Page 308 - Contributed Paper Session (CPS) - Volume 6

CPS1937 Xu Sun et al.
th
ij, entry in E is the proportion of ties in the network that go from the
community in row i to the community in column j. In this case, each entry is
th
the proportion of respondents in the ij, cell. The diagonal elements in E define
the share of within-community ties. To compare this to the overall distribution
of both within- and between-community ties, the modularity function is
defined as = () − || ||, where Tr( ) is the trace of matrix E and ‖ ‖
2
indicates the sum of the elements of matrix.
Intergenerational mobility tables are typically square because they have the
same social classes for parents and respondents. As such, the table is
amenable to spectral partitioning as discussed above. There is, however, a
problem that is the resulting community structure would only row (partition
parental social class) because it does not recognize column (respondent social
class) is a distinct mode of the data. Fortunately, standard multi-mode
generalizations allow row and column (parental social class and respondent
social class) to be dually represented in the same community partition. To do
so, one simply includes the mobility table and its transpose in a larger block
off-diagonal matrix (Wasserman & Faust, 1994). Denote a mobility table as M.
The spectral decomposition of the following matrix would allow the rows and
the columns to be dually represented in sub-sequent community partitions:
T
= 0 . Here M is notation for the transpose of matrix M and 0 is notation
0
for a matrix of zeroes.
When the genspectrum decomposition is applied to typical social networks
(e.g., person-to-person networks), the expected network is invariably an
“independence” model; that is, each entry in the expected network is the row
sum times the column sum divided by the number of relations in the network.
More generally, the approach taken herein is to remove the effects of
“independence” model of mobility compute the community structure of
residual variation, and determine the effects of community membership on
social mobility patterns by parameterizing a within-community effect in the
design or “model” matrix. In every empirical case examined, the effect of
community membership is quite large while being very parsimonious (i.e., 1
degree of freedom).

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