Pascal Maillard

Franck Tessier
email: franck.tessier@univ-rennes1.fr

Emmanuelle Orhan

François Cheviré

Roger Marchand

Thermal Ammonolysis Study o fthe Rare-Earth Tantalates RTaO4

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Oxides are not only useful precursors for the synthesis of oxynitrides, they constitute also an excellent basis of comparison for their characterization which may be considered as the study of modifications brought by the nitrogen/oxygen substitution. Attractive properties directly related to the role played by nitrogen can result from this N/O substitution [1][2][3] .

The behavior of the rare-earth tantalates RTaO4 (R = La→Yb, Y) heated in flowing ammonia at 900-950°C has been investigated previously as leading to perovskite and/or pyrochlore oxynitrides depending on the size of the rare-earth 4 . In particular, Pors et al. have described formation of R2Ta2O5N2 pyrochlores for R = Nd→Yb and Y, the stability of which increases for smaller elements so that single phases could be isolated from R = Er.

In the present work, as part of a continuing study of rare-earth and transition metal oxynitrides exploring in particular chimie douce approaches, we have revisited the crystal chemistry of the R-Ta-O-N oxynitrides. While the stoichiometry R2Ta2O5N2 and the pyrochlore structure were confirmed for the larger rare-earths (R = Nd→Gd), we have evidenced, using X-ray and neutron diffraction, a different structure-type for the smaller rareearths which give rise, actually, to fluorite-type phases, as illustrated here with R = Ho, Er, Yb and Y.

A particular interest was devoted to elaborate reactive precursors by involving complexation-calcination methods, typically the amorphous citrate route. Among several advantages, this method can induce formation of metastable phases, for example the cubic polymorph of the rare-earth tungstates R6WO12, which normal synthesis requires temperatures over 1200°C 5 . However, its essential interest lies in an improved reactivity towards ammonolysis which makes possible a decrease in the nitridation temperature onset and, consequently, an extension of the oxynitride thermal stability domain.

The R-Ta-O-N oxynitrides with fluorite structure we have isolated form solid solution domains with variable nitrogen content and powder color. Previous results obtained from rare-earth tungstates have shown interest of such composition domains for colored pigments applications, where it is possible to modify the powder color and hold it to a well-defined shade by only adjusting the nitrogen enrichment 3,6 .

Furthermore, it should be noted that, in addition to luminescence properties 7,8 , the rare-earth tantalates RTaO4 show photocatalytic activity for water splitting 9 . Corresponding colored oxynitride compositions are potentially of interest for visible-light-driven photocatalysts, as already observed for tantalum oxynitrides [10][11][12] .

Experimental

Preparation of oxide precursors. RTaO4 tantalates were prepared both by a conventional ceramic route and by taking advantage of a chimie douce process -the amorphous citrate route -which gives the oxide precursors an enhanced homogeneity and reactivity towards ammonolysis.

Solid state route. The starting ternary oxides RTaO4 were prepared by heating at 1400°C, in a muffle furnace, appropriate mixtures of rare-earth oxide and Ta2O5. Three cycles of 12 hours heating with intermediate grinding were often necessary to obtain X-ray pure phases.

Amorphous citrate route. Among numerous chimie douce-type processes which have been developed to prepare oxide powders in order to improve their quality (purity, chemical homogeneity, etc.) and their reactivity, the process involving citric acid as a complexing agent was preferentially used. It is not, strictly speaking, a classic sol-gel process in the usual sense that the gel is not formed by a metal-oxygen-metal network, but rather from calcination of metal-organic complexes, thus producing ultrafine reactive powders with an excellent chemical homogeneity 3 . Rare-earth oxides separately dissolved in concentrated hydrochloric acid (37 %, Merck) and a tantalum oxalate solution (H.C. Starck, [Ta2O5] = 175 g L -1 ) were used as starting materials. Citric acid (C6H8O7,  99 %, Merck) dissolved in a minimum amount of water was added to each solution in the proportion of one mole per cation valence, the addition being followed by a 20 min stirring step at 120°C. As the complexation of cations by citric acid is improved at pH  7, the acidic solutions were neutralized by ammonia solution (25 %, Merck) 13 . The solutions were then mixed and the resulting solution stirred at 150°C for 20 min to promote polymerization. The liquid was then progressively heated up to 250°C, leading after 5 h to an expanded black solid residue. This solid was finally ground and calcined at 600°C in air in an alumina crucible until total elimination of carbon.

Thermal ammonolysis. Nitridation reactions were carried out in alumina boats placed inside an electric furnace through which ammonia gas flowed with a flow rate of 40-50 L h -1 . The temperature was raised in the 900-950°C range at a heating rate of 10°C min -1 . Generally after 15 h reaction time, the furnace was switched off and the nitrided powders were allowed to cool to room temperature under pure nitrogen atmosphere.

X-ray diffraction. XRD powder patterns were recorded using a Philips PW3710 diffractometer operating with Cu K  radiation (  = 1.5418 Å). X'PERT softwares -Data

Collector and Graphics, and Identify -were used, respectively, for pattern recording, analysis and phase matching. The lattice parameters were refined using Dicvol. 14 Neutron diffraction. Neutron diffraction experiments were performed on the two-axis powder diffractometer 3T2 at the Orphée 14 MW reactor of the CEA Saclay (France), using  = 1.2251 Å as the neutron wavelength. The diffraction pattern of the oxynitride powder was recorded over the angular range 6°< 2 < 120°in steps of 0.05°. The crystal structure of the oxynitride was refined using the Rietveld method with Fullprof 15 .

Elemental analysis. Nitrogen and oxygen contents were determined with a LECO TC-436 analyzer using the inert gas fusion method. Nitrogen was measured as N2 by thermal conductivity and oxygen as CO2 by infrared detection. The apparatus was calibrated using N2 and CO2 gas (purity  99.95 %) as well as -TaN as a nitrogen standard [START_REF] Dopita | Defect and Diffusion Forum[END_REF] .

Specific surface area.

A Flowsorb II 2300 Micromeritics apparatus was used to determine the specific surface area of the powders by the single point BET method. Before measurement, the samples were outgassed under He/N2 flow between 100 and 200°C for 30 min.

Results and Discussion

Two groups of rare-earth tantalates RTaO4 can be distinguished from a crystallographic viewpoint, according to the rare-earth size. With R = La→Pr, they show a BaMnF4-type structure and a LaTaO4-type structure, both related to perovskite 17,18 . The tantalates containing smaller rare-earths (R = Nd→Lu, Y) are characterized by a fergusonitetype monoclinic unit cell at low temperature and a tetragonal one at high temperature, which form corresponds to a scheelite-type phase usually obtained through a reversible phase transition at temperatures higher than 1300°C 19 .

