* Transitivity and Global clustering coefficient
- transitive relation: a~b and b~c => a~c
- network transitivity: vertex u is connected to v, and v is connected to
w, then u is connected to w
- perfect transitivity in a network would imply that the network is a
collection of isolated cliques
- many real networks show partial transitivity: if u and v are connected,
and v and w are connected, then the probability that u and w are connected
is high
- quantification: the fraction of paths of length two that are closed
C = (# closed paths of length two) / (# paths of length two)
(Also called as the global clustering coefficient)
- paths from u to v and from v to u are counted differently
- C can be written in terms of number of triangles
C = (# triangles * 6) / (# paths of length two)
- a connected triple is defined as a set of three nodes and two edges
C = (# triangles * 3) / (# connected triples )
- Most social networks have high clustering coefficients (i.e. much higher
than c/n where c is the average degree)
- clustering coefficients for directed networks are less used
- global clustering coefficient and the average clustering coefficient
in general have different values
* Vertex similarity/equivalence
- Structural equivalence
- depends on the number of common neighbors
- cosine similarity : rows corresponding to the two vertices are
treated as vectors, and the dot product is taken
- Jaccard coefficient : (# common neighbors) / (# distinct neighbors)
- Regular equivalence
- vertices are similar if they have neighbors which themselves are
similar
- as such, doesn't assign high self-similarity
- modification leads to paths of even length only
- Katz similarity : take paths of all lengths!
- Katz centrality is then the sum of the Katz similarity of a vertex
with all the vertices in the network