Increased Connectedness As a
Function of Organized
Inequality in Directed
Networks:
Illustrated by the Network of
Placements Among U.S. Sociology
Departments
Scott L. Feld, Purdue University and
Michael G. Bisciglia, LSU
• Connectedness is Important
• Inequality Makes Connections
• This paper is about 4 particular structures
of inequality, and 3 particular types of
indirect connections in directed networks.
• In directed networks, each node has an indegree, the
number of ties directed towards the node, and an
outdegree, the number of ties directed away from the
node.
• In a closed network, the mean outdegree and mean
indegree are necessarily equal to one another and equal
to the total number of ties divided by the number of
nodes.
• However, outdegrees and indegrees can have very
different amounts of variation from one another.
The four properties of structure of inequality:
1)Inequality of Outdegree. The variance of outdegree is an indicator and
the standard deviation divided by the mean is a standardized indicator of
inequality of outdegree in the network.
2)Inequality of Indegree. The variance of indegree is an indicator and
the standard deviation divided by the mean is a standardized indicator of
inequality of indegree in the network.
3)Correlation of indegree and outdegree. The correlation between
indegree and outdegree over nodes is a normalized indicator of the
extent to which nodes that receive more ties also send more ties.
4)Correlation of outdegrees. The correlation between outdegrees of
nodes that are connected by ties is a normalized indicator of the extent to
which nodes that send more ties tend to send ties to nodes that send more
ties.
Implications of These Properties for The
Overall Pattern of Relationships:_
Undifferentiated Network That Has None of the Designated Properties
Figure 1: Contrasting Networks With and Without the Designated Properties
An Undifferentiated Network That Has None of the Designated Properties
A Differentiated Network That Has All of the Designated Properties
This second network is a core-periphery situation. According to
Borgatti and Everett (1999), the defining features of a core-periphery
situation are a dense core, a less dense connection between core and
periphery, and a still less dense periphery. The network displayed here
meets these conditions.
We suggest that a network that has all four properties that we describe
will always have the form of core-periphery with the high outdegree
nodes in the core.
However,
a)there are networks that have some of these properties that are not coreperiphery networks, and
b)there are core-periphery networks that do not have all these properties.
The four properties together seem to be sufficient but not necessary for a
core-periphery network.
These logical connections require further investigation.
We use the language of family and kinship to facilitate discussion of the
various types of short paths.
The primitive direct relation:
Parents
A–>B
Three indirect relations:
Siblings
B<--A-->C
Grandparents
A–>B–>C
Uncle/Aunt
C<--A-->B–>D
These are the two and three step relations that we consider most likely to
be consequential.
We do not consider any in-law relations or great-grandparent relations.
The salience of indirect relations is ultimately an empirical issue in each
context.
Numbers of sibling relationships
(1) s = x2 + o2
s = mean number of sibling relations per node
x = the mean number of ties per node
o2 = variance in outdegree among nodes
Numbers of grandparent to grandchild relations
(2) g = x2 + rio * i * o
g = mean number of grandparent relations per node
x = the mean number of ties per node
rio = correlation between outdegree and indegree across nodes
i = standard deviation of indegree among nodes
o = standard deviation of outdegree among nodes
Numbers of uncle/aunt to niece/nephew relations
(3) u = s * g / x + rout * s * g * x
u = the mean number of uncle/aunt relations per node
s = mean number of sibling relations per node
g = mean number of grandparent relations per node
rout = the correlation of outdegrees over the set of ties
s = the standard deviation in the number of siblings of all
ties
g = the standard deviation in the number of grandparent to
grandchild relations mediated by all ties
We use the network among PhD granting departments of sociology in
the United States as an illustration of how the properties of inequality in
networks is related to the connectedness of networks.
We specifically examine the network among the training institutions that
is created by the graduates of training organizations who currently staff
other training organizations.
Data from the 2001 Guide to Graduate Departments of Sociology,
published by the American Sociological Association.
We examine the network among 111 PhD schools formed by the 1,881
faculty who received their own PhDs from another of these same 111
schools.
A node is a sociology department at a university.
Its indegree is the size of the department.
Its outdegree is the number of placements of its alumni as faculty in
other departments.
The mean department faculty size is equal to the mean number of
department placements; in the system of sociology departments, these
means are both 16.95.
Variation in Numbers of Placements
The placements of these schools in other departments range widely from
0 to 138 with a mean of 16.95 and a standard deviation of 25.89.
The top 5 schools accounted for 30% of the placements. These were
Wisconsin, Chicago, University of California, Berkeley, Michigan, and
Harvard.
Variation in Department Size
The sizes range from 4 to 50 with a mean of 16.95 and a standard
deviation of 7.52.
Correlation of indegree and outdegree
Larger departments have more placements; this correlation is +0.39.
Correlation of outdegrees
There is a strong tendency for all departments to place their alumni in
departments that make fewer placements than they make (see Feld,
Bisciglia, and Ynalvez, 2004).
Nevertheless, departments that make more placements tend to place their
alumni in other departments that make more placements; r = +0.213.
Variation in Placements and the Number of Sibling Relations
There are 233% more sibling relations among departments than if there
was no variation in outdegree; that is more than 3 times the number of
sibling relations expected with no variation in placements.
Variation in Placements and Size, Correlation Between Placements
and Size, and the Number of Grandparent Relations
There aret 26% more grandparent to grandchild relations would be
expected with no variation in indegree or outdegree or no correlation
between indegree or outdegree.
All Four Properties and the Number of Uncle/Aunt Relations
The large numbers of sibling and grandparent relations imply large
numbers of uncle/aunt relations, over 4 times what would otherwise be
expected. In addition, the correlation of outdegrees increases that
another 24%. The number of uncle/aunt to niece/nephew relations is
more than 5 times what would be expected in the absence of this
structure of inequality.
Inequality is Magnified with Connectivity
Our central conclusion has been that the prominent structure of
inequality increases the connectedness of the system as a whole by
increasing the frequencies of consequential short indirect relations
among the nodes.
However, the increased connectedness that results from inequality is not
equally shared by all the nodes. Rather, the nodes that have more direct
parent to child connections tend to have more of every other type of
connection that is built on those parent child connections. We have not
presented any analytic results that show a formal connection between the
four properties of inequality in direct relations and the inequality of
expressing these various indirect relations. However, the empirical
results show much greater inequality in participation in many indirect
relations than in the direct relations.
Summary and Conclusions
We have presented several properties that we expect to be typical of
networks of training organizations. These are inequality of outdegree,
inequality of indegree, correlation of outdegrees and indegree, and
correlation of outdegrees. We show that when these properties are
present, the result is a much larger set of connections in the network than
would otherwise be present.
We use our particular example of the network of sociology PhD granting
departments to illustrate the extent of the properties and of the
consequent connections in the network in one situation.
We suggest that networks among all kinds of training organizations are
generally characterized by the structures of inequality that we have
described that lead to high levels of connectedness.
Empirical research is required to explore the extent to which these
indirect connections have important consequences in terms of both
overall increased communication and influence, and especially
disproportionate influence of certain actors.
Finally, we suggest that similar types of implications should be explored
for other types of directed networks; e.g. networks of affect among
persons, citations among journals, and officer directors among
corporations.
Table 1: Frequencies of Indirect Relations Among Sociology Departments
Parents
A–>B
Siblings
B<--A-->C
Grandparents A–>B–>C
Uncle/Aunt C<--A-->B–>D
PARENTS
A–Top 5
Bottom 45
B–Top 5
Bottom 45
SIBLINGS
GRANDPARENTS UNCLE/AUNT
563(29.9%)
65,423(62.0%)
17,164(42.6%)
1,962,130(70.2%)
78( 4.1%)
262( 0.2%)
541( 1.3%)
1,792( 0.1%)
125( 6.6%)
9,575( 9.1%)
14,833(36.8%)
1,121,783(40.1%)
626(33.3%)
27,465(26.0%)
1,173( 2.9%)
60,121( 2.2%)
SAME AS B
2,498( 6.2%)
300,954(10.8%)
SAME AS B
13,283(32.9%)
638,666(22.9%)
C–Top 5
Bottom 45
D–Top 5
182,964( 6.5%)
Bottom 45
TOTAL
111
876,820(31.4%)
1881
105605
40313
2794824