As described previously 20 , thermal ammonolysis of LaTaO4 prepared by the ceramic route only results in a distorted perovskite-type oxynitride, LaTaON2. Its crystal structure was refined recently by X-ray and neutron diffraction from a well-crystallized powdered sample resulting from a mineralisator-assisted ammonolysis 21 fergusonites showed coexistence with the perovskite of another phase, less rich in nitrogen, assumed to be a pyrochlore R2Ta2O5N2 3,4 . Its stability increasing as the rare-earth ionic radius decreases, it could be obtained as a single phase with R = ErYb and Y. The corresponding cubic lattice parameters of the so-called "pyrochlores" are listed in Table 1, going from 10.56

Å for R = Nd to 10.235 Å for R = Yb 4 . RTaON2 perovskites with R = Nd and Sm crystallize in an orthorhombic GdFeO3-type unit cell 20 . We have shown in this study that GdTaON2

presents the same deformation, unlike it was previously observed (cubic unit cell, a = 4.022(1) Å) 20 . Unit cell parameters have been refined to a = 5.552(3) Å , b = 5.690(2) Å and c = 7.923(3) Å (after XRD pattern in Fig. 1). The citrate route was also advantageously utilized to prepare the gadolinium oxynitride phase which was unambiguously identified as a pyrochlore (Fig. 2). Starting from an X-ray amorphous GdTaO4 precursor calcined at 650°C, the reaction under ammonia flow at 900°C for only 5 h produced an orange powder with a refined cubic unit cell parameter a = 10.42(1) Å (Table 1). Its elemental analysis gave a nitrogen content (3.72 ± 0.06 wt.%) slightly higher than the calculated value (3.57 wt.%) for Gd2Ta2O5N2, indicating presumably beginning formation of the nitrogen-rich perovskite oxynitride GdTaON2 (N wt.% = 7.33). The perovskite formation was confirmed for higher reaction times by the presence of extra weak peaks, as clearly observed in Fig. 2 after 15 h nitridation at 900°C.

In the case of smaller rare-earths Ho, Er, Yb and Y, we have studied the influence of the preparation method of the oxide precursors on the nature of the ammonolysis products.

RTaO4 tantalates (R = Ho, Er, Yb and Y) prepared by the ceramic route were nitrided at 950°C. Under these conditions, the reaction kinetics is quite slow, requiring 5 nitridation cycles -15 h each -to totally eliminate the starting oxide from the final product. Fig. 3a displays a representative X-ray powder pattern of that series obtained after ammonolysis of HoTaO4. No single oxynitride was isolated due to the presence of extra peaks belonging to a perovskite phase. While we can confirm unambiguously the pyrochlore structure for the larger rare-earths (for ex. Nd or Sm), X-ray powder patterns for smaller R elements (R = HoYb, Y) are indexed in a fluorite-type unit cell with 1/8 the volume (space group Fm 3 m).

Comparison between Figures 2 and 3a

shows that the pyrochlore-type X-ray powder pattern differs from a fluorite one by an additional low intensity peak (hkl: 311) located at 2θ ~28.8°( Cu Kα) which requires to double the fluorite cubic unit cell parameter. It could be thought logical a priori not to find that peak in the case of Ho, Er or Yb, as their atomic number (Z = 67, 68 and 70, respectively) is relatively close to that of tantalum (ZTa = 73), but in the case of Y (ZY = 39), this peak should be present if a cationic order existed, like in an A2B2X7 pyrochlore-type arrangement. Actually, from dysprosium onwards, the oxynitride phase which forms is really a fluorite, not a pyrochlore. Several attempts were carried out at lower temperature (925°C) to only have the fluorite, but this temperature corresponds more or less to the reaction onset. On the other hand, an ammonolysis carried out at 1000°C induced a mixed-valent state of tantalum as indicated by the black color of the resulting powder.

Mixtures were obtained also with the other R elements (R = Er, Yb and Y) studied, evidencing the limit of the ceramic route: the starting oxides are well crystallized powders with low specific surface area, and consequently low reactivity.

On the contrary, a complexation-calcination method could synthesize fine powders, potentially able to react with ammonia at lower temperature 22 . Illustrated by HoTaO4, the citrate route yields a black expanded solid which gives after progressive calcination up to 600°C an X-ray amorphous diffraction powder pattern. Heatings performed at higher temperature, up to 900°C, improve the crystallization of the powder, without reaching, however, a better crystallization degree than that shown in Fig. 4, where a comparison is made with the diffraction profile of the same composition, HoTaO4, prepared by a conventional route. Let us observe that the citrate route does not lead to the expected fergusonite, but to the high temperature scheelite polymorph. This phase being metastable, its heating at higher temperature than ~900°C induces rapidly the phase transition scheelite → fergusonite, as the latter is thermodynamically stable up to ~1300°C. Scheelite phases ErTaO4, YbTaO4 and YTaO4 were isolated in the same way at low temperature. As illustrated in Fig. 3b with holmium, these scheelite-type tantalates (R = Ho, Er, Yb and Y) can be nitrided as low as 900°C and transformed easily into pure fluorite oxynitride phases. Only one 15 h treatment in flowing ammonia was enough to prepare dark yellow oxynitride powders. The proportion of incorporated nitrogen increases as a function of nitridation temperature and time, thus delimiting in each case an oxynitride solid solution domain (Table 2). This progressive nitrogen enrichment is balanced by an increasing number of vacancies.

As the electroneutrality rule requires that 3 O 2-are replaced by 2 N 3-, according to the following equation: 3 O 2-= 2 N 3-+ 1 □, one vacancy is created per two introduced nitrogen atoms. It results in a slight decrease in the cubic unit cell parameter value. Let us note that after long nitridation times (> 90 h) at 900°C, extra weak peaks belonging to a perovskitetype oxynitride phase became detectable.

Another clear indication of the existence of continuous solution domains was the powder color after reaction, as the progressive nitrogen enrichment resulted in a continuous color variation related to the shift of the absorption edge position in the Visible 6 . If we choose the case of yttrium which does not show any interference with f-type transitions, the color of the YTa(O,N,□)4 powders varies from yellow (after 15 h) to brown for the most substituted compositions (after 90 h).

Furthermore, we have observed that the synthesis of the fluorite oxynitrides was easier when starting from a scheelite precursor than from a fergusonite one. Two relevant arguments can be set out to explain the reaction kinetics, the morphology of the precursors and their crystal structure. On one hand, the reactivity of the "citrate route" powder is correlated with a high specific surface area (≥ 15 m 2 g -1 ), compared to a "ceramic route" powder (< 2 m 2 g -1 ). The ammonolysis consists of a solid-gas reaction where the nitrogen species have to diffuse deep inside the grains of powder to produce an homogeneous nitrided phase, while corresponding oxygen is removed as water vapor. Consequently, high specific surface areas will lead more effectively to powders homogeneously nitrided to the core. On the other hand, from a structural viewpoint the fluorite structure looks closer to the scheelite structure which can be described as resulting from the stacking of two fluorite unit cells [START_REF] Galasso | Structure and Properties of Inorganic Solids[END_REF] .

The change from fluorite to scheelite amounts to a cationic ordering and a global shift of the anions towards Ta atoms (in our case).

To sum up, we can state that cationic order (or the pyrochlore structure) exists for the large rare-earths when the disparity of ionic radius is really significant, while a disordered RTa(O,N,)4 phase (or a defect fluorite structure) is formed with the smaller elements, from Ho, or even Dy. It is surprising, however, to index the XRD powder patterns in a small fluorite unit cell, which implies necessarily a single cationic position when the ionic radii (in eightfold coordination, after Shannon [START_REF] Shannon | [END_REF] ) are rather different: Ta 5+ : 0.74 Å; Dy 3+ , Ho 3+ →Yb 3+ :

1.027 Å, 1.015 Å→0.985 Å (Y 3+ : 1.019 Å). Actually, a structure refinement by the Rietveld method, performed with the yttrium phase, produced abnormally high values of isotropic atomic displacement parameters, as an indication of a different site occupation from an unit cell to another. The resulting superstructure may be due to ordering between anions and vacancies, or to a shift from their ideal position.

