Graph Theory provides a framework for the mathematical representation of complex networks. A graph is an ordered pair comprising a set of vertices (nodes) together with a set of edges (also called arcs or links), which are twoelement subsets of V. The network structure is usually defined by the adjacency matrix, which defines which two nodes are connected by assigning a 1 to the corresponding element of the matrix; otherwise, the value in the matrix is 0 if there is no connection between the two nodes. As described, this type of graph is unweighted and undirected. A graph is weighted if a value (weight) is assigned to each edge representing properties of the connection like cost, reliability, capacities, etc. For example, the matrix of physical distances is often used in conjunction with the adjacency matrix to describe a network also with respect to its spatial dimension [18,19].

In this paper, we take for exemplification of the analyses proposed the 380 kV Italian power transmission network (IPTN) (Figure 1). This network is a branch of the high-voltage-level transmission network, which can be modeled as a graph of (generators and distributors) connected by M = 171 links [20,21], defined by its adjacency matrix A whose entries are 1 if there is an edge joining node i to node j or 0 otherwise. In Figure 1, the generators, i.e. hydro and thermal power plants, are represented by squares whereas the distribution substations are represented by circles.

Modularity is ubiquitous in many networks of scientific and technological interest, ranging from the World Wide Web to biological networks [7,22]. As a result, it is often possible to identify groups of elements that are highly interconnected with each other, but have only a few links to components outside of the group to which they belong to. These communities usually combine into each other in a hierarchical manner [7], in which nodes form groups and then join the groups of groups, and so forth, starting from the lowest levels of organization (individual nodes) up to the level of the entire system. This suggests the development of a hierarchical structure to describe a complex network system at different levels of resolution, with the aim of managing the complexity of the system more effectively.

A successive Unsupervised Spectral Clustering Algorithm (USCA) [23], which is invariant to cluster shapes and densities and simple to implement, has been adopted in this study to build the hierarchical structure of the IPTN system. Cluster analysis aims at recognizing natural groups within classes of entities [24]. The problem is to assign categories to unlabelled data, encouraging the search of implicit information in the network structure encoded in its graph [25]. Consequently, modularity patterns within a complex network system can be revealed without a priori knowledge of their existence. The detailed description of different clustering methods is beyond the scope of this article. For a systematic and synthetic review, the reader is encouraged to look at [24-26].

The USCA makes use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before Fuzzy c-Means (FCM)—clustering in fewer dimensions. Schematically, it is performed by the steps [23] in Table 1.

In the first step, the Laplacian matrix is calculated from the similarity (affinity) matrix as follows. The input similarity matrix is of size and its generic element represents the similarity between nodes i and j in the network. The diagonal components are set to 1 and the matrix is symmetric. The degree matrix is the diagonal matrix with diagonal entries defined by

Then, the normalized graph Laplacian matrix can be obtained:

where and is the identity matrix of size.

By recursively operating the USCA on the data of the IPTN presented in Section 2.1 above, a 5-levels hierarchical structure of the system is constructed which contains the complete system at the top and individual elements at the bottom (the top panel of Figure 2 gives out the structure of the hierarchy, detailed in the first 3 levels).

Based on the hierarchy structure resulting from the successive application of USCA, artificial networks can be

Table 1. Unsupervised spectral clustering algorithm.

defined at each layer. The artificial network at level l of the hierarchy is described as a graph with, whereis the number of levels of the hierarchy. We use to represent the artificial node i

at level l, which corresponds to a cluster of real network nodes. Artificial nodes are connected by artificial links

composed by those actual network links connecting (in parallel) the actual nodes in the clusters forming the artificial nodes,

.

The connection pattern between artificial nodes at level l is illustrated by an adjacency matrix whose element

ifi.e. if in the artificial nodes and there is at least one actual link connecting two actual nodes, and 0 otherwise.

Figure 2 presents the hierarchy structure of the IPTN system and the artificial networks associated with the first 3 levels of the hierarchy. At the top of the hierarchy (i.e. l = 1), the network is a single unit, i.e. one artificial node, which consist of all actual nodes. At the second level, we have

and

with.

The integer that is indicated in the Figure in proximity of the generic i-th artificial node indicates the number of actual nodes which compose it, e.g. is representative of a group of 38 actual network nodes. Note that at the bottom of the hierarchy, we find the original network, i.e. each artificial node is an actual node and each artificial edge corresponds to an actual link.

The hierarchical model offers different levels of resolution at the different levels of the hierarchy. The artificial networks at the top of the hierarchy contain limited detail information of the local connectivity patterns (in the limit, only one node represents the whole network at the first level of the hierarchy); as we move down the hierarchy, more local information enters the model, at the expense of an increase in the dimension of the network. These characteristics can be leveraged efficiently to manage the complexity of a complex network system.

