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Fiedler-Based Characterization and Identification of Leaders in Semi-Autonomous Networks

Evyatar Matmon, Daniel Zelazo

TL;DR

The paper addresses identifying leader nodes in semi-autonomous consensus networks from observed dynamics. It develops a spectral characterization based on the grounded Laplacian $ar{L}_{11}$ and its Fiedler vector $v_F$, and connects this to the notion of relative tempo to relate steady-state velocities to $v_F$. A data-driven algorithm reconstructs $v_F$ from velocity measurements without topology knowledge, enabling leader identification. Numerical experiments demonstrate effective separation and leader recovery, illustrating practical potential for uncovering hidden leadership structures from dynamics.

Abstract

This paper addresses the problem of identifying leader nodes in semi-autonomous consensus networks from observed agent dynamics. Using the grounded Laplacian formulation, we derive spectral conditions that ensure the components of the Fiedler vector associated with leader and follower nodes are distinct. Building on the foundation, we emply the notion of relative tempo from prio works as an observable quantity that relates agents' steady-state velocities to the Fiedler vector. This relationship enables the development of a data-driven algorithm that reconstructs the Fiedler vector - and consequently identifies the leader set - using only steady-state velocity measurements, without requiring knowledge of the network topology. The proposed approach is validated through nuerical examples, demonstrating how spectral properties and relative tempo measurements can be combined to reveal hidden leadership structures in consensus networks.

Fiedler-Based Characterization and Identification of Leaders in Semi-Autonomous Networks

TL;DR

The paper addresses identifying leader nodes in semi-autonomous consensus networks from observed dynamics. It develops a spectral characterization based on the grounded Laplacian and its Fiedler vector , and connects this to the notion of relative tempo to relate steady-state velocities to . A data-driven algorithm reconstructs from velocity measurements without topology knowledge, enabling leader identification. Numerical experiments demonstrate effective separation and leader recovery, illustrating practical potential for uncovering hidden leadership structures from dynamics.

Abstract

This paper addresses the problem of identifying leader nodes in semi-autonomous consensus networks from observed agent dynamics. Using the grounded Laplacian formulation, we derive spectral conditions that ensure the components of the Fiedler vector associated with leader and follower nodes are distinct. Building on the foundation, we emply the notion of relative tempo from prio works as an observable quantity that relates agents' steady-state velocities to the Fiedler vector. This relationship enables the development of a data-driven algorithm that reconstructs the Fiedler vector - and consequently identifies the leader set - using only steady-state velocity measurements, without requiring knowledge of the network topology. The proposed approach is validated through nuerical examples, demonstrating how spectral properties and relative tempo measurements can be combined to reveal hidden leadership structures in consensus networks.

Paper Structure

This paper contains 6 sections, 3 theorems, 39 equations, 5 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM STATEMENT
  3. Leader-Identifiable Graphs
  4. A Leader Identification Algorithm
  5. NUMERICAL EXAMPLE
  6. CONCLUSIONS

Key Result

Proposition 1

The semi-normalized adjacency matrix $\hat{A}(\mathcal{G})$ has real, simple, greatest eigenvalue equal to $1$. Furthermore, the corresponding eigenvector is $v_F$, the Fiedler vector of $\bar{L}_{11}$.

Figures (5)

  • Figure 1: Conceptual overview of the network identification problem: (a) the interaction graph is unknown; (b) trajectories are observed; (c) after identification, the edges are recovered.
  • Figure 2: Example of augmented graph $\mathcal{\bar{G}}$. Leaders (green) receive inputs from external sources (red).
  • Figure 3: Example of a graph sequence where leaders (green) remain fixed and followers (blue) become progressively more connected. The leader degrees remain constant across all graphs, and leaders are never directly connected.
  • Figure 4: Sensing graph of Example 1.
  • Figure 6: Relative tempo for the network in Example 1. Note the gap between leaders and follower components. The blue curve corresponds to the followers, while the green curve represents the leaders. The black dashed lines is the true value of the Fiedler vector.

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Definition 1: Graph Sequence
  • Example 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof