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FlipDyn in Graphs: Resource Takeover Games in Graphs

Sandeep Banik, Shaunak D. Bopardikar, Naira Hovakimyan

TL;DR

A dynamic game model extending the FlipDyn framework to a graph-based setting, where each node represents a dynamical system, which results in a hybrid dynamical system where the discrete state governs the continuous state evolution and the corresponding state cost.

Abstract

We present \texttt{FlipDyn-G}, a dynamic game model extending the \texttt{FlipDyn} framework to a graph-based setting, where each node represents a dynamical system. This model captures the interactions between a defender and an adversary who strategically take over nodes in a graph to minimize (resp. maximize) a finite horizon additive cost. At any time, the \texttt{FlipDyn} state is represented as the current node, and each player can transition the \texttt{FlipDyn} state to a depending based on the connectivity from the current node. Such transitions are driven by the node dynamics, state, and node-dependent costs. This model results in a hybrid dynamical system where the discrete state (\texttt{FlipDyn} state) governs the continuous state evolution and the corresponding state cost. Our objective is to compute the Nash equilibrium of this finite horizon zero-sum game on a graph. Our contributions are two-fold. First, we model and characterize the \texttt{FlipDyn-G} game for general dynamical systems, along with the corresponding Nash equilibrium (NE) takeover strategies. Second, for scalar linear discrete-time dynamical systems with quadratic costs, we derive the NE takeover strategies and saddle-point values independent of the continuous state of the system. Additionally, for a finite state birth-death Markov chain (represented as a graph) under scalar linear dynamical systems, we derive analytical expressions for the NE takeover strategies and saddle-point values. We illustrate our findings through numerical studies involving epidemic models and linear dynamical systems with adversarial interactions.

FlipDyn in Graphs: Resource Takeover Games in Graphs

TL;DR

A dynamic game model extending the FlipDyn framework to a graph-based setting, where each node represents a dynamical system, which results in a hybrid dynamical system where the discrete state governs the continuous state evolution and the corresponding state cost.

Abstract

We present \texttt{FlipDyn-G}, a dynamic game model extending the \texttt{FlipDyn} framework to a graph-based setting, where each node represents a dynamical system. This model captures the interactions between a defender and an adversary who strategically take over nodes in a graph to minimize (resp. maximize) a finite horizon additive cost. At any time, the \texttt{FlipDyn} state is represented as the current node, and each player can transition the \texttt{FlipDyn} state to a depending based on the connectivity from the current node. Such transitions are driven by the node dynamics, state, and node-dependent costs. This model results in a hybrid dynamical system where the discrete state (\texttt{FlipDyn} state) governs the continuous state evolution and the corresponding state cost. Our objective is to compute the Nash equilibrium of this finite horizon zero-sum game on a graph. Our contributions are two-fold. First, we model and characterize the \texttt{FlipDyn-G} game for general dynamical systems, along with the corresponding Nash equilibrium (NE) takeover strategies. Second, for scalar linear discrete-time dynamical systems with quadratic costs, we derive the NE takeover strategies and saddle-point values independent of the continuous state of the system. Additionally, for a finite state birth-death Markov chain (represented as a graph) under scalar linear dynamical systems, we derive analytical expressions for the NE takeover strategies and saddle-point values. We illustrate our findings through numerical studies involving epidemic models and linear dynamical systems with adversarial interactions.

Paper Structure

This paper contains 11 sections, 3 theorems, 55 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem Formulation
  4. FlipDyn-G for General Problem
  5. Saddle-point value of any node
  6. NE takeover strategies of the FlipDyn-G game
  7. FlipDyn-G for scalar LQ Problems
  8. Dual Deter FlipDyn-G game
  9. Conclusion
  10. Acknowledgments.
  11. Disclosure of Interests.

Key Result

lemma thmcounterlemma

Under Assumption ast:general_costs, the saddle-point value of the FlipDyn-G game eq:obj_def_E at any time $k \in \mathcal{K}$, subject to the FlipDyn dynamics eq:FD_u_compact and continuous state dynamics eq:CS_u is given by: where $y_{k}^{\alpha_{k}*}$ and $z_{k}^{\alpha_{k}*}$ correspond to NE takeover policies obtained upon solving the zero-sum matrix defined by $\Xi_{k+1}^{\alpha_{k}}$ as a l

Figures (7)

  • Figure 1: A directed multigraph consisting of 3 nodes. At time $k = 1$, the FlipDyn state is $\alpha_1 = 1$. The actions of both players are $\{ 1,2,3 \}$.
  • Figure 2: A graph consisting of $N$ nodes .
  • Figure 3: (a) An epidemic model represented as a graph with four nodes. The FlipDyn states of the graph are susceptible (S), Infected (I), Recovered (R), and Deceased (D). (b) Saddle-point parameters for each node $\alpha = \{\text{S,I,R,D}\}$, over time $k$, with horizon length $L = 20$.
  • Figure 4: For the node $\alpha = \text{I}$, the NE policy of the (a) defender and (b) adversary, where $y_{k}^{\text{I}}(\alpha), z_{k}^{\text{I}}(\alpha), \alpha = \{\text{S,I,R,D}\}$ corresponds to the probability of selecting the takeover node $\alpha$, given $\alpha_k = \text{I}$.
  • Figure 5: A stock market Markov chain model represented as a graph with three nodes. The FlipDyn states of the graph are Bull (Bu), Bear (Br), and Stagnant (St).
  • ...and 2 more figures

Theorems & Definitions (6)

  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • remark thmcounterremark
  • theorem thmcountertheorem
  • proof
  • remark thmcounterremark