More accurate anionic positions were expected from a neutron diffraction study where the respective neutron scattering lengths of yttrium, tantalum, oxygen and nitrogen are 7.75, 6.91, 5.805 and 9.36 fm [START_REF]Neutron data booklet[END_REF] . Fig. 5, Tables 3 and4 give details of the neutron diffraction refinement. The observations made by XRD are confirmed: a cubic fluorite structure, no superstructure peaks and high atomic displacement parameters. In addition, a wavy background, not present in the XRD pattern, denotes a more complex structure where anions and vacancies may be ordered. Therefore, the small fluorite unit cell corresponds only to an apparent structure. Unfortunately, the sample cristallinity was not enough to get reliable results from our attempts of high resolution electron microscopy.

  . It was solved in the C2/m space group (a = 8.0922(31) Å , b = 8.0603(2) Å, c = 5.7118(2) Å,  = 134.815(1)°), showing an ordered anionic sublattice. From R = Nd  Dy, Ho, thermal ammonolysis of the RTaO4

Fig. 1 .Fig. 2 .Fig. 3a .Fig. 3b .

 123a3b Fig. 1. XRD powder pattern of GdTaO2N after heating GdTaO4 at 1000°C under ammonia flow (4x15 h cycle).

Fig. 4 .

 4 Fig. 4. Comparison between the XRD powder patterns of scheelite-type HoTaO4 (citrate route, 900°C) and fergusonite-type HoTaO4 (ceramic route, 1400°C).

Fig. 5 .

 5 Fig. 5. Neutron diffraction powder pattern of the YTa(O,N,□)4 fluorite phase (N wt.% = 4.66).

Thermal Ammonolysis Study of the Rare-Earth Tantalates RTaO 4

	Pascal Maillard, Franck Tessier*, Emmanuelle Orhan, François Cheviré
	and Roger Marchand
	UMR CNRS 6512 "Verres et Céramiques", Institut de Chimie de Rennes,
	Université de Rennes 1, 35042 Rennes Cedex, France

Table 1 .

 1 Competition between pyrochlore (R2Ta2O5N2) and perovskite (RTaON2) structures within the R-Ta-O-N systems. (parameters after4,20 , except for *: this study). From Dy toYb, pyrochlore and defect-fluorite solid solutions parameters, determined in this study, are related by a factor 2.

	R	La	Nd	Sm	Gd	Dy	Ho	Y	Er	Yb
	Pyrochlore unit cell (Å)	-	10.56	10.501	10.44 10.42(1) *	10.34	10.327	10.301	10.282	10.235
	Perovskite unit
	cell (Å)

Table 2 .

 2 RTa(O, N, □)4 fluorite-type solid solutions: nitridation and unit cell parameters.

	R	Thermal ammonolysis conditions	N wt.%	Experimental formulation	Cubic unit cell parameter (Å)
	Ho	15h, 900°C 6 x 15h, 950°C	2.08 3.24	HoTaO3.10N0.60□0.30 HoTaO2.61N0.93□0.46	5.170(2) 5.1572(6)
	Y	15h, 900°C 6 x 15h, 950°C	3.56 4.66	YTaO2.76N0.83□0.41 YTaO2.39N1.08□0.53	5.1675(5) 5.1555(2)
	Er	15h, 900°C 6 x 15h, 950°C	2.97 3.19	ErTaO2.83N0.78□0.39 ErTaO2.62N0.92□0.46	5.156(2) 5.1507(7)

Table 3 .

 3 Details of Rietveld refinement for the YTa(O,N,□)4 fluorite phase (N wt.% = 4.66).

	.
	a (Å)	5.15268 (6)
	V (Å 3 )	136.8
	Space group	Fm 3 m
	Z	4
	 (Å)	1.2251
	No. of reflections	22
	Profile function	TCH 25
	Rp	28.6
	Rwp	15.2
	Rexp	12.91
	 2	1.39

Table 4 .

 4 Fractional atomic coordinates and isotropic displacement parameters for the YTa(O,N,□)4 fluorite phase (N wt.% = 4.66).

	Atom Position x y z Occupation Biso (Å 2 )
	Y	4a	0 0 0	0.5	2.11
	Ta	4a	0 0 0	0.5	2.11
	O	8c	1/4 1/4 1/4	0.53	6.95
	N	8c	1/4 1/4 1/4	0.31	6.95

Acknowledgments

The authors are grateful to Françoise Bourée (Laboratoire Léon Brillouin, Saclay, France) for recording the neutron diffraction data, as well as to Thierry Roisnel (Université de Rennes 1, France) for his advices about Fullprof and WinPlotr softwares.


Yi-Ping Fang

Nicola Pedroni

Senior Member, IEEE Enrico Zio

Resilience-Based Component Importance Measures for Critical Infrastructure Network Systems

In this paper, we propose two metrics, i.e., the optimal repair time and the resilience reduction worth, to measure the criticality of the components of a network system from the perspective of their contribution to system resilience. Specifically, the two metrics quantify: 1) the priority with which a failed component should be repaired and re-installed into the network and 2) the potential loss in the optimal system resilience due to a time delay in the recovery of a failed component, respectively. Given the stochastic nature of disruptive events on infrastructure networks, a Monte Carlo-based method is proposed to generate probability distributions of the two metrics for all of the components of the network; then, a stochastic ranking approach based on the Copeland's pairwise aggregation is used to rank components importance. Numerical results are obtained for the IEEE 30-bus test network and a comparison is made with three classical centrality measures.

C OMPLEXITY of critical infrastructures (CIs), such as power grids, the Internet, transportation networks, and so forth, is increasing. Disruptive events, whether they are malevolent attacks, natural disasters, or human-caused accidents, can have significant impacts on these real-world complex networks composed of numerous interconnected functional and structural elements.

Justifiably, then, critical infrastructure protection (CIP) has become a priority for all nations [START_REF] Zio | The Monte Carlo Simulation Method for System Reliability and Risk Analysis[END_REF]. The focus has been traditionally placed on physical protection and asset hardening [START_REF] Lewis | Critical Infrastructure Protection in Homeland Security: Defending a Networked Nation[END_REF]- [START_REF] Bush | Homeland Security Presidential Directive-7 (HSPD-7)[END_REF]. However, in recent years, lessons learned from some catastrophic accidents have pushed part of the focus on the concept of "resilience" [START_REF] Pursiainen | The challenges for European critical infrastructure protection[END_REF], [START_REF] Obama | Presidential Policy Directive 21: Critical Infrastructure Security and Resilience[END_REF]. The outcomes of the 2005 World Conference on Disaster Reduction (WCDR) witness the significance of introducing the term "resilience" into the disaster discourse, giving birth to a new culture of disaster response [START_REF] Cimellaro | Framework for analytical quantification of disaster resilience[END_REF]. Consequently, government policy has also evolved to encourage efforts that would allow assets to continue operating at some level or quickly return to full operation after the occurrence of disruptive events [START_REF] Moteff | Critical Infrastructure Resilience: The Evolution of Policy and Programs and Issues for Congress[END_REF].

