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Dynamic Centrality Measures for Water Distribution Network Hydraulics

MirSaleh Bahavarnia, Salma M. Elsherif, Ahmad F. Taha

arXiv:2609.10800v1eess.SYcs.SYmath.OC

TL;DR

WDN vulnerability measures based only on topology overlook network dynamics, while the paper seeks to rank influential pipes using both. It linearizes a WDN flow model and applies a vulnerability vector to pipe-flow states; the paper reports extensions for input types, insignificant components, and node rankings. The measure is intended to support vulnerability assessment and prioritization of maintenance and operational efforts.

  • Problem

    Graph-theoretic centrality measures rank components by topology while overlooking WDN dynamics, leaving a need for a measure that incorporates both.

    Source: §I · p. 2

  • Method

    The paper linearizes a WDN model around equilibrium and uses a control-theoretic vulnerability vector to rank pipe-flow states by their influence under input changes.

    Source: §I · p. 2

  • Results

    The paper reports an analysis of input types and timing effects, a graph-theoretic characterization of zero-vulnerability components, and a node measure interpreted as a dynamic version of degree.

    Source: §I · p. 3

  • Takeaways & Limitations

    The vulnerability vector identifies influential pipes to help engineers understand WDN vulnerability and prioritize maintenance and operational efforts.

    Source: §I · p. 2

    Tuning some parameters requires trial and error, including the time-horizon length parameter T.

    Source: §VI · p. 14

Abstract

from arXiv · show

Water distribution networks (WDNs) are susceptible to various failures, including but not limited to human errors, cyber-attacks, and network modifications, necessitating the vulnerability analysis of WDNs. Graph-theoretic centrality measures---as a main class of centrality measures---aim to rank the network components solely based on their influence (i.e., criticality) on the WDN topology in the case of input changes, while overlooking the WDN dynamics. To overcome such a limitation, this paper uses a control-theoretic centrality measure to identify the network's most and least influential pipes in WDNs, by simultaneously incorporating the dynamics and topology of the WDN. First, given a WDN modeled by nonlinear differential-algebraic equations (NDAEs) consisting of transient flow dynamics as differential equation (DE) and conservation of water mass as algebraic equation (AE) and considering the pipe flow rates as states of the state-space (SS) representation, we extract a linearized system modeled by linear ordinary differential equations (LODEs) around the equilibrium (i.e., steady) flow rate vector. Second, treating pipe flow rates as SS nodes (states), we introduce a node centrality-based measure, namely \textit{vulnerability vector (VV)}, to rank the network pipes based on their influence on the dynamics and topology of the WDN in the case of input changes. In particular, the network's most and least influential pipes can be identified through such a centrality-based approach. This enables water engineers to understand the WDN's vulnerability better and effectively prioritize the maintenance and operational efforts on the most influential pipes within the WDN.

I. INTRODUCTION AND PAPER CONTRIBUTIONS

The paper develops a control-theoretic centrality measure for ranking influential pipes by combining WDN dynamics and topology, addressing limits of topology-only rankings. It derives this measure from a linearized flow-state model and extends the analysis to input types, insignificant components, and node rankings.

I. INTRODUCTION AND PAPER CONTRIBUTIONS

  • The approach addresses the gap between graph-theoretic rankings that omit hydraulic dynamics and hydraulic simulations that can be data-dependent and computationally cumbersome.The paper frames these measure classes as complementary: graph-theoretic measures are efficient but less accurate first approximations, while simulation-based measures provide a more comprehensive understanding.Source: §I · pp. 1–2
  • The paper adapts a control-theoretic centrality measure to WDNs, where state-space nodes represent pipe flows rather than network junctions.Unlike the earlier general-network formulation, the proposed measure examines responses to step, pulse, and impulse inputs.Source: §I · p. 2

A. Preliminaries on NDAE and LODE Models of WDNs

  • Vulnerability analysis starts from a WDN model with transient flow dynamics and water-mass conservation, then linearizes it around the equilibrium flow-rate vector with pipe rates as states.The resulting linear ordinary differential equation model is used to quantify pipe influence.Source: §I · p. 2, A. Preliminaries on NDAE and LODE… · p. 3

