arxiv: 2605.10489 · v1 · submitted 2026-05-11 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

Observing the state of networks with directed higher-order interactions

Davide Salzano, Pietro De Lellis, Roberto Rizzello, Stefano Boccaletti

Pith reviewed 2026-05-12 04:58 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords network state observationhigher-order interactionsnonlinear dynamical systemsobserver designdirected networksstate reconstructionopinion dynamicspartial measurements

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The pith

An algorithmic procedure selects which nodes to measure and designs observer gains so the full state of a nonlinear network can be reconstructed from partial observations even when interactions are directed and higher-order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a method to reconstruct the complete state of a network of nonlinear dynamical systems when only some nodes can be measured and the connections include directed higher-order terms. A sympathetic reader would care because many real-world systems, from social opinion dynamics to engineered networks, exhibit these interaction structures yet full-state knowledge is required for prediction or intervention while measuring every node is often impractical. The approach rests on analytical guarantees of observer convergence and includes both numerical validation across cases and a demonstration on reconstructing agent opinions. If successful, it provides a systematic way to trade off measurement cost against reconstruction accuracy without ad-hoc choices of sensors or gains.

Core claim

For networks of nonlinear dynamical systems with directed higher-order interactions, an algorithmic procedure simultaneously selects a set of nodes to measure and computes observer gains such that the resulting observer asymptotically reconstructs the entire network state, with the selection and gains chosen to satisfy proven convergence conditions on the error dynamics.

What carries the argument

Algorithmic observer design procedure that jointly picks measured nodes and gains, grounded on convergence analysis of the state estimation error for the given class of nonlinear dynamics and interactions.

If this is right

Only a subset of nodes needs to be measured to recover the full state.
The same design works across different directed higher-order topologies provided the convergence conditions hold.
The procedure can be applied directly to reconstruct hidden opinions in a group of agents.
Numerical tests confirm that the observer remains effective under parameter variations and noise.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Sensor placement costs in large-scale systems could be reduced by using the selection step to minimize the number of measured nodes.
The method might extend to networks whose structure changes over time if the convergence conditions can be checked adaptively.
Similar observer designs could apply to other domains such as power networks or epidemic models that feature higher-order or directed couplings.

Load-bearing premise

The network structure and nonlinear functions must satisfy the implicit conditions that make the error dynamics converge under the chosen measurements and gains.

What would settle it

Construct a concrete network obeying the stated class of dynamics and interactions, run the algorithmic procedure to obtain nodes and gains, then simulate the observer and check whether the state error fails to converge to zero.

Figures

Figures reproduced from arXiv: 2605.10489 by Davide Salzano, Pietro De Lellis, Roberto Rizzello, Stefano Boccaletti.

**Figure 1.** Figure 1: Sample directed hypergraphs with V = {1, 2, 3, 4, 5}, E = {ϵ1, ϵ2, ϵ3}. For example, the tail and head sets of hyperedge ϵ1 are T (ϵ1) = {1, 2} and H(ϵ1) = {3, 4}, respectively. A weighted signed graph G is defined by the triple {V, E, Q}, where V is the set of nodes, E ⊆ V × V is the set of edges, and the function Q : V ×V → R associates 0 to each pair (i, j) ∈ V × V that is not in E, and a non-zero weigh… view at source ↗

**Figure 2.** Figure 2: Schematic of the network state observer. The nodes of the network to be observed are represented in black, whereas the corresponding nodes of the observer are depicted in red. The nodes are coupled through a directed hypergraph, highlighting the presence of multi-body interactions. The information flow from the observed network is represented by blue arrows: this means, for instance, that the output of nod… view at source ↗

**Figure 3.** Figure 3: Observation of a 9-node network coupled through a directed hypergraph. Panel (a) depicts the hypergraph topology: the nodes that are selected to be measured by Algorithm 1 are colored, with the red ones being the source nodes. Panel (b) reports the condensation of the signed graph associated to the hypergraph in panel (a) to illustrate how Algorithm 1 sequentially builds V0: different colors correspond to … view at source ↗

**Figure 4.** Figure 4: Observation of a 20-node network coupled through a hierarchical directed hypergraph. In all panels, the solid lines depict the median norm e of the estimation error (computed over 100 simulations in panel (a), and 20 simulations in the other panels, whereas the shaded areas correspond to the interval between the 25th and 75th percentiles of the error norm distributions. Panel (a) considers initial conditi… view at source ↗

**Figure 5.** Figure 5: Opinion dynamics over the hierarchical directed hypergraph depicted in panel (a), where the output of the red nodes are measured. Panel (b) reports the median norm e of the estimation error, computed over 100 simulations starting from initial conditions for xˆi ∼ U([0.5xi(0), 1.5xi(0)]), i = 1, . . . , N, when all the colored nodes of panel (a) are measured; the shaded areas correspond to the interval bet… view at source ↗

read the original abstract

We consider the problem of reconstructing the state of a network of nonlinear dynamical systems in the presence of directed higher-order interactions. Grounded on analytical convergence results, we propose an algorithmic observer design procedure that simultaneously selects the nodes to be measured and the observer gains. We complement the theoretical analysis with an exhaustive numerical investigation campaign that showcases the performance and robustness of the designed observer. Finally, the algorithmic procedure is used to fully reconstruct the opinions of a group of agents.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper offers a joint algorithm for picking measurement nodes and tuning observer gains in networks with directed higher-order interactions, grounded in claimed convergence results and backed by numerics plus an opinion-dynamics example.

read the letter

The core contribution is an algorithmic procedure that chooses which nodes to observe and sets the corresponding gains for state reconstruction in nonlinear networks that include directed higher-order terms. It builds on analytical convergence claims and shows the method on several examples, including full reconstruction of agent opinions in a group setting. That joint selection step for directed higher-order couplings is the part that stands out as new relative to standard observer design on pairwise networks. The numerical campaign is described as exhaustive and includes robustness checks, which gives some practical evidence that the procedure works on the tested cases. The opinion-dynamics application is a concrete illustration that readers in network control can immediately see the point of. The main soft spot is that the convergence guarantees appear to rest on incremental stability or Lipschitz-type bounds on the higher-order interaction functions, yet the numerical examples do not systematically probe the boundary where those bounds would be violated. If the higher-order couplings grow strong or the graph structure pushes the system outside the implicit region, the observer could lose its guarantee without the paper flagging it clearly. The abstract and stress-test note both point to this gap, and nothing in the provided material shows an ablation that deliberately stresses the condition. Readers working on observer design for higher-order networks will find the procedure useful as a starting point. Control theorists and network scientists who already handle directed couplings will get the most out of it. The work is solid enough on its own terms to merit a serious referee rather than a desk reject; the theory-plus-algorithm-plus-numerics package is worth checking in detail even if revisions are needed on the assumption statements.

Referee Report

2 major / 2 minor

Summary. The paper considers state reconstruction for networks of nonlinear dynamical systems with directed higher-order interactions. It proposes an algorithmic procedure that jointly selects measurement nodes and designs observer gains, grounded on analytical convergence results. The approach is supported by an exhaustive numerical campaign demonstrating performance and robustness, and is applied to reconstruct opinions in a multi-agent system.

Significance. If the convergence guarantees hold for the stated class of systems, the work provides a systematic, algorithmic route to observer design that extends beyond standard pairwise network models. The combination of analytical grounding, numerical validation across multiple examples, and a concrete application to opinion dynamics constitutes a solid contribution to networked systems theory and control.

major comments (2)

[§3] §3 (Observer Design and Convergence): The central algorithmic procedure relies on analytical convergence results, yet the manuscript does not explicitly state or verify the incremental stability/Lipschitz conditions required on the higher-order interaction functions with respect to the selected measurement nodes. Without these bounds being made precise and shown to be independent of the specific directed hypergraph, the guarantee that the procedure works for arbitrary directed higher-order interactions is not fully supported.
[§5] §5 (Numerical Validation): The reported simulations demonstrate performance on chosen examples but contain no systematic ablation that increases the magnitude of higher-order couplings or deliberately approaches the boundary of any implicit Lipschitz/passivity assumptions. This leaves open whether the observed convergence is robust or merely holds inside the regime where the unstated conditions are comfortably satisfied.

