Economics: Always a Small World?

Has our profession been a Small World since the 2000s or earlier? An assesment on the network of formal and informal collaboration.

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In a popular study of co-author networks in Economics, Goyal, van der Leij and Moraga-González (2006) show that our profession is a Small World only since the 1990s. I repeat their analysis on our network of intellectual collaboration.

A Small World Network – introduced by Watts and Strogatz (1998) – is a very handy representation of the world: Many short distance links, few long distance links. Think of many groups well connected intra-group but sparsely connected inter-group. While most people do not know each other, they are connected indirectly via hubs. Whether a network displays small-world properties or not is relevant for reaching consensus among agents, maintaining power grids and fighting diseases (this is the motivating example in Watts and Strogatz (1998)).

A social network is said to have small-world properties, if the average shortest distance between all nodes does not grow too fast (at most Ο(log N)) as the number of nodes N increases. For their study, Goyal, van der Leij and Moraga-González (2006) reformulate this mathematical requirement into four necessary and sufficient conditions:

  1. number of nodes is large as compared to the average number of neighbors
  2. There exists a giant component (in which a large share of nodes is somehow connected via possibly intermediate steps)
  3. Average distance between nodes is small
  4. Clustering – share of one’s neighbors that are also neighbors themselves – is high

Goyal, van der Leij and Moraga-González (2006) argue that the world of Economics is becoming smaller, precisley by examining small world properties. The networks they study are three co-author networks, one for each decade 1970-1979, 1980-1989, and 1990-1999. Over time, they show that our profession has become more connected.

The question is whether co-author networks are the right network representing our profession.

Laband and Tollison (2000) as well as we in our papers have shown that a large extend of intellectual collaboration is of informal nature. That is, researchers other than authors contribute to the production of science. For research articles, these informal collaborators are acknowledged typically on the frontpage of an article.

I examine networks, each rebuild from full research articles published between 1997 and 2011 in six major finance journals (The Journal of Finance, The Review of Financial Studies, the Journal of Financial Economis, the Journal of Financial Intermediation, the Journal of Money, Credit & Banking, and the Journal of Banking and Finance). Each network is five years long, so that we have three networks: 1997-2001, 2002-2006, and 2007-2011. In this network two academics are connected when either they have jointly published an article, or one acknowledges the other, or both. For comparison I build a classical co-author network. The following tables present the relevant small-world figures:

Table 1: Statistics of the co-author network
1997-2001 2002-2006 2007-2011
Total nodes 1,825 2,683 4,225
Degree:
  Average 1.78 1.93 2.31
  Standard deviation 1.40 1.56 1.86
Giant component:
  Size 161 672 1,560
  Percentage 9% 25% 37%
Second-largest component 58 23 23
Isolated nodes:
  Number 220 257 267
  Percentage 12% 10% 6%
Clustering coefficient 0.37 0.39 0.48
Distance in giant component:
  Average 10.13 12.88 10.74
  Standard deviation 6.44 5.05 3.65

The figures are quite in line with Table 1 in Goyal, van der Leij and Moraga-González (2006), except for the existence of a giant component, which only emerged in the 2000s – and even there it is debateable whether a share of less than 40% constitutes a giant component.

Let’s add acknowledgements:

Table 2: Statistics of the network of intellectual collaboration
1997-2001 2002-2006 2007-2011
Total nodes 4,754 6,532 9,624
Degree:
  Average 7.80 8.68 10.73
  Standard deviation 12.53 13.93 18.54
Giant component:
  Size 4,507 6,234 9,156
  Percentage 95% 95% 95%
Second-largest component 11 11 12
Isolated nodes:
  Number 21 18 25
  Percentage <1% <1% <1%
Clustering coefficient 0.48 0.49 0.54
Distance in giant component:
  Average 2.35 2.22 1.92
  Standard deviation 0.99 0.90 0.79

We can immediately see that accounting for author-commenter links increases almost all numbers (except those for distance, which decrease): The the networks’ connectivity improves dramatically. Most notably the giant component, which connects virtually all nodes in the network. This implies information can reach every node in the network, which it could not in the co-author network. It also flows much faster, as the lower distances show.

One caveat is in order though, and that is the inflation of links between authors and commenters. We generally do not know which author spoke to which commenter on a multi-authored article. Therefore we assume a link between every author and every commenter. This inflates degree strongly distance somewhat, so these figures should be digested with caution. And while it has an unknown effect onclustering, too many edges do not affect the size and share of the giant component.

We also look at a much smaller network of only 6 journals and three years, while Goyal, van der Leij and Moraga-González (2006) study a network of an unreported number of journals (they use the EconList database) and 10 years. We believe however that our claim remains valid: The inclusion of informal links between economists is the right way to go because it accounts common information flows.

References