I. INTRODUCTION AND PAPER CONTRIBUTIONS

  • Vulnerability vectors rank pipes by their influence on WDN dynamics and topology, identifying the most and least influential pipes for maintenance and operational prioritization.The measure is node-centrality-based because pipe flow rates are state-space nodes, but it functions as an edge-centrality measure by ranking pipes.Source: §I · p. 2
  • Theoretical expressions for vulnerability vectors support analysis without computationally expensive exact evaluation of variations under different inputs.The paper also investigates how input types and timing parameters affect pipe influence in benchmark WDNs.Source: §I · pp. 2–3
  • The paper proposes an iterative graph-theoretic procedure to characterize components with zero vulnerability and a node centrality measure interpreted as a dynamic version of degree.Numerical simulations examine input types and timing parameters; the paper also reports a comparison with graph-theoretic edge betweenness centrality.Source: §I · pp. 2–3

1) Conversion of the NDAE model to an LDAE model:

This section describes conversion of the nonlinear differential-algebraic WDN model into a linearized algebraic model around equilibrium. It also outlines equilibrium-flow computation and the subsequent conversion to a linear ordinary differential equation model.

1) Conversion of the NDAE model to an LDAE model:

  • The NDAE model is linearized around the equilibrium flow-rate and input vectors to obtain an LDAE representation.The following conversion step produces an LODE model for quantifying pipe influence.Source: 1) Conversion of the NDAE model to an… · pp. 4–5
  • The equilibrium flow-rate vector is computed by setting the equilibrium input to zero and iteratively solving the model for given fixed-node heads and external demand.The passage specifies use of the Newton-Raphson method.Source: 1) Conversion of the NDAE model to an… · p. 4
  • The Newton-Raphson procedure uses an iteration-specific Jacobian and a stopping criterion based on an error tolerance.The supplied passages do not state the tolerance value.Source: 1) Conversion of the NDAE model to an… · p. 4

2) Conversion of the LDAE model to an LODE model:

The paper converts the WDN’s differential-algebraic model into an ordinary differential model using the mass-conservation constraint’s null space. It frames the task as ranking pipes by their influence on both network dynamics and topology under input changes.

2) Conversion of the LDAE model to an LODE model:

  • The mass-conservation constraint is rewritten using E = C_v^T C_v, whose null space is represented by eigenvectors U2; expressing flow deviations as q̄(t) = U2q̃(t) yields LODE dynamics.The reduced states q̃(t) represent net imbalance flows that act as decoupled virtual fluid circuits.Source: 2) Conversion of the LDAE model to an… · p. 5
  • The resulting state matrix F is Hurwitz, and the LODE dynamics depend on external demand, network topology, pipe inertia, pipe dimensions, and fixed-head nodes.The equilibrium flow vector q* also depends on those network and hydraulic quantities.Source: 2) Conversion of the LDAE model to an… · p. 5

B. Problem Statement

  • The problem is to rank WDN pipes by their criticality to network dynamics and topology when inputs change.The questions also ask whether VV can be derived without exact-variation evaluation, whether influence depends on input type or timing, and whether insignificant components have a graph-theoretic interpretation.Source: B. Problem Statement · p. 5
  • Pipe flow rates serve as state-space nodes, while input changes are applied through virtual valves to assess pipe influence.Step and pulse inputs are used for the main derivations; impulse inputs are also considered for abrupt changes that may cause failures.Source: B. Problem Statement · p. 5, 2) Conversion of the LDAE model to an… · p. 5

III. VV COMPUTATION

The vulnerability vector assigns each pipe a score based on its state response to a shifted step or pulse input. For shifted steps, the paper gives a computational expression using an RMS-based operator and a Lyapunov equation.

III. VV COMPUTATION

  • Each pipe’s VV entry vm measures its influence under a shifted step or pulse input, with α, μ, and η specifying input magnitude, start time, and pulse duration.The most- and least-influential pipes are identified by the maximum and minimum VV entries, respectively.Source: §III · p. 6
  • The paper selects computational operators to keep VV finite and informative, noting that a supremum-based alternative requires solving a non-convex optimization problem.The RMS-based operator is computationally preferred to that alternative.Source: §III · p. 6, A. VV Computation for the Shifted Step… · p. 6

A. VV Computation for the Shifted Step Input

  • For shifted step inputs, an RMS-based operator provides finite VV values, and Proposition 1 expresses the resulting vector using a Lyapunov-equation solution.The solution WT−μ can be interpreted as an observability Gramian and computed with Lyapunov solvers.Source: A. VV Computation for the Shifted Step… · p. 6

B. VV Computation for the Shifted Pulse Input

For shifted pulse inputs, the paper derives VV expressions using two computational operators. The associated Lyapunov solution can be interpreted as an observability Gramian.

