# Connected Component Analysis on an Undirected Graph (Scripts) 1.0

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## Connected Component Analysis on an Undirected Graph (Scripts) Publisher's description

### Connected component analysis on undirected graphs, with thresholding and connectivity constraints.

Connected component analysis on undirected graphs, with thresholding and connectivity constraints.
Tristan Ursell, (c) 2012
Connected component analysis on an undirected graph, with various
thresholding and connectivity constraints.

[groups,orphans] = graph_analysis(W);
[groups,orphans] = graph_analysis(W,'field',value,...);
[groups,~] = graph_analysis(W,'field',value,...);

W is the N x N adjaceny matrix for a symmetric graph. Thus W should be a symmetric matrix, if it is not, this function will give an error. Self connections (i.e. diagonal elements) are not allowed and will be removed automatically. The values of the parameters below are applied as a union set, that is, all original elements must meet all of the conditions specified by the parameters to be included in a group.

If only W is given, then all components with W>0 will be analyzed and grouped, with the default constraints.

'min_conn' (1 <= min_conn <= max_conn) specifies the minimum degree of connectivity (not including itself) for any element in W to be included in a group. The default 'min_conn' value is 1.

'max_conn' (min_conn <= max_conn <= N) specifies the maximum degree of connectivity for any element in W to be included in a group. The default 'max_conn' value is N.

'min_like' is the minimum likelihood value for an element in W to be included in any group. The default value is 0. The 'likelihood' is not a probability, and hence is not bounded between zero and one, but the ratio of likelihoods should be equal to the ratio of probabilities of two states.

'max_rank' (1 <= max_rank <= N) is the number of highest likelihood values to use in forming connected groups. For instance, if max_rank = 3, and a row of W is:

1 3 6 4 4 5 6 8 0 3 1

then the matrix row will become:

0 0 1 0 0 0 1 1 0 0 0

If a row of W is:

1 3 6 4 4 6 6 6 0 3 1

with max_rank = 1,2,3 or 4, the row becomes:

0 0 1 0 0 1 1 1 0 0 0

with max_rank = 5, the same row becomes:

0 0 1 1 1 1 1 1 0 0 0

The default value of 'max_rank' is N.

'plot' with value 1 will generate plots of the grouping algorithm as it creates block diagonal groups in from top left to bottom right in W.

The output 'groups' is a structure array with fields:

groups(i).num_els = number of elements in group i.
groups(i).block = sub-block identity of group i.
groups(i).elements = elements of W that are in group i.
groups(i).degrees = degrees of connection for each element in group i.
orphans = elements of W that were not in any group.

The number of distinct groups is length(groups).

Example with a block diagonal random matrix W:

W=blkdiag(rand(50,50),rand(100,100),rand(200,200),rand(300,300));
W=(W+W')/2;

figure;
imagesc(W)
axis equal tight
xlabel('Elements of W')
ylabel('Elements of W')

%more inclusive connections, less inclusive probability
[groups,orphans]=graph_analysis(W,'min_like',0.95,'min_conn',3,'plot',1);

%less inclusive connections, more inclusive probability
[groups,orphans]=graph_analysis(W,'min_like',0.9,'min_conn',6,'plot',1);

#### System Requirements:

No special requirements.
Program Release Status: New Release
Program Install Support: Install and Uninstall

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