"Resilience" comes from the Latin word "resilio" that literally means "to leap back" and denotes a system attribute characterized by the ability to recover from challenges or disruptive events. The Merriam-Webster dictionary defines resilience as "the ability to recover from or adjust easily to misfortune or change." In this view, systems should not only be reliable, i.e., having an acceptably low failure probability, but also resilient, i.e., having the ability to optimally recover from disruptions of the nominal operating conditions [START_REF] Zio | Reliability engineering: Old problems and new challenges[END_REF], [START_REF] Ouyang | Review on modeling and simulation of interdependent critical infrastructure systems[END_REF].

In this context, the present paper addresses the issue of quantifying the importance of components in contributing to the resilience of a critical infrastructure. Component importance measures (CIMs) have been thoroughly studied in the field of reliability theory and risk analysis. Various analytical and empirical CIMs have been proposed in the literature, e.g., Birnbaum [START_REF] Birnbaum | On the importance of different components in a multicomponent system[END_REF], Fussell-Vesely [START_REF] Fussell | How to calculate system reliability and safety characteristics[END_REF], Reliability Achievement/Reduction Worth [START_REF] Gandini | Importance and sensitivity analysis in assessing system reliability[END_REF], [START_REF] Levitin | Generalized importance measures for multistate elements based on performance level restrictions[END_REF], Barlow-Proschan [START_REF] Barlow | Importance of system components and fault tree events[END_REF], Natvig [START_REF] Natvig | New light on measures of importance of system components[END_REF], and their extensions [START_REF] Ramirez-Marquez | Composite importance measures for multi-state systems with multi-state components[END_REF]- [START_REF] Fang | Hierarchical modeling by recursive unsupervised spectral clustering and network extended importance measures to analyze the reliability characteristics of complex network systems[END_REF], [START_REF] Natvig | Measures of component importance in repairable multistate systems-A numerical study[END_REF], [START_REF] Natvig | Multistate Systems Reliability Theory with Applications[END_REF], [START_REF] Natvig | Measures of component importance in nonrepairable and repairable multistate strongly coherent systems[END_REF]. CIMs have been shown valuable in establishing direction and prioritization of actions related to an upgrading effort (e.g., reliability improvement) in system design, or in suggesting the most efficient way to operate and maintain system status. However, none of the existing classical CIMs based on the reliability concept are directly applicable to the post-disaster phase, since there is no scope to exhibit reliability after the occurrence of system failure.

The role that a component plays in a network system has been measured by various so-called centrality measures, looking from the point of view of the complex interaction and communication flow in the network [START_REF] Borgatti | Centrality and network flow[END_REF], [START_REF] Kröger | Vulnerable Systems[END_REF]. Classical topological centrality measures are the degree centrality [START_REF] Nieminen | On the centrality in a graph[END_REF], [START_REF] Freeman | Centrality in social networks conceptual clarification[END_REF], the closeness centrality [START_REF] Freeman | Centrality in social networks conceptual clarification[END_REF]- [START_REF] Wasserman | Social Network Analysis: Methods and. Applications[END_REF], the betweenness centrality [START_REF] Freeman | Centrality in social networks conceptual clarification[END_REF], and the information centrality [START_REF] Latora | A measure of centrality based on network efficiency[END_REF]. They specifically rely on topological information to qualify the importance of a network component. Additionally, Freeman et al. [START_REF] Freeman | Centrality in valued graphs: A measure of betweenness based on network flow[END_REF] proposed a flow betweenness centrality measure based on the idea of maximum network flow; Newman [START_REF] Newman | A measure of betweenness centrality based on random walks[END_REF] suggested a random walk betweenness measure that counts essentially all paths between vertices and which makes no assumptions of optimality; Jenelius et al. [START_REF] Jenelius | Importance and exposure in road network vulnerability analysis[END_REF] proposed several vulnerability-based importance measures for transportation networks; Hines and Blumsack [START_REF] Hines | A centrality measure for electrical networks[END_REF] introduced an "electrical centrality" measure for electrical networks by taking into account the electrical topology of the network; Zio and Piccinelli [START_REF] Zio | Randomized flow model and centrality measure for electrical power transmission network analysis[END_REF] provided a randomized flow model-based centrality measure specifically for electrical networks; Zio and Sansavini [START_REF] Zio | Component criticality in failure cascade processes of network systems[END_REF] introduced component criticality measures from the cascade failure process point of view, for general network systems. Nevertheless, none of these analyses takes into account the dynamics of system recovery from the effects of a disruptive event.

Resilience-based metrics of component criticality with respect to their influence on the overall resilience of the system (i.e., on the system's ability to quickly recover from a disruptive event) can be helpful for preparing an efficient component repair checklist in the event of system failure [START_REF] Natvig | Measures of component importance in repairable multistate systems-A numerical study[END_REF]. Natvig et al. introduced a dual extended Natvig measure for repairable systems: according to this measure, the components that are considered important are those whose repair reduces significantly the expected time of residence of the system in the worst states [START_REF] Natvig | Measures of component importance in repairable multistate systems-A numerical study[END_REF]. Hence, this dual Natvig measure is a resilience measure for multistate components in a multistate system. A dual extension of the Barlow-Proschan measure has also been suggested for multistate repairable systems, based on the probability that the repair of the th component is the cause of a system state improvement, given that this has occurred [START_REF] Natvig | Measures of component importance in nonrepairable and repairable multistate strongly coherent systems[END_REF].

Recently, Barker et al. [START_REF] Barker | Resilience-based network component importance measures[END_REF] introduced two resilience-based network component importance metrics. Although the resilience definition, which the importance metrics rely on, actually embraces the temporal dimension of system recovery, it can be considered "memoryless": in the sense that the system resilience metric at a given time does not take into account the information about the restoration behavior before time . Thus, it may happen that different restoration curves with different levels of favorability have the same value of resilience . In addition, the two metrics introduced in [START_REF] Barker | Resilience-based network component importance measures[END_REF] seek to quantify the effect that the disruption, rather than the recovery (behavior), of an individual component has on the system resilience, implying that the resilience improvement is achieved by actions related to system protection and fortification efforts in system design, although they have also been applied to compare network recovery strategies [START_REF] Baroud | Importance measures for inland waterway network resilience[END_REF].

In this study, a new definition of system resilience and a resilience optimization framework are first presented, based on which we then introduce two network components importance measures, namely, the optimal repair time and the resilience reduction worth, useful for prioritizing restoration activities. The two measures quantify: 1) the priority with which a failed component should be repaired and re-installed into the network and 2) the potential loss in the optimal system resilience due to a time delay in the recovery of a failed component, respectively. A stochastic ranking technique proposed by Barker et al. [START_REF] Barker | Resilience-based network component importance measures[END_REF], which is based on the Copeland's pairwise aggregation [START_REF] Merlin | Copeland method II: Manipulation, monotonicity, and paradoxes[END_REF], is applied in this paper to rank the components' criticalities.

