given sufficient time, all node states would become +1, revealing very little about correlations. Therefore we assign a +1 state to a node if the author cites a paper within the observation period. If during the next period he or she does not cite it, it will flip back to \(-1\).

In what follows, we follow the formalism as described by Ver Steeg and Galstyan (2011), which, for an individual time step, also closely resembles the study of Bell scenarios by semidefinite programming in quantum information theory (Navascués et al. 2007). Suppose we are looking at a pair of authors, A for Alice and B for Bob. Let \(\alpha _+\) be the probability that node A flips from \(+\) to −, and \(\alpha _-\) the probability of the reverse transition. The initial probability of being in the \(+\) state is \(\alpha _0\). We define the same probabilities for B with \(\beta _+, \beta _-\) and \(\beta _0\). The state of node A at time step t is \(A_t\), and the sequence \(A_{1:T}\) denotes the states until some time step T; similarly for B. Further suppose that A depends on some hidden variable \(R_A\) and B on \(R_B\). A random variable E depends on both hidden variables and it represents edges between time steps, that is, E describes our graph structure.

The probability of a sequence of possible transitions is as follows:
$$\begin{aligned} P(A_{1:T}|R_A)&= \alpha _+^{F_+(A)}\alpha _-^{F_-(A)}(1-\alpha _-)^{S_-(A)}(1-\alpha _+)^{S_+(A)}\\&\quad \alpha _0^{1/2(1+A1)}(1-\alpha _0)^{1/2(1-A1)}, \end{aligned}$$
(1)
where \(F_\pm\) and \(S_\pm\) are counters of the transitions:
$$\begin{aligned} F_\pm&= \sum _{t=1}^{T-1}\frac{1}{4}(1\pm A_t)(1-A_{t+1}A_t).\\ S_\pm&= \sum _{t=1}^{T-1}\frac{1}{4}(1\pm A_t)(1+A_{t+1}A_t). \end{aligned}$$
Similarly for B. Let \(x=(\alpha _0, \alpha _+, \alpha _-, \beta _0, \beta _+, \beta _-)\) be the parameter vector.
We are ready to move towards a geometric description of the problem. Let us take observables \(O_j(A, B)\) on A and B—these can be the indicator functions of all possible outcomes, for instance. We define the expectation values of these observables as
$$\begin{aligned} y_j = \sum _{R_A, R_B} P(R_A, R_B|E)f_j(x), \end{aligned}$$
(2)
where
$$\begin{aligned} f_j(x) = \sum _{A,B}P(A_{1:T}|R_A)P(B_{1:T}|R_B)O_j(A,B). \end{aligned}$$
The constraints on the variables are such that they must be probabilities, therefore we have
$$\begin{aligned} K = \{x\in {\mathbb {R}}^6: g_i(x)=x_i(1-x_i)\ge 0, i=1,\ldots ,6\}. \end{aligned}$$
(3)
The equalities in \(y_j\) together with the constraints in K are all polynomials. If there is a hidden variable model, the constraints can be satisfied. If not, the problem is infeasible and we must reject the hidden variable model.

Identifying the feasibility of this problem is a hard task, and we provide a relaxation. This relaxation will approximate the feasible set from the outside: that is, if the relaxation is an infeasible problem, the original one too must be infeasible. Therefore by the same relaxation one can reject hidden variable models.

To explain how it works, suppose we are interested in finding the global optimum of the following constrained polynomial optimization problem:
$$\begin{aligned} \min _{x\in {\mathbb {R}}^n}f(x) \end{aligned}$$
such that
$$\begin{aligned} g_i(x) \ge 0, i=1,\ldots ,r \end{aligned}$$
Here f and \(g_i\) are polynomials in \(x\in {\mathbb {R}}^n\). We can think of the constraints as a semialgebraic set \({\mathbf {K}}=\{x\in {\mathbb {R}}^n: g_i(x) \ge 0, i=1,\ldots ,r\}\). Lasserre’s method gives a series of semidefinite programming (SDP) relaxations of increasing size that approximate this optimum through the moments of x (Lasserre 2001). For polynomial optimization problems of noncommuting variables this amounts to the exclusion of hidden variable theorems in networked data, and that we can verify the strength of observed correlations.
Even in this formulation, there is an implicit constraint on a moment: the top left element of the moment matrix is 1. Given a representing measure, this means that \(\int _{\mathbf {K}} {\mathrm {d}}\mu =1\). It is actually because of this that a \(\lambda\) dual variable appears in the dual formulation:
$$\begin{aligned} \max _{\lambda , \sigma _0} \lambda \end{aligned}$$
such that
$$\begin{aligned}&f(x) - \lambda = \sigma _0 + \sum _{i=1}^r \sigma _i g_i\\&\sigma _0, \sigma _i\in \Sigma {[x]}, {\mathrm {deg}}\sigma _0\le 2d. \end{aligned}$$
In fact, we can move \(\lambda\) to the right-hand side, where the sum-of-squares (SOS) decomposition is, \(\lambda\) being a trivial SOS multiplied by the constraint \(\int _{\mathbf {K}} {\mathrm {d}}\mu\), that is, by 1.

