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Generalized Blockmodeling
Samantha Lam, Jeffrey Chan, Conor Hayes
Digital Enterprise Research Institute, National University of Ireland, Galway
samantha.lam@deri.org, jkc.chan@deri.org, conor.hayes@deri.org
1. Introduction
As online social network data become increasingly
available and popular, there is a ongoing need to
analyze and model them in a scalable manner. To
understand these large networks, it is useful to be able
to reduce and summarize them in terms of their
underlying structure. Popular approaches include
community finding [1] and blockmodeling [2], both of
which aim to group the strongly associated vertices
together. Our research is focused on the latter approach.
2. Generalized blockmodeling
Generalized blockmodeling [3] decomposes a
network into partitions and assigns a relation type to
each pair of partition (called a block), which describes
the relationship between the partitions. A major
component to the generality of this method comes from
the use of regular equivalence as a defining feature of
the blocks. The previously well-studied structural
equivalence proved to be somewhat too restrictive to
describe real-world networks which led to the
proposition of regular equivalence [3].
Up to now, the generalized blockmodel analysis of
social networks has not received much attention, partly
due to the computational demands of the existing
algorithms. Therefore, we have designed approaches
based on genetic algorithms (GA) and simulated
annealing (SA) to fit generalized blockmodels. We
have found both approaches are at least two orders of
magnitude faster than the existing method.
3. Improving algorithms
In [3], they proposed the greedy KL-based approach
to fit blockmodels. This algorithm considers the
solution neighborhood of each vertex, and then greedily
makes a move that minimizes the objective cost. A
neighborhood move as a vertex moves from one
partition to another, and the swapping of two vertices in
different partitions were also considered. However,
there was no description as to how to optimize the
blocks types themselves. Therefore, we also introduce
an additional step, where the blocks types are optimized
after the partitions are optimized.
3.1. Results
We evaluate the efficiency and optimization
performance of three algorithms – the proposed KLbased,
and two aforementioned ones, SA and GA. To
measure the scalability and optimizing ability of the
algorithms, we generated synthetic datasets using a
community generating algorithm and he results indicate
that for larger networks, if speed is important, than the
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SA algorithm should be used, but if accuracy is more
important, than the GA should be used.
To demonstrate the importance of increasing the
scalability of fitting generalized blockmodels, we fitted
blockmodels to the Enron and flight route datasets that
could be not be fitted before because of the limitations
of the KL algorithm.
We used the GA algorithm to explore the Enron
dataset over three time periods - prior, during and after
the crisis. As a guide to the communications between
the employees we used results found by [4] to help us
construct our blocks. The best-fitted blockmodels
summarized the roles, the key relationships (block
types) between the different roles.
For the flight route dataset, we decided to use
European airlines as it was shown that a hub-andspoke/hierarchy
model existed . For this data, we found
that an addition of a 'density' block to be a fruitful
addition to the nine types specified by [3].
4. Improving the objective function
The current definition of the objective function
proposed by [3] is somewhat naïve. It is essentially a
simple count/percentage of the number of deviations of
an element from its ideal block. We are currently
investigating methods to improve upon this definition.
5. Future work
We aim to investigate additional ways to measure
and rank discovered blockmodels, such as improving
the objective function and defining other block types.
6. References
[1] A. Clauset, M.E.J. Newman, and C. Moore, “Finding
community structure in very large networks”, Phys. Rev. E,
American Physical Society, Vol. 7 Issue 6, 2004, 066111.
[2] S. Wasserman, and K. Faust, “Social network analysis:
Methods and applications”, Cambridge university press, 1994
[3] P. Doreian, V. Batagelj, and A. Ferligoj, “Generalized
Blockmodeling”, Cambridge university press, 2005
[4] J. Diesner. and T.L. Frantz, and K.M. Carley,
“Communication networks from the Enron email corpus “It's
always about the people. Enron is no different”,
Computational & Mathematical Organization Theory,
Springer, Vol. 1 No. 3, 2005, pp. 201-228