As a case study, the IEEE 30-bus test network is considered: the criticalities of the components computed by the proposed indicators are compared with those produced by three classical measures of betweenness centrality [START_REF] Freeman | Centrality in valued graphs: A measure of betweenness based on network flow[END_REF], [START_REF] Newman | A measure of betweenness centrality based on random walks[END_REF], [START_REF] Kinney | Modeling cascading failures in the North American power grid[END_REF].

It is noted that the differences of the resilience-based CIMs proposed in the present paper with respect to that in [START_REF] Barker | Resilience-based network component importance measures[END_REF] mainly fall into the following two aspects: 1) the concept and definition of system resilience which the CIMs rely on takes into account the cumulative restored system functionality and 2) the focus of the proposed resilience-based CIMs is to quantify the effect that the recovery, rather than the disruption, of individual components has on the global system resilience, for most effective system (recovery) operation.

The remainder of this paper is organized as follows. Section II provides the general framework of the study, recalling the definition of system resilience and the resilience optimization model. In Section III, two measures of component criticality for system resilience, and a simulation methodology for their calculation and ordering are presented. Section IV illustrates the calculation of the proposed metrics on the IEEE 30-bus test network: the obtained components rankings are compared with those produced by classical betweenness centrality measures. Concluding remarks are drawn in Section V.

II. METHODOLOGICAL BACKGROUND: SYSTEM RESILIENCE DEFINITION AND OPTIMIZATION FOR INFRASTRUCTURE NETWORK SYSTEMS

This section provides the definition of system resilience and the resilience optimization framework, which serve as methodological background for the resilience-based component importance measures that will be discussed in Section III.

A. System Resilience Definition

As illustrated in Fig. 1, a quantifiable and time-dependent system performance function (also called system level delivery function or figure-of-merit) is the basis for the assessment of system resilience [START_REF] Cimellaro | Framework for analytical quantification of disaster resilience[END_REF], [START_REF] Barker | Resilience-based network component importance measures[END_REF], [START_REF] Baroud | Importance measures for inland waterway network resilience[END_REF]. It has a nominal value under nominal operating conditions. The system operates at this level until suffering a disruptive event at time . The disruption generally deteriorates system performance to some level at time . Then, recovery action is started, affecting and improving system performance until it achieves, at a later time , a targeted level of performance that could be the same, close to, or better than original system performance , for which recovery is considered completed. The dotted curve in Fig. 1 denotes the targeted system performance if not affected by disruption, which is generally evolving due to the dynamic nature of service demand and system upgrading; in this study, it is assumed to be equal to and remain invariant, for simplicity of illustration. In addition, it is noted that various strategies exist for recovery activities, and system performance is ultimately a function of recovery decisions. The period of is generally considered as the recovery time [START_REF] Cimellaro | Framework for analytical quantification of disaster resilience[END_REF]. Let be the resilience of a system at time . In its basic form, describes the cumulative system functionality that has been restored at time , normalized by the expected cumulative system functionality supposing that the system has not been affected by disruption during this time period and given as [START_REF] Zio | The Monte Carlo Simulation Method for System Reliability and Risk Analysis[END_REF] Graphically, is quantified by the ratio of the area with diagonal stripes to the area of the shaded part , as shown in Fig. 2.

Note that the formulation in (1) focuses mainly on the recoverability dimension of resilience and is in the range of [0, 1]. when , which means that a system has not recovered from its disrupted state (i.e., there has been no "resilience" action); when , which corresponds to the ideal case where a system recovers to its target state immediately after disruption. This resilience quantification is consistent with the original meaning of the concept of resilience and is capable of measuring at the same time the magnitude and rapidity of system recovery action. More importantly, this definition of system resilience is not memoryless since it considers cumulatively the system functionality restored, differently from [START_REF] Barker | Resilience-based network component importance measures[END_REF].

B. System Resilience Optimization

A disruptive event could impact one or more components of an infrastructure network system. In the case of multiple component failures, a systemic recovery action should be undertaken with the order of failed components to repair such that system resilience is maximal, i.e., to achieve optimal (restored) cumulative system functionality over the recovery time considered.

A variety of frameworks of optimization for post-disaster recovery of an infrastructure network system can be designed, focusing on different aspects of the restoration strategy, e.g., the order of repair of the damaged components, where and how to allocate repair resources and so forth. This study focuses on the significance of the recovery of a component with respect to the resilience of the system. Consequently, the optimization is designed to find the optimal order of repair of the set of failed components with the objective of achieving maximum system resilience over the restoration time horizon.

The mathematical model for the resilience optimization concerns a network comprising a set of nodes connected by a set of links or arcs . The network nodes are distinguished in supply nodes , transshipment nodes , and demand nodes

. Each arc has an associated capacity , each supply node has a supply capacity per time unit and each demand node has a demand per time unit. Network flow is delivered from supply nodes to demand nodes respecting the flow capacities of the links and supply/demand capacities of the nodes. The performance of the network is evaluated by determining the maximum amount of flow that can be received by the demand nodes. Formally, the system performance function is defined as [START_REF] Lewis | Critical Infrastructure Protection in Homeland Security: Defending a Networked Nation[END_REF] where represents the amount of flow received by demand node at time .

Disruptions happen and create damages to nodes and/or links in the network, which is modeled as removal of a subset of arcs, , from the network. 1 The arcs in set are viewed as non-operational immediately after the disruption. System performance achieves its minimum value at this time, which is seen as the initial stage of system restoration that we focus on (we set at this time for computational convenience, i.e., ). The recovery optimization framework aims at identifying the subset of links in to repair and the order, in which the links should be repaired so as to achieve maximum system resilience over the restoration horizon . In this study, link repairs are assumed to be discrete tasks and only a single arc can be repaired at a given time period. Thus, discrete time periods are considered, hereafter, instead of the continuous one in the resilience definition of (1). In addition, we do not model in detail the mechanisms and procedures according to which a single failed component is repaired; rather, the focus is on the identification of the exact (optimal) time when the disrupted arcs should be brought back online and, thus, on the temporal sequence of the restoration actions on all the failed components. Obviously, the inclusion of possibly different repair times for different components may produce a different component ranking. However, this would be automatically accounted for in the procedure by the inclusion of proper "hard" constraints and, thus, it would not impair the applicability and generality of the approach . By combining (1) and (2), system resilience to be maximized at time is given by (3) 1 For nodes, they can be converted to equivalent arcs by introducing additional arcs and nodes into the network, i.e., by "splitting" a node into two nodes and an arc.

