Cross-affiliation collaboration and power laws for research output of institutions: Evidence and theory from top three finance journals

1 INTRODUCTION

Recent decades have witnessed the fast growth of research collaboration in social sciences (see, e.g., Buyalskaya et al., 2021; Moody, 2004; Van Noorden, 2015). In Economics, in particular, Goyal et al. (2006) pioneer coauthor networks of economists and demonstrate that the social distance between economists who publish in journals declines significantly during 1970–2000. The three coauthor networks of economists, one for each decade, display small-world properties of Watts (1999). In additional to formal collaboration through coauthorship, Laband and Tollison (2000) point out that informal intellectual collaboration through commenting on a paper or discussing it at workshops, seminars, or conferences is also commonplace in Economics and to a lager extent than that in Biology. Both formal and informal collaboration have positive effects on research productivity1 (see, e.g., Ductor, 2015; Ductor et al., 2014; Rose & Georg, 2021).

This paper investigates a new means to further promote collaboration, that is, cross-affiliation in which researchers expand their scope of collaboration by affiliating with multiple institutions at the same time.2 Here, we focus on the widely-acknowledged top-three general interest journals in Financial Economics, namely, the Journal of Finance (JF, founded in 1946), the Journal of Financial Economics (JFE, founded in 1974), and the Review of Financial Studies (RFS, founded in 1986), which serve as clear exemplifications. We study top journals for two reasons. On one hand, top journal publications have a powerful influence on the impact of the underlying research. They also carry vast weight in the career paths of academic researchers, the recruitment decisions of research institutions, and the allocation of research funding. Hence we may intend first to obtain results regarding top journals. On the other hand, space in top journals is limited. Competencies in assembling research teams, communicating with peer experts, and acquiring research resources (e.g., data accesses, hardware and software, funding, visibility, and credibility) across institutions play a more and more important role in top journal publications. Intense competition3 forces researchers to develop new forms of collaboration. And top journals constitute natural sites where these new forms of collaboration emerge.

We hand-collect the author and affiliation information for all articles published in the top finance journals. Figure 1 presents the patterns of collaboration in these journals during 1980–2016.4 Panel A shows the average number of coauthors by aggregating the three journals. Coauthorship displays steady, approximately linear growth observed earlier in other fields, such as Economics (Card & DellaVigna, 2013) and Neuroscience (Barabási et al., 2002). Panel B shows the new form of collaboration through cross-affiliation. We plot here the average number of institutional affiliations reported per author by aggregating the three journals.5 The number lingers at a low level before 1995 but starts to ramp up thereafter. In 22 years, the average number of institutions per author climbs from 1.1 to 1.3. Put differently, on average, one out of nine authors is affiliated with multiple institutions before 1995, while the ratio increases to one out of three at 2016. Panel C shows the average number of institutions per author separately for each journal. The same trend is prevalent across all three.

Institutions are fundamental units in which research activities are organized and conducted. Their important role in academic research is manifest in, for example, resource allocation, infrastructure provision, and prestige effect in value attribution and funding mechanisms.6 The new form of collaboration through cross-affiliation imparts fresh insights into the growth of research output of institutions gauged by their top journal publications. In this study, we find that research institutions self-organize into a scale-free state and their top finance journal publications are characterized by power laws in the upper tail. The upper tail retains 6.0% of the totally 828 institutions at the end of 2016, while these most prolific institutions produce 61.3% of the totally 13,548 top journal publications. The upper tail distribution is highly-skewed: the mean number of publications is 166 while the most productive institution, NBER (National Bureau of Economic Research), contributes 1105 publications. The power laws govern institutions of diverse nature and scattered across geographic regions and time of establishment, indicating that these specific traits of institutions may be of less importance in driving the dynamics of their research output. The power laws are robust when considered separately for each of the last 6 years during our sample period. This result enables us to make further inference on the network since it seems to have reached a steady scale-free state. The power laws are also robust when considered separately for each of the three journals. This result reassures us that the scale-invariance is not caused by some journal-specific policies.

