Infrastructure Resilience Curves: Performance Measures and Summary Metrics

1 Introduction

• A resilience curve tracks a selected performance measure over time, while a summary metric reduces the entire curve to a scalar for comparison.Curves typically show performance declining after a disruption and then recovering; summary metrics quantify key curve dimensions.Source: §1 · pp. 2–3
• This survey treats resilience curves broadly, encompassing cascading failure, degraded response, non-increasing recovery, and cases where full restoration is impossible.Its synthesis and recommendations are intended to encompass all resilience curve forms.Source: §1 · pp. 2–3
• Performance measures and summary metrics should reflect stakeholder interests, but their suitability may vary across systems and scenarios.A measure or metric that represents stakeholder interest in one system and scenario may not accurately quantify that interest in another.Source: §1 · p. 3
• Measure and metric choices can change recommendations, yet the literature lacks general guidance, partly because selection is an analytical burden.Different measures or metrics can yield significantly different recommendations, while the literature does not often address their selection.Source: §1 · pp. 3–4
• The survey focuses on resilience-curve measures and metrics, excluding broader infrastructure resilience and non-curve metrics.It presents a broad survey of critical-infrastructure resilience literature while limiting its scope to synthesis and application of curve measures and metrics.Source: §1 · p. 4
• The survey aims to clarify adoption and implementation through taxonomies of performance measures and summary metrics, alongside guidance on selection and communication.Its roadmap also includes measure normalization and metrics derived from ensembles of metric functions, measures, and scenarios.Source: §1 · p. 4

2.1 Methodology

• The search used Web of Science and filtered for performance-versus-time curves, excluding event-axis, cumulative-loss, and internal-state-only illustrations.Source: §2.1 · p. 5
• Reviewers also added referenced publications cited as sources for conceptual and qualitative resilience-curve approaches.Source: §2.1 · p. 5

2.2 Sources

• 273 publications formed the survey scope: 184 survived filtering from 1,384 accessible candidates among 1,518 search results, and 89 more were added from references.Source: §2.2 · p. 5
• The collection spanned 2003–2020, included conceptual curves without specified measures or metrics, and comprised journal articles, conference papers, book sections, magazine articles, and a report.Source: §2.2 · p. 5

3 Performance Measures

• Performance normalization is framed through three schemes: static, endogenous, and exogenous.Source: §3 · p. 6
• The taxonomy defines domain-agnostic characteristics to support discussion of performance measures for different applications rather than prescribing domain-specific formulations.Source: §3 · p. 6

3.1 Categorization of Performance Measures

• Availability measures capture system capacity or functionality, while productivity measures quantify service provided and are most appropriate when demand changes with infrastructure condition.Availability is most appropriate when demand is constant or independent of system availability; productivity reflects supply, capacity, and demand and may be expressed relative to demand or as shortfalls.Source: §3.1 · pp. 7–8
• Quality measures describe service character and can reveal tradeoffs between steady-state and disrupted performance that availability or productivity measures may miss.Their use is more complex because analysis must model how supply and demand affect service quality.Source: §3.1 · pp. 8–9

3.2 Multiple and Ensemble Measures

• Monotonic candidate measures can still recommend different restoration strategies, while measures across multiple subsystems may diverge and their tradeoffs remain underexplored.The section identifies open-ended stakeholder interpretation, separate summary metrics, and ensemble metrics as approaches to handling multiple measures.Source: §3.2 · p. 9
• Ensemble measures can use shared or converted units, including production, material, and transportation costs or translating sector “inoperability” into financial impact, –.A transportation-and-energy conversion using fixed parameters prompted a call for more sophisticated models and data.Source: §3.2 · p. 10
• Some unitless ensemble measures combined components without deliberate weighting, using ratios, summed deficits, or products across infrastructure domains.Examples include water delivery capacity relative to shortest path length, petroleum consumption and storage deficits, and a power resilience index combining four system attributes.Source: §3.2 · p. 10
• Weighting changed investment valuations across five stakeholders and five measures, and affected restoration decisions in an integrated network measure.The integrated network measure combined topology-based measures for power, radio, and communication.Source: §3.2 · p. 10
• Ensemble-measure weighting should reflect stakeholder values, yet weight selection and validation remain underexplored; some publications proposed customer-priority or criticality weighting but gave no recommended values or methods –.Source: §3.2 · p. 10

