sna 3a: centrality
DESCRIPTION
SNA 3A: Centrality. Lada Adamic. is counting the edges enough?. Stanford Social Web (ca. 1999). network of personal homepages at Stanford. different notions of centrality. In each of the following networks, X has higher centrality than Y according to a particular measure. indegree. - PowerPoint PPT PresentationTRANSCRIPT
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SNA 3A: Centrality
Lada Adamic
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is counting the edges enough?
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Stanford Social Web (ca. 1999)
network of personal homepages at Stanford
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indegree
In each of the following networks, X has higher centrality than Y according to a particular measure
outdegree betweenness closeness
different notions of centrality
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review: indegree
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trade in petroleum and petroleum products, 1998, source: NBER-United Nations Trade Data
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Quiz Q:
Which countries have high indegree (import petroleum and petroleum products from many others) Saudi Arabia Japan Iraq USA Venezuela
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review: outdegree
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trade in petroleum and petroleum products, 1998, source: NBER-United Nations Trade Data
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Quiz Q:
Which country has low outdegree but exports a significant quantity (thickness of the edges represents $$ value of export) of petroleum products Saudi Arabia Japan Iraq USA Venezuela
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trade in crude petroleum and petroleum products, 1998, source: NBER-United Nations Trade Data
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Undirected degree, e.g. nodes with more friends are more central.
Assumption: the connections that your friend has don't matter, it is what they can do directly that does (e.g. go have a beer with you, help you build a deck...)
putting numbers to it
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divide degree by the max. possible, i.e. (N-1)
normalization
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Freeman’s general formula for centralization (can use other metrics, e.g. gini coefficient or standard deviation):
How much variation is there in the centrality scores among the nodes?
maximum value in the network
centralization: skew in distribution
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CD = 0.167
CD = 0.167
CD = 1.0
degree centralization examples
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example financial trading networks
high in-centralization: one node buying from many others
low in-centralization: buying is more evenly distributed
real-world examples
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In what ways does degree fail to capture centrality in the following graphs?
what does degree not capture?
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Stanford Social Web (ca. 1999)
network of personal homepages at Stanford
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Brokerage not captured by degree
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constraint
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constraint
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intuition: how many pairs of individuals would have to go through you in order to reach one another in the minimum number of hops?
betweenness: capturing brokerage
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Where gjk = the number of shortest paths connecting jk gjk(i) = the number that actor i is on.
Usually normalized by:
number of pairs of vertices excluding the vertex itself
betweenness: definition
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betweenness on toy networks non-normalized version:
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betweenness on toy networks non-normalized version:
A B C ED
A lies between no two other vertices B lies between A and 3 other vertices: C, D, and E C lies between 4 pairs of vertices (A,D),(A,E),(B,D),(B,E)
note that there are no alternate paths for these pairs to take, so C gets full credit
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betweenness on toy networks non-normalized version:
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betweenness on toy networks non-normalized version:
A B
C
E
D
why do C and D each have betweenness 1?
They are both on shortest paths for pairs (A,E), and (B,E), and so must share credit: ½+½ = 1
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Quiz Question What is the betweenness of node E?
E
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Lada’s old Facebook network: nodes are sized by degree, and colored by betweenness.
betweenness: example
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Quiz Q: Find a node that has high betweenness but
low degree
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Quiz Q: Find a node that has low betweenness but
high degree
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closeness
What if it’s not so important to have many direct friends?
Or be “between” others
But one still wants to be in the “middle” of things, not too far from the center
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need not be in a brokerage position
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Closeness is based on the length of the average shortest path between a node and all other nodes in the network
Closeness Centrality:
Normalized Closeness Centrality
closeness: definition
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A B C ED
closeness: toy example
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closeness: more toy examples
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Quiz Q:
Which node has relatively high degree but low closeness?