The fact that requirements for role assignments are often not satisfied in application data, gives rise to the idea to formulate role assignments as a so-lution of an optimization problem, i. e., to compute the equivalence relation which (under certain constraints) differs the least from the imposed compat-ibility requirement. Such an approach, which is mostly taken from [36] will be presented here. For sake of simplicity, we consider only graphs with edge weights equal to one in this section.
LetG= (V, E) be a graph with adjacency matrixA. IfP ={C1, . . . , Ck} is a partition of V and Ci and Cj two of its classes, then the submatrix of A corresponding to the edges that connect vertices from Ci with vertices from Cj is called ablock and is denoted byA[Ci, Cj] orE(Ci, Cj). If, for instance, P is a structural equivalence for G, then all blocks are either constantly one or constantly zero. If P is a regular equivalence, then all blocks are either constantly zero, or have at least one non-zero entry in each row and each column. The approach from [36] allows more general types of role assignments.
We denote with Πk the set of all vertex partitions that have exactly k classes. An optimization problem is given by
• a set of ideal blocks B and
• a specification of the local error d(E(Ci, Cj), B) which measures the difference between the block E(Ci, Cj) and an ideal blockB ∈ B If the above two points are fixed, then the global error of a partition P =
{C1, . . . , Ck}is defined by
Φ(P) = X
i,j
minB∈Bd(E(Ci, Cj), B) ,
and the problem is to determine a partition P ∈ Πk that minimizes Φ(P).
This model for generalized blockmodeling can be specialized to types of role assignments that have already been introduced in this chapter.
Structural equivalence. For the optimization problem associated with structural equivalence, the set of ideal blocks consists of all blocks (of ar-bitrary dimension) that are either constantly zero or constantly one. The local errord(E(Ci, Cj), B) counts the numbers of ones inE(Ci, Cj) if B is an all-zero block and d(E(Ci, Cj), B) counts the numbers of zeros in E(Ci, Cj) if B is an all-one block. The global error function Φ(P) hence counts for all blocks either the number of ones if the block has more zeros than ones or the number of zeros if the block has more ones than zeros and sums up these counts over all k2 blocks.
Regular equivalence. The ideal blocks of the problem associated with regular equivalence are either constantly zero or have the property that no row is constantly zero (i. e., has at least one entry that is one) and no column is constantly zero. For computing the least local error minB∈Bd(E(Ci, Cj), B) one has to count the number of rows and columns that are constantly zero and the number of ones in the block E(Ci, Cj) and take the smaller number of these two. Since it is N P-complete to decide whether a graph admits a regular equivalence with exactly k equivalence classes (see Theorem 2.3.19) the optimization problem associated with regular equivalence is N P-hard as well.
Other block types. In addition to the block types for structural and regular equivalence, Ferligojet al.[36] propose several other ideal block types.
• Row-dominant and column-dominant. An ideal row-(column-)dominant block contains at least one row (column) that is constantly one. This implies that there is at least one vertex in the row-(column-)class that is connected to all vertices in the column-(row-)class. The local error is given by the minimum number of zeros in a row (column).
• Row-regular and column-regular. An ideal row-(column-)regular block has no row (column) that is constantly zero. The local error is given by the number of rows (columns) that are constantly zero.
• Row-functional and column-functional blocks have in every row (col-umn) exactly one entry that is one. Thus, the associated block defines a function from the row (column) class to the column (row) class. The local error adds up for all rows (columns) the absolute value of the difference between its number of ones and one.
The set of ideal blocks B is not restricted to consist purely of one of the above types. For instance, B could contain the set of structural blocks and the set of row-functional blocks if this seems to be appropriate for the analysis.
Pre-specified blockmodeling. In pre-specified blockmodeling the block type that best fits a given block E(Ci, Cj) can not be chosen from a set but is fixed a priori. Pre-specified blockmodeling is N P-hard since it can be specialized to theN P-hard problem of determining a regular equivalence that yields a pre-specified role graph (compare Theorem 2.3.19).
An optimization algorithm. Ferligoj et al. [36] propose a local opti-mization algorithm to compute an approximative solution for optiopti-mizational blockmodeling.
1. Chose an initial partitionP (e. g., a random partition).
2. If in the neighborhood of P there exists a better partition P0, then repeat Step 2 with P0.
Here the neighborhood of a partition is defined by two transformations: in-terchanging two vertices from different clusters, and moving a vertex from one cluster to another (making sure that the number of clusters stays the same). They propose repeating this procedure several hundred times with different random initial partitions.
Summary for Optimizational Blockmodeling
Several types of equivalence relations (for instance structural and automor-phic equivalence or equitable partitions) suffer from the fact that their re-quirements are often not satisfied in application data. Obviously, the opti-mizational approach overcomes this drawback: a partition that optimizes a certain requirement always exists. A major drawback of this approach is that the solution is computationally intractable in general and the behavior of the proposed local optimization algorithm is not well understood. Furthermore,
an optimal solution may be highly sensitive to small changes in the input data or may even not be unique.
Structural similarities, defined in Sect. 3.2, take a different approach by relaxing the partitions rather than the compatibility requirement.