Tutorial 9: Paths & Ego Networks
Beyond a single shortest path between two named nodes (covered in
Visualization via nwpath), this tutorial covers three ways of
characterizing a network’s overall path structure and each node’s own local neighborhood.
Geodesics
nwgeodesic computes the shortest-path distance between every pair of nodes at once, saving
the result as a new network:
. nwwebuse florentine, nwclear
. nwgeodesic flomarriage, name(flodist) generate(ecc)
----------------------------------------
Network name: flomarriage
Network of shortest paths: flodist
----------------------------------------
Nodes: 16
Symmetrized : (already undirected)
------------------------------------
Paths: 120
Unconnected paths: 15
Average shortest path length: (not defined)
Diameter: (not defined)
Radius: (not defined)
Average shortest path length, diameter, and radius all come back undefined here because
flomarriage has an isolated node (pucci, met already in
Manipulating Networks) with no path to anyone. nwsummarize on the
new distance network shows every pairwise distance directly:
. nwsummarize flodist, mat
--------------------------------------------------
Network name: flodist
Network id: 3
Directed: false
Valued: false
Two-mode: false
Nodes: 16
Selfloop: false
Edges: 105
Minimum value: 1
Maximum value: 5
Density: 2.49
Temporal: false
[symmetric]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
+---------------------------------------------------------------------------------+
1 | . |
2 | 2 . |
3 | 2 2 . |
4 | 4 2 3 . |
5 | 3 3 1 2 . |
6 | 3 1 3 3 4 . |
7 | 3 1 3 1 3 2 . |
8 | 4 2 4 2 4 3 1 . |
9 | 1 1 1 3 2 2 2 3 . |
10 | 3 3 3 5 4 4 4 5 2 . |
11 | 4 3 2 1 1 4 2 3 3 5 . |
12 | . . . . . . . . . . . . |
13 | 2 2 2 2 2 3 2 3 1 3 2 . . |
14 | 2 2 2 4 3 3 3 4 1 1 4 . 2 . |
15 | 3 3 2 1 1 4 2 3 2 4 1 . 1 3 . |
16 | 2 2 2 2 3 3 1 2 1 3 3 . 1 2 2 . |
+---------------------------------------------------------------------------------+
Row/column 12 (pucci) is entirely missing, confirming it’s unreachable from everyone else.
generate() (optional here — nwgeodesic’s main output is the distance network itself) adds each
node’s eccentricity: the length of its own single longest shortest path.
. gsort -ecc _nwnode
. list _nwnode ecc in 1/5
+--------------------+
| _nwnode ecc |
|--------------------|
1. | bischeri 5 |
2. | lamberteschi 5 |
3. | pazzi 5 |
4. | peruzzi 5 |
5. | acciaiuoli 4 |
+--------------------+
On a valued network nwgeodesic and nwpath use the tie values as distances by default; the
option binary gives the number of steps instead (see valued networks).
Which network types each command accepts is tabulated on the page
network types.
Reachability
Sometimes the exact distance doesn’t matter — only whether a path exists at all. nwreach
collapses the geodesic calculation down to a plain 0/1 network:
. nwreach flomarriage, sym name(floreach)
. nwsummarize floreach, mat
--------------------------------------------------
Network name: floreach
Network id: 4
Directed: false
Valued: false
Two-mode: false
Nodes: 16
Selfloop: false
Edges: 105
Minimum value: 0
Maximum value: 1
Density: .875
Temporal: false
[symmetric]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
+---------------------------------------------------------------------------------+
1 | . |
2 | 1 . |
3 | 1 1 . |
4 | 1 1 1 . |
5 | 1 1 1 1 . |
6 | 1 1 1 1 1 . |
7 | 1 1 1 1 1 1 . |
8 | 1 1 1 1 1 1 1 . |
9 | 1 1 1 1 1 1 1 1 . |
10 | 1 1 1 1 1 1 1 1 1 . |
11 | 1 1 1 1 1 1 1 1 1 1 . |
12 | 0 0 0 0 0 0 0 0 0 0 0 . |
13 | 1 1 1 1 1 1 1 1 1 1 1 0 . |
14 | 1 1 1 1 1 1 1 1 1 1 1 0 1 . |
15 | 1 1 1 1 1 1 1 1 1 1 1 0 1 1 . |
16 | 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 . |
+---------------------------------------------------------------------------------+
Node 12 (pucci) is the one row/column of all zeros — the reachability network makes an isolated
node visually obvious even in a network too large to eyeball directly for connectivity.
Ego networks
nwego looks at each node’s own local neighborhood in isolation: how many alters does it have
(sizevar()), and how interconnected are those alters with each other, independent of their
(by definition, universal) ties back to the node itself (densvar()). nwego matches its results
to the nodes by name, so it does not matter that the gsort above changed the order of the
observations:
. nwego flomarriage, sizevar(egosize) densvar(egodensity)
----------------------------------------
Network name: flomarriage
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
egosize | 16 2.5 1.505545 0 6
egodensity | 11 .2181818 .2192734 0 .6666667
. gsort -egosize _nwnode
. list _nwnode egosize egodensity in 1/8
+---------------------------------+
| _nwnode egosize egoden~y |
|---------------------------------|
1. | medici 6 .0666667 |
2. | guadagni 4 0 |
3. | strozzi 4 .3333333 |
4. | albizzi 3 0 |
5. | bischeri 3 .3333333 |
|---------------------------------|
6. | castellani 3 .3333333 |
7. | peruzzi 3 .6666667 |
8. | ridolfi 3 .3333333 |
+---------------------------------+
medici has by far the largest ego network (6 alters) but one of the lowest densities among
them (0.067 — its alters barely know each other) — the same brokerage pattern already visible in
Centrality’s betweenness results, now visible directly as a size/density
trade-off. peruzzi, by contrast, has a smaller ego network (3 alters) that’s densely
interconnected (0.667) — its contacts mostly already know each other, leaving less room for
peruzzi itself to broker between them.