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Resilient Control of Dynamic Flow Networks Subject to Stochastic Cyber-Physical Disruptions

Yu Tang, Li Jin

TL;DR

This work develops a resilience framework for dynamic flow networks under stochastic cyber-physical disruptions by modeling the system as a piecewise-deterministic Markov process and deriving stability criteria via tailored Lyapunov functions. It introduces throughput-based resilience metrics MECC and EMCC and shows how open-loop, mode-dependent, and density-dependent controls can achieve these targets under various observation and storage assumptions. The analysis yields computable lower bounds on resiliency scores and demonstrates improvements through numerical examples on representative networks. The results offer practical guidance for designing resilient routing in transportation and communication networks facing recurring disruptions, with potential extensions to cyclic and larger-scale systems. Through the combination of stochastic modeling, monotone dynamics, and control design, the paper connects fundamental throughput concepts to actionable resiliency-by-design strategies.

Abstract

Modern network systems, such as transportation and communication systems, are prone to cyber-physical disruptions and thus suffer efficiency loss. This paper studies network resiliency, in terms of throughput, and develops resilient control to improve throughput. We consider single-commodity networks that admit congestion propagation. We also apply a Markov process to model disruption switches. For throughput analysis, we first use insights into congestion spillback to propose novel Lyapunov functions and then exploit monotone network dynamics to reduce computational costs of verifying stability conditions. For control design, we show that (i) for a network with infinite link storage space, there exists an open-loop control that attains the min-expected-cut capacity; (ii) for a network with observable disruptions that restrict maximum sending and/or receiving flows, there exists a mode-dependent control that attains the expected-min-cut capacity; (iii) for general networks, there exists a closed-loop control with throughput guarantees. We also derive lower bounds of resiliency scores for a set of numerical examples and verify resiliency improvement with our method.

Resilient Control of Dynamic Flow Networks Subject to Stochastic Cyber-Physical Disruptions

TL;DR

This work develops a resilience framework for dynamic flow networks under stochastic cyber-physical disruptions by modeling the system as a piecewise-deterministic Markov process and deriving stability criteria via tailored Lyapunov functions. It introduces throughput-based resilience metrics MECC and EMCC and shows how open-loop, mode-dependent, and density-dependent controls can achieve these targets under various observation and storage assumptions. The analysis yields computable lower bounds on resiliency scores and demonstrates improvements through numerical examples on representative networks. The results offer practical guidance for designing resilient routing in transportation and communication networks facing recurring disruptions, with potential extensions to cyclic and larger-scale systems. Through the combination of stochastic modeling, monotone dynamics, and control design, the paper connects fundamental throughput concepts to actionable resiliency-by-design strategies.

Abstract

Modern network systems, such as transportation and communication systems, are prone to cyber-physical disruptions and thus suffer efficiency loss. This paper studies network resiliency, in terms of throughput, and develops resilient control to improve throughput. We consider single-commodity networks that admit congestion propagation. We also apply a Markov process to model disruption switches. For throughput analysis, we first use insights into congestion spillback to propose novel Lyapunov functions and then exploit monotone network dynamics to reduce computational costs of verifying stability conditions. For control design, we show that (i) for a network with infinite link storage space, there exists an open-loop control that attains the min-expected-cut capacity; (ii) for a network with observable disruptions that restrict maximum sending and/or receiving flows, there exists a mode-dependent control that attains the expected-min-cut capacity; (iii) for general networks, there exists a closed-loop control with throughput guarantees. We also derive lower bounds of resiliency scores for a set of numerical examples and verify resiliency improvement with our method.

Paper Structure

This paper contains 37 sections, 13 theorems, 96 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related work
  4. Our contributions
  5. Dynamic flow network with cyber-physical disruptions
  6. Flows, control laws and disruptions
  7. Sending/receiving flows
  8. Control laws
  9. Disruption modes
  10. PDMP model
  11. Stability and resiliency
  12. Stability and invariant set
  13. Resiliency, throughput and min-cut capacities
  14. Resiliency analysis
  15. Main results
  16. ...and 22 more sections

Key Result

Theorem 1

Consider an acyclic network satisfying Assumptions asm_fault.1-asm_fault.3. Suppose that the network admits a demand $\alpha$ and an invariant set $\mathcal{X}^\mu\subseteq\prod_{e\in\mathcal{E}}[\underline{x}_e^\mu, \bar{x}_e^\mu]$ under a control law $\mu:\mathcal{S}\times\mathcal{X}\to\mathbb R_{ where $\{p_{s}|s\in\mathcal{S}\}$ is the steady-state probability distribution of disruption modes

Figures (6)

  • Figure 1: A single-origin-single-destination network: we denote by $\mathcal{E}_e^-$ (resp. $\mathcal{E}_{\sigma_e}^-$) the set of incoming links of link $e$ (resp. node $\sigma_e$), by $\mathcal{E}_e^+$ (resp. $\mathcal{E}_{\tau_e}^+$) the set of outgoing links of link $e$ (resp. node $\tau_e$), by $\mathcal{A}_e^-$ the set of links upstream of link $e$, and by $\mathcal{A}_e^+$ the set of links downstream of link $e$.
  • Figure 2: Examples of sending/receiving flows.
  • Figure 3: Wheatstone bridge with $\mathcal{E}_{\mathrm{inf}}=\{e_o,e_5\}$ denoted by dotted lines: $\mathcal{B}_{e_o}^+=\{e_1,e_2\}$, $\mathcal{A}_{e_o,\mathrm{inf}}^+=\{e_5\}$, $\mathcal{A}_{e_o,\mathrm{fin}}^+=\{e_1,e_2,e_3,e_4\}$, $\mathcal{A}_{e_5,\mathrm{inf}}^-=\{e_o\}$ and $\mathcal{A}_{e_5,\mathrm{fin}}^-=\{e_1,e_2,e_4\}$.
  • Figure 4: Classification of incoming links $\mathcal{E}_v^-$ and outgoing links $\mathcal{E}_v^+$ for $e\in\mathcal{E}_{\mathrm{inf}}$ and $v\in\mathcal{V}$: (a) a general case, (b) classification for link $e_o$ and node $v_1$, given $\mathcal{E}_{v_1}^-=\{e_1\}$, $\mathcal{E}_{v_1}^+=\{e_2, e_3\}$, $\mathcal{A}_{e_o}^-=\varnothing$ and $\mathcal{B}_{e_o}^+=\{e_1,e_2\}$, (c) classification for link $e_5$ and node $v_1$, given $\mathcal{E}_{v_1}^-=\{e_1\}$, $\mathcal{E}_{v_1}^+=\{e_2, e_3\}$, $\mathcal{A}_{e_5}^-=\{e_o,e_1,e_2,e_4\}$ and $\mathcal{B}_{e_5}^+=\varnothing$.
  • Figure 5: Classification of incoming links $\mathcal{E}_v^-$ and outgoing links $\mathcal{E}_v^+$$e\in\mathcal{E}_{\mathrm{inf}}$ and $v\in\mathcal{V}$ in Theorem 2.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Definition 1: Locally responsive control laws
  • Definition 2: Stability & Instability
  • Definition 3: Invariant set
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Proposition 2
  • Theorem 3: Max-flow min-expected-cut
  • ...and 8 more