B. VV Computation for the Shifted Pulse Input

  • A supremum-based operator is another possible choice for pulse inputs, but its computation requires a non-convex optimization problem.The paper contrasts this option with RMS-based operators.Source: B. VV Computation for the Shifted… · p. 7
  • For shifted pulse inputs, Propositions 2 and 3 give VV expressions using two computational operators, including an RMS-based operator and an operator taking the maximum RMS across two time intervals.The input is a positive step followed by a delayed negative step, parameterized by its start time and pulse duration.Source: B. VV Computation for the Shifted… · p. 7
  • The pulse-input expression uses Wη, the positive-definite solution of a Lyapunov equation, which the paper interprets as an observability Gramian.The paper also provides a final-value expression for the VV.Source: B. VV Computation for the Shifted… · p. 7

IV. VV EFFECTIVENESS, ABRUPT INPUT CHANGES, AND INSIGNIFICANT COMPONENTS

The section assesses VV against quasi-exact input variations, examines abrupt impulse changes, and identifies insignificant pipes using VV. Because exact NDAE variations are computationally costly, the analysis instead evaluates LODE variations, which approximate the NDAE behavior.

IV. VV EFFECTIVENESS, ABRUPT INPUT CHANGES, AND INSIGNIFICANT COMPONENTS

  • The VV-based analysis is intended for offline planning and risk assessment, not real-time control, and is presented as scalable to networks with thousands of nodes and pipes.The authors cite screening many threat vectors over weeklong horizons and producing data for day-ahead scheduling or post-incident mapping as computational needs.Source: §IV · pp. 7–8

A. Effectiveness Validation of VVs

  • VV provides a first-order, computationally efficient means of ranking pipe contributions to input-induced variation, while exact NDAE variations require costly iterative solves and repeated simulations.The paper calls the comparison variations “quasi-exact” because they are computed from the linearized LODE model rather than the nonlinear NDAE model.Source: §IV · p. 7, A. Effectiveness Validation of VVs · p. 8

B. Abrupt Input Changes Scenario

  • With an equilibrium initial state and Hurwitz F + F^T, the impulse-response norm decreases after time μ and reaches its supremum at μ.This yields Jm(α, T) = |α|∥Ge_Mm∥ under the stated condition.Source: B. Abrupt Input Changes Scenario · p. 8

C. VV-based Network’s Insignificant Components (Pipes)

The section relates VV-insignificant pipes to graph structure and gives an iterative procedure for finding a subset of them. The identified subset provides a lower bound on the total number of insignificant pipes.

C. VV-based Network’s Insignificant Components (Pipes)

  • A zero edge weight is sufficient for the corresponding pipe to have VV zero, and the graph-theoretic procedure identifies a subset of such pipes.The procedure iteratively removes qualifying edges and includes their indices in the subset.Source: C. VV-based Network’s Insignificant… · p. 9
  • A degree-1 node with varying total head implies its pipe has VV zero; an adjacent degree-2 varying-head node also implies its pipe has VV zero.These conditions identify VV-insignificant pipes through Proposition 4.Source: C. VV-based Network’s Insignificant… · p. 9
  • The identified subset’s size is a lower bound on the total number of VV-insignificant pipes.The section denotes the subset’s cardinality as nIC and the total number as nIC.Source: C. VV-based Network’s Insignificant… · p. 9

V. NUMERICAL SIMULATIONS

The simulations apply the VV procedure to 80 benchmark WDNs and examine influential pipes and nodes. In the 66th network, the most influential pipe is identified, while the least-influential pipe is one of multiple insignificant pipes.

