6. DEVELOPMENT AND TESTING OF NETWORK THEORY APPROACHES
6.3 Primary Influence Vertices
Primary influence PI is a statistical property of a vertex that was specifically devised for this study to measure the power or authority that a vertex has over other vertices in a weighted directed network. It is acknowledged that there are other statistics that are used to measure the power of vertices in networks, such as degree centrality, as explained in Section 2.1.4.
Primary influence counts the number of vertices in a network over which a given vertex vi has power, or authority. A vertex vi has a power over other vertices, v ϵ [G(v)], if and only if all the highest weighted incoming arcs of v originate from vi, or from another vertex over which vi has power. Alternatively, it can be said that a vertex has power over other vertices when the following four criteria are met: The four criteria are illustrated using Figure 6.3. The numbers inside the vertices and on the arcs are vertex numbers and arc weights, respectively.
1. If there is only one arc that terminates into a receiver vertex, the sender vertex has primary influence over the receiver vertex. For example (see Figure 6.3), one arc (v5v12) terminates into v12. Therefore v5 has primary influence over v12.
2. If there are more than one arcs that terminate into a receiver vertex, then the sender vertex of the arc with the highest weight has primary influence over the receiver vertex. Arcs from v1, v2 and v3 in Figure 6.3 terminate into v4. The respective weights of the arcs are five, three and seven. In this case v3 has primary influence over v4. 3. If vi and vj are among the sender vertices of arcs that terminate into vk where w(vivk) =
w(vjvk), and vivk and vjvk have highest weights than the rest of the arcs that terminate into vk, then neither vi nor vj has primary influence over vk. However, if vi has primary influence over vj, then vi will inherit primary influence over vk as well. For example (see Figure 6.3), vertex v10 receives arcs from v8 and v9. The two arcs have the same weight and, hence, neither v8 nor v9 has primary influence over v10. However, vertex v7
receives arcs from v6 and v11 with similar weights. Vertex v11 has a primary influence
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over v6, and hence v11 has primary influence over v7 as well, while v6 has no primary influence over vertex v7.
4. If vi has primary influence over vj and vj has primary influence over vk, then vi has primary influence over vk. For example (see Figure 6.3) v3 has primary influence over v4 and v4 has primary influence over v5. Vertex v5 in turn has primary influence over v12. Therefore v3 has primary influence over v4, v5 and v12.
Figure 6.3 A network for illustrating the primary influence criteria
The primary influence vertex concept has its basis in the TOC and network theory. The Theory of Constraints indicates that a constraint is the lowest performing component in a system (see Chapter 3). If the TOC philosophy is extended to an individual vertex vi in a weighted cause-and-effect network, then the vertex that is sending the highest weighted arc to vi would be the constraint to the performance of vi. Also, according to the TOC philosophy, if two vertices vi and vj are among the sender vertices of arcs that terminate into vk, where w(vivk) = w(vjvk), and vivk and vjvk have the highest weights compared to the other arcs that terminate into vk, then neither vi nor vj would be the constraint to the performance of vk. This would be so because by solving either vi or vj, the performance of vk would still remain the same. Criterion number 4 is based on the concept of “reach” (see Section 4.1.5), which allows
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a vertex to extend its primary influence beyond its first out-neighbour via heavily weighted pathways.
The number of vertices over which a vertex vi has a primary influence is called its primary influence index, IPI(vi). The primary influence indices of the vertices in the network in Figure 6.3 are shown in Table 6.3. Custom written software was used to compute the primary influence indices of the vertices.
Table 6.3 Primary influence indices for the network in Figure 6.3 Name of vertex Primary influence index
v1 0
v2 0
v3 3
v4 2
v5 1
v6 0
v7 0
v8 0
v9 0
v10 0
v11 2
v12 0
Four vertices v3, v4, v5 and v11 have primary influence indices of at least one, while the remainder have primary influence indices of zero. The vertices that have primary influence indices of at least one are called primary influence vertices. Conversely, the vertices that have primary influence indices of zero are called non-primary influence vertices. Primary influence vertices of a network represent the driver factors in the system that the network represents.
The average primary influence for a network is calculated in this study as the quotient of the sum of all the IPI(vi) in a network and the number of primary influence factors in the network. It must be mentioned that is calculated like this for practical reasons. The number of non-primary influence vertices (i.e. the vertices whose IPI = 0) in a network can be large.
Thus, a that is calculated as the quotient of the sum of all the IPI(vi) in a network and the number of factors in the network could be too small to be of significance for further analyses (see Equation 6.1 in Section 6.5) The value for the network in Figure 6.3 is two.
The primary influence concept can be used for diagnosing problems in complex systems and for devising strategies for improving the performance of such systems. The primary influence
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was specifically devised to address two phenomena that generally occur in complex systems.
The first phenomenon is multiple causalities. This phenomenon occurs when a factor has more than one cause. Examples of multiple causalities can readily be found in ISSPSs. For example, under loading, aged equipment and less preventative maintenance may be some of the causes of low transport efficiency. However, it may be difficult to determine which of the factors drives low transport efficiency.
The second phenomenon that can be addressed by the PI concept is the knock-on effect (or ripple effect) of a causal factor. A vertex directly affects its 1st out-neighbours. Additionally, a vertex may indirectly affect higher order out-neighbours (e.g. 2nd, 3rd, ..., nth) through a series of paths (Pn). Some vertices affect large parts of a network via heavily weighted pathways.
An example of knock-on effect in ISSPSs is a flood. Floods may directly or indirectly cause many problems in an ISSPS, such as the lodging of sugarcane, an increase in the percentage of soil in sugarcane, road damage, low transport efficiency, low availability of sugarcane at mills and low sugarcane quality.
The primary influence concept can be used to identify the factors on which interventions aimed at improving the performance of the system should be targeted. The primary influence concept may offer a method for identifying important factors in complex systems and a way for developing techniques that can be used to identify the factors on which interventions aimed at improving the performance of systems must be targeted. This is well aligned with the TOC philosophy.