where is the target system performance. The variables of the resilience optimization problem include: 1) continuous variables , and , that denote the flow moving from node to node through link at time unit ; 2) continuous variables , , that represent the amount of flow received by demand node at time unit , and 3) binary state variables , and , such that if arc is operational and if arc is not operational at time unit . We are interested in optimizing the resilience over the whole restoration process: thus, the timespan is the total recovery time, defined as the period necessary to restore the system functionality to the same level as the original system. Consequently, the formulation of the resilience optimization problem is as follows: The objective (4) is to maximize the system resilience over the time horizon of recovery. Constraints (5)-( 9) are typical network flow constraints over the links and supply/demand nodes in the network in period . They ensure that: the flow generated at a supply node does not exceed its supply capacity [START_REF] Bush | Homeland Security Presidential Directive-7 (HSPD-7)[END_REF]; the amount of net injected flow at a transshipment node is zero [START_REF] Pursiainen | The challenges for European critical infrastructure protection[END_REF]; the amount of net injected flow at a demand node is equal to the received flow at the node [START_REF] Obama | Presidential Policy Directive 21: Critical Infrastructure Security and Resilience[END_REF] while not exceeding its requested demand [START_REF] Cimellaro | Framework for analytical quantification of disaster resilience[END_REF]; the flow on an operational link does not exceed its capacity and there is no flow passing through an arc if the arc is failed [START_REF] Moteff | Critical Infrastructure Resilience: The Evolution of Policy and Programs and Issues for Congress[END_REF]. Constraint [START_REF] Zio | Reliability engineering: Old problems and new challenges[END_REF] ensures that once an arc has been restored at time , it will keep operational thereafter. Finally, constraint [START_REF] Ouyang | Review on modeling and simulation of interdependent critical infrastructure systems[END_REF] ensures that only a single arc can be repaired at any given timeslot. This resilience optimization above defined is a typical mixed integer programming (MIP) problem. A commercial optimization solver Cplex [START_REF]Cplex Optimization Studio IBM ILOG[END_REF] is used in this study for its solution. It is noted that this resilience optimization model is only applied for the purpose of illustration of resilience-based component importance metrics. More complex optimization models (e.g., taking into account the cost and duration of repairing a particular failed link) can be adopted in other application cases. Besides, a specific application involving the 400-kV French Power Transmission Network of the proposed resilience (restoration) optimization problem has been presented in [START_REF] Fang | Critical infrastructure protection by advanced modelling, simulation and optimization for cascading failure mitigation and resilience[END_REF]. While the present study mainly focuses on the definition of resilience-based CIMs and the analysis of their properties, as mentioned before.

III. MATH RESILIENCE-BASED CIMS FOR INFRASTRUCTURE NETWORK SYSTEMS

A. CIM Definition

As described in Section II-B, the analysis concerns a network comprising a set of nodes and a set of links . The binary state variable of arc at time is defined by , . The initial impact experienced by the network after a disruptive event at time is represented by the removal of a subset of arcs from the network, setting , . We introduce the failure probability of arc under event , [START_REF] Fussell | How to calculate system reliability and safety characteristics[END_REF] Equation ( 13) describes how individual components (links) are initially affected by a disruptive event . Section II-B explains how these failed components optimally recover from the disruption state following the event. Finally, (1) incorporates these dimensions to quantify system resilience.

When considering component criticality in a resilience setting, we are interested in understanding: 1) the optimal time to repair the failed components in order to maximize system resilience and 2) the effect that the timely recovery of the components have on the overall resilience of the system. These concepts are at the basis of the definition of the two resilience-based importance measures here proposed.

Given a particular initial failure state, the optimal repair time of a failed arc is defined as the time when the arc is restored to operating status so to maximize the system resilience over the recovery time , given by [START_REF] Gandini | Importance and sensitivity analysis in assessing system reliability[END_REF] It can be obtained by first solving the MIP problem (4)-( 12) and then computing it based on the state variables of the arc as [START_REF] Levitin | Generalized importance measures for multistate elements based on performance level restrictions[END_REF] The timespan for restoration is chosen as the time period necessary to restore the system functionality to the same level as the original system. It is noted that the optimal repair time offers an explicit quantification of the priority that should be given to the reparation and installation of arc into the network. Low values of indicate higher priority of being repaired and re-installed into the network, i.e., higher ranking of the component in the repair checklist.

To account for the delay in the restoration of a particular link , a resilience reduction worth (RRW) metric is introduced as ( 16)

where represents the optimal system resilience at restoration time , corresponds to the optimal system resilience at time if link cannot be repaired until time

, where is the delay with respect to its optimal repair time given by [START_REF] Gandini | Importance and sensitivity analysis in assessing system reliability[END_REF]. Equation ( 16) quantifies the potential (normalized) loss in optimal system resilience due to a delay in the repair of link . This metric is comparable to the so-called reliability reduction worth [START_REF] Espiritu | Component criticality importance measures for the power industry[END_REF], which measures the potential damage caused to the system reliability by the failure of a particular component. It can provide valuable information to guide the recovery process of a particular component. Components with high values of should be given high priority in the restoration process, e.g., be assigned adequate restoration resources to avoid delays that would have a more significant impact on system restoration.

It is noted that the resilience-based CIMs in [START_REF] Barker | Resilience-based network component importance measures[END_REF] are introduced to quantify the effect that the disruption, rather than the recovery (behavior), of an individual component has on the system resilience (represented by the system recovery time). Specifically, the first metric in [START_REF] Barker | Resilience-based network component importance measures[END_REF] measures the contribution that the disruption of link has on the system recovery time, weighted by the ratio of system service loss due to the disruption effect on link to the maximum loss among all of the links. The second metric in [START_REF] Barker | Resilience-based network component importance measures[END_REF] quantifies how the system recovery time is improved if link is invulnerable. Both definitions imply that the resilience improvement is achieved by actions related to system protection and fortification efforts in system design. However, the two resilience-metrics introduced in the present paper measure the effect that the recovery of an individual component has on the global system resilience, by quantifying: 1) the priority with which a failed component should be repaired and re-installed into the network and 2) the potential loss in the optimal system resilience due to a time delay in the recovery of a failed component, respectively. Hence, our resilience-based metrics are able to suggest the most effective way for system operation, i.e., to help the implementation of: 1) recovery schedule planning and 2) restoration resources allocation.

B. Methodology for Component Importance Ordering

Ordering network links recovery on the basis of the values of the criticality measures described above, i.e., the optimal repair time and resilience reduction worth (fixed ), requires quantifying the effect of timely repairing these links on the overall resilience of the system. Given the stochastic nature of disruptive events in terms of components failures after the event, the resilience-based criticality measures introduced are not represented by deterministic values, but rather by probability distributions. Therefore, given a network under a disruptive event , we first apply a Monte Carlo-based method to generate distributions of optimal repair time and resilience reduction worth for all of the links in the network. Then, we rank links importance using a Copeland's pairwise aggregation-based stochastic approach proposed by Barker et al. [START_REF] Barker | Resilience-based network component importance measures[END_REF]. The detailed steps of the algorithm are as follows.

1) A network is initially operating with a given parameters setting: flow demand of all of the demand nodes in , supply capacity of all of the supply nodes in and link capacity for all of the network arcs in .

2) A failure configuration of the network is randomly sampled on the basis of the failure probabilities of each arc in the system given by ( 13), under a disruptive event at initial time . The operation state variables of failed links are set to 0, i.e., , .

3) The resilience optimization model of ( 4)-( 12) is applied and solved by Cplex to obtain the optimal strategy of network recovery, i.e., the optimal repair time for each failed arc . 4) In order to evaluate the second importance measure , for each failed arc , the additional constraint that the restoration of arc should not be accomplished earlier than (i.e., ) is added to the optimization model of ( 4)- [START_REF] Birnbaum | On the importance of different components in a multicomponent system[END_REF]. Then, is obtained by solving this "modified" optimization model by Cplex. Finally, the resilience reduction worth for each arc is recorded. 5) To account for the stochasticity of the disruptive event in terms of arcs failures, repeat Step 2 to Step 4 for a chosen number of iterations, generating probability distributions for and , for all of the links in the network. 6) Given the distributions of (resp., ) for each arc , perform a stochastic ranking of links according to ascending (resp., descending) values (see Section III-C).