V. NUMERICAL SIMULATIONS

  • The 80 benchmark WDN simulations use fixed reservoir heads of 65 m, and the VV procedure sorts pipe values to identify the most and least influential pipes.The procedure first solves for an equilibrium flow and constructs the system matrices used in VV computation.Source: §V · pp. 9–10

A. Network’s Most and Least Influential Components

  • In the 66th WDN, pipe 251, connecting nodes 252 and 364, is most influential; pipe 375 is one least-influential pipe.Figure 2 plots VV values by pipe index and marks the most influential pipe and one representative least-influential pipe.Source: A. Network’s Most and Least… · p. 10, Figure 2 · p. 10
  • The 66th WDN has 67 insignificant pipes, so its marked least-influential pipe is not unique.The iterative procedure gives a lower bound of 66 for the total of 67 insignificant pipes.Source: A. Network’s Most and Least… · p. 10, Figure 2 · p. 10
  • The simulations also identify the network’s most and least influential nodes using the VV-based approach.Additional numerical simulations for the node results are given in Appendix F.Source: A. Network’s Most and Least… · p. 10

B. Parametric Dependency of the VV-based Influences

VV-based pipe rankings can depend on input timing and on the equilibrium initialization, while the least-influential pipe is stable across the tested timing variations. The reported correlations with quasi-exact variations are strongest for non-impulse inputs near equilibrium, whereas graph-theoretic edge betweenness does not effectively quantify pipe influence in the comparison.

1) Dependency of the VV-based influences on the time horizon length:

  • Varying the time horizon leaves the least-influential pipe unchanged, but can change the most-influential pipe for shorter horizons under shifted step and impulse inputs.The most-influential pipe was not sensitive to the horizon for shifted pulse input in the reported case.Source: 1) Dependency of the VV-based… · p. 11

2) Dependency of the VV-based influences on the shifting parameter:

  • Varying the shifting parameter leaves the least-influential pipe unchanged, while larger shifts can change the most-influential pipe for shifted step and impulse inputs.The most-influential pipe was not sensitive to the shift for shifted pulse input in the reported case.Source: 2) Dependency of the VV-based… · p. 11

3) Dependency of the VV-based influences on the pulse length parameter:

  • Varying pulse length leaves the least-influential pipe unchanged, but can change the most-influential pipe at smaller pulse lengths.The reported parameter dependency can differ across benchmark WDNs; for the 62nd network, the most-influential index changed repeatedly with horizon length.Source: 3) Dependency of the VV-based… · p. 11

4) Dependency on the initialization point:

  • For linearized dynamics, accurate identification of influential components is theoretically tied to accurate equilibrium-state computation.The authors report that larger initialization perturbations make misidentification of the most-influential component more likely in some cases.Source: 4) Dependency on the initialization… · pp. 11–12
  • For the 66th WDN, up to 15% equilibrium-state perturbation retained the most-influential component in all tested states, while 50% perturbation retained it in 87.6%.The study used 50 randomly perturbed states per perturbation level and averaged results over 10 replicates.Source: 4) Dependency on the initialization… · p. 12

C. Correlation between the Quasi-Exact Variations and VVs

  • VV correlates with quasi-exact variations across the tested scenarios, particularly for non-impulse inputs; those correlations deteriorate as non-equilibrium initialization deviates further from equilibrium.Shifted pulse input with S2[·] performed best and shifted impulse input with S3[·] worst in VV effectiveness; the reported correlations were more representative near equilibrium for non-impulse inputs.Source: C. Correlation between the Quasi-Exact… · p. 12
  • All reported correlations between VV and normalized edge betweenness centrality are negative, and the most- and least-influential pipes differ between the measures.The authors conclude that graph-theoretic edge betweenness does not effectively quantify pipe influence because it overlooks WDN dynamics.Source: C. Correlation between the Quasi-Exact… · p. 13

VI. CONCLUDING REMARKS

The paper presents the vulnerability vector (VV) as a control-theoretic centrality measure for identifying influential WDN pipes while incorporating network dynamics and topology. It also gives mathematical expressions for VV, examines input and timing dependence, and describes extensions to insignificant components and nodes.

