Notes on Pyramids
The ownership network is defined by dyadic, asymmetric ties among firms, where
Own(a,b) is a tie between firms a and b, where b is owned by a. As such, we consider the ownership network in graph theory terms as an oriented, one-mode graph without cycles. This means that there’s only one type of entity (we don’t distinguish between firms and owners), and we don’t expect firms to be owned by firms which are owned by them.
In the abovementioned network, a node which has no parent is considered a top node or Pyramid root . In Figure A we see that nodes B, A and C are top nodes.
A Pyramid is defined by a top node T (there are as many pyramids in a network as root nodes), and all reachable nodes from it, also called descendants. In graph theory, this is known as a rooted tree . [Reference, I think Knuth]
Fig.A, a one mode, directed network
Fig.B, the network decomposed into pyramids.
From the example above, we extract three pyramids of roots A, B and C.
We start by defining the size of a pyramid p as the number of nodes contained in it. We also define the depth (alternatively known as height ) of a pyramid as the length of the path between the pyramid root and the furthest node in the pyramid. The set of nodes at a given depth is known as the level . Root nodes are considered to be at level 0.
As such, pyramid A is of size 3 and depth 2, pyramid B is of size 6 and depth 2, while pyramid C is of size 7 and depth 3.
The closer a node sits to the root, the more relevant it will be in the pyramidal structure. As such, we define level weight of level l as lw ( l )
 d
 d l

1
, where d is the pyramid depth.
We can further normalize this value to obtain a measure in the [0, 1] range. Thus the normalized level weight is lw norm
( l )
 d  i

1 d d

 l l

1 i

1
With these definitions we can now calculate the relevance of a specific node within a pyramid. We call this measure a node’s weight, and we defined it as the level’s normalized weight over the number of siblings or nodes at the same level . We define it as w ( k )
 lw norm
( l k
) siblings k
, where l is node’s k level. k

We measure a pyramid’s independence by summing the node’s weight of all nodes in a pyramid which only belong to the given pyramid.
I ( p )
 n
 p k


1

!
k
 p : w ( k )
Numeric example:
Let’s calculate these measures for the pyramid of root C. We see that nodes I, J and K belong only to this pyramid.
This pyramid ’s depth is 3, and thus the level and normalized level weights will be:
Level Level
1
2
3
Normalized Nodes at
Weight
(3 – 1 + 1) / 3 = 1
Level Weight
(3 – 1 + 1) / 6 = .5 level l
3
(3 – 2 + 1) / 3 = .66 (3 – 2 + 1) / 6 = .33 2
(3 – 3 + 1) / 3 = .33 (3 – 3 + 1) / 6 = .16 1
Node Weight at Level l
.5 / 3 = .16
.33 / 2 = .16
.16 / 1 = .16
Independent
Nodes at level L
1 (node I)
1 (node J)
1 (node k)
Independence = 1 * .16 + 1 * .16 + 1 * .16 = .48
Depth : indicates how entrenched? a pyramid is.
Breadth : what does it indicate?
Other things we might care about:
- Participation index (see airport paper)
- What is the threshold to call a pyramid as such?
- How many nodes in a country belong in pyramids?
- How many pyramids does this pyramid connect to?
- How far up in the pyramid do the shared nodes come in?