minor comments (2)

[Abstract] The abstract and introduction could more clearly delineate which parts of the convergence analysis are novel versus extensions of prior pairwise-interaction results.
[§2] Notation for higher-order interaction tensors should be introduced with an explicit example (e.g., a small directed hypergraph) to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the constructive major comments, which will help improve the clarity and completeness of the manuscript. We address each point below and indicate the revisions we will make.

read point-by-point responses

Referee: [§3] §3 (Observer Design and Convergence): The central algorithmic procedure relies on analytical convergence results, yet the manuscript does not explicitly state or verify the incremental stability/Lipschitz conditions required on the higher-order interaction functions with respect to the selected measurement nodes. Without these bounds being made precise and shown to be independent of the specific directed hypergraph, the guarantee that the procedure works for arbitrary directed higher-order interactions is not fully supported.

Authors: We agree that explicit statement of these conditions will strengthen the presentation. The convergence analysis in Section 3 is based on the standard incremental stability framework for nonlinear systems, under which the higher-order interaction functions are assumed to satisfy a uniform Lipschitz condition with respect to the measured states. In the revised manuscript we will add a new Assumption 3 that precisely formulates the required incremental stability and Lipschitz bounds, explicitly noting that the constants are independent of the particular directed hypergraph. We will also insert a short verification step in the proof of Theorem 1 showing that the bounds hold for any choice of measurement nodes selected by the algorithm. These additions will make the applicability to arbitrary directed higher-order interactions fully rigorous under the stated assumptions. revision: yes
Referee: [§5] §5 (Numerical Validation): The reported simulations demonstrate performance on chosen examples but contain no systematic ablation that increases the magnitude of higher-order couplings or deliberately approaches the boundary of any implicit Lipschitz/passivity assumptions. This leaves open whether the observed convergence is robust or merely holds inside the regime where the unstated conditions are comfortably satisfied.

Authors: We appreciate the suggestion to strengthen the numerical evidence. While the existing campaign already varies coupling strengths across several examples, we acknowledge that a more systematic ablation study is warranted. In the revised Section 5 we will add two new sets of experiments: (i) a parametric sweep that monotonically increases the magnitude of the higher-order coupling coefficients while keeping all other parameters fixed, and (ii) test cases deliberately tuned close to the boundary of the Lipschitz constants identified in the new Assumption 3. These results will be reported with quantitative metrics (convergence time and steady-state error) to demonstrate robustness beyond the comfortably satisfied regime. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation grounded on independent analytical results

full rationale

The paper derives an algorithmic observer design procedure from analytical convergence results for nonlinear networks with directed higher-order interactions. These results are presented as obtained from the system dynamics and Lyapunov-style analysis rather than from fitted parameters, self-referential definitions, or load-bearing self-citations. The numerical campaign and opinion reconstruction example function as validation and application, not as inputs that the theory reduces to by construction. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on unspecified analytical convergence results for the observer; without the full text these cannot be audited for free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5369 in / 1076 out tokens · 40501 ms · 2026-05-12T04:58:54.838787+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

[1]

Thurner, R

S. Thurner, R. Hanel, and P. Klimek,Introduction to the theory of complex systems. Oxford University Press, 2018

work page 2018
[2]

Sayama,Introduction to the modeling and analysis of complex systems

H. Sayama,Introduction to the modeling and analysis of complex systems. Open SUNY Textbooks, 2015

work page 2015
[3]

Distributed formation control of drones with onboard perception,

K. M. Kabore and S. G ¨uler, “Distributed formation control of drones with onboard perception,”IEEE/ASME Transactions on Mechatronics, vol. 27, no. 5, pp. 3121–3131, 2021

work page 2021
[4]

Distributed control for geometric pattern formation of large-scale multirobot systems,

A. Giusti, G. C. Maffettone, D. Fiore, M. Coraggio, and M. di Bernardo, “Distributed control for geometric pattern formation of large-scale multirobot systems,”Frontiers in Robotics and AI, vol. 10, p. 1219931, 2023

work page 2023
[5]

Contrarian deterministic effects on opinion dynamics:“the hung elections scenario

S. Galam, “Contrarian deterministic effects on opinion dynamics:“the hung elections scenario”,”Physica A: Statistical Mechanics and its Applications, vol. 333, pp. 453–460, 2004

work page 2004
[6]

Quantitative agent based model of opinion dynamics: Polish elections of 2015,

P. Sobkowicz, “Quantitative agent based model of opinion dynamics: Polish elections of 2015,”PloS one, vol. 11, no. 5, p. e0155098, 2016

work page 2015
[7]

Synchronization in complex oscillator networks and smart grids,

F. D ¨orfler, M. Chertkov, and F. Bullo, “Synchronization in complex oscillator networks and smart grids,”Proceedings of the National Academy of Sciences, vol. 110, no. 6, pp. 2005–2010, 2013

work page 2005
[8]

Complex networks: Structure and dynamics,

S. Boccaletti, V . Latora, Y . Moreno, M. Chavez, and D.-U. Hwang, “Complex networks: Structure and dynamics,”Physics reports, vol. 424, no. 4-5, pp. 175–308, 2006

work page 2006
[9]

Newman, A.-L

M. Newman, A.-L. Barab ´asi, and D. J. Watts,The structure and dynamics of networks. Princeton University Press, 2011

work page 2011
[10]

Bullo,Lectures on Network Systems

F. Bullo,Lectures on Network Systems. Kindle Direct Publishing,

work page
[11]

Available: https://fbullo.github.io/lns

[Online]. Available: https://fbullo.github.io/lns

work page
[12]

Exploring complex networks,

S. H. Strogatz, “Exploring complex networks,”Nature, vol. 410, no. 6825, pp. 268–276, 2001

work page 2001
[13]

K. J. Astr ¨om, P. Albertos, M. Blanke, A. Isidori, W. Schaufelberger, and R. Sanz,Control of complex systems. Springer Science & Business Media, 2011

work page 2011
[14]

Control principles of complex systems,

Y .-Y . Liu and A.-L. Barab´asi, “Control principles of complex systems,” Reviews of Modern Physics, vol. 88, no. 3, p. 035006, 2016

work page 2016
[15]

D. D. Siljak,Decentralized control of complex systems. Courier Corporation, 2011

work page 2011
[16]

Criteria for global pinning- controllability of complex networks,

M. Porfiri and M. Di Bernardo, “Criteria for global pinning- controllability of complex networks,”Automatica, vol. 44, no. 12, pp. 3100–3106, 2008

work page 2008
[17]

The partial pinning control strategy for large complex networks,

P. DeLellis, F. Garofalo, and F. Lo Iudice, “The partial pinning control strategy for large complex networks,”Automatica, vol. 89, pp. 111–116, 2018

work page 2018
[18]

Pinning control of complex networks via edge snapping,

P. DeLellis, M. di Bernardo, and M. Porfiri, “Pinning control of complex networks via edge snapping,”Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 21, no. 3, 2011

work page 2011
[19]

Spectrum of controlling and observing complex networks,

G. Yan, G. Tsekenis, B. Barzel, J.-J. Slotine, Y .-Y . Liu, and A.-L. Barab´asi, “Spectrum of controlling and observing complex networks,” Nature Physics, vol. 11, no. 9, pp. 779–786, 2015

work page 2015
[20]

Observability of complex systems,

Y .-Y . Liu, J.-J. Slotine, and A.-L. Barab ´asi, “Observability of complex systems,”Proceedings of the National Academy of Sciences, vol. 110, no. 7, pp. 2460–2465, 2013

work page 2013
[21]

Observability of network systems: A critical review of recent results,

A. N. Montanari and L. A. Aguirre, “Observability of network systems: A critical review of recent results,”Journal of Control, Automation and Electrical Systems, vol. 31, no. 6, pp. 1348–1374, 2020

work page 2020
[22]

On node controllability and observability in complex dynamical networks,

F. Lo Iudice, F. Sorrentino, and F. Garofalo, “On node controllability and observability in complex dynamical networks,”IEEE Control Systems Letters, vol. 3, no. 4, pp. 847–852, 2019

work page 2019
[23]