Our results provide an important regularity for theories of research output growth, which target the empirical power laws documented here. We propose a model of collaboration network that explains the scale invariance, based on further examinations into the structural reasons for the emergence of the scale-free properties. Specifically, a research institution enters the network as a node when it first publishes on a top finance journal and its degree henceforth equals its total number of publications. Two institutions are linked if they are affiliated to by at least one author of an article published in a top finance journal.7 The deep collaboration in financial research is revealed by a giant component of the network, which consists of 99.5% of all the institutions at the end of 2016.8 The dynamics of the network system are driven by nonlinear growth through enlarging coauthorship and cross-affiliation, as well as linear preferential attachment by incrementing total publications of institutions at a rate proportional to their current counts.9 The two identified mechanisms are generic, regardless of the types (universities, research centers, banks, etc.) and other characteristics (location and time of establishment) of the institutions. We consider three channels to allocate new degrees/top journal publications: (i) publications brought about by new nodes; (ii) publications allocated randomly to existing nodes; and (iii) publications allocated preferentially to existing nodes. The model remains tractable and predicts a power law for the stationary degree distribution in the upper tail. Furthermore, the power exponent is inversely related to the proportion of preferential allocation. Using the estimated power law exponents, we compute that 87.0% of the total publications originate from success-breeds-success,10 which engenders the high skewness in the distributions of the institutions' research output. On the contrary, the model also predicts that accelerated growth of collaboration, through coauthorship and cross-affiliation, increases the power exponent, and hence works to restore the homogeneity of the institutions.

There are several related studies on cross-affiliation. Katz and Martin (1997) point out that coauthorship is only a partial indicator of collaboration and further consider interinstitutional and international collaboration for adequacy. Hottenrott and Lawson (2017) show an increase in cross-affiliation in three countries and three scientific fields. Matveeva and Ferligoj (2020) demonstrate a similar increase in Russia. Hottenrott et al. (2021) provide a systematic study of cross-affiliation in 40 countries and 26 scientific fields. They focus on authors and articles with multiple affiliations and suggest a core-periphery network for the countries. All these studies do not examine the effect of cross-affiliation on the distribution publications among the institutions.

Power laws are an important empirical regularity for a wide spectrum of natural and social phenomena (Dorogovtsev & Mendes, 2003; Jackson, 2008). Beginning with Pareto (1896), well-documented power laws in Economics and Finance include individual wealth/income, international trade, city size, stock-market activity, firm size, CEO compensation, and supply of regulations (see Gabaix, 2009, and the references therein). In academia, the number of papers written by individual scientists, the number of coauthors of mathematicians, and the number of citations of papers also follow power laws (Newman, 2003). This paper provides evidence from Finance Economics that research output of institutions obeys power laws.

The literature on research collaboration through the lens of social networks starts with Newman (2001b, 2001c, 2001d).11 Barabási et al. (2002) extend Newman's analyses of static networks and examine the dynamics and evolution of coauthor networks. Goyal et al. (2006) treat Economics as a unified discipline and investigate coauthor networks of all journal articles included in the EconLit database. Fafchamps et al. (2010) find that the distance between two authors in the network is negatively related with the probability that they will form research teams in the future. van der Leij and Goyal (2011) examines the strength of strong ties in the network. Ductor et al. (2014) and Ductor (2015) find that coauthorship increases research productivity. Several studies investigate subfields of Economics including Evolutionary Economics (Dolfsma & Leydesdorff, 2010), Econometrics (Andrikopoulos et al., 2016), Population Economics (Brown & Zimmermann, 2017), and Financial Economics. For coauthor networks12 in Financial Economics, Fatt et al. (2010) explore coauthorship in JF. Andrikopoulos and Trichas (2018) study coauthorship in the Journal of Corporate Finance (JCF), which is the oldest academic journal specializing in corporate finance research. Exceptionally, the two authors give an example of the ego-network of the City University of Hong Kong, although they do not further examine the collaboration network of institutional affiliations of financial economists who publish in JCF. Li et al. (2018) find scale-free features in interinstitutional scientific collaboration networks in materials science. However, they do not examine network growth through collaboration, especially cross-affiliation. And they do not provide a theoretic link between the scale-free property and cross-affiliation. We characterize the degree distribution, collaboration patterns, and growth mechanisms of the collaboration network of research institutions in Financial Economics. We propose a model of network growth to explain the emergence of scaling in the network.

The contribution of the model analysis is threefold. First, we provide a realistic model which encompasses a rich set of dynamic patterns of network growth. Although some of these features have been separately examined in previous studies, we synthesize them with a tractable model and closely examine the interaction between the different model mechanisms. Second, we extend the analysis of Dorogovtsev and Mendes (2001b, 2003), who study exact power function form of network growth, to general polynomial growth, as required by the collaboration patterns in financial research. Our modeling approach could potentially be applied to investigate a wider range of networks. Third, we exploit our network model to develop a quantitative understanding towards the allocation of publications in financial research, and demonstrate the important role played by collaboration, through both coauthorship and cross-affiliation, in shaping the research output of institutions.

The rest of the paper is organized as follows. Section 2 presents the definition of a power law and describes the data and econometric approaches. Section 3 discusses the empirical results. Section 4 proposes an explanation of the empirical power laws, based on an accelerated network model. Section 5 concludes.

2 POWER LAWS FOR RESEARCH INSTITUTIONS

2.2 Definitions and empirical methods

A power law, also referred to as a scaling law or a Pareto law, for a random variable x is a distribution described by the density function

()

where is the power exponent and is the lower bound to the power law behavior. To avoid divergence as , we require . The power exponent α controls the speed of decay of the density, with a lower corresponding to a lower speed of decay and thus a longer upper tail. The exponent also determines the convergence of all moments of the distribution. Interestingly, for , all moments of the distribution are infinite. And the nth moment exists if and only if . On the other hand, a power law usually does not describe the whole distribution of a random variable, but governs its tail behavior. Therefore, besides the technical reason, the lower bound is a threshold to determine whether an observation belongs to the tail, or enters the phase of scaling. Only observations in the tail can then be applied to calibrate the associated power law. The countercumulative distribution function is defined to be . We have

()

Due to the form of the countercumulative distribution function, it is sometimes referred to as a rank-size rule.

A power law model and the lower bound to the power law behavior can be determined adaptively. Given the cutoff for points in the upper tail, we get the measured data with observations. For our study, one observation is the number of publications of an institution during a certain sample period. We estimate as the slope of the OLS log-log regression (Gabaix & Ibragimov, 2011):

()

where is an adjustment to the rank . is typical in the literature while is optimal. The downward adjustment of 1/2 to the rank reduces the small-sample bias to the leading order for the log-log regression.16 The asymptotic standard error of is computed at , not the conventional OLS standard error that is nullified by the positive autocorrelation introduced by the ranking procedure into the residuals.

In our study, refers to the number of publications of institutions, and is the density function of x. That is, can be approximated by dividing the proportion of institutions with publications within a bin containing by the width of the bin. And the countercumulative distribution in Equation (2) is the proportion of institutions with publications larger than or equal to x. Note that the rank of an institution, , normalized by the total number of institutions, , is the proportion of institutions with publications larger than or equal to the publications of this institution, that is, . Take logs of both sides of Equation (2), we can derive Equation (3), where the denominator as a normalization factor is absorbed by the regression constant.

The lower bound now hinges on a tradeoff: Too high will leave out legitimate data points thus increase finite sample biases, while too low introduces biases by mixing with nonpower-law data. We adopt the approach proposed by Clauset et al. (2009), which chooses to minimize the distance between the empirical distribution of the data and the fitted power law. We measure the distance using the Kolmogorov–Smirnov statistic, , which is most common for nonmormal data:

()

where and are the cumulative distribution functions of the data and the fitted power law model, respectively. Our method avoids the subjectivity in applying some upper quantile or visual checks to fix as often done in the literature (see, e.g., Beirlant et al., 2004; Drees et al., 2000), while other more subtle methods require estimation of extra parameters (Embrechts et al., 1997).

A two-parameter distribution, such as the lognormal, may provide a better fit for the data due to the added curvature from the free parameter (Eeckhout, 2004). To test potential deviations from a power law, we employ the Gabaix–Ibragimov test (Gabaix & Ibragimov, 2008) which augments the OLS regression with a quadratic term:

()

where . recenters such that stays the same, irrespective of the quadratic term. The standard error of the test statistic is computed at . We reject the null of an exact power law if and only if is statistically different from zero.

3 EMPIRICAL FINDINGS

We show in Panel A of Figure 2 the distribution of top finance journal publications accumulated to the end of 2016 for all institutions. We plot the rank of the institutions against their total publications in log–log coordinates. We impose the optimal downward adjustment of 1/2 to the rank as suggested by Gabaix and Ibragimov (2011). The upper tail of the distribution falls onto a straight line that hints at a power law, or a rank-size rule, as dictated by Equation (2).

Applying the method discussed in the previous section, we retain the 50 most prolific institutions in the upper tail, with the estimated cutoff of . We report the complete list of the 50 institutions together with their publication records in Table 1. Although the 50 institutions account for merely 6.0% of the entire sample of institutions that had published in top finance journals during our sample period, they contribute a predominant 61.3% of total publications. The distribution of the 50 institutions is highly right-skewed, with a mean number of publications of 166 and a median of 112; notably, NBER is solely responsible for 1105 publications.