3.3 Performance Normalization

• Normalization facilitates comparisons, but showing normalized values alone can obscure important context; presenting both actual and normalized values addresses this limitation.Unnormalized values are appropriate when actual performance provides key context or the nominal reference is unclear or irrelevant.Source: §3.3 · p. 11
• Normalization uses static, exogenous, or endogenous references: respectively, a fixed value, a time-varying scenario-independent baseline, or a scenario-responsive target.The manuscript notes that denominator changes can affect normalized performance as much as changes in actual performance.Source: §3.3 · p. 11
• Availability measures were normalized almost exclusively against fixed nominal values, emphasizing restoration to full function, while static normalization of quality measures was less straightforward.For quality measures, normalized performance may exceed one when the reference is typical performance; other reference choices can yield steady-state values below one.Source: §3.3 · p. 12
• For productivity measures, a fixed reference assumes constant service demand, so flattening real demand variation can affect normalized performance and analytical recommendations.Service demand can vary in steady-state and disrupted conditions.Source: §3.3 · p. 13
• Exogenous normalization captures natural demand variation but not interactions between the scenario and demand, and normalized performance may consequently exceed one.The authors caution that such outcomes should be anticipated when using exogenous normalization for productivity measures.Source: §3.3 · p. 14
• Endogenous normalization applies across all three measure categories, is useful when full recovery is infeasible, and is especially promising for productivity, but requires more modeling effort.The target may respond to the hazard or system dynamics, and its underlying mechanisms may require substantial modeling.Source: §3.3 · pp. 14–15
• General recommendations for endogenously normalized measures remain underexplored; when dynamic demand complicates productivity normalization, quality-measure normalization is likewise challenged.The passages frame this as a general assertion about demand dynamics and normalization.Source: §3.3 · p. 15
• Endogenous quality normalization can accommodate service standards that vary across scenario phases; the unquantified resilience prism distinguishes community goals over days, weeks, and years.The timing of transitions between goals may be influenced by dynamic service demand, even when demand is not captured within ℛ(t).Source: §3.3 · p. 15
• Travel-time measures require origin-destination demand data, and Chang et al. ruled out travel delay because of modeling challenges.Source: §3.3 · p. 15
• Quantitatively, ℛ(t) would be impossible or undesired for most productivity measures; quality ℛ0 may reflect typical or expected performance, with adaptation resulting in a new ℛ0 in the next scenario.Source: §3.3 · p. 15

4 Summary Metrics

• Summary metrics map resilience curves to scalar values, and the review defines six categories with examples and selection guidance.Metrics enable comparisons across scenarios and configurations; the survey found no consensus on a best metric, instead linking metrics to desired attributes.Source: §4 · pp. 15–16
• Only one publication compared threshold- and integral-based traffic-system metrics, and each led to a different recommendation.The review reports that the survey otherwise found no consensus on a best metric.Source: §4 · p. 16
• Metrics may use actual performance P(t) or normalized performance p(t), incorporate a reference value, and require choices about units, milestones, and normalization.The formulation considerations include performance units and reference values, time units and endpoints, performance-times-time units and control time, and weighting or conversion for combined measures.Source: §4 · p. 16, §4.1 · p. 17

