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BottleNeck CentralityDefinition
For each node v in the undirected graph, construct a tree T_{v} of shortest paths from that node to
all other nodes in the graph. For a node v, n_{v} is the number of nodes that are directly or
indirectly connected to node v (i.e. the tree T_{v} contains n_{v}
nodes). So extract all nodes w on the above defined tree T_{v}
of shortest paths from node v, such that more than nv/4 paths
from v to other nodes in the tree meet at node w. Nodes w
extracted in this way represent ‘bottle necks’ of the shortest
path tree T_{v} rooted at node v, since at least n_{v}/4 paths of the
n_{v}node tree T_{v} ‘meet’ at w. For every node v of the graph,
construct these shortest path trees T_{v} rooted at v, and
extracted their ‘bottle neck’ nodes. Note that the same node
may be a ‘bottle neck’ of different shortest path trees. So
counted in how many shortest path trees each of the extracted
‘bottle neck’ nodes appeared [PRŽULJ, N., 2004].
P_{s}(v) = 1 if more than V(T_{s})/4 paths from node s to other nodes in T_{s} meet at the vertex v, otherwise P_{s}(v) = 0. For each node v in an interaction network, a tree of shortest paths starting from v is constructed. Taking v as the root of the tree T_{v}, the weight of a node w in the tree T_{v} is the number of descendants of w, that is to say, equal to the number of shortest paths starting from v passing through w. A node w is called a bottleneck node in T_{v} if the weight of w is no less than n/4, where n is the number of nodes in T_{v}. The score of node w, BN(v), is defined to be the number of node v such that w is a bottleneck node in T_{v} [LIN, C.Y. 2008]. References