We classify the 50 institutions according to several criteria in Table 2. The comprehensiveness of our sample is revealed by the scattering of the institutions across type, geographic region, and time of establishment. For instance, the majority of the most productive institutions, as expected, are universities and colleges. Meanwhile, we find that three research bureaus/centers (NBER, CEPR [Centre for Economic Policy Research], and ECGI [European Corporate Governance Institute]), the Federal Reserve Board, and the Federal Reserve Bank of New York also make their way into the upper tail of the distribution. It is clear from the table that the present study broadly includes all institutions that embark on academic research in Financial Economics as “research institutions”.

We present the distribution of the 50 most prolific institutions in panel B of Figure 2. A power law model describes the data well (for over two decades of institutions and from to publications). We report the OLS regression results in Table 3. For the three journals combined, the regression is as high as 98.6%. The power exponent is estimated at 1.640 with a standard error of 0.328. As a robustness check, we repeat the analysis for the preceding 5 years, from 2011 to 2015. The results are presented in Table 4. The power exponent was fairly stable, even though the average number of publications increased dramatically over time. The complex system of institutions, with diverse natures and worldwide origins, seems to have reached a scale-free stationary state.

Table 3. Power law exponents for institutions' top finance journal publications. Estimates are from OLS regressions with standard errors in parentheses.

	All Journals	JF	JFE	RFS
	1.640	1.490	1.689	1.615
	(0.328)	(0.298)	(0.338)	(0.323)
	0.986	0.988	0.982	0.961

Table 4. Power law exponents for institutions' top finance journal publications over a 5-year period. Note the stability in regardless of increasing mean number of publications.

Year	Mean number of publications
2015	155	1.632 (0.326)	0.985
2014	146	1.616 (0.323)	0.984
2013	135	1.610 (0.322)	0.980
2012	123	1.572 (0.314)	0.973
2011	113	1.541 (0.308)	0.965

To explore the unanimity of the power law behavior, we redo the graph of the upper tail distribution for each journal separately in panel C of Figure 2. Since the three journals publish different total numbers of articles, we normalize each institution's publications in one journal by the average number of publications in that journal. If the same power law permeates all three journals, we expect their graphs to collapse onto one another after normalization. This is confirmed in panel C of Figure 2. Moreover, Table 3 shows that, for each journal separately, the regression is no less than 96.1%, and the power exponent ranges from 1.490 to 1.689 with no statistically significant differences among those estimates given the standard errors.

We present in Table 5 the results of the Gabaix-Ibragimov test for deviations from our proposed power law models. For all the three journals, separately or combined, we see that is not significantly different from zero given the standard errors. We fail to significantly improve model performance by introducing curvature to the models.

Table 5. Tests of the power law models to empirical data. is the Gabaix-Ibragimov statistic with standard errors in parentheses.

	All journals	JF	JFE	RFS
	0.102	0.109	0.182	0.224
	(0.269)	(0.222)	(0.285)	(0.261)

For a final robustness check, we apply the well-known tail-index estimator of Hill (1975) to our empirical data. The estimator is

()

with a standard error of .17 We use the Kolmogorov–Smirnov test of the empirical CDF against the Hill's estimate to detect potential deviations from power laws. We present the results for model estimation in panel A and model tests in panel B of Table 6. We confirm that, although some variations in the results are unavoidable when we apply different estimation methods, we find no qualitative differences in the results under either the OLS log–log regression or Hill's tail-index estimator approach. Specifically, the discrepancies in the power exponent estimates are not statistically significant given the standard errors for the power law models considered. Furthermore, the Kolmogorov–Smirnov test cannot reject the power law models at a conventional significance level.

Table 6. Power law models by Hill's tail-index estimator. is Hill's estimator with standard errors is in parentheses. KS-Stat is the Kolmogorov–Smirnov test statistic with p-values is in parentheses.

	All journals	JF	JFE	RFS
Panel A: Hill's tail-index estimator
	1.489	1.208	1.516	1.313
(SE)	(0.205)	(0.166)	(0.208)	(0.180)
Panel B: Kolmogorov–Smirnov test
KS-Stat	0.065	0.131	0.123	0.124
(p-value)	(0.968)	(0.300)	(0.373)	(0.357)

In sum, the proposed power laws seem to succinctly capture growth in the research output of institutions. We now turn to examine the dynamics and evolution of collaboration patterns in Financial Economics. Due to the diversity in the nature of the institutions and the robustness of the power law behavior, we expect generic mechanisms, less attached to traditional value of institutions such as funding and service, that engender the scaling laws documented here. We will provide a model of network growth which explicitly reveals the impact of cross-affiliation on the distribution of publications.