We normally think of the constraints that define \({\mathbf {K}}\) as a collection of \(g_i(x)\) polynomial constraints underlying a semialgebraic set, and then in the relaxation we construct matching localizing matrices. We can, however, impose more constraints on the moments. For instance, we can add a constraint that \(\int _{\mathbf {K}} x{\mathrm {d}}\mu = 1\). All of these constraints will have a constant instead of an SOS polynomial in the dual.

This SDP hierarchy and the SOS decomposition have found extensive use in analyzing quantum correlations (Navascués et al. 2007; Pironio et al. 2010), and given the notion of local hidden variables in studying nonlocality, there is a natural extension to studying causal structures in general (Ver Steeg and Galstyan 2011). For a static latent homophily model, we are interested in the following SOS decomposition:
$$\begin{aligned} \max _{b, \sigma _i(x)} b\hat{y} \end{aligned}$$
(4)
such that
$$\begin{aligned} 1-bf(x)&= \sigma _0 + \sum _i \sigma _i(x)g_i(x)\\ \sigma _i&\in \Sigma [x], \end{aligned}$$
where \(\hat{y}\) contains the observables extracted from the data, and f(x) and \(g_i(x)\) encode our model. If this problem is infeasible, we can rule out a local hidden variable model as imposed by the constraints.

Corpus

Table 1

The number of published entries, along with total number of citations, mean number of citations, and first year of inclusion in the WoS index is found in the table

Ord

Journal

Recs

Citations

Mean citations

Mean citations per year

First year

1

Journal of the American Society for Information Science and Technology

2494

22958

9.21

1.11

2001

Journal of the American Society for Information Science

2977

39593

13.3

0.57

1970

American Documentation

780

4347

5.57

0.11

1956

Journal of Documentary Reproduction (United States)

     

2

Journal of Informetrics

420

3714

8.84

1.69

2007

3

Scientometrics

3637

38202

10.5

0.94

1978

Journal of Research Communication Studies

119

137

1.15

0.03

1978

4

Information Systems Research

649

25817

39.78

3.19

1994

5

MIS Quarterly

1071

70899

66.2

4.54

1981

6

College and Research Libraries

5156

12144

2.36

0.12

1956

7

Journal of the American Medical Informatics Association

4260

40687

9.55

0.95

1994

8

Library and Information Science Research

1209

6198

5.13

0.4

1984

Library Research (United States)

     

9

Annual Review of Information Science and Technology

550

7269

13.22

0.82

1966

10

Journal of Documentation

3700

18437

4.98

0.26

1945

11

Journal of Health Communication

1233

10570

8.57

0.99

1997

12

Journal of Information Science

1379

7802

5.66

0.29

1979

Information Scientist (United Kingdom)

     

Institute of Information Scientists. Bulletin (United Kingdom)

     

13

International Journal of Geographical Information Science

1299

14635

11.27

1.09

1997

International Journal of Geographical Information Systems

311

6547

21.05

0.99

1991

14

Journal of Information Technology

612

5613

9.17

0.8

1993

15

Library Quarterly

4603

6200

1.35

0.07

1956

16

Journal of the Medical Library Association

1104

4275

3.87

0.44

2002

Bulletin of the Medical Library Association

3639

10255

2.82

0.11

1956

17

Empty

     

18

Arxiv Digital Libraries (cs.DL)

     

19

Information and Management

1702

31902

18.74

1.52

1983

Systems Objectives Solutions

63

274

4.35

0.13

 

Information Management

200

25

0.13

0

1983

Management Datamatics (Netherlands)

     

Management Informatics (Netherlands)

     

IAG Journal (Netherlands)

     

20

Reference Librarian

     

Total number of records

43167

 

11.53

0.88

 

Longitudinal data were collected from Web of Science, using the journal indices WoS-Extended, SSCI, and AHCI between 1945 and 2013 (Table 1). The collection consists of the full set of published items in 20 high impact journals found in the database. 43168 items where collected in total, comprising of 22784 articles (52.4%), 10270 book reviews (23.8%), 2325 editorial material papers (5.4%), and 1898 proceedings papers (articles) (4.4%).