The terminal-pair or node-pair reliability (TPR) problem amounts to determining the probability of successful communication between a specified source node and a terminal node in a network, given the probability of success of each link and node in the network. Let us introduce a binary vector to represent the state of the network, i.e. the state of each of its edges and the state of each of its nodes, where if edge is operating and 0 otherwise (for node). For simplicity of illustration, we assume that nodes cannot fail, while edges can (thus is no longer considered hereafter). The state of the network is defined as being non-failure if the specified terminal-pair is connected by at least one path of operating edges; otherwise it is failure. All possible failure states are included in the subset of the set containing all possible scenarios (failure and non-failure). An inclusive TPR analysis requires considering all elements in. We then define the TPR as:

where  is a binary function which indicates the connection availability between node-pair s and d (1 = connection; 0 = no-connection). Let us assume that each edge has associated a probability of being operating and a probability of being failed; then, the TPR of the network can be calculated as:

where represents the state of network edge and is the set of failed edges for a given state. Note that the implicit assumption underpinning Equation (4) is that the network edges are independent.

When the computational cost of the network is high (it grows exponentially with the number of network components), then, the artificial network at a suitable level of the hierarchy can be leveraged to carry out the TPR. At the generic level of the hierarchy, the artificial link connecting nodes (clusters) and is composed by actual network links in parallel,

;

then, the reliability of the artificial edge at level l can be calculated by:

where indicates the failure probability of the actual link that in the real network connects nodes and.

Various algorithms to solve the classic TPR problem have been reported in literature, with various computational efficiencies [29-31]. A so-called Modified Dotson algorithm [30], which has been claimed and tested to subdue others in computational time, is used here for the TPR assessment based on the hierarchical modeling. The failure probability of the transmission lines in the IPTN system is computed based on outage statistics provided in [32], by assuming that the edge failure probability is proportional to its length with an average failure rate occ/100 mile-year, and average outage duration time t = 64.81 hours/occ.

In Figure 3 right-panel, the connection reliability between nodes 1 and 127 in the IPTN network system (left panel in Figure 3) is shown as resulting from evaluations at each of the five levels of the hierarchical model described in the previous section. The right panel of Figure 3 gives the probabilities of connectivity failure between nodes 1 and 127 from level 2 to level 5 (top) and

the computational time needed for the analysis (bottom); the values have been normalized with respect to the maximum values of connectivity failure probability and computational time, which occur at the bottom of the hierarchy (level 5) corresponding to the whole network. The result at the first level has not been shown since its value is simply 0, i.e., node 1 and 127 are in a single unit and will not disconnect. One can see that the difference between the actual and estimated failure probabilities decreases as the assessment moves downs to the bottom of the hierarchy, balanced by the computation time which instead increases significantly. The decision maker can obtain satisfying estimations of the failure probability at a hierarchical level of lower complexity, e.g. level 3, thus saving significantly in computation time.

Component importance measures are widely used in system engineering to identify components within the system that most significantly influence the system behavior with respect to reliability, risk and/or safety. The indications drawn are valuable for establishing direction and prioritization of actions, related to reliability improvement during system design and optimization of operation and maintenance.

A well known IM is the so called Birnbaum IM defined as (with reference to system reliability, as the system performance indicator) [14]:

where is the Birnbaum Importance (BI) of component i; represents the reliability of the system; is the reliability of component i ; is the system reliability calculated assuming that component i is perfectly operating and the system reliability in the opposite case of component i failed. The BI measures the significance of component i to system reliability by the rate at which system reliability improves with the reliability of component i. As shown in Equation (6), the BI of component i does not depend on itself, so that two components i and j may have a similar value although they have different reliability values and, respectively; this could be seen as a limitation of BI.

The Criticality Importance (CI) measure overcomes the above limitation by considering component unreliability [17]. It is defined as:

where is the unreliability of component i and is the system unreliability. Now, a less reliable component is more critical than another one with same value of BI.

Fuessell & Vesely [15] proposed an alternative importance measure according to which the importance of a component in the system depends on the number and on the order of the cut sets in which it appears [17]. Most commonly used as a risk reduction indicator, Fuessell & Vesely Importance (FVI) quantifies the maximum decrement in system reliability caused by a particular component being failed:

The previously proposed IMs (BI, CI and FVI) are functionally different. They evaluate subtly different properties of the system behavior, and therefore, are often used in a complementary fashion to infer different information. To apply the IMs for analyzing a network system such as the IPTN, it is necessary to extend the definition of the IMs to account for the multiple terminal or node pairs (e.g. generator-distributor pairs) where connectivity defines the network functionality.

