In generalized blockmodeling, the

blockmodeling Blockmodeling is a set or a coherent framework, that is used for analyzing social structure and also for setting procedure(s) for partitioning (clustering) social network's units ( nodes, vertices, actors), based on specific patterns, which form ...

is done by "the translation of an equivalence type into a set of permitted block types", which differs from the conventional blockmodeling, which is using the indirect approach. It's a special instance of the direct blockmodeling approach. Miha Matjašič, Marjan Cugmas and

Aleš Žiberna Aleš Žiberna is a Slovene statistician, whose specialty is network analysis. His specific research interests include blockmodeling, multivariate analysis and computer intensive methods (e.g., computer simulations, resampling methods). Curr ...

, blockmodeling: An R package for generalized blockmodeling, ''Metodološki zvezki'', 17(2), 2020, 49–66. Generalized blockmodeling was introduced in 1994 by

Patrick Doreian Patrick Doreian is an American mathematician and social scientist, whose specialty is network analysis. His specific research interests include blockmodeling, social structure and network processes. Doreian, Professor Emeritus from the Univers ...

Vladimir Batagelj Vladimir Batagelj (born June 14, 1948 in Idrija, Yugoslavia) is a Slovenian mathematician and an emeritus professor of mathematics at the University of Ljubljana. He is known for his work in discrete mathematics and combinatorial optimization, p ...

and

Anuška Ferligoj Anuška Ferligoj is a Slovenian mathematician, born August 19, 1947 in Ljubljana, Slovenia, whose specialty is statistics and network analysis. Her specific interests include multivariate analysis (theory and application in social sciences, m ...

Definition

Generalized blockmodeling approach is a direct one, "where the optimal partition(s) is (are) identified based on minimal values of a compatible criterion function defined by the difference between empirical blocks and corresponding ideal blocks". At the same time, the much broader set of block types is introduced (while in conventional blockmodeling only certain types are used). The conventional blockmodeling is inductive due to nonspecification of neither the clusters or the location of block types, while in generalized blockmodeling the

blockmodel Blockmodel (sometimes also block model) in blockmodeling (part of network science) is defined as a multitude of structures, which are obtained with: * identification of all vertices (e.g., units, nodes) within a cluster and at the same time repre ...

is specified with more detail than just the permition of certain block types (e.g., prespecification). Further, it's possible to define departures from the permitted (ideal) blocktype, using

criterion function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...

. Using local optimization procedure, firstly the initial clustering (with specified number of

cluster may refer to: Science and technology Astronomy * Cluster (spacecraft), constellation of four European Space Agency spacecraft * Asteroid cluster, a small asteroid family * Cluster II (spacecraft), a European Space Agency mission to study th ...

s is done, based on random creation. How the clusters are neighboring to each other, is based on two transformations: 1) a

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

is moved from one to another cluster or 2) a pair of vertices is interchanged between two different clusters. This process of transformation steps is repeated many times, until only the best fitting partitions (with the minimized value of the criterion function) are kept as

s for the future exploration of the network. Different types of generalized blockmodeling are: * generalized binary blockmodeling, * generalized valued blockmodeling and * generalized homogeneity blockmodeling.

Benefits

According to

, the benefits of generalized blockmodeling, are as follows: * usage of explicit criterion function, compatible with a given type of equivalence, results to in-built measure of fit, which is integral to the establishment of the blockmodels (in conventional blockmodeling, there is no compelling and coherent measures of fit); * partitions, based on generalized blockmodeling, regularly outperform and never perform less well than the partitions, based on conventional approach; * with generalized blockmodeling it's possible to specify new types of blockmodels; * this potentially unlimited set of new block types also results in permittion of inclusion of substantively driven blockmodels; * in generalized blockmodeling, the specification of the block types and the location of some of them in the blockmodel is possible; * researcher can speficy which (pair of) vertices must be (not) clustered together; * this approach also allows the imposition of penalties, resulting into identification of empirical null blocks without inconsistencies with a corresponding ideal null block.

Problems

According to Doreian, the benefits of generalized blockmodeling, are as follows: * unknown sensitivity to particular data features, * examination of

boundary problem A boundary problem in analysis is a phenomenon in which geographical patterns are differentiated by the shape and arrangement of boundaries that are drawn for administrative or measurement purposes. The boundary problem occurs because of the loss o ...

s, * computationally burdensome, which results in a constraint regarding practical network size (generalized blockmodeling is thus primarily used to analyse smaller networks (below 100 units)), * identifying structure from incomplete network information, * most of generalized blockmodeling is based on binary networks, but there is also development in the field of valued networks, * criterion function is minimized for a specified blockmodel, with results in issues of evaluating statistically, based on the structural data alone, * problems regarding three dimensional network data, * problems regarding the evolution of fundamental network structure.

Book

The book with the same title, ''Generalized blockmodeling'', written by Patrick Doreian, Vladimir Batagelj and Anuška Ferligoj, was in 2007 awarded the Harrison White Outstanding Book Award by the Mathematical Sociology Section of

American Sociological Association The American Sociological Association (ASA) is a non-profit organization dedicated to advancing the discipline and profession of sociology. Founded in December 1905 as the American Sociological Society at Johns Hopkins University by a group of fif ...

References

Selected bibliography

, Vladimir Batagelj,