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NETWORK > ROLES > EXACT > OPTIMIZATION

PURPOSE Optimizes a cost function that gives an approximate measure of the degree to which a partition corresponds to automorphically equivalent sets using a tabu search.

DESCRIPTION Two vertices u and v of a labeled graph G are automorphically equivalent if all the vertices can be relabeled to form an isomorphic graph with the labels of u and v interchanged. Given a partition of the network then the partition divides the adjacency matrix into blocks. For an automorphic partition the cell values for a row or column within a block will have the same distribution of values. An approximate measure of the extent to which these blocks conform to automorphic equivalence is given by the following procedure. For each block calculate the variance of the sum of squares of each row and the variance of the sum of squares of each column. The approximate automorphic cost is the sum of all these variances for every block. The routine attempts to optimize this cost function to try and find the best partition of the vertices into a specified number of blocks.

PARAMETERS

Input dataset:
Name of file containing network to be analyzed. Data type: Valued graph.

Number of blocks (Default = 2).
Number of groups or blocks into which the vertices are to be assigned.

Are diagonal values valid (Default = NO).
Whether diagonals are to be included in cost function.

For binary data:  convert to geodesic distances (Default = NO).
No performs an analysis on raw adjacency matrix.
Yes converts the adjacencies to distances and uses this as the input data. If there is no path connecting two vertices then the distance of n is used, where n is the number of vertices in the network.

Maximum # of iterations in a series (Default = max(2,n/3)).
The algorithm starts from an arbitrary partition and attempts to decrease the cost by taking the steepest descent. If the cost cannot be reduced then the algorithm continues its search in the neighborhood of the current partition. This search direction is a mildest ascent direction and from there new search directions are explored. This exploration only continues for a fixed number of iterations in a series. If no improvement is made after the fixed number of iterations the algorithm terminates with the current minimum. Increasing the parameter gives a more exhaustive and therefore slower search. The recommended default value is automatically entered on the form once the input data has been selected.

Length of time in penalty box (Default = 10).
If the algorithm makes an ascending step then it is possible that the best possible descending step is the reverse of the direction just taken. This parameter prohibits a move along the reverse direction for a set number of steps. The larger the value the more difficult it will be to come back to a previously explored local minimum, however it will also be more difficult to explore the vicinity of that minimum. The default of 10 has been shown experimentally to be the most useful.

Number of random starts (Default = 10 - 2logn).
The whole procedure is repeated with a different initial partition. The best of these are then selected as a minimum.

Random Number Seed
The random number seed generates the initial partition. UCINET generates a different random number as default each time it is run. This number should be changed if the user wishes to repeat the analysis with different initial configurations. The range is 1 to 32000.

Output Partition Dataset (Default = 'ABMPart').
Name of output file to contain a partition indicator vector. This vector has the form (k1,k2,...ki...) where ki assigns vertex i to block ki, so that (1 1 2 1 2) assigns vertices 1, 2 and 4 to block 1 and 3 and 5 to block 2. This vector is not displayed in the LOG FILE.

Output Indicator Dataset: (Default = 'ABMSets').
Name of file which contains a block by actor incidence matrix.  A 1 in row i column j indicates that actor j is a member of block i.  This matrix is not displayed in the LOG FILE.

LOG FILE The value of the cost function or Fit.

List of blocks. Each block is labeled and is specified by the vertices it contains.

The blocked adjacency matrix. The rows and columns of the original adjacency matrix are permuted into blocks. The adjacency matrix is displayed in terms of the matrix blocks it contains.

TIMING Each iteration of the tabu search algorithm is O(N^2).

COMMENTS Care should be taken when using this routine.

The algorithm seeks to find the minima of the cost function. Even if successful this result may still have a high value in which case the blocking may not conform very closely to automorphic equivalence. In addition there may be a number of alternative partitions that also produce the minimum value;  the algorithm does not search for additional solutions. Finally it is possible that the routine terminates at a local minima and does not locate the desired global minima.

To test the robustness of the solution the algorithm should be run a number of times from different starting configurations. If there is good agreement between these results then this is a sign that there is a clear split of the data into the reported blocks.

REFERENCES Glover F (1989). Tabu Search - Part I. ORSA Journal on Computing 1, 190-206.

Glover F (1990). Tabu Search - Part II. ORSA Journal on Computing 2, 4-32.