Variation of the underlying network
4.4 Effects on the scope hierarchy
The scope size distribution suggests that characteristic scopes persist over a certain range of network modifications and that their sizes scale approxi-mately linearly with the size of the network. In this section, the structure of the scope hierarchy itself is analyzed in dependence of a changing underlying network.
In chapter 3 it has been shown that an artificial network containing all possible reactions results in a simple and intuitive hierarchy where all scopes represent a certain set of building blocks. When a certain number of reactions is randomly removed, the scope hierarchy gets transformed into a new struc-ture, where characteristic scopes, represented by nodes with large degree, still characterize a specific set of building blocks, while others are already too loosely connected to make use of their building block content and are represented by nodes with a small degree or by sink nodes. In the following, the transition between these two hierarchies is evaluated by observing the scope hierarchies of a network when consecutively more and more reactions are deleted. This process is then continued to a case where all reactions are removed.
reactions deleted
scope size
0 1000 2000 3000 4000
0 500 1000 1500 2000 2500
Figure 4.3: Effect of multiple reaction deletions on the sizes of random multi scopes. The shading scales with the number of scopes having a certain scope size and were calculated on a network reduced by a certain number of reac-tions. Here, for each number of deleted reactions (step size 25), 25 random networks were generated and for each of these networks 400 scopes of 30 random seed compounds were calculated. Therefore, the minimal scope size is 30.
Figure 4.4 shows 10 selected steps of such a process for an artificial net-work defined by N(A,B,C,D,E)=(4,3,2,1,1) which contains 239 compounds and in its unreduced form 3816 reactions. Figure 4.5 gives characteristic values of the graphs. The unreduced network yields the expected simple hierarchy.
Many seed compounds result in the same scopes, i.e. are interconvertible.
Interestingly, a large number of reactions can be removed before this hier-archy is substantially changed. In the next phase, the number of nodes and ranks in the graph increases. The scopes of the original simple hierarchy per-sist and become the so called characteristic scopes, while new, less connected scopes appear, resulting from seed compounds which are, due to reaction deletion, not anymore able to reach one of the characteristic scopes. The characteristic scopes persist over a large number of deletion steps.
At some point, the number of ranks in the hierarchies becomes again smaller while the number of nodes approaches the total number of com-pounds, which means that there are almost no interconvertible compounds anymore. Also, the number isolated nodes strongly increases. These effects are due to a beginning disintegration of the network whose ability to do
con-E D
AB3EA4BE A2E A4B2
A4B2E ABD
ABDE ABC
ABC2AB3C
A2C A2B3
A3C A4B A4CA3C2
ABCE
A2B2CDA3C2EA4BEA4B3E
ABC2E
B2DE B2C B2CEB2CD
B2CDE
A2BC2DE A2B2CDA3C2EA3BCDEA4BEA4B3E
ABC2E
AB2D AB2CD
AB2C2D
AB3C2 A2BC A3BD A3C2D A2CD
A3B2D
B2DE B2C B2CE B2CD
B2CDE
A2BDEA2B2CD A2B2C2E A2B3C2DE
A3CDE
AB3CD A3BCDA2B3C2D A4C2D A3DA4BD A4BC A3CDA3B2C2 A3BC2 A2B3C
A4B2CD
A4CA4CEA4CDA4CDE
A4C2
A4BC2DE A4B2CDE A4B3C2A4B3CE
A4B3C2E
Figure 4.4: Dependence of the scope hierarchy of an artificial network with N(A,B,C,D,E)=(4,3,2,1,1) (239 compounds, maximal 3816 reactions) on the num-ber r of remaining reactions in the network.
versions is strongly reduced. This eventually also leads to a disappearance of the characteristic scopes which is consistent with the effect observed for the scope size distributions in figure 4.3. Eventually, the network disintegrates completely when all reactions are removed.
The source-sink connectivity (figure 4.5) gives an idea of the ability of the network to do conversions. In a complete network there exists a scope containing all building blocks and therefore all other scopes are included in this top scope. For the hierarchy graph that means that there exist one source and as many sinks as there exist building blocks and the source is connected to all sinks. The source-sink connectivity is defined as the quotient of the number of all connected source-sink-pairs and the number of all in principle possible source-sink-pairs, or more precisely
csd = p+ni
(ns+ni)(nd+ni), (4.8) wherepis the number of connected source-sink-pairs (not considering isolated nodes),nsthe number of sources,ndthe number of sinks andnithe number of isolated nodes, which can be interpreted as source and sinks at the same time and therefore influence the source-sink connectivity in the above described way. For the hierarchy of the complete network this connectivity is 1. In the completely disintegrated network all compounds can only be converted into themselves, which means that the hierarchy contains only isolated nodes.
Therefore the source-sink connectivity is n/n2, with n being the number of compounds in the network, i.e. for the analyzed network 0.0042. For all cases between these two extrema the source-sink connectivity monotonously decreases with decreasing numbers of reactions in the network.
A similar procedure can analyze the effect of network modifications on the scope hierarchy of the KEGG network. However, it should be noted, that the addition of new artificial reactions to the KEGG network is methodically difficult. Therefore, only a further reduction from the present network is analyzed. It is apparent, that the KEGG network is not a complete network in the sense that it is able to perform all possible conversions. It can therefore be expected that the process on the KEGG network only covers part of the process shown on the artificial network.
Figure 4.6 shows the graph characteristics of the hierarchies of the KEGG network during the reduction process. When comparing with the character-istics of the artificial network (figure 4.5) it becomes clear, that the KEGG network starts somewhere in the middle of the process observed on the arti-ficial network. Afterwards, the reduction processes proceed similar.
Apparently, the original KEGG network has already been reduced to a point where further reaction deletions have a significant influence on the
1 10 100 1000 10000 number of reactions
0 50 100 150 200 250 300
quantity
1 10 100 1000 10000
number of reactions 0
2 4 6
ranks
0,001 0,01 0,1 1
source/sink connectivity
Figure 4.5: Characteristics of the scope hierarchies in depen-dence of the number of remaining reactions in the network with N(A,B,C,D,E)=(4,3,2,1,1). Left graph: the total number of nodes (cir-cle), the number of source nodes (squares), the number of sink nodes (triangles) and isolated nodes (x). Right graph: number of ranks in the hierarchy (triangles) and the source-sink connectivity (circles), i.e. num-ber of connected source/sink pairs divided by the product of source and sink vertices. It should be noted that the reduction process actually pro-ceeds from right to left since the number of reactions in the network is decreased.
10 100 1000 10000
number of reactions 0
1000 2000 3000 4000 5000
quantity
10 100 1000 10000
number of reactions 0
5 10
ranks
0,0001 0,001
source/sink connectivity
Figure 4.6: Characteristics of the scope hierarchies in dependence of the number of remaining reactions in the KEGG network. Left graph: the total number of nodes (circle), the number of source nodes (squares), the number of sink nodes (triangles) and isolated nodes (x). Right graph:
number of ranks in the hierarchy (triangles) and the source/sink connec-tivity (circles), i.e. number of connected source-sink pairs divided by the product of source and sink vertices.
structure of the hierarchy, but where the ability of performing conversions is still large enough to show characteristic scopes. This seems to be a reasonable compromise between a network having too many redundant reactions and a network which only has a limited ability to perform conversions.