Specializing such extension for the analysis of the importance of components of the IPTN system, we introduce the Extended Birnbaum Importance (EBI) measure as the average of all BI values obtained considering all possible Generator-Distributor pairs reliabilities in the network system:

where and are the number of generators and distributors in the network respectively; and are sets of node generators and distributors respectively; is the TPR between node s and node d; and represent the terminal pair reliabilities between node s and node d, in the condition that component i is perfectly operating and completely failed, respectively.

Similarly, we can define Extended Criticality Importance (ECI) and Extended Fussell & Vesely Importance (EFVI) measures:

where is the Extended Criticality Importance (ECI) measure of component i and is the Extended Fussell & Vesely Importance measure.

The EIMs introduced have been calculated for the IPTN system at different levels of the hierarchical model of the system developed. For the evaluation, an artificial node functions as a generator as long as there is at least one actual generator node within it; otherwise it is simply a distributor.

Tables 2 and 3 report the results of the importance assessment (EBI, EFVI are given in Table 2 and ECI in Table 3) for the artificial edges of the network at level 2 of the hierarchy. For EBI and EFVI, all components in the artificial network have the same importance rank, but with slight differences between EBI and EFVI values, and the artificial edge {2-4} is the most important in the artificial network (see the bottom panel of Figure 2). This is due to the fact that this artificial edge is the only possible link between a generator in artificial node and the distributors in other artificial nodes, and thus its disconnection would cause a large-scale generator-distributor connectivity failure. The rank based on the ECI is different from that of EBI and EFVI, and the most important artificial edge is {3-4}; the difference lies in the definition, as discussed before: EBI depends only on the structure of the system and not on the reliability of the considered component, whereas ECI takes the unreliability of the component into consideration, and in fact, the artificial edge {3-4} is made of only one actual edge with relatively high probability of failure, which leads to the highest ECI value.

By combining the indications of EBI and ECI, it is advisable to offer advices to the decision maker for the purpose of system maintenance and operation optimization [10]. When EBI & EFVI is high and ECI is low like in the case of artificial edge {2-4}, the system safety can be improved by protecting against failure of each component, e.g., adding alternative edges between artificial node and node (or). For the case of low EBI & EFVI and high ECI (artificial edge {3-4}), the decision maker should invest in improvements of the component itself, to decrease the failure probability.

Tables 4 and 5 report the evaluation results at level 3 of the hierarchy. Fictitious edge {4-9}, composed by actual edges {110-111, 112-114, 107-109}, has highest EBI and EFVI values but relatively low ECI value (ranked 15th among all 17 artificial edges), indicating that the system reliability is highly sensitive to its failure, whereas the component itself is relatively reliable. On the contrary, the artificial edge {1-10} composed by only one actual edge {64-78} is highly unreliable itself, and its EBI and EFVI values are both ranked 8th among all 17 edges. It is important to pay attention to these artificial edges with both relatively high EBI & EFVI ranks and ECI ranks, which means not only that their failures cause a significant deterioration of the system reliability but also that they are vulnerable themselves. In this respect, by combining Tables 4 and 5, we find that artificial edges {1-11} (whose actual network link is {71-83}), {6-10} (which is composed by actual link {76-79}), and {10-12} (which is composed by actual links {75-88, 80-95}) are the three artificial edges most critical for the system reliability.

The bold edges in Figure 4 represent the edges of the actual network system which have resulted most critical based on the extended importance measure evaluation carried out at level 3 of the hierarchy model. These edges should be paid special attention. For links {110-111, 112-114, 107-109}, improving the defense in depth against

Table 2. EBI and EFVI at level 2 of the hierarchical model.

Table 3. ECI at level 2 of the hierarchical model.

Table 4. EBI and EFVI at level 3 of the hierarchical model.

Table 5. The results of ECI assessment at level 3 of the hierarchical model.

their failures is advisable to improve the reliability of the system, while for links {64-78, 71-83, 76-79, 80-95, 75-88}, the edge unreliability should also be mitigated.

Tables 6 and 7 report the results of the EIMs evaluation at level 4 of the IPTN hierarchical model. It turns out that artificial edge {7-11} (corresponding to actual link {119-122}) has the highest EBI and EFVI values and artificial edge {1-22} (corresponding to actual link {64-78}) has the highest ECI rank and relatively high EBI and EFVI ranks, indicating its criticality to system reliability.

Finally, Table 8 reports the computation times required for the calculations of the EIMs at different levels in the hierarchy: as expected, the more we go down in the hierarchy the higher the computation time.