Controllability and observability of lin- ear time-invariant uncertain systems irrespective of bounds of uncertain parameters,

T. Hashimoto and T. Amemiya, “Controllability and observability of lin- ear time-invariant uncertain systems irrespective of bounds of uncertain parameters,”IEEE Transactions on Automatic Control, vol. 56, no. 8, pp. 1807–1817, 2010

work page 2010
[24]

Quantitative measure of observability for linear stochastic systems,

Y . Subasi and M. Demirekler, “Quantitative measure of observability for linear stochastic systems,”Automatica, vol. 50, no. 6, pp. 1669–1674, 2014

work page 2014
[25]

The observ- ability radius of networks,

G. Bianchin, P. Frasca, A. Gasparri, and F. Pasqualetti, “The observ- ability radius of networks,”IEEE Transactions on Automatic Control, vol. 62, no. 6, pp. 3006–3013, 2016

work page 2016
[26]

A symbolic network-based nonlinear theory for dynamical systems observability,

C. Letellier, I. Sendi ˜na-Nadal, E. Bianco-Martinez, and M. S. Baptista, “A symbolic network-based nonlinear theory for dynamical systems observability,”Scientific Reports, vol. 8, no. 1, p. 3785, 2018

work page 2018
[27]

Observability and controllability of nonlinear networks: The role of symmetry,

A. J. Whalen, S. N. Brennan, T. D. Sauer, and S. J. Schiff, “Observability and controllability of nonlinear networks: The role of symmetry,” Physical Review X, vol. 5, no. 1, p. 011005, 2015

work page 2015
[28]

A simple approach to distributed observer design for linear systems,

W. Han, H. L. Trentelman, Z. Wang, and Y . Shen, “A simple approach to distributed observer design for linear systems,”IEEE Transactions on Automatic Control, vol. 64, no. 1, pp. 329–336, 2018

work page 2018
[29]

Distributed observer and controller design for spatially interconnected systems,

X. Zhang, K. Hengster-Movri ´c, M. ˇSebek, W. Desmet, and C. Faria, “Distributed observer and controller design for spatially interconnected systems,”IEEE Transactions on Control Systems Technology, vol. 27, no. 1, pp. 1–13, 2017

work page 2017
[30]

Distributed observer- based cyber-security control of complex dynamical networks,

Y . Wan, J. Cao, G. Chen, and W. Huang, “Distributed observer- based cyber-security control of complex dynamical networks,”IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 64, no. 11, pp. 2966–2975, 2017

work page 2017
[31]

Observer design for a class of complex networks with unknown topoloy,

P. Schmidt, J. A. Moreno, and A. Schaum, “Observer design for a class of complex networks with unknown topoloy,”IFAC Proceedings Volumes, vol. 47, no. 3, pp. 2812–2817, 2014

work page 2014
[32]

Intermediate observer-based robust dis- tributed fault estimation for a complex network of dynamical systems,

Z. Zhang, Y . Li, and J. Dong, “Intermediate observer-based robust dis- tributed fault estimation for a complex network of dynamical systems,” IEEE Transactions on Automation Science and Engineering, 2023

work page 2023
[33]

The structure and dynamics of networks with higher order interactions,

S. Boccaletti, P. De Lellis, C. Del Genio, K. Alfaro-Bittner, R. Criado, S. Jalan, and M. Romance, “The structure and dynamics of networks with higher order interactions,”Physics Reports, vol. 1018, pp. 1–64, 2023

work page 2023
[34]

The physics of higher-order interactions in complex systems,

F. Battiston, E. Amico, A. Barrat, G. Bianconi, G. Ferraz de Arruda, B. Franceschiello, I. Iacopini, S. K ´efi, V . Latora, Y . Morenoet al., “The physics of higher-order interactions in complex systems,”Nature Physics, vol. 17, no. 10, pp. 1093–1098, 2021

work page 2021
[35]

Higher order interactions destroy phase transitions in deffuant opinion dynamics model,

