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NETWORK > SUBGROUPS > N-CLIQUES

PURPOSE Find all n-cliques in a network.

DESCRIPTION An n-clique of an undirected graph is a maximal subgraph in which every pair of vertices is connected by a path of length n or less. These are found using an adapted version of the Bron and Kerbosch (1973) algorithm.  The routine will also provide an analysis of the overlapping structure of the n-cliques.  This analysis gives information on the number of times each pair of actors are in the same n-clique and gives an hierarchical clustering based upon this information.

PARAMETERS
Input dataset:
Name of file containing network to be analyzed. Data type: Graph.

Value of N: (Default = 2)
All members of an n-clique are connected by a path of length n or less.  A value of 1 would give all Luce and Perry cliques;  the maximum value of N-1 would give the components of the graph.

Minimum Size: (Default = 3)
This gives the smallest group size which is to be considered an n-clique. The range is 1 to N.

Analyze pattern of overlaps?  (Default = YES).
Yes means that an analysis of n-clique overlap will be performed. This includes the construction of an n-clique co-membership matrix, and an hierarchical clustering which is saved in a partition indicator matrix as described below.
No restricts the analysis to identifying n-cliques only.
 
Diagram Type: (Default = 'Tree diagram')
When analyzing the overlap the clustering diagram can either be a Tree Diagram or a Dendrogram.

(Output) n-clique indicator matrix: (Default = 'NClqSets').
Name of file which contains a n-clique by actor incidence matrix.  A 1 in column i row j indicates that actor j is a member of n-clique i.  This matrix is not displayed in the LOG FILE.

(Output) Co-membership matrix: (Default = 'NClqOver').
Name of file which contains n-clique overlap matrix described in LOG FILE below.  Note that if no analysis of pattern overlaps was chosen then this file is not created.

(Output) Partition indicator matrix: (Default = 'NClqPart').
Name of file which contains partition indicator matrix derived from overlap analysis.  The partition indicator matrix corresponds to the hierarchical clustering displayed in the LOG FILE. A value of k in a column labeled i and row j means that actor j is in partition k and is in i n-cliques with every other member of partition k.  Actor k is always a member of partition k, and is a representative label for the group.


LOG FILE Number of n-cliques found.
List of n-cliques, labeled - each n-clique is specified by the vertices it contains.

The following output is also produced if YES was inserted on the form in reply to the question 'Analyze pattern of overlaps?' The first part of the output will be the tree diagram or dendrogram corresponding to the single link clustering of the n-clique overlap matrix. In the n-clique overlap matrix  a value of k in row i column j means that vertices i and j occurred in the same n-clique k times.  The ith diagonal entry gives the number of n-cliques which contain i.

The tree diagram (or a dendrogram) re-orders the actors so that they are located close to other actors in similar clusters. The level at which any pair of actors are aggregated is the point at which both can be reached by tracing from the start to the actors from right to left. The scale at the top gives the level at which they are clustered and corresponds to the number of overlaps. The diagram can be printed or saved. Parts of the diagram can be viewed by moving the mouse to the split point in a tree diagram or the beginning of a line in the dendrogram and clicking. The first click will highlight a portion of the diagram and the second click will display just the highlighted portion. To return to the original right click on the mouse. There is also a simple zoom facility simply change the values and then press enter. If the labels need to be edited (particularly the scale labels) then you should take the partition indicator matrix into the spreadsheet editor remove or reduce the labels and then submit the edited data to Tools>Dendrogram>Draw. 

   
Behind the diagram is a window containing the number of n-cliques and a list as specified above. This is followed by a clustering diagram representing the same clustering as the tree diagram (or dendrogram).  The columns are rearranged and labeled.  A '·' in row label i column label j means that vertex j was not in i n-cliques with any other vertex.  An 'X' indicates that vertex j was in i n-cliques with all vertices on the same row as j which can be found by tracing across that row without encountering a space.

TIMING Algorithm is exponential.

COMMENTS Usually only 2-n-cliques or 3-n-cliques are of significance.

REFERENCES Luce R (1950).  Connectivity and generalized n-cliques in sociometric group structure.  Psychometrika 15, 169-190.

Bron C and Kerbosch J (1973).  Finding all n-cliques of an undirected graph.  Comm of the ACM 16, 575-577.