VI. CONCLUDING REMARKS

  • VV incorporates WDN dynamics and topology to identify the network’s most and least influential pipes.The paper relates this integration to the time-dependent response and to model terms that depend on the incidence matrix.Source: §VI · p. 13
  • Propositions 1–3 provide mathematical expressions for VV in terms of network dynamics and topology, avoiding exact evaluation of input-associated variations.The authors state these expressions facilitate vulnerability analysis without computationally expensive exact evaluations.Source: §VI · p. 13
  • VV-based pipe influence depends on input type and timing parameters, with different benchmark WDNs showing different parameter dependencies and sensitivities.The simulations examine time horizon T, shifting parameter µ, and pulse length η.Source: §VI · p. 13
  • An iterative graph-theoretic procedure characterizes VV-based insignificant components and yields a lower bound on their total number.The paper also reports a graph-theoretic interpretation for insignificant pipes and nodes.Source: §VI · p. 13
  • A VV-based node centrality measure identifies influential WDN nodes and is interpreted as a dynamic version of degree.The paper frames this as an additional measure induced from VV.Source: §VI · p. 13

APPENDIX A CONSTRUCTION OF THE NDAE REPRESENTATION OF

The appendix derives the WDN NDAE representation from transient pipe-flow dynamics and water-mass conservation. It then establishes a shifted-step-input expression using a uniquely solvable Lyapunov equation.

APPENDIX A CONSTRUCTION OF THE NDAE REPRESENTATION OF

  • The NDAE representation is obtained by combining transient pipe-flow dynamics with water-mass conservation at nodes with varying total heads.The derivation differentiates the conservation constraint and uses invertibility associated with the connected subgraph before defining B.Source: Appendix A · p. 14

APPENDIX B PROOF OF PROPOSITION 1

  • For the shifted step input, χ1 is computed by solving a Lyapunov equation, which has a unique solution because F is Hurwitz.The appendix further establishes that this solution is positive definite.Source: Appendix B · pp. 14–15

APPENDIX C PROOF OF PROPOSITION 2

The appendices derive VV expressions for shifted pulse inputs using auxiliary quantities and Lyapunov equations. They specify the matrices used in the third proposition’s expression and identify χ1 as a unique positive-definite Lyapunov solution.

APPENDIX C PROOF OF PROPOSITION 2

  • For the shifted pulse input, the proof computes β1 and β2 using expressions involving the state matrix F and pulse timing parameters.The β1 derivation introduces χ1 through a Lyapunov equation; the proof states its solution is unique and positive definite.Source: Appendix C · p. 15
  • The shifted-pulse proof gives a vectorized formula for χ1 and then computes β1 and β2.The formula uses F, the identity matrix, and pulse length η.Source: Appendix C · p. 15

APPENDIX D PROOF OF PROPOSITION 3

  • For Proposition 3, χ2, χ3, and χ4 are specified in terms of F and η, while χ1 is the unique positive-definite solution of a Lyapunov equation.The proof substitutes these quantities into the expression involving pK_m.Source: Appendix D · p. 15

APPENDIX E PROOF OF PROPOSITION 4

The appendices characterize insignificant components through relations involving B and the incidence matrix, then extend VV-based influence ranking to nodes. Figures also show least-influential pipes fixed across tested conditions in two benchmark networks while most-influential pipes are examined across input and timing settings.

APPENDIX E PROOF OF PROPOSITION 4

  • For a degree-1 node, the proof identifies an associated pipe satisfying B e_M_m = 0; an adjacent degree-1 and degree-2 node pair can identify another such pipe.The derivation uses CvB = 0 and the corresponding incidence-matrix rows.Source: Appendix E · pp. 15–16

APPENDIX F VV-BASED NETWORK’S MOST AND LEAST INFLUENTIAL

  • The VV-based node measure aggregates neighboring-node quantities and ranks nodes by sorting its values; it is interpreted as dynamic degree.The passage also states that zero-valued nodes are insignificant and that the procedure provides a lower bound on their total number.Source: Appendix F · p. 16
  • For the 66th benchmark WDN, node 276 is most influential and node 376 is one least influential node; the least-influential node can be non-unique because its measure is zero.The reported count is 66 insignificant nodes, and the figure shows one representative least-influential node.Source: Appendix F · p. 16, Figure 6 · p. 16

DEPENDENCY OF THE VV-BASED INFLUENCES

  • The figures track the most-influential pipe across tested timing parameters and input/operator settings; the least-influential pipe remains the 375th in the 66th network and the 339th in the 62nd.The plotted parameters are time horizon T, shifting parameter µ, and pulse length η where specified.Source: Figure 8 · p. 17, Figure 9 · p. 17
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