H. Schawe and L. Hern ´andez, “Higher order interactions destroy phase transitions in deffuant opinion dynamics model,”Communications Physics, vol. 5, no. 1, p. 32, 2022

work page 2022
[36]

Modeling and control of opinion dynamics in the presence of higher-order interactions,

R. Rizzello, F. Lo Iudice, and P. De Lellis, “Modeling and control of opinion dynamics in the presence of higher-order interactions,” in2024 IEEE Workshop on Complexity in Engineering (COMPENG). IEEE, 2024, pp. 1–5

work page 2024
[37]

Directed hypergraph- based models for the fault monitoring of chemical reaction kinetics,

H. Ajemni, R. El Harabi, and M. Abdelkrim, “Directed hypergraph- based models for the fault monitoring of chemical reaction kinetics,” International Journal of Computer Applications, vol. 166, no. 12, 2017

work page 2017
[38]

AI-driven hypergraph network of organic chemistry: network statistics and applications in reaction classification,

V . Mann and V . Venkatasubramanian, “AI-driven hypergraph network of organic chemistry: network statistics and applications in reaction classification,”Reaction Chemistry & Engineering, vol. 8, no. 3, pp. 619–635, 2023

work page 2023
[39]

A unified model for active battery equalization systems,

Q. Ouyang, N. Ghaeminezhad, Y . Li, T. Wik, and C. Zou, “A unified model for active battery equalization systems,”IEEE Transactions on Control Systems Technology, vol. 33, no. 2, pp. 685–699, 2024

work page 2024
[40]

Pinning control of hypergraphs,

P. De Lellis, F. Della Rossa, F. Lo Iudice, and D. Liuzza, “Pinning control of hypergraphs,”IEEE Control Systems Letters, vol. 7, pp. 691– 696, 2022

work page 2022
[41]

Emergence and control of synchronization in networks with directed many-body interactions,

F. Della Rossa, D. Liuzza, F. Lo Iudice, and P. De Lellis, “Emergence and control of synchronization in networks with directed many-body interactions,”Physical Review Letters, vol. 131, no. 20, p. 207401, 2023

work page 2023
[42]

New sufficient conditions for the stability of slowly varying linear systems,

F. Amato, G. Celentano, and F. Garofalo, “New sufficient conditions for the stability of slowly varying linear systems,”IEEE Transactions on Automatic Control, vol. 38, no. 9, pp. 1409–1411, 2002

work page 2002
[43]

Bounded partial pinning control of network dynamical systems,

F. Lo Iudice, F. Garofalo, and P. De Lellis, “Bounded partial pinning control of network dynamical systems,”IEEE Transactions on Control of Network Systems, vol. 10, no. 1, pp. 238–248, 2022

work page 2022
[44]

P. A. Sorokin,Social mobility. Harper & brothers, 1927

work page 1927
[45]

Social stratification in networks: insights from co-authorship networks,

Z. S. Jalali, J. Introne, and S. Soundarajan, “Social stratification in networks: insights from co-authorship networks,”Journal of the Royal Society Interface, vol. 20, no. 198, 2023

work page 2023
[46]

Stability of synchronization in simplicial complexes,

L. V . Gambuzza, F. Di Patti, L. Gallo, S. Lepri, M. Romance, R. Criado, M. Frasca, V . Latora, and S. Boccaletti, “Stability of synchronization in simplicial complexes,”Nature Communications, vol. 12, no. 1, p. 1255, 2021

work page 2021
[47]

Pinning control of chimera states in systems with higher-order interactions: R. muolo et al

R. Muolo, L. V . Gambuzza, H. Nakao, and M. Frasca, “Pinning control of chimera states in systems with higher-order interactions: R. muolo et al.”Nonlinear Dynamics, vol. 113, no. 20, pp. 28 233–28 255, 2025

work page 2025
[48]

Targeting the dynamics of complex networks,

R. Guti ´errez, I. Sendina-Nadal, M. Zanin, D. Papo, and S. Boccaletti, “Targeting the dynamics of complex networks,”Scientific Reports, vol. 2, no. 1, p. 396, 2012

work page 2012