Network’s Connectivity Dynamic Modelling using a Topological Binary Model: Critical Transitions Concept
1. Network Overall Connectivity
Following the notations of the graph theory, a graph G(N,E) is composed of N nodes (vertices) connected through a set of E edges (links). The set of edges E
contains all existing links in the network. Formally, the link l(i,j) denotes a direct link
) ,
( ji while l{ }i,j denotes an indirect link ( ji, ), between the two nodes i and j.in the paper, we will denote links without brackets such as li,j. The distinction between
direct and indirect link is signalled by the order of the corresponding tensor describing the link, as will be explain later.
There are many useful metrics to measure graph connectivity. Among the well-known are degree distribution (Barabasi, 1999), characteristic path length (Watts, 1998), graph diameter [8] and clustering coefficient (Albert, 2002). These measures provide a useful set of statistics for comparing power grids with other graph structures.
The “graph diameter” is amongst the basic notations of the graph theory. It does particularly interest us. For any pairs of nodes i and j ∈N, let δi,j denotes the path
between i and j. The diameter D of the graph G is defined as the max of all δi,j,
{
i j N}
D=maxδi,j/ , ∈ . D is the highest of the lowest paths.
In this paper, a concept derived from the diameter D is used and measured using a
metric called the “nominal connectivity order” and is explained in the following.
2.
Network Connectivity Order
The “connectivity order” of a network is a metric proposed to measure the global connectivity of a graph.
In §2.1, the notion of the 1st order “binary connectivity tensor” is established based on the “adjacency matrix” from graph theory.
In §2.2, the process of determining the higher order “binary connectivity tensors” is explained.
In §2.3, the notion of the network nominal connectivity order, NCO, is introduced. 2.1 Network Binary Topological Description
Following the graph theory, we use the “adjacency matrix- A ” to describe the topology of a given network such as: 1
, j
i
e = 1 if nodes i and j are directly connected, otherwise 1 , j i e = 0
(
i,j∈N)
. The exponent 1 in 1 , j ie denotes that it is a 1st order
connectivity element, i.e. it describes a direct link between the nodes i and j.
The topological mapping of the network, presented in Figure 1, is given in Table 1. The 1st order mapping represents the network as it should be in its nominal operability state. It is the nominal operability state after the design specifications, accepted by the operator and approved by other stakeholders. As one can see, not all the nodes are directly connected. However, all the nodes are still connected but at “higher connectivity orders”. The idea, now, is how to determine in systematic way these existing higher connectivity orders.
2.2 Network Higher Connectivity Orders
Many nodes are not connected at the 1st order level, i.e. not directly connected. They have 1
, j
i
e =0. However, they are connected at higher orders, determined as following.
Let n+1 ij
u
(
i,j∈N)
be the connectivity tensor describing the(
)
thn+1 connectivity order between nodes and is determined following after (Eid, 2012) and (Eid, 2013), as following: n lj il n ij e e u +1= 1• (1) Where, n lj il e e1• n mj im n j m m i n j i n j i n j i e e e e e e e e e e • + • + • + + • + • = − ( −1) 1 1 ) 1 ( 3 1 3 2 1 2 1 1 1 ... .
Once, n+1 ij
u is determined, one proceeds to the determination of 1 , + n j i e as following: 1 + n ij e = > = = = = = + + 0 1 0 0 0 1 1 n ij n ij u if u if j i if , and n=1,2,3,... (2)
We, then, proceed to determining the minimum connectivity order of each couple of nodes, ni,j, i.e. to determine the minimum value of n at which the value of the binary
tensor n j i
e, switches to one for each couple in the net.
One can follow the evolution of the binary connectivity tensor in the tables from (1-a) to (1-e) related to the network described in Figure 1. As an example, for the couple of nodes 4 and 6, the minimum connectivity order is 3, i.e. 1 0
6 , 4 = e and 2 0 6 , 4 = e but 1 6 , 4 = n
e for all n≥3. Each couple of nodes has, then, its characteristic minimum connectivity order.
Having designed a systematic process to determine higher connectivity order tensors and the characteristic minimum connectivity order of each of nodes, we are going to define a metric to measure the network overall connectivity in the following section, §2.3.
2.3 Network Nominal Connectivity Order
Having determined the minimum connectivity order of each couple of nodes ni,j N
j
i, ∈ , one may be at that stage interested in establishing a measure of the network overall connectivity state.
The approach proposes a metric for measuring the network overall connectivity and denote it by the “Nominal Connectivity Order-NCO”. The NCO is the lowest connectivity order, min
{
ni,j,i, j∈N}
, at which each node is connected to all theothers at which each node is connected to all others. Higher is the NCO, lower is the network connectivity quality.
The network overall connectivity quality decreases with the increase of the connectivity order and inversely. The network overall connectivity state is, also, directly related to the “network operability/performance state” that can be defined as “the likelihood” of the network to be in nominal operation mode at instant “t”. Ultimately, the highest operability is attended when 1 1
=
ij
e for all the nodes.
The NCO is the connectivity order at which the network’s operability complies with the design requirements & specifications and consented by the operator and other concerned stakeholders. The network given in Fig.1 has a NCO equal to five while the corresponding graph diameter is three. The network contains 10 nodes and 15
edges. If each node was directly connected with all the others, the network would have had 45 edges. At a connectivity order equal to five, each node is connected with all the others. At that level of connectivity, each node sees the 14 other nodes, in the network described in Figure 1.
Once the NCO concept is well established and determined, one can proceed to the determination of the “critical transitions”.