Collective decision-making dynamics in hypernetworks

Angela Fontan; Silun Zhang

REVIEW 3 major objections 5 minor 33 references

Collective decision-making dynamics in hypernetworks

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Higher-order interactions turn a single decision threshold into a bistable window.

desk verdict A clean h=2 extension with a plausible pitchfork-unfolding mechanism, but the bistable interval is not actually proven: the decisive stability lemma is deferred and the normal form is only local. read the letter →

arxiv 2509.05182 v1 pith:XSZVZEES submitted 2025-09-05 math.OC cs.MAcs.SYeess.SY

Angela Fontan , Silun Zhang This is my paper · ORCID

classification math.OCcs.MAcs.SYeess.SY MSC 34C2305C6593D05

keywords hypernetworkshigher-orderinteractionscollectivedecision-makingpitchforkbifurcationbistabilityconsensusequilibriasaturatingnonlinearitiesmultiagentsystems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies how three-agent (triadic) interactions change a networked community's ability to make a collective decision. In the pairwise-only model there is one critical level of social effort: below it the community stays deadlocked at zero opinion, above it a consensus decision appears. The paper tries to prove that adding triadic interactions unfolds this sharp transition: a whole interval of social effort appears in which the deadlock and a nontrivial consensus are both stable, so the final outcome depends on initial conditions. If correct, this gives a mechanism by which higher-order social ties can make a community's decision history-dependent rather than predetermined by the strength of social pressure.

What carries the argument

The carrying object is the scalar function g(eps,pi) = -(1+alpha)eps + pi(psi_u(eps)+alpha psi_u(eps)^2), where alpha measures the ratio of triadic to pairwise influence and psi_u is the common saturating sigmoid. Its tangency condition defines pi*_1, and under the paper's symmetry assumptions any consensus equilibrium must solve g(eps,pi)=0. The second load-bearing object is the Lyapunov-Schmidt reduction of the full hypernetwork at the pitchfork point, whose normal form ydot = (pi-pi_1)y + kappa_1 y^3 + kappa_2 y^2 with kappa_1<0, kappa_2>0 encodes the unfolding: the negative cubic gives the pitchfork shape, and the positive quadratic is exactly the symmetry-breaking caused by triadic inte

What would settle it

Construct a five-node hypernetwork satisfying the paper's Assumption 1 and Assumptions 2(i)-(ii) but with triadic tensors rescaled so that 1_n^T A_i^(3)1_n is not proportional to (A^(2)1_n)_i; simulate (5) with the same sigmoid and run a numerical continuation in pi. If the first saddle-node does not occur at the pi*_1 predicted by (10), or if a nontrivial consensus equilibrium exists without solving g(eps,pi)=0, then the paper's explicit bistability interval and consensus reduction would fail outside Assumption 2(iii).

Watch

Extended reading notes

Core claim

The paper claims that for a hypernetwork of order two, with cooperative saturating nonlinearities, the presence of triadic interactions breaks the symmetry of the pairwise pitchfork bifurcation and creates an interval (pi*_1, pi_1) of bistability. Below pi*_1 the origin—interpreted as deadlock—is globally asymptotically stable; above pi_1 it is unstable; in between, the origin and a stable nontrivial consensus equilibrium coexist, separated by an unstable equilibrium. The key quantitative results are an explicit threshold pi*_1 defined by the tangency of g(eps,pi) = -(1+alpha)eps + pi(psi_u(eps)+alpha psi_u(eps)^2), and a Lyapunov-Schmidt normal form ydot = (pi-pi_1)y + kappa_1 y^3 + kappa_2

Load-bearing premise

The central quantitative results rely on the assumption that every agent's total triadic influence is exactly a common multiple alpha of its pairwise influence, together with the assumption that all agents use the same sigmoid nonlinearity; if this proportionality fails, the explicit