Centrality

“How central is this node?” has more than one answer — each of nwcommands’ six centrality commands captures a different notion of importance. This tutorial runs all six on the same network, the Florentine marriage network, so the different rankings can be compared directly.

Degree

nwdegree counts a node’s direct ties:

. nwwebuse florentine, nwclear

. nwdegree flomarriage, generate(deg)
----------------------------------------
  Network name: flomarriage
----------------------------------------
    Degree distribution

        deg |      Freq.     Percent        Cum.
------------+-----------------------------------
          0 |          1        6.25        6.25
          1 |          4       25.00       31.25
          2 |          2       12.50       43.75
          3 |          6       37.50       81.25
          4 |          2       12.50       93.75
          6 |          1        6.25      100.00
------------+-----------------------------------
      Total |         16      100.00

   Degree centralization:: .267

Betweenness

nwbetween counts how often a node sits on the shortest path between two other nodes — a measure of “brokerage” rather than direct connectivity:

. nwbetween flomarriage, generate(btw)
----------------------------------------
  Network name: flomarriage
----------------------------------------
    Betweenness centrality

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
         btw |         16        9.75    12.30056          0       47.5

Closeness

nwcloseness measures how short a node’s paths are to everyone else in the network:

. nwcloseness flomarriage, generate(clo far near)
----------------------------------------
  Network name: flomarriage
----------------------------------------
    Closeness centrality

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
         clo |         15    .4447452    .0801896   .3061225         .6
         far |         16      32.625    10.73235          0         49
        near |         15    .0296497     .005346   .0204082        .04

clo has one fewer observation than far - the isolated node (pucci, with degree 0 above) has no path to anyone, so closeness is undefined for it even though farness (. for an unreachable pair contributes nothing) still computes.

Eigenvector

nwevcent scores a node higher when its neighbors are themselves well-connected, not just by raw neighbor count:

. nwevcent flomarriage, generate(evc)
----------------------------------------
  Network name: flomarriage
----------------------------------------
    Eigenvector centrality
.   (calculated on symmetrized network)

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
         evc |         16    .2189135    .1246942   1.90e-34   .4303081

PageRank

nwpagerank is the stationary distribution of a random surfer who follows outgoing ties (with probability damping(), default 0.85) or jumps to a uniformly random node otherwise. Unlike eigenvector centrality, it works natively on directed networks and always produces a unique, well-defined score even where eigenvector centrality would refuse:

. nwpagerank flomarriage, generate(pgr)
(16 missing values generated)
----------------------------------------
  Network: flomarriage
  Damping: .85
----------------------------------------

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
         pgr |         16       .0625    .0322572    .009901   .1443735

Katz

nwkatz extends degree centrality by also counting nodes reachable at distance 2, 3, … away, penalizing each by how far away it is:

. nwkatz flomarriage, generate(ktz)
----------------------------------------
  Katz centrality (shortest-path distance-decay)
  Network name: flomarriage
       Network has been symmetrized for calculation.
----------------------------------------
       Alpha: 1

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
         ktz |         16      13.125         3.5          0         14

The measures don’t always agree

Sort by betweenness, and medici comes out clearly on top - well ahead of the next node, and by a wider margin than its degree alone would suggest:

. gsort -btw

. list _nwnode deg btw in 1/5

     +---------------------------+
     |  _nwnode   deg        btw |
     |---------------------------|
  1. |   medici     6       47.5 |
  2. | guadagni     4   23.16667 |
  3. |  albizzi     3   19.33333 |
  4. | salviati     2         13 |
  5. |  ridolfi     3   10.33333 |
     +---------------------------+

medici does have the highest degree here too, but the gap to second place is far larger for betweenness (47.5 vs. 23.2, roughly double) than for degree (6 vs. 4) - consistent with the historical account of the Medici as brokers sitting between otherwise-separate factions, not merely the family with the most marriage ties. Looking at every measure for medici at once:

. format clo evc pgr ktz %6.3f

. list _nwnode deg btw clo evc pgr ktz if _nwnode == "medici"

     +-------------------------------------------------------+
     | _nwnode   deg    btw     clo     evc     pgr      ktz |
     |-------------------------------------------------------|
  1. |  medici     6   47.5   0.600   0.430   0.144   14.000 |
     +-------------------------------------------------------+

medici tops every measure here - but that won’t always be true for every node in every network. Which measure is “right” depends entirely on what kind of influence you’re actually asking about: direct access (degree), control over information flow between others (betweenness), efficiency of reach (closeness), prestige-by-association (eigenvector, PageRank), or penalized total reach (Katz).