The selection process was conducted by using four different journal rankings. The reason for using multiple source rankings was to minimize the impact of perspective, where, for example, the JCR ranking for Information and Library Studies contains journals from the Information Systems area, however that would not count as (core) LIS journals by practioners in the field. The ranking schemes used were JCR 2012, JCR 1997 (the oldest one found readily in the WoS platform), Google top publications (H5-Index), and Elsevier SCImago Rank 2012. Journal rank data and citation data were collected on January 20, 2014.

The inclusion of publication years 2013 and 2014 is not complete, since it is generally acknowledged that WoS has not received the underlying data until late spring the year after publication. Since the dataset is used for information based research and not for performance based evaluation, inclusion of as much as possible material was deemed more important than completeness.

To rank the journals, in all four lists the 20 top journals were scored from 20 to 1, so that the top journal earned 20 points and the last one earned 1 point. Then the points from each of the occurring journals in the four rankings were added and the journals were listed again based on their combined score for Table 1.

For every selected journal title, the title was run against the Ulrichs Periodicals Directory to identify title changes during the span of the journal’s publishing history. In all, 33 versions of the titles were searched for in WoS. Of these, 24 titles were found in the database.

The number of published entries, along with total number of citations, mean number of citations, and first year of inclusion in the WoS index are presented in Table 1. The coauthor network has 45904 nodes and 78418 edges.

An illustrative example

We decided to conduct an experiment with a semi-synthetic example to verify whether such a network of citations allows for the exclusion of latent hidden variables. For this case, to design a model of influence, the graph had to be directed, whereas a coauthor network is typically undirected. To establish directions in the graph, we considered a pairwise asymmetric relationship between authors in which one of the authors is ‘dominant’. To this end we considered the following two alternatives:
  1. 1.

    The more dominant author is the one with more citations. As in our corpus every author pair has the same number of citations, this option was not viable and was therefore discarded;

     
  2. 2.

    The more dominant author has a higher degree in the graph of the coauthor network because he or she had more coauthors in the past. This enabled us to direct the graph.

     

We assumed that the network structure does not evolve over time. Taking the directed coauthor network graph in consideration, we assigned a state to each node, and set its value randomly with \(\pm 1\) with equal probability.

Once this initialization was done, we had to simulate influence. We randomly picked a pair, and the nondominant author copied the state of the dominant one. In a time step, we did M such random picks, where M is the number of edges. This gave sufficient opportunity for the graph to flip most of its nodes if necessary. We created two more time slices on top of the initial one. Using these time slices, we could calculate the statistics \(P(A_{1:T}B_{1:T}|E=1)\) with \(T=3\), where \(E=1\) meant that there was a directed edge from author A to author B.

With this random initialization, one can detect if, given a particular graph structure, there is a possibility of latent homophily at all. We used metaknowledge1 to work with the citation network (McIlroy-Young and McLevey 2015), Ncpol2sdpa2 to generate the SDP relaxations (Wittek 2015), and Mosek3 to solve the SDP. The computational details are available online.4 Taking the observables \(O_j(A,B)\) as the indicator function and a level-3 relaxation of the Lasserre hierarchy, the SDP solver detects any dual infeasibility. In turn, such an infeasibility means that the SOS decomposition does not exist and we can rule out latent homophily as the source of correlations with a high degree of confidence.

Static latent homophily in the coauthor network: results and discussion

As a joint probability distribution, one obtains 64 possible combinations of outcomes, because for each author and time period, the outcome is binary, and given two authors and three time periods, we obtain this number. We observe all possible outcomes on this sample. We used the same \(O_j(A,B)\) observable as in the semi-synthetic example, i.e. the indicator function, and a level-3 relaxation of the Lasserre hierarchy.

