Quantification of networks using local quantifiers
For very small networks, sometimes visualizing it to draw conclusions works.
However, raw network data are difficult to interpret when a network is large.
Thus, we want to define quantitative measures to make sense of huge data.
* Degree distribution
- p_k : fraction of vertices with degree k
- alternatively, it is the probability that a randomly chosen vertex has
degree k
- degree distribution doesn't determine the whole structure
- hubs and right-skewed degree distributions
- for directed networks, there are two degree distributions: in and out
- even better, define p_{jk} to be the probability that a node has
in-degree j and out-degree k
* Local and average clustering coefficients
- triangles exist in many real networks
- C_i : probability that two neighbors of i are connected
- in many real networks C_i decreases with k_i
- structural holes and the influence of a vertex
- average clustering coefficient = sum(C_i) / n
* Centralities
- how important is a vertex?
- influential spreaders
- influential papers
- important websites
- degree
- more neighbors => higher importance
- eigenvector
- not all neighbors are equivalent
- importance of a vertex is higher if it's neighbours are themselves
important
- awards a number of points proportional to the centrality scores of
the neighbors
- proportional to the leading eigenvector
- problems with directed graphs:
- two types of eigenvectors
- could be zero even when many vertices point to a vertex