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The complexity of large-scale network systems made of a large number of nonlinearly interconnected components is a restrictive facet for their modeling and analysis. In this paper, we propose a framework of hierarchical modeling of a complex network system, based on a recursive unsupervised spectral clustering method. The hierarchical model serves the purpose of facilitating the management of complexity in the analysis of real-world critical infrastructures. We exemplify this by referring to the reliability analysis of the 380 kV Italian Power Transmission Network (IPTN). In this work of analysis, the classical component Importance Measures (IMs) of reliability theory have been extended to render them compatible and applicable to a complex distributed network system. By utilizing these extended IMs, the reliability properties of the IPTN system can be evaluated in the framework of the hierarchical system model, with the aim of providing risk managers with information on the risk/safety significance of system structures and components.

Critical infrastructures are engineered distributed systems which provide the fundamental support to modern Industry and society. Examples are computer and communication systems, power transmission and distribution systems, rail and road transportation systems, oil/gas systems and water distribution systems. Failures of such systems can have multiple, transnational impacts of significant size [1-3]. Hence, identifying and quantifying the reliability and vulnerability of such systems is crucial for designing the adequate protections, mitigation and emergency actions against failures [

These systems are exposed to multiple hazards and threats, some of which are even unexpected and emergent, and consist of a large number of elements whose interactions are not easily modeled and quantified, so that a complete analysis by exhaustive treatment cannot be pursued. As a result, the performance and reliability assessment of such “complex” systems has proved to be a non-trivial task in practice.

Recent studies suggest that many real complex network systems exhibit a modularized organization [4,5]. In many cases, these modularized structures are found to correspond to functional units within networks (ecological niches in food webs, modules in biochemical networks) [

In the analysis of systems with respect to their failure behavior, Importance Measures (IMs) are used to identify the weak points and quantify the impact of component failures [10,11]. IMs provide numerical indicators to determine which components are most important for system reliability improvement or most critical for system failure. Many different IMs have been proposed in the literature [12,13], among which classical and relevant statistics are Birnbaum [

measures can be applied directly to complex network systems, because of the distributed character of functionality and service that they provide.

The purpose of this paper is twofold: firstly to propose a scheme of recursive clustering to obtain a hierarchical modeling framework associated with different variedsize grained virtual networks; then to introduce Extended Importance Measures (EIMs) which are compatible with the distributed characteristics of complex network systems, to evaluate the components importance in the framework of the hierarchical system representation.

The remainder of this paper is organized as follows: Section 2 presents the methodology of hierarchical modeling, taking the structure of the 380 kV Italian Power Transmission Network (IPTN) as an example for illustration; in Section 3, the basic terminal-pair connection reliability problem is first introduced, based on which the traditional IMs are extended and then calculated for the IPTN system; conclusions are drawn in Section 4.

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) (

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 [

A successive Unsupervised Spectral Clustering Algorithm (USCA) [

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 [

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

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

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.

and

with.

The integer that is indicated in the

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.

It is known that most network reliability problems are NP-hard and therefore there is a significant gap between theoretical analysis and the ability to compute different reliability parameters for large or even moderate network systems [

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 [

In

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) [

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 [

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 [

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

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 [

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

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,

The modeling and analysis of complex network systems is a non-trivial task. Related decision-making regarding reliability and vulnerability is limited by computational resources.

In this work, we have introduced a framework for hierarchical modeling of complex network systems, which leads to the definition of different varied-size grained artificial networks. The construction of the hierarchical model is obtained by a recursive unsupervised spectral clustering method. The hierarchical model thereby obtained provides a multi-scaled representation of the original network system, with more detailed information but high complexity at the lower levels of the hierarchy, and simplified structure but relatively low complexity at the higher levels. The availability of different scales of modeling resolution allows a flexible management of the analysis, at the level of details desired for its purposes. The 380 kV Italian Power Transmission Network (IPTN) has been taken as an illustration.

Furthermore, Importance Measures (IMs) such as Birnbaum, Fuessell & Vesely and Criticality, have been extended for application to the terminal-pair reliability problem in complex distributed network systems.

The calculation of the extended IMs at different levels of the hierarchical system modeling has demonstrated the effectiveness of the proposed hierarchical modeling, with the IM-ranking of the IPTN elements offering insights on how to improve the system against failures of most critical elements.

The authors are thankful to Dr. Giovanni Sansavini of Politecnico di Milano for supporting the work with rele-

vant information on the network structure used as reference system and to Dr. Yanfu Li of Supelec for fruitful discussions.