# cohesive.blocks

0th

Percentile

##### Calculate Cohesive Blocks

Calculates cohesive blocks for objects of class igraph.

Keywords
graphs
##### Usage
cohesive.blocks(graph, labels = TRUE)blocks(blocks)
blockGraphs(blocks, graph)
cohesion(blocks)
hierarchy(blocks)
parent(blocks)
plotHierarchy(blocks,
layout=layout.reingold.tilford(hierarchy(blocks), root=1), ...)
exportPajek(blocks, graph, file, project.file = TRUE)
maxcohesion(blocks)## S3 method for class 'cohesiveBlocks':
print(x, \dots)
## S3 method for class 'cohesiveBlocks':
summary(object, \dots)
## S3 method for class 'cohesiveBlocks':
length(x)
## S3 method for class 'cohesiveBlocks':
plot(x, y, colbar = rainbow(max(cohesion(x))+1),
col = colbar[maxcohesion(x)+1], mark.groups = blocks(x)[-1], ...)
##### Arguments
graph
For cohesive.blocks a graph object of class igraph. It must be undirected and simple. (See is.simple.)

For blockGraphs and exportPajek

labels
Logical scalar, whether to add the vertex labels to the result object. These labels can be then used when reporting and plotting the cohesive blocks.
blocks,x,object
A cohesiveBlocks object, created with the cohesive.blocks function.
file
Defines the file (or connection) the Pajek file is written to.

If the project.file argument is TRUE, then it can be a filename (with extension), a file object, or in general any king of connection object.

project.file
Logical scalar, whether to create a single Pajek project file containing all the data, or to create separated files for each item. See details below.
y
The graph whose cohesive blocks are supplied in the x argument.
colbar
Color bar for the vertex colors. Its length should be at least $m+1$, where $m$ is the maximum cohesion in the graph. Alternatively, the vertex colors can also be directly specified via the col argument.
col
A vector of vertex colors, in any of the usual formats. (Symbolic color names (e.g. red, blue, etc.) , RGB colors (e.g. #FF9900FF), integer numbers referring to the current palette. By d
mark.groups
A list of vertex sets to mark on the plot by circling them. By default all cohesive blocks are marked, except the one corresponding to the all vertices.
layout
The layout of a plot, it is simply passed on to plot.igraph, see the possible formats there. By default the Reingold-Tilford layout generator is used.
...
Additional arguments. plotHierarchy and plot pass them to plot.igraph. print and summary ignore them.
##### Details

Cohesive blocking is a method of determining hierarchical subsets of graph vertices based on their structural cohesion (or vertex connectivity). For a given graph $G$, a subset of its vertices $S\subset V(G)$ is said to be maximally $k$-cohesive if there is no superset of $S$ with vertex connectivity greater than or equal to $k$. Cohesive blocking is a process through which, given a $k$-cohesive set of vertices, maximally $l$-cohesive subsets are recursively identified with $l>k$. Thus a hiearchy of vertex subsets is found, whith the entire graph $G$ at its root. The function cohesive.blocks implements cohesive blocking. It returns a cohesiveBlocks object. cohesiveBlocks should be handled as an opaque class, i.e. its internal structure should not be accessed directly, but through the functions listed here.

The function length can be used on cohesiveBlocks objects and it gives the number of blocks. The function blocks returns the actual blocks stored in the cohesiveBlocks object. They are returned in a list of numeric vectors, each containing vertex ids.

The function blockGraphs is similar, but returns the blocks as (induced) subgraphs of the input graph. The various (graph, vertex and edge) attributes are kept in the subgraph.

The function cohesion returns a numeric vector, the cohesion of the different blocks. The order of the blocks is the same as for the blocks and blockGraphs functions.

The block hierarchy can be queried using the hierarchy function. It returns an igraph graph, its vertex ids are ordered according the order of the blocks in the blocks and blockGraphs, cohesion, etc. functions.

parent gives the parent vertex of each block, in the block hierarchy, for the root vertex it gives 0.

plotHierarchy plots the hierarchy tree of the cohesive blocks on the active graphics device, by calling igraph.plot.

The exportPajek function can be used to export the graph and its cohesive blocks in Pajek format. It can either export a single Pajek project file with all the information, or a set of files, depending on its project.file argument. If project.file is TRUE, then the following information is written to the file (or connection) given in the file argument: (1) the input graph, together with its attributes, see write.graph for details; (2) the hierarchy graph; and (3) one binary partition for each cohesive block. If project.file is FALSE, then the file argument must be a character scalar and it is used as the base name for the generated files. If file is basename, then the following files are created: (1) basename.net for the original graph; (2) basename_hierarchy.net for the hierarchy graph; (3) basename_block_x.net for each cohesive block, where x is the number of the block, starting with one.

maxcohesion returns the maximal cohesion of each vertex, i.e. the cohesion of the most cohesive block of the vertex.

The generic function summary works on cohesiveBlocks objects and it prints a one line summary to the terminal.

The generic function print is also defined on cohesiveBlocks objects and it is invoked automatically if the name of the cohesiveBlocks object is typed in. It produces an output like this: Cohesive block structure: B-1 c 1, n 23 '- B-2 c 2, n 14 oooooooo.. .o......oo ooo '- B-4 c 5, n 7 ooooooo... .......... ... '- B-3 c 2, n 10 ......o.oo o.oooooo.. ... '- B-5 c 3, n 4 ......o.oo o......... ... The left part shows the block structure, in this case for five blocks. The first block always corresponds to the whole graph, even if its cohesion is zero. Then cohesion of the block and the number of vertices in the block are shown. The last part is only printed if the display is wide enough and shows the vertices in the blocks, ordered by vertex ids. o means that the vertex is included, a dot means that it is not, and the vertices are shown in groups of ten. The generic function plot plots the graph, showing one or more cohesive blocks in it.

##### Value

• cohesive.blocks returns a cohesiveBlocks object.

blocks returns a list of numeric vectors, containing vertex ids.

blockGraphs returns a list of igraph graphs, corresponding to the cohesive blocks.

cohesion returns a numeric vector, the cohesion of each block.

hierarchy returns an igraph graph, the representation of the cohesive block hierarchy.

parent returns a numeric vector giving the parent block of each cohesive block, in the block hierarchy. The block at the root of the hierarchy has no parent and 0 is returned for it.

plotHierarchy, plot and exportPajek return NULL, invisibly.

maxcohesion returns a numeric vector with one entry for each vertex, giving the cohesion of its most cohesive block.

print and summary return the cohesiveBlocks object itself, invisibly.

length returns a numeric scalar, the number of blocks.

##### concept

Structurally cohesive blocks

##### References

J. Moody and D. R. White. Structural cohesion and embeddedness: A hierarchical concept of social groups. American Sociological Review, 68(1):103--127, Feb 2003.

graph.cohesion

##### Aliases
• cohesive.blocks
• cohesiveBlocks
• blocks
• blockGraphs
• cohesion
• hierarchy
• parent
• plotHierarchy
• exportPajek
• maxcohesion
• plot.cohesiveBlocks
• summary.cohesiveBlocks
• length.cohesiveBlocks
• print.cohesiveBlocks
##### Examples
## The graph from the Moody-White paper
mw <- graph.formula(1-2:3:4:5:6, 2-3:4:5:7, 3-4:6:7, 4-5:6:7,
5-6:7:21, 6-7, 7-8:11:14:19, 8-9:11:14, 9-10,
10-12:13, 11-12:14, 12-16, 13-16, 14-15, 15-16,
17-18:19:20, 18-20:21, 19-20:22:23, 20-21,
21-22:23, 22-23)

mwBlocks <- cohesive.blocks(mw)

# Inspect block membership and cohesion
mwBlocks
blocks(mwBlocks)
cohesion(mwBlocks)

# Save results in a Pajek file
exportPajek(mwBlocks, mw, file="/tmp/mwBlocks.paj")

# Plot the results
if (interactive()) {
plot(mwBlocks, mw)
}

## The science camp network
camp <- graph.formula(Harry:Steve:Don:Bert - Harry:Steve:Don:Bert,
Pam:Brazey:Carol:Pat - Pam:Brazey:Carol:Pat,
Holly   - Carol:Pat:Pam:Jennie:Bill,
Bill    - Pauline:Michael:Lee:Holly,
Pauline - Bill:Jennie:Ann,
Jennie  - Holly:Michael:Lee:Ann:Pauline,
Michael - Bill:Jennie:Ann:Lee:John,
Ann     - Michael:Jennie:Pauline,
Lee     - Michael:Bill:Jennie,
Gery    - Pat:Steve:Russ:John,
Russ    - Steve:Bert:Gery:John,
John    - Gery:Russ:Michael)
campBlocks <- cohesive.blocks(camp)
campBlocks

if (interactive()) {
plot(campBlocks, camp, vertex.label=V(camp)\$name, margin=-0.2,
vertex.shape="rectangle", vertex.size=24, vertex.size2=8,
mark.border=1, colbar=c(NA, NA,"cyan","orange") )
}
Documentation reproduced from package igraph, version 0.6.5-2, License: GPL (>= 2)

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