Dividing an organization using clustering of weighted graphs
Modularity versus mincut
Centrality versus modularity and mincut
An example graph
Consider the problem of dividing a graph into two parts. The links
in the graph have a weight. When the graph is split some links will
be cut. We want to minimize the sum of the weights of the links that are cut.
This is called the minimal cut of the graph.
The graph could model an organization. The nodes could be
actors in the organization and the weights could represent the importance
of their relations. When a reorganization requires a division,
the minimal cut could be the best way to allocate the
actors to the two new parts of the organization.
The tree of minimal cuts
The so called min-cut algorithm of Gomory-Hu  finds the minimal cut
between all pairs of nodes in a graph. The result of this algoritm is a tree
with the same nodes as the graph, but where each link represents a cut in the
original graph. The Gomory-Hu tree of the example graph is shown in the figure on the left.
Example: When we remove the (red) link between A5 and A8 from this tree, two subtrees remain:
A4,A5,A6,A7 on the right side and A8,A9,A1,A2,A3 on the left side.
The weight of the link we just removed is 4. This happens to be precisely
the sum of the weights of the links that are cut by this partitioning in the original graph.
It is the minimal cut between the nodes A5 and A8.
The best split
This is the minimal cut we just described. The sum of the weights
of the cut links is 4. In the Gomory Hu tree it is the (red) link between the nodes
A5 and A8
Newman  defined a quantity called modularity. It ranges from 0 (bad) to 1 (best)
and is a measure for the internal coherence of a given clustering of nodes in a graph.
In a random graph the modularity is close to zero for all clusterings.
In a graph with no links between clusters (only links within clusters)
the modularity approaches 1.
A search algorithm can look for the highest modularity over all possible clusterings.
For 2 clusters this algorithm happens to produce the same result as the min-cut algorithm
in the case of the current example graph.
For 3 clusters the optimal clustering is shown in the figure on the left.
The modularity is 0.4
 R. E. Gomory, T. C. Hu
Multi-Terminal Network Flows
Journal of the Society for Industrial and Applied Mathematics, Vol. 9, No. 4 (Dec., 1961), pp. 551-570
 M. E. J. Newman
Fast algorithm for detecting community structure in networks
Phys. Rev. E 69, 066133 (2004)