4.1 Notation, Curve Milestones, and Control Times

• Static normalization that does not distinguish p(t) from P(t) may hinder extending metrics to exogenous baselines and endogenous performance targets.Some publications instead defined metrics in terms of a potentially time-varying reference R(t), even when implementing it as static.Source: §4.1 · p. 17
• Setting t0 at disruption can suit curves that degrade at the same time, but risks conflating hazard exposure with system performance and undervaluing resistance, detectability, and prediction.The control interval defines the period of interest for assessing summary metrics.Source: §4.1 · pp. 17–18
• Milestones commonly mark phase transitions, but they may be absent, additional, or unclear, so metric-defining milestones should be explicit and validated across trajectories.The literature’s common milestones are not necessarily essential, comprehensive, or unambiguous.Source: §4.1 · pp. 17–18
• Three common terminal-time choices are expected or mean recovery, maximum recovery, and stakeholder-specific considerations; these choices can truncate curves, and maximum recovery is undefined without full recovery.Maximum recovery time may be determined post hoc from Monte Carlo runs or from knowledge of underlying dynamics; stakeholder-specific endpoints may aid communication.Source: §4.1 · p. 18
• A fourth terminal-time choice is the lifecycle of probabilistic hazards, which evaluates hazard generation over time and may require discount rates and consideration of post-recovery adaptation.Each disruption scenario can still be evaluated with summary metrics described in the section.Source: §4.1 · p. 18

4.2 Magnitude-based Metrics

• Magnitude-based metrics quantify performance at a specific milestone or point in time and include residual performance, depth of impact, and restored performance.Source: §4.2 · p. 18
• Residual performance records performance after disruption at a selected milestone, while depth of impact is its complement, 1 - p(t_d); either may use actual or normalized units.Source: §4.2 · pp. 18–19
• Residual metrics require an explicit milestone—stakeholders may care about post-hazard, post-cascade, or minimum performance—and misaligned milestones or endogenous targets can make the minimum poorly reflect robustness.Source: §4.2 · p. 19
• Restored performance measures performance after recovery is complete relative to the initial level; it can capture partial recovery or post-recovery improvements, with calculation dependent on reference and normalization.Source: §4.2 · p. 19

4.3 Duration-based Metrics

• Duration metrics quantify intervals between milestones, including disruption, recovery, absorb, endure, and hazard-exposure-to-disruption periods.Source: §4.3 · pp. 19–20
• Disruption-duration metrics generally have a clear start, but recovery milestones can be ambiguous, and recovery time is undefined if full recovery never occurs.Specified thresholds, such as restoration to 95% of daily transit ridership, can define recovery duration.Source: §4.3 · p. 20

4.4 Integral-based Metrics

• Integral metrics combine performance and time as cumulative impact—the integrated performance shortfall from a reference—or its complement, cumulative performance.Source: Figure 3 · p. 20, §4.4 · p. 20
• The review uses “cumulative performance” rather than “resilience” because that broad label has been used for other metrics and factors, and no single summary metric fully describes a system.Source: §4.4 · p. 21
• Normalizing both performance and time was the most common integral approach, producing a unitless metric.Source: §4.4 · p. 21
• Integral metrics commonly use a global control duration, but this may obscure recovery detail or confuse comparisons between curves with different milestones.A fixed interval may be preferable when comparing curves with different milestones.Source: §4.4 · p. 21
• Straightforward integral metrics assume all performance-by-time units are equally valuable, while limited alternatives explore milestone weighting, nonlinear value, and separate pre- and post-disruption evaluations.These alternatives point to opportunities to better reflect stakeholder value across a scenario.Source: §4.4 · pp. 21–22

4.5 Rate-based Metrics

• Derivative measures were called agility, or local resilience,; failure rate was linked to resistance and adaptive capability during graceful degradation, and recovery rate to restorative capability.Source: §4.5 · p. 22
• For negative-derivative formulations, higher values with lower magnitude are preferred, whereas absolute-value formulations prefer lower values.Failure rates were calculated using curve derivatives or linear and piecewise-linear approximations; the sign-versus-absolute-value distinction matters when metrics are used in an ensemble.Source: §4.5 · p. 22
• Recovery-rate metrics used derivatives, linear or piecewise-linear approximations, and exponential-fit decay constants; higher values indicate steeper recovery.Two novel extensions included applying arctan to the linear approximation and a formulation providing J∈[0,1].Source: §4.5 · p. 22
• Rate-based metrics require clear milestones: recovery measures may span recovery alone or combined endurance and recovery, while ambiguous milestones can make implementation challenging.Many metrics were defined for curves with immediate recovery, where t_s=t_d.Source: §4.5 · p. 22