We used different splits over the corpus to analyze the network at different granularity. In the most basic split, the sample corpus factorized in three periods with the following distribution:

Period

Number of papers

1945–1968

4104

1968–1991

12293

1991–2014

26770

Clearly, the earliest period was the sparsest. The SDP solver detected dual infeasibility, therefore we could rule out latent homophily as the single source of correlations. On this time scale, however, assuming that the network remained static is unrealistic. Therefore, we repeated the test with a span of 30, 10, and 5 years.

For the 30- and the 10-year spans, we analyzed every subsequent fifth year as the starting year. Due to sparse data in the first years, all analysis in this part started with 1949. Thus, for instance, we analyzed 1949–1979, followed by 1954–1984, and so on. This gave us a total of twenty time intervals, with only one case, the 10-year period of 1949–1959 allowing the possibility of latent homophily.

For the 5-year intervals, we started with 1959, again, for reasons of data sparsity. Then we analyzed intervals starting with every third year, so, for instance, 1959–1964, followed by 1962–1967, and so on. This gave us another seventeen data points, with only two intervals, 1959–1964 and 1965–1970, not being able to rule out latent homophily.

Our result indirectly confirms that ‘contagion' in the practice of citation is a distinct possibility. If citation patterns continue spreading, over time everybody will cite more or less the same papers. This in turn explains the phenomenon of Sleeping Beauties (Ke et al. 2015): since dominant authors do not cite such articles, everybody else ignores them.

Secondly, we recall that in its simplest form, Bell’s theorem states that no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics, i.e. it rules out such variables as a viable explanation of quantum mechanics. Therefore we hypothesized that if we can find entanglement in our data, with local hidden variables as their source ruled out, patterns in the sample must be quantum-like for non-obvious reasons. Ruling out Bell inequalities as the source of entanglement in our results points to such non-classical correlations at work in the dataset.

Conclusions

Citation and coauthor networks offer an insight into the dynamics of scientific progress. To understand this dynamics, we treated such a network as the representation of a causal structure, a logical process captured in a graph, and inquired from a causal perspective if authors form groups primarily due to their prior shared interest, or if their favourite topics are ‘contagious’ and spread through co-authorship. Following an algebraic geometric methodology that relies on a sequence of semidefinite programming (SDP) relaxations, we analyzed a sample citation network for the impact of latent hidden variables. Using the SDP relaxations, we were able to rule out latent homophily, or shared prior interest as the source of correlations, hinting at that citation patterns in fact spread.

Statistical sampling on the author pairs was akin to making repeated measurements with bipartite Bell scenarios in quantum mechanics. The finding that shared prior interest as a latent variable cannot account on its own for citation patterns calls for a related analysis into the nature of ‘contagious’ influences including fashionable topics, reputation etc., affecting the outcome. This confirmation and the algebraic geometric framework to compute it are novel concepts in scientometrics. We hope this work will act as a stepping stone for further research.

Footnotes

Notes

Acknowledgements

Peter Wittek and Sándor Darányi were supported by the European Commission Seventh Framework Programme under Grant Agreement Number FP7-601138 PERICLES. The dataset was compiled by Nasrine Olson and Gustaf Nelhans (University of Borås).

