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The structure of collaboration in the Journal of Finance

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Abstract

This paper studies the structure of collaboration in the Journal of Finance for the period 1980–2009 using publication data from the Social Sciences Citation Index (SSCI). There are 3,840 publications within this period, out of which 58% are collaborations. These collaborations form 405 components, with the giant component capturing approximately 54% of total coauthors (it is estimated that the upper limit of distinct JF coauthors is 2,536, obtained from the total number of distinct author keywords found within the study period). In comparison, the second largest component has only 13 members. The giant component has mean degree 3 and average distance 8.2. It exhibits power-law scaling with exponent α = 3.5 for vertices with degree ≥5. Based on the giant component, the degree, closeness and betweenness centralization score, as well as the hubs/authorities score is determined. The findings indicate that the most important vertex on the giant component coincides with Sheridan Titman based on his top ten ranking on all four scores.

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Correspondence to Choong Kwai Fatt.

Appendix I

Appendix I

Consider a graph G = (V, E) where E is the set of edges connecting vertices defined in vertex set V. The construction of a binary network model (Krichel and Bakkalbasi 2006) based on G requires that each e ij  ∈ E encodes the presence or absence of a connection between vertex i and j. For the case of a directed graph: we set the edge weight \( {e_{ij} = 1} \) if a link exists from vertex i to j, and \( {e_{ij} = 0} \) if i and j are unconnected (i ≠ j). For the case of an undirected graph: \( {e_{ij} = e_{ji} = 1} \) if vertex i and j are connected (i ≠ j), while \( {e_{ij} = e_{ji} = 0} \) if unconnected. For both directed and undirected graphs, we set \( {e_{ii} = 0} \) so that G does not contain any loops.

Degree centrality

The degree centrality of vertex v is simply given by the number of edges incident upon it. Suppose that there are n vertices in vertex set V, then the degree centralization is defined by the following formula (Freeman 1979):

$$ {C_{D} \left( v \right) = {\frac{deg\left( v \right)}{n - 1}}},{\text{ where }}deg\left( v \right) = {\text{ degree of vertex}}\,v.$$
(1)

Closeness centrality

The closeness centrality of vertex v is defined as the average number of steps required to reach every other reachable vertex in the graph. Specifically, it is the inverse of the mean geodesic distance (length of shortest paths) to/from all the other vertices in the graph, as defined by the following formula (Freeman 1979):

$$ {C_{C} \left( v \right) = {\frac{n - 1}{{\sum\nolimits_{i \ne j} d \left( {i,j} \right)}}}},$$
(2)

where d(i, j) = distance between vertex i and j.

Betweenness centrality

The betweenness centrality of vertex v is defined as the number of geodesics (shortest paths) on the graph that pass through it. Its value can be computed by the following formula (Freeman 1979):

$$ {C_{B} \left( v \right) = \sum\limits_{{ i \ne v \ne j \in V \atop i \ne j}} {{\frac{{\sigma_{ij} \left( v \right)}}{{\sigma_{ij} }}}} },$$
(3)

where σ ij (v) is the number of shortest paths from vertex i to j that pass through v, while σ ij is the number of shortest paths from vertex i to j. The betweenness centralization is given by the betweenness centrality divided by \( {\left( {n - 1} \right)\left( {n - 2} \right)} \) for directed graphs and \( {\frac{1}{2}\left( {n - 1} \right)\left( {n - 2} \right)} \) for undirected graphs.

HITS algorithm: hubs/authorities score

Hyperlink-Induced Topic Search, or HITS (Kleinberg 1998), is a link analysis algorithm originally designed to rank webpages by using the method of eigenvector centrality (Bonacich 1972). HITS assigns two scores to each vertex on graph G: a hub score y i and an authority scorex i . The underlying logic behind the method is that a good authority is cited by many good hubs, while a good hub cites many good authorities. This mutual reinforcement between authority and hub vertices can be represented by two operations I and O. The I operation updates the x-weights (authorities score) as follows.

$$ {x_{i} \leftarrow \sum\limits_{{j :\left( {j,i} \right) \in E}} {y_{j} } } $$
(4)

The O operation updates the y-weights (hubs score) as follows.

$$ {y_{i} \leftarrow \sum\limits_{{j :\left( {i,j} \right) \in E}} {x_{j} } } $$
(5)

In matrix representation, these two operations can be written succinctly as:

$$ {I\left( \cdot \right) = L^{T} ,O\left( \cdot \right) = L.} $$
(6)

By recursively updating the x- and y-weights, the authority and hub scores of each vertex eventually converge at their final values. At the tth iteration, we obtain the following expressions:

$$ {\begin{gathered} x^{{\left( {t + 1} \right)}} = I\left( {O\left( {x^{\left( t \right)} } \right)} \right) = L^{T} Lx^{\left( t \right)} \\ y^{{\left( {t + 1} \right)}} = O\left( {I\left( {y^{\left( t \right)} } \right)} \right) = LL^{T} y^{\left( t \right)} \\ \end{gathered} }.$$
(7)

The final solutions x*, y* are the principal eigenvectors of L T L (authority matrix) and LL T (hub matrix), which are the singular decomposition of L (Ding et al. 2002). For undirected graphs, L is symmetric and therefore \( {L^{T} L = LL^{T} = L^{ 2} } \) (Shafer et al. 2006).

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Fatt, C.K., Ujum, E.A. & Ratnavelu, K. The structure of collaboration in the Journal of Finance. Scientometrics 85, 849–860 (2010). https://doi.org/10.1007/s11192-010-0254-0

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