4 A MODEL OF ACCELERATED NETWORKS

A variety of processes can explain the emergence of scaling (see, e.g., Barabási & Albert, 1999; Ferrer i Cancho & Solé, 2003; Gabaix, 1999).18 We find empirical support for the network model of Barabási and Albert (1999), who pioneer two generic mechanisms: One is growth, that is, new nodes continue entering the network; the other is preferential attachment, also referred to as success-breeds-success, popularity-is-attractive, rich-get-richer, or the Matthew effect, that is, a node with a higher degree will enjoy a cumulative advantage and acquire even more links in the future.19

Specifically, we treat institutions with at least one top finance journal publication as nodes in our collaboration network. Therefore, an institution joins the network when it publishes for the first time in the top finance journals. And we treat the total number of publications of a node as the degree of the node. In concert with our data acquisition method, we increase the degree of a node by one per citation of the corresponding institutional affiliation by an author in an article. We plot in Figure 3 the structure of the collaboration network at the end of 2016. Two nodes are connected if the corresponding two institutions work together in at least one publication during 1995–2016. Notably, the network has a giant component that includes 99.5% of the nodes, which reveals the close linkage among the academic society for financial research. The most connected node is NBER, which has collaborated with 300 other institutions during our sample period. In contrast, the average number of collaborating institutions for all nodes in the network is 13.5.20 Hence the average number of collaborators is small relative to that of the most connected “star.”21

4.1 Growth patterns

We first examine the growth of the collaboration network of financial research institutions. We plot in Panel A of Figure 4 the number of nodes newly joining the network. We see that the growth of the nodes is stable, with certain fluctuations around the financial crisis of 2008. Panel B of Figure 4 presents the fraction of publications contributed by the new nodes for each year from 1995 to 2016. This proportion again stays relatively stable, with an average of 6.0% from 2000 to 2016.

We then investigate the allocation of new links (publications) to the nodes. Exact power laws follow only from linear preferential attachment (Krapivsky et al., 2000). That is, the probability for an existing node to acquire new links () is proportional to its degree (), or . Exploiting the network maps at the end of each year from 1995 to 2016, we use the method provided in Jeong et al. (2003) to measure preferential attachment. We provide detailed results in Appendix A. We find that can be well approximated by , where is estimated at 1.00 with a standard error of 0.08. Thus, linear preferential attachment seems to be a good match for data.

In the following, we develop a network model that captures three additional features of the collaboration network. First, expanding coauthorship (panel A of Figure 1) and cross-affiliation (panel B of Figure 1) generate nonlinear growth in total publications. Thus, instead of linear growth considered by Barabási and Albert (1999), we incorporate accelerated growth following Dorogovtsev and Mendes (2001b, 2003). Second, we include internal links because 94.0% of the total publications come from collaboration among already existing nodes (Ghoshal et al., 2013). Third, although preferential attachment enables institutions successful in historic records to produce even more research output in the future, we randomly allocate a fraction of new publications to the existing nodes, allowing research opportunities to strike institutions by sheer chance (Ghoshal et al., 2013).22

4.2 An evolving network with accelerated growth

Consider discrete time , with time interval . Assume that there is one node in the network at time 0 and that one new node enters the network at each . Then there are nodes in the network at time , excluding the new node entering at the time.23 Let denote the publication rate at time . Hence is the total number of new publications during the time interval . These publications are distributed among all nodes in the network at time . We assume that, a faction of the total publications is distributed preferentially to the existing nodes, a fraction is distributed randomly (with equal probability ) to the existing nodes, and a fraction is allocated to the new node. The constants , and are nonnegative and sum to one.24 Preferential and random allocations are independent events.

Let denote the degree at time of a node entering at time s. Notice that

()

The four conditional probabilities in the above equation can be evaluated by linear preferential attachment, random allocation, and independence of the events.

Let denote the probability that the node has degree at time . A new node entering at time obtains the degree . Thus, . We have the following master equation for degree probabilities of individual nodes.

()

where is the cumulative degrees of all nodes at . It is worth mentioning that , that is, the degree of a node at time is not greater than the cumulative input of degrees until time . Rearrange the terms of Equation (8), we have:

To exploit the tractability of the rate-equation approach (Dorogovtsev et al., 2000; Krapivsky & Redner, 2002), we let go to zero as time and degree vary continuously. Suppose the degree rate function is continuous. The continuous-time approximation of the master Equation (8) is obtained as follows. For ,

()

where is a generalized probability density function representing the density of the degree distribution at time of the node s. The initial condition is for , where is the delta function with the properties for , and for a nonnegative constant .

Multiplying by and integrating from 0 to with respect to for the above equation, we obtain:

()

Denote the average degree . Then we obtain the evolution equation for :

()

with boundary condition .

The integration is the total number of nodes with degree at time t, and is the number of nodes at time , excluding the newly coming one. Hence the degree distribution at time is given by . Actually, the distribution can be recovered from by the relation

()

where is a solution to the equation . The first equality is due to the fact that the solution is actually a delta function for some and the second equality is according to the general property of the delta function.

We now turn to solve Equation (11). A general solution for is

()

By the boundary condition can be identified and we obtain

()

Recall that is the average degree at time of the nodes entering at time s. The above equation indicates that a lower corresponds to a higher for a fixed .

Let be a solution to the equation . Then . Taking partial derivative with respect to , it follows that

()

Hence

where satisfies

()

Next, we introduce an assumption on the rate function to evaluate . Suppose that is a nonnegative continuous function defined on .

Assumption.There exist constants , such that for all .

For example, or for some positive constant , satisfies the assumption.25 Clearly, when the assumption is satisfied, is such a function that , and all exist. We write if a continuous function satisfies the assumption.26 We can directly show that if .

Using the above assumption, we can estimate the right-hand side of Equation (14). We find that

()

where

Thus, and

()

The second equality in the above equation holds when is large and has the same or a higher order than of . Note that the total degree has an order of thus the average degree has an order of . Finally, it follows from Equations (15) and (12) that the degree distribution at time is

()

or

()

when is large. Therefore, the degree distribution obeys a power law in the upper tail with exponent

()

To ensure the existence of , we require or .

5 CONCLUSION

In this paper, we document the recent fast development of cross-affiliation of researchers who publish in the top finance journals from 1995 to 2016, and the resulting highly nonlinear growth of research output following mixed forms of collaboration. We show that scale-free power laws characterize the distributions of worldwide institutions' top finance journal publications. We propose an accelerated network model to explain the scale-invariance. We investigate the growth mechanisms of the network and find that linear preferential allocation of new publications plays a key role in the emergence of the scaling behavior. We consider three distinct ways by which new publications are generated, in particular, through new nodes (institutions publishing for the first time in top finance journals), random allocation, and preferential allocation. The last channel accounts for a dominant 87.0% of all publications. Our theoretic analysis also demonstrates that higher-order growth of the network, possibly originating from new forms of collaboration, reduces the dispersion in the institutions' cumulative research output.

The power law distributions we focus on here may characterize other fields and extend to a broader set of journals as well. Cross-field examinations of the scaling behavior and the role played by preferential attachment in the allocation of publications could undoubtedly enrich our understanding of production of scientific knowledge of the institution level. We can also examine the inverse relationship between order of growth in collaboration and heterogeneity of research institutions through cross-subfields and even cross-disciplinary studies. Furthermore, due to the obvious relations between the authors and their institutional affiliations, an interesting future direction is to reconcile the small-world properties of the coauthor network discovered on the individual researcher level and the scale-free properties of the collaboration network on the institution level. Last but not least, we have demonstrated the effect of collaboration on the distribution of research output, while we have not touched on the efficiency of such scale-free networks as a way of organizing research activities, and the robustness of such networks under attack (e.g., the accidental demise of a major research institution). These works await comprehensive data to be gathered, and we leave them to future studies.

AUTHOR CONTRIBUTIONS

Hui Dong: Conceptualization; data curation; formal analysis; funding acquisition; writing—review and editing; writing—review and editing. Dan Luo: Conceptualization; data curation; formal analysis; funding acquisition; methodology; project administration; writing—original draft; writing—review and editing. Xudong Zeng: Conceptualization; formal analysis; funding acquisition; methodology; writing—original draft; writing—review and editing. Zhentao Zou: Conceptualization; data curation; formal analysis; funding acquisition; methodology; writing—original draft.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest.

ETHICS STATEMENT

None declared.