References

  1. Aral, S., Muchnik, L., & Sundararajan, A. (2009). Distinguishing influence-based contagion from homophily-driven diffusion in dynamic networks. Proceedings of the National Academy of Sciences, 106(51), 21544–21549.CrossRefGoogle Scholar
  2. Asano, M., Basieva, I., Khrennikov, A., Ohya, M., Tanaka, Y., & Yamato, I. (2012). A quantum-like model of Escherichia coli’s metabolism based on adaptive dynamics. In Proceedings of QI-12, 6th International Quantum Interaction Symposium, (pp. 60–67).Google Scholar
  3. Bar-Ilan, J. (2008). Informetrics at the beginning of the 21st century: A review. Journal of Informetrics, 2(1), 1–52.CrossRefGoogle Scholar
  4. Bell, J. (1964). On the Einstein-Podolsky-Rosen paradox. Physics, 1(3), 195–200.Google Scholar
  5. Blacoe, W., Kashefi, E., & Lapata, M. (2013). A quantum-theoretic approach to distributional semantics. In Proceedings of NAACL-HLT-13, Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, (pp. 847–857).Google Scholar
  6. Borgman, C. L., & Furner, J. (2005). Scholarly communication and bibliometrics. Annual Review of Information Science and Technology, 36(1), 2–72.CrossRefGoogle Scholar
  7. Börner, K., Penumarthy, S., Meiss, M., & Ke, W. (2006). Mapping the diffusion of scholarly knowledge among major US research institutions. Scientometrics, 68(3), 415–426.CrossRefGoogle Scholar
  8. Bruza, P., & Woods, J. (2008). Quantum collapse in semantic space: Interpreting natural language argumentation. In Proceedings of QI-08, 2nd International Symposium on Quantum Interaction.Google Scholar
  9. Bruza, P. D., Widdows, D., & Woods, J. (2009). A quantum logic of down below. In K. Engesser, D. Gabbay, & D. Lehmann (Eds.) Handbook of Quantum Logic and Quantum Structures, vol. 2.Google Scholar
  10. Busemeyer, J., & Bruza, P. D. (2012). Quantum models of cognition and decision. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  11. Christakis, N. A., & Fowler, J. H. (2007). The spread of obesity in a large social network over 32 years. New England Journal of Medicine, 357(4), 370–379.CrossRefGoogle Scholar
  12. Cohen, T., Widdows, D., Schvaneveldt, R., & Rindflesch, T. (2010). Logical leaps and quantum connectives: Forging paths through predication space. In Proceedings of QI-10, 4th Symposium on Quantum Informatics for Cognitive, Social, and Semantic Processes, (pp. 11–13).Google Scholar
  13. Cronin, B., & Overfelt, K. (1994). The scholar’s courtesy: A survey of acknowledgement behaviour. Journal of Documentation, 50(3), 165–196.CrossRefGoogle Scholar
  14. Cronin, B., & Sugimoto, C. R. (2014). Beyond bibliometrics: Harnessing multidimensional indicators of scholarly impact. Cambridge: MIT Press.Google Scholar
  15. Darányi, S., & Wittek, P. (2012). Connecting the dots: Mass, energy, word meaning, and particle-wave duality. In Proceedings of QI-12, 6th International Quantum Interaction Symposium, (pp. 207–217).Google Scholar
  16. de Solla Price, D. J. (1965). Networks of scientific papers. Science, 149(3683), 510–515.CrossRefGoogle Scholar
  17. Einstein, A., Podolsky, B., & Rosen, N. (1935). Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47(10), 777.CrossRefMATHGoogle Scholar
  18. Garfield, E. (2009). From the science of science to scientometrics visualizing the history of science with HistCite software. Journal of Informetrics, 3(3), 173–179.CrossRefGoogle Scholar
  19. Garfield, E., Pudovkin, A. I., & Istomin, V. S. (2003). Why do we need algorithmic historiography? Journal of the American Society for Information Science and Technology, 54(5), 400–412.CrossRefGoogle Scholar
  20. Garfield, E., Sher, I., & Torpie, R. (1964). The use of citation data in writing the history of science. Report 99, The Institute for Scientific Information.Google Scholar
  21. Gilbert, N. G. (1977). Referencing as persuasion. Social Studies of Science, 7(1), 113–122.CrossRefGoogle Scholar
  22. Haven, E. (2015). Financial payoff functions and potentials. In Proceedings of QI-14, 8th International Conference on Quantum Interaction, (pp. 189–195).Google Scholar
  23. Hicks, D. (2012). Performance-based university research funding systems. Research Policy, 41(2), 251–261.CrossRefGoogle Scholar
  24. Kaplan, N. (1965). The norms of citation behavior: Prolegomena to the footnote. American Documentation, 16(3), 179–184.CrossRefGoogle Scholar
  25. Ke, Q., Ferrara, E., Radicchi, F., & Flammini, A. (2015). Defining and identifying sleeping beauties in science. Proceedings of the National Academy of Sciences, 112(24), 7426–7431.CrossRefGoogle Scholar
  26. Kessler, M. M. (1963). Bibliographic coupling between scientific papers. American Documentation, 14(1), 10–25.CrossRefGoogle Scholar
  27. Khrennikov, A. (2010). Ubiquitous quantum structure: From psychology to finance. Berlin: Springer Verlag.CrossRefMATHGoogle Scholar
  28. Lasserre, J. (2001). Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11(3), 796–817.MathSciNetCrossRefMATHGoogle Scholar
  29. Ma, L., Krishnan, R., & Montgomery, A. L. (2015). Latent homophily or social influence? An empirical analysis of purchase within a social network. Management Science, 61(2), 454–473.CrossRefGoogle Scholar
  30. McCain, K. W. (1986). Cocited author mapping as a valid representation of intellectual structure. Journal of the American Society of Information Science, 37(3), 111–122.CrossRefGoogle Scholar
  31. McIlroy-Young, R., & McLevey, J. (2015). metaknowledge: Open source software for social networks, bibliometrics, and sociology of knowledge research. ON: Waterloo.Google Scholar
  32. Mitroff, I. I. (1974). Norms and counter-norms in a select group of the Apollo Moon scientists: A case study of the ambivalence of scientists. American Sociological Review, 39(4), 579–595.CrossRefGoogle Scholar
  33. Mugur-Schächter, M. (2014). On the concept of probability. Mathematical Structures in Computer Science, 24(03), e240309.MathSciNetCrossRefMATHGoogle Scholar
  34. Navascués, M., Pironio, S., & Acín, A. (2007). Bounding the set of quantum correlations. Physical Review Letters, 98(1), 10401.CrossRefGoogle Scholar
  35. Pearl, J. (2009). Causality: Models, Reasoning and Inference. New York: Cambridge University Press.CrossRefMATHGoogle Scholar
  36. Pironio, S., Navascués, M., & Acín, A. (2010). Convergent relaxations of polynomial optimization problems with noncommuting variables. SIAM Journal on Optimization, 20(5), 2157–2180.MathSciNetCrossRefMATHGoogle Scholar
  37. Rosengren, K. E. (1968). Sociological aspects of the literary system. Stockholm: Natur och Kultur.Google Scholar
  38. Sandstrom, P. (2001). Scholarly communication as a socioecological system. Scientometrics, 51(3), 573–605.CrossRefGoogle Scholar
  39. Small, H. (1973). Co-citation in the scientific literature: A new measure of the relationship between two documents. Journal of the American Society of Information Science, 24(4), 265–269.CrossRefGoogle Scholar
  40. van Eck, N. J., & Waltman, L. (2010). Software survey: VOSviewer, a computer program for bibliometric mapping. Scientometrics, 84(2), 523–538.CrossRefGoogle Scholar
  41. van Eck, N. J., & Waltman, L. (2014). Visualizing bibliometric networks. In Y. Ding, R. Rousseau, & D. Wolfram (Eds.), Measuring Scholarly Impact (pp. 285–320). Springer Science + Business Media.Google Scholar
  42. Vanclay, J. K. (2012). Impact factor: Outdated artefact or stepping-stone to journal certification? Scientometrics, 92(2), 211–238.CrossRefGoogle Scholar
  43. Ver Steeg, G., & Galstyan, A. (2011). A sequence of relaxations constraining hidden variable models. In Proceedings of UAI-11, 27th Conference on Uncertainty in Artificial Intelligence, (pp. 717–726).Google Scholar
  44. Ver Steeg, G. L. (2015). Bell inequalities for complex networks. Technical report, University of Southern California.Google Scholar
  45. White, H. D., & Griffith, B. C. (1981). Author cocitation: A literature measure of intellectual structure. Journal of the American Society of Information Science, 32(3), 163–171.CrossRefGoogle Scholar
  46. Widdows, D., & Cohen, T. (2009). Semantic vector combinations and the synoptic gospels. In Proceedings of QI-09, 3rd International Symposium on Quantum Interaction, (pp. 251–265).Google Scholar
  47. Wittek, P. (2015). Algorithm 950: Ncpol2sdpa—sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting variables. ACM Transactions on Mathematical Software, 41(3), 21.MathSciNetCrossRefMATHGoogle Scholar
  48. Wittek, P., Lim, I. K., & Rubio-Campillo, X. (2013). Quantum probabilistic description of dealing with risk and ambiguity in foraging decisions. In Proceedings of QI-13, 7th International Quantum Interaction Symposium, (pp. 296–307).Google Scholar
  49. Zahedi, Z., Costas, R., & Wouters, P. (2014). How well developed are altmetrics? a cross-disciplinary analysis of the presence of “alternative metrics” in scientific publications. Scientometrics, 101(2), 1491–1513.CrossRefGoogle Scholar
  50. Ziman, J. (2000). Real science. What it is, and what it means. Cambridge: Cambridge University Press.CrossRefGoogle Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.ICFO-The Institute of Photonic SciencesBarcelona Institute of Science and TechnologyCastelldefelsSpain
  2. 2.University of BoråsBoråsSweden

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