C. Stochastic Ranking

In order to rank network links according to the distribution of their optimal repair time (or resilience reduction worth ) obtained at step 6) of the algorithm above, an approach based on the Copeland's pairwise aggregation method [START_REF] Merlin | Copeland method II: Manipulation, monotonicity, and paradoxes[END_REF] is proposed. The Copeland's method (CM) is a simple nonparametric Condorcet method used in the political field (voting) that does not require any information about decision maker preference and operates on a multi-indicator matrix formed by objects characterized by attributes [START_REF] Pomerol | Multicriterion Decision in Management: Principles and Practice[END_REF]. CM relies on pairwise comparisons between objects in the candidate pool, and the so-called Copeland score is defined for each object as the difference between the number of times that this object beats the other objects and the number of times that it is beat by other objects.

The CM-based ranking approach applied here corresponds to a modification proposed by Al-Sharrah [START_REF] Al-Sharrah | Ranking using the Copeland score: A comparison with the Hasse diagram[END_REF]. It first examines the CDF of a given variable for all the candidates, e.g., the CDF of ; then, it compares the CDF of two candidates under analysis, i.e., links and , with respect to specific attributes of the CDF: for example, attribute may represent the th percentile. Subsequently, a quantity is calculated based on a pairwise comparison between links and with respect to (percentile) of the corresponding distributions , as where the sentence means that dominates with respect to the ranking rule of the variable considered, i.e., for , while if is considered, while symbol " " represents the opposite meaning.

if if if ( 17
)
is initialized at zero for the first (percentile) and ( 16) is iterated through all attributes (percentiles). Then, the Copeland score for each link is defined as [START_REF] Andrews | Birnbaum's measure of component importance for noncoherent systems[END_REF] This Copeland score is finally used to rank all of the links: the higher , the higher the contribution of link to the overall resilience of the network.

IV. CASE STUDY

A. Resilience-Based Criticality Measures on the IEEE 30-Bus Test System

The IEEE 30-bus test system [START_REF]Power System Test Case Archive[END_REF] is taken as a reference case study for the proposed resilience-based component importance measure approach. This system (Fig. 3) represents a portion of the American Electric Power System and is composed of 30 buses connected by 41 transmission lines. To carry out the analysis, each system component is transposed into a node or edge of the representative topological network, as shown in Fig. 4. Three different physical types of nodes are considered: generator nodes (where the electricity flow is fed into the network), demand nodes (where customers are connected), and transfer or transmission nodes (without customers or sources).

The simulation procedure introduced in Section III-B is then used to rank each component of the IEEE 30-bus network according to the criticality metrics introduced. In normal condi- tions, the network is assumed to operate under the following parameters setting: the generation capacity is identical for all generation nodes and equal to 60, in arbitrary units (a.u.); the flow demands are 20 a.u. for all load nodes; the values of the transmission capacities are 20 a.u. for all of the network links. The homogeneous assignments of generation capacity, demand, and link capacity are here applied for the purpose of identifying the resilience criticalities of all of the network arcs stemming from their different topological connections. For the same reason, a constant failure probability is assumed for all of the network links under disruptive event . The roulette wheel selection method [START_REF] Lipowski | Roulette-wheel selection via stochastic acceptance[END_REF] is used in step 2 for sampling a failure configuration by selecting a failed link at each spin until a certain number of arcs are selected. Fig. 5 illustrates the cumulative distribution functions (CDFs) of for five representative links ( , , , and

), obtained at step 5) of the procedure by applying the simulation algorithm proposed in Section III-B (for samples). The figure illustrates the probability that is less than or equal to a target value . It can be seen that the optimal repair time associated with link , i.e., , will never be larger than 5 (square-line curve in Fig. 5). Moreover, the curve for link always "dominates" the other curves. Therefore, this link should have the highest priority to be repaired in order to maximize system resilience.

However, considering e.g., links (circle line) and (triangle line) in Fig. 5, it is not evident which one "dominates" the other, due to the intersection of their CDF curves. Thus, the CM-based ranking approach introduced in Section III-C is applied to rank the importance of the links. Fig. 6 reports the Copeland scores of all the 41 links in the Fig. 5. Cumulative probability distributions of the optimal repair time for five representative links. Fig. 6. Copeland score ranking of the optimal repair time for all IEEE 30-bus network links. IEEE 30-bus network, ordered in descending order, with link having the highest score, followed by links , , and so forth. Furthermore, Fig. 7 graphically illustrates the Copeland score of the optimal repair time for all IEEE 30-bus network links, where links with higher values of Copeland score are represented as thicker and darker edges. It is shown that two types of links are more important in terms of : 1) the links which connect the generator nodes with the other two types of nodes (transmission nodes and demand nodes), e.g., links , , , etc., and 2) the links which are the only ones connected to demand nodes, e.g., link

. The restoration of these types of links is most likely able to augment the total amount of flow received by the demand nodes of the network: thus, high priority should be given to these links when considering the repair order of the failed links.

Figs. 8 and9 illustrate the results based on the resilience reduction worth for all the links and for a delay time 3 units, i.e., the Copeland score ranking and its graphical representation, respectively. It is noted that different sets of values of the resilience reduction worth for all network links can be obtained under different values of delay ; however, the ranking of for all the links by the Copeland method will keep consistent in our study. It is shown that is the most critical link in terms of , i.e., a delay in its restoration would cause the largest reduction in system resilience among all the network links; thus, adequate resources should be given to make sure of its timely restoration. Besides, it is noted that the links with high Copeland scores in terms of the optimal repair time also have high Copeland score ranking in terms of the resilience reduction worth : the correlation coefficient between the two Copeland scores is for .

B. Comparison With Betweenness Centrality Measures

Betweenness centrality indices have been introduced as measures of component importance in a network, taking into account the different ways in which a component interacts and communicates with the rest of the network [START_REF] Freeman | Centrality in social networks conceptual clarification[END_REF], [START_REF] Zio | Randomized flow model and centrality measure for electrical power transmission network analysis[END_REF]. A classical centrality measure is the topological betweenness centrality introduced in the social network field, which is based on the idea that a component is central if it is lies between many other components, in the sense that it is traversed by many of the shortest paths connecting pairs of nodes [START_REF] Freeman | Centrality in social networks conceptual clarification[END_REF], usually called as shortest path betweenness. The topological betweenness centrality of a given link in a supply-demand differentiated network is given by [START_REF] Kinney | Modeling cascading failures in the North American power grid[END_REF] (

) 19
where is the number of topological shortest paths between supply nodes and demand nodes, and is the number of supply-demand shortest paths passing though link .

To account for the issue that in some cases flow may not follow the ideal geodesic paths from supply to demand nodes, a betweenness centrality measure based on the idea of maximum network flow has been proposed [START_REF] Freeman | Centrality in valued graphs: A measure of betweenness based on network flow[END_REF], usually known as flow betweenness. The measure counts all independent paths that carry information when a maximum flow is pumped between each pair of vertices. The flow betweenness of a component is defined as the amount of flow through it when the maximum flow is transmitted from source to demand , averaged over all and . It is quantitatively defined as [START_REF] Freeman | Centrality in valued graphs: A measure of betweenness based on network flow[END_REF] (20) where is the maximum flow from a source node to a demand node and is the maximum flow from to that passes through link . In practical terms, however, neither of the two betweenness measures introduced above is realistic. Both count only a small subset of possible paths between vertices, and both assume some kind of optimality in information transmission (shortest paths or maximum flow). Therefore, a new betweenness measure that counts essentially all paths between vertices and which makes no assumptions of optimality has been suggested, called random walk betweenness [START_REF] Newman | A measure of betweenness centrality based on random walks[END_REF]. This measure is based on random walks between vertex pairs and asks, in essence, how often a given component will fall on a random walk between another pair of vertices. Roughly speaking, the random walk betweenness of a link is equal to the number of times that a random walk starting at and ending at passes through the link along the way, averaged over all and . Let be the current flow from to , through link . Then, the random walk betweenness of a link is defined as [START_REF] Borgatti | Centrality and network flow[END_REF] We are interested in comparing the ranking results of our resilience-based component importance measures to these betweenness centrality indices, i.e., shortest path betweenness, flow betweenness and random walk betweenness for the proposed IEEE 30-bus network. Fig. 10 shows the values of the Copeland scores for the optimal repair time (left panel) and for the resilience reduction worth (right panel) plotted with respect to the shortest path betweenness for all of the links of IEEE 30-bus network. No obvious correlation can be identified from the figures. Actually, the correlation coefficients between , , and are and , respectively. Similarly, Fig. 11 plots the relationship between the Copeland scores for the optimal repair time (left panel) and the resilience reduction worth (right panel) with the flow betweenness ; Fig. 12 shows the same scatterplots with respect to the random walk betweenness .

The correlation coefficients are , , and , respectively. These results show that the betweenness centrality indices (e.g., shortest path betweenness, flow betweenness and random walk betweenness) do not capture the component criticality with respect to resilience for the recovery of the IEEE 30-bus network. This implies that these centrality measures (which are calculated under normal operation condition) are not applicable to guide the system restoration after a disruptive event, e.g., to prepare an efficient component repair priority checklist in the event of system failure.

V. CONCLUSION

This paper primarily contributes two metrics to measure the criticality of network components from the perspective of their contribution to system resilience, defined as the cumulative system functionality that has been restored at time , normalized by the expected cumulative system functionality supposing that the system has not been affected by disruption during this time period.

The first resilience-based component importance measure, i.e., the optimal repair time in [START_REF] Gandini | Importance and sensitivity analysis in assessing system reliability[END_REF], offers an explicit quantification of the priority that should be given to arc to be repaired and re-installed into the network. Lower values of indicate higher priority, i.e., higher rank in the component repair checklist for system restoration in the event of system failure. The second resilience-based component importance measure, i.e., the resilience reduction worth , quantifies the potential loss in optimal system resilience due to a delay in the repair time of link . This measure can provide valuable information to guide the recovery process of a particular component: components with high values of should be given high priority to their timely restoration, e.g., be assigned with adequate restoration resources.

Given the stochastic nature of disruptive events on an infrastructure network, a Monte Carlo-based method has been proposed to generate distributions of optimal repair time and resilience reduction worth for all the components in the network; then, a stochastic ranking approach based on the Copeland's pairwise aggregation method has been applied to rank components importance.

The results of the two measures applied to the IEEE 30-bus test network demonstrate some non-obvious and meaningful conclusions about the contributions of certain links to the resilience of the network. It is shown that two types of links are most important in terms of : 1) the links which connect generator nodes with the other two types of nodes (transmission nodes and demand nodes), e.g., links , , etc., and 2) the link which is the only arc connecting to demand nodes, i.e., link . The restoration of these types of links is most likely able to augment the total amount of flow received by the demand nodes of the network so that high priority should be given to these links in the reparation list. Besides, those links with high Copeland scores in terms of also have high Copeland scores ranking in terms of the resilience reduction worth : actually, the correlation coefficient between the two quantities is . It is noted that the differences of the resilience-based CIMs proposed in the present paper with respect to that in [START_REF] Barker | Resilience-based network component importance measures[END_REF] mainly fall into the following two aspects: 1) the concept and definition of system resilience which the CIMs rely on is an non-memoryless one in this study by taking into account the cumulative restored system functionality and 2) the focus of the proposed resilience-based CIMs is to quantify the effect that the recovery, rather than the disruption, of individual components has on the global system resilience, thus, are valuable in suggesting the most effective way for system (recovery) operation.

Finally, it is shown that the classical betweenness centrality indices, such as the shortest path betweenness, flow betweenness and random walk betweenness, do not capture resilience criticality as do the resilience-based measures and . Actually, the existing centrality indices and the CIMs in the reliability engineering literature are not appropriate to help implement resilience planning because they do not take into account system recovery time. Instead, the two measures proposed in the present paper provide insights useful for practical restoration activities of infrastructure networks after suffering a disruptive event.

It is noted that in this study we consider a system suffering a specific type of events e (e.g., earthquakes of a certain magnitude) and rank the component importance by a stochastic ranking method. The results are valuable to help implement the system restoration if this type of event happens on the system. However, it might be possible that an infrastructure system is affected by different types of events concurrently (e.g., earthquake and tsunami). Future studies will concentrate on the application of the resilience-based component importance measures to different types of infrastructure networks subject to (possibly different types of correlated) disruptive events, in order to further demonstrate the practical effectiveness of the measures.

Fig. 1 .

 1 Fig. 1. Generic system performance transition curve under the occurrence of disruption.

Fig. 2 .

 2 Fig. 2. Conceptual illustration of the proposed resilience measurement.

Fig. 3 .

 3 Fig. 3. Single-line diagram of the IEEE 30-bus test system.

Fig. 4 .

 4 Fig. 4. Graph representation of the IEEE 30-bus test system. The dark grey circles labeled with G represent the generator nodes, the white circles labeled with T represent transmission nodes, and the light grey circles labeled with D represent the demand nodes.

Fig. 7 .

 7 Fig. 7. Graphical illustration of the Copeland scores of the optimal repair time for all IEEE 30-bus network links. Links with higher value of Copeland score are represented as thicker and darker edges.

Fig. 8 .

 8 Fig. 8. Copeland score ranking of the resilience reduction worth for all IEEE 30-bus network links.

Fig. 9 .

 9 Fig. 9. Graphical illustration of the Copeland scores of the resilience reduction worth for all IEEE 30-bus network links. Links with higher values of Copeland score are represented as thicker and darker edges.

Fig. 10 .

 10 Fig. 10. Scatterplot of the Copeland scores of (a) the optimal repair time and (b) resilience reduction worth with the shortest path betweenness for the links of the IEEE 30-bus network.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Scatterplot of the Copeland scores of (a) the optimal repair time and (b) resilience reduction worth with flow betweenness for the links of the IEEE 30-bus network.

ACKNOWLEDGMENT

The authors would like to thank the anonymous referees for providing helpful comments and suggestions for our paper.