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CPS1937 Xu Sun et al.
mobility table may be understood to be a two-mode network, where one
mode is parental social class and the other mode is respondent social class.
What are shared between these two modes are members, or the count of
people who have a given social class and have parents of a given social class.
It is often assumed that the same social classes are represented for both
parents and respondents, which implies that both modes share the same
number of nodes, or in this case, categories of social class. Newman (2010)
provided an approach for the analysis of social networks is the identification
of communities or cohesive subgroups. Here, a community refers to a subset
of nodes which share relations at above expected rates. Community detection
has a rich history in computer science and the social sciences (Wasserman &
Faust, 1994), but until Girvan and Newman (2002) this problem is brought to
the attention of the general scientific community.
In this paper, We use eigenspectrum decomposition approach Newman’s
(2006a,b) because it is easily applicable to intergenerational mobility tables,
has been generalized to multi-mode networks, such as mobility tables, and is
highly efficient and accurate relative to other solutions to the community
finding problem. Newman’s (2006a) eigenspectrum approach is elegantly
simple. One begins with a relational matrix that is denoted by A, defines a
matrix of expected cell counts that is denoted by P (an “independence” model),
subtracts P from A to yield B, which is called the residuals matrix, and finally
one computes the eigenspectrum decomposition of the residuals matrix. The
eigenspectrum of the residual matrix, B, sheds light on the structure of A
(Newman, 2006a,b). Newman has shown that the signs of the entries in the
eigenvector associated with the largest eigenvalue partition the nodes into an
optimal two community split. Subsequent splits into more than two
communities may be determined by examining the signs of the entries in the
second leading eigenvector, and so on.
Another development with respect to community structures was to define
the quality function that is unfortunately also called modularity Newman
&Girvan (2004). Modularity (denoted by Q) indicates the strength of, or
variance explained by, a community structure discovered by the community
finding algorithms. That is, it provides a bench-mark with which to compare
possible solutions to the community structure. Larger values of Q indicate that
larger shares of the relations in the data are within communities; hence larger
values indicate a better fit to the data. The specific formula for modularity was
generalized for the eigenspectrum approach by Newman (2006b). Thus, the
eigenspectrum decomposition of the residual matrix, B, can be used to identify
possible solutions to the community structure, and the quality function
modularity can be used to identify which solution is “best”.
The formula for the modularity function is the intuitive. Define a number-
of-communities by number-of-communities matrix, which is denoted E. The

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