4.6 Threshold-based Metrics

• Thresholds can mark nonlinear interpretive transitions in performance or time and shifts in stakeholder value or the application of summary metrics.Performance thresholds identify levels below which performance is unacceptable; recovery-time thresholds set when objectives should be achieved.Source: §4.6 · p. 23
• Threshold-based metrics comprise adherence measures that classify performance against critical or recovery-time limits, and modified measures that adjust metrics using thresholds.Adherence may mean staying above a critical performance threshold, recovering within a time threshold, or meeting both.Source: §4.6 · p. 23
• Examples include zeroing below-threshold metrics,, residual capacity relative to a critical threshold, recovery-time-adjusted impact, and brittleness from below-threshold impact [287], [288].Source: §4.6 · p. 23

4.7.1 Ensembles of Summary Metric Functions

• Metric ensembles combine distinct metrics from one curve and are generally unitless, except when converted to financial units or combined with non-curve considerations.Examples include translating cumulative impact and recovery time into financial units or weighting cumulative impact and recovery costs,,,.Source: §4.7.1 · p. 23
• Validation of metric-ensemble forms was underexplored: products can reach zero if any constituent is zero, a behavior stakeholders must judge against their goals.Sums and products can preserve directionality between constituent metrics and their ensemble while behaving differently at extremes.Source: §4.7.1 · p. 24

4.7.2 Ensembles of Performance Measures

• Every reported measure-ensemble instance used an integral-based metric; common-unit measures could be summed, while dissimilar units required normalized metrics for combination.The review identifies further opportunities for ensemble metrics across performance measures.Source: §4.7.2 · p. 24
• Reported weighting approaches reflected decision-maker preferences, subsystem interdependence, time-varying needs, and historical cross-sector correlations,,.Weighting may be needed to represent stakeholder preferences when combining performance measures.Source: §4.7.2 · p. 24

4.7.3 Ensembles of Scenarios

• Scenario ensembles were more often reduced to a single metric; showing constituent curves, error bars, or confidence intervals made summaries clearer.An expected trajectory can be mistaken for a specific system trajectory when presented alone.Source: §4.7.3 · p. 25
• Integral-based summaries of an “expected trajectory” may match expected integral metrics, but can misrepresent magnitude-, duration-, rate-, and threshold-based metrics.The expected trajectory may not correspond to any curve in the ensemble; curve-based descriptive statistics and likelihoods of specific scenarios are alternatives.Source: §4.7.3 · p. 25

4.8 Summary Functions

• Unlike scalar summary metrics, 21 publications implemented summary functions evaluable at any point in a scenario.Examples include local resilience as the curve derivative, and a space-time dynamic measure defined as normalized cumulative performance since disruption.Source: §4.8 · p. 26
• Recovery ratio was the most common summary function and measures performance improvement at any time relative to the curve’s minimum.Henry and Ramirez-Marquez established its commonly used form, value of resilience, Я(t).Source: §4.8 · p. 26
• The metric provides Я(t_d)=0 in all scenarios, but this is not “zero resilience”; Я(t)=1 does not indicate the system is “fully resilient”.Source: §4.8 · p. 26
• Recovery ratio may support restoration sequencing and component-importance estimation,,,,.Its usual implementation presumes availability measures and a fixed ℛ0, although the definition can be extended to performance targets ℛ(t).Source: §4.8 · pp. 26–27

5.1 Performance Measure Selection and Implementation

• Availability focuses on infrastructure, productivity incorporates service supply and demand, and quality captures service character while potentially expanding intervention options.Quality measures can help compare trade-offs between steady-state and disrupted performance, but their suitability depends on the system and scenario.Source: §5.1 · p. 27
• Availability measures and static normalization can be inappropriate when they overvalue excess capacity and undervalue dynamic, real-time resilience.Measure selection should reflect stakeholder goals and system realities rather than modeling convenience.Source: §5.1 · p. 28
• Productivity measures should generally be primary, with dynamic service demand assumed and fixed or exogenous baselines used only when justified.The review notes that productivity measures are often extensions of availability-based analyses, while dynamic performance targets and normalization remain underexplored.Source: §5.1 · p. 28
• Ensemble performance measures can obscure differences among constituent measures, and their formulations and weighting require deliberate consideration.The review identifies these issues as generally unexplored and advises addressing them whenever an ensemble measure is used.Source: §5.1 · p. 29

5.2 Summary Metric Selection and Implementation

• Summary metrics should reflect stakeholder goals and curve trajectories, with integral-based calculations using deliberate control intervals rather than shifting curve milestones as boundaries.Empirical or simulated curves may not match expected shapes, so metric definitions should accommodate trajectory variety.Source: §5.2 · p. 29
• Metric definitions should account for milestones that may be absent and be clearly articulated and validated to avoid hiding unexpected consequences.Some metrics, such as recovery duration, may be undefined for unrecoverable systems; static-normalization assumptions may also constrain an analysis’s applicability.Source: §5.2 · pp. 29–30
• Ensemble summary metrics face challenges similar to ensemble measures, including underexplored methods for determining and validating them and potentially endogenous weighting.The manuscript also identifies alternative ways of combining metrics across curves or across multiple metrics of an individual curve as generally underexplored.Source: §5.2 · p. 30
• Time and performance thresholds remain underexplored, may influence other metrics nonlinearly, and can themselves vary as a scenario unfolds.Thresholds can also serve as stand-alone metrics, but incorporating them adds modeling complexity.Source: §5.2 · p. 30
• Performance targets, thresholds, and weighting may be endogenous, making their dynamics with infrastructure, stakeholder behavior, and stakeholder goals an area needing further focus.Understanding and representing these factors requires stakeholder engagement and validation.Source: §5.2 · p. 30, §5.4 · p. 31

5.3 Communication of Results

• Analysts may avoid labeling performance measures and summary metrics as “resilience,” since stakeholders’ interpretation matters and such metrics do not describe resilience by themselves.The passage distinguishes a measure at one point in time and a metric from a single curve trajectory from resilience as a broader consideration.Source: §5.3 · p. 30
• Normalized values alone can obscure whether changes reflect the performance measure or its target and may minimize differences in actual values across measures.The discussion suggests presenting normalized and unnormalized results together; normalized reporting is challenged by performance targets and endogenous normalization.Source: §5.3 · p. 30
• For scenario ensembles, distributions of summary metrics reveal overlap, variance, and skew, whereas an “expected trajectory” risks misleading stakeholders and yielding objectively incorrect metrics.The authors recommend describing ensembles through summary-metric distributions and descriptive statistics rather than collapsing curves into a single expected trajectory.Source: §5.3 · pp. 30–31

5.4 Improving the Practice of Resilience Analysis

• Few surveyed publications included empirical analysis, and those focused on well-documented historical events, so empirical analysis and calibration warrant further research.The authors call for direct, validated, actionable analysis of practical systems and problems rather than new models and metrics merely for illustration.Source: §5.4 · p. 31
• Assumptions of non-decreasing recovery, availability measures, and static normalization can make strategies appear only “less good” rather than “bad,” though real systems may violate them.Under these assumptions, curves are monotonic transformations of one another, eliminating the chance of diverging results.Source: §5.4 · p. 31

• Performance measures are grouped by increasing modeling complexity as availability, productivity, and quality; spatial variation or multiple stakeholders may require multiple measures or ensembles.Source: §6 · p. 32
• Normalization schemes are static, exogenous, or endogenous; productivity may require dynamic targets tied to infrastructure states, stakeholder goals, and behaviors.Source: §6 · p. 32
• The six summary-metric categories are magnitude, duration, integral, rate, threshold, and ensembles; metric choice and specification matter because common assumptions can affect communication of resilience.Examples include instantaneous performance loss and non-decreasing recovery.Source: §6 · p. 32
• For ensembles of curves, report descriptive statistics for each curve’s metric rather than deriving metrics from an “expected trajectory” curve.Source: §6 · p. 32
• Because infrastructure systems are socio-technical while prior literature emphasizes technical aspects, future research should engage stakeholders in defining performance and validate how interpretations affect recommendations.Source: §6 · p. 32