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Optimal pinning control of directed hypergraphs

Fabio Della Rossa, Davide Liuzza, Francesco Lo Iudice, Pietro De Lellis

Abstract

Identifying the nodes that must be directly controlled to steer a network along a desired trajectory remains an open problem for digraphs, and even more so for hypergraphs. In this manuscript, we investigate network systems coupled via directed hypergraphs and consider a broad class of individual dynamics and coupling configurations, extending the definition of type II networks originally formulated for digraphs. For this class of networks with higher-order interactions, we establish necessary and sufficient conditions under which a pinning selection locally ensures successful control. Building on these analytical results, we propose a greedy heuristic for pinning control selection, which demonstrably outperforms existing methods.

Optimal pinning control of directed hypergraphs

Abstract

Identifying the nodes that must be directly controlled to steer a network along a desired trajectory remains an open problem for digraphs, and even more so for hypergraphs. In this manuscript, we investigate network systems coupled via directed hypergraphs and consider a broad class of individual dynamics and coupling configurations, extending the definition of type II networks originally formulated for digraphs. For this class of networks with higher-order interactions, we establish necessary and sufficient conditions under which a pinning selection locally ensures successful control. Building on these analytical results, we propose a greedy heuristic for pinning control selection, which demonstrably outperforms existing methods.
Paper Structure (19 sections, 5 theorems, 27 equations, 5 figures, 2 tables)

This paper contains 19 sections, 5 theorems, 27 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Matrix notation
  4. Directed hypergraphs
  5. Signed graphs ahmadizadeh2017eigenvalues
  6. Pinning controllability of network systems on hypergraphs
  7. Network model and control question
  8. Convergence analysis
  9. Limit spectrum of $M=L+\kappa P$
  10. Select observations
  11. Selection of the pinning hyperedges
  12. Numerical applications
  13. Leader-follower consensus
  14. Pinning synchronization of Lorenz systems
  15. Individual dynamics
  16. ...and 4 more sections

Key Result

Proposition 1

if $\Lambda_{\max}:=\max_{\lambda\in\sigma(M(\kappa))} \Lambda(\lambda)<0$, then the controlled network eq:controlled_network is locally asymptotically controlled to the pinner's trajectory.

Figures (5)

  • Figure 1: Directed three-body nearest neighbor hypergraph.
  • Figure 2: Leader-follower consensus dynamics over a 7-node three-body nearest-neighbor hypergraph. In panel (a), nodes 1, 3, and 5 are controlled through standard, pairwise pinning; in panel (b), each pinning hyperedge corresponds to measuring the average state of two nodes, whereby $\mathcal{H}(\varepsilon_1^p)=\{1,2\}$, $\mathcal{H}(\varepsilon_2^p)=\{3,4\}$, $\mathcal{H}(\varepsilon_3^p)=\{5,7\}$. The left panels depict the topology of the controlled network, whereas the right panels report the state dynamics when $x_i(0)=i,\,i=1,\ldots,7$, $x_{p0}=8$, $\sigma_\varepsilon=1$ for all $\varepsilon\in\mathcal{E}_c$, and $\kappa=5$, with a thick dashed line identifying the pinner's trajectory.
  • Figure 3: Network topology of the controlled network considered in Example 2.
  • Figure 4: Consensus dynamics in the uncontrolled network \ref{['eq:uncontrolled_network']} (with $f=0$, $g(z)=z$, $n=1$). Panel (a) depicts the network topology, which is a 6-node directed three-body nearest neighbor, whereas panel (b) reports the state dynamics when $x_i(0)=i,\,i=1,\ldots,6$ and $\sigma_\varepsilon=1$.
  • Figure 5: Networks of $N=100$ Lorenz systems coupled through a directed ER hypergraph with parameters $p=0.01$ and $o=4$. Panels (a) and (b) superimpose the eigenvalues of $L$ and $M$ to the Master Stability Function obtained for $g(z)=\arctan z$, respectively. Each eigenvalue is associated to a white cross, some of them are repeated, and some other (associated with negative values of $\Lambda$) are not in the plot as they have real or imaginary part larger than $40$. Panels (c) and (d) depict the time evolution of the first state variable of each system and of the node error norm, respectively.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Corollary 1
  • Theorem 1
  • proof
  • Definition 3
  • Corollary 2
  • proof
  • ...and 3 more