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Decentralized State-Dependent Markov Chain Synthesis with an Application to Swarm Guidance

Samet Uzun, Nazim Kemal Ure, Behcet Acikmese

TL;DR

This work addresses decentralized design of Markov chains to achieve a target steady-state distribution for swarm guidance without relying on connectivity of the communication graph. It introduces a state-dependent consensus protocol and a DSMC algorithm that updates the Markov matrix $M(k)$ using local errors, yielding exponential convergence to the prescribed distribution with $M(k)$ remaining column-stochastic and approaching the identity as convergence is reached. A modified DSMC with a shortest-path augmentation handles recurrent and transient states to further improve convergence. In probabilistic swarm guidance, DSMC outperforms prior homogeneous and time-inhomogeneous schemes, offering faster convergence, fewer transitions, and robustness to agent addition/removal, with quantization error bounded by $q_N(v)\le m/(4N)$. The approach provides a scalable, decentralized framework for Markov-chain synthesis in dynamic networks and has practical impact on large-scale swarm control and distributed stochastic optimization.

Abstract

This paper introduces a decentralized state-dependent Markov chain synthesis (DSMC) algorithm for finite-state Markov chains. We present a state-dependent consensus protocol that achieves exponential convergence under mild technical conditions, without relying on any connectivity assumptions regarding the dynamic network topology. Utilizing the proposed consensus protocol, we develop the DSMC algorithm, updating the Markov matrix based on the current state while ensuring the convergence conditions of the consensus protocol. This result establishes the desired steady-state distribution for the resulting Markov chain, ensuring exponential convergence from all initial distributions while adhering to transition constraints and minimizing state transitions. The DSMC's performance is demonstrated through a probabilistic swarm guidance example, which interprets the spatial distribution of a swarm comprising a large number of mobile agents as a probability distribution and utilizes the Markov chain to compute transition probabilities between states. Simulation results demonstrate faster convergence for the DSMC based algorithm when compared to the previous Markov chain based swarm guidance algorithms.

Decentralized State-Dependent Markov Chain Synthesis with an Application to Swarm Guidance

TL;DR

This work addresses decentralized design of Markov chains to achieve a target steady-state distribution for swarm guidance without relying on connectivity of the communication graph. It introduces a state-dependent consensus protocol and a DSMC algorithm that updates the Markov matrix using local errors, yielding exponential convergence to the prescribed distribution with remaining column-stochastic and approaching the identity as convergence is reached. A modified DSMC with a shortest-path augmentation handles recurrent and transient states to further improve convergence. In probabilistic swarm guidance, DSMC outperforms prior homogeneous and time-inhomogeneous schemes, offering faster convergence, fewer transitions, and robustness to agent addition/removal, with quantization error bounded by . The approach provides a scalable, decentralized framework for Markov-chain synthesis in dynamic networks and has practical impact on large-scale swarm control and distributed stochastic optimization.

Abstract

This paper introduces a decentralized state-dependent Markov chain synthesis (DSMC) algorithm for finite-state Markov chains. We present a state-dependent consensus protocol that achieves exponential convergence under mild technical conditions, without relying on any connectivity assumptions regarding the dynamic network topology. Utilizing the proposed consensus protocol, we develop the DSMC algorithm, updating the Markov matrix based on the current state while ensuring the convergence conditions of the consensus protocol. This result establishes the desired steady-state distribution for the resulting Markov chain, ensuring exponential convergence from all initial distributions while adhering to transition constraints and minimizing state transitions. The DSMC's performance is demonstrated through a probabilistic swarm guidance example, which interprets the spatial distribution of a swarm comprising a large number of mobile agents as a probability distribution and utilizes the Markov chain to compute transition probabilities between states. Simulation results demonstrate faster convergence for the DSMC based algorithm when compared to the previous Markov chain based swarm guidance algorithms.

Paper Structure

This paper contains 14 sections, 8 theorems, 44 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Main Contributions
  4. Organization and Notation
  5. A Consensus Protocol with State-Dependent Weights
  6. Decentralized State-Dependent Markov Chain Synthesis
  7. The DSMC Algorithm
  8. The Modified DSMC Algorithm
  9. An application of the DSMC Algorithm on Probabilistic Swarm Guidance
  10. Probabilistic Swarm Guidance
  11. Synthesis of the Markov Matrix
  12. Numerical Simulations
  13. Conclusion and Future Works
  14. Acknowledgement

Key Result

Lemma 1

Assume that Condition con:pos_w holds. If $e(k) \neq \bm{0}$, then for each $i$ such that $e[i](k) \geq 0$, $\exists j$ such that $e[j](k) < 0$, where $A_{a_w}^n[i,j](k)\neq 0$ for some exponent $n \in \mathbb{Z}_+$.

Figures (6)

  • Figure 1: Vertices of the graph and corresponding values are represented by circles. The indices of the vertices are shown in their corners. Edges of the vertices are represented by lines and weights of the edges are represented on the sides of the edges. Green and red lines represent the edges that are implied by Condition \ref{['con:pos_w']}\ref{['item:c1a']} and Condition \ref{['con:pos_w']}\ref{['item:c1b']}, respectively. Black lines represent the remaining edges of the graph.
  • Figure 2: Representation of the distribution of the swarm for the time-steps $0$, $250$, $251$, and $750$, respectively. There are $400$ ($20 \times 20$) bins and $5000$ agents in the operational region at the beginning of the simulation. The agents converge to the "E" letter in $250$ time-steps and approximately $1/3$ agents are removed from the operational space. Then, the remaining agents converge to the desired distribution again in $500$ time-steps.
  • Figure 3: Comparison of change of the total variation and the number of transitions with time for the algorithms. In this case, the adjacency matrix only allows agents to transition to $1$-step away bins.
  • Figure 4: Comparison of change of the total variation and the number of transitions with time for the algorithms. In this case, the adjacency matrix allows agents to transition to $10$-step away bins.
  • Figure 5: Representation of the distribution of the swarm for several time-steps. There are $600$ ($20 \times 30$) bins and $20000$ agents in the operational region at the beginning of the simulation. Swarm distribution converges to a different multimodal Gaussian distribution in every $40$ time-step except the time-steps between $160$ and $240$. Agents converge to the $4$ different multimodal Gaussian distributions up to the $160^{th}$ time-step. At the $161^{th}$ time-step, approximately $1/3$ agents are removed and the remaining agents converge to the same desired distribution in $40$ time-steps. Then, $7500$ agents are uniformly added to the operational region at $201^{th}$ time-step and all agents converge to the same desired distribution again in $40$ time-steps. After the $240^{th}$ time-step, agents converge to the $3$ new multimodal Gaussian distributions up to the $360^{th}$ time-step.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Definition 3
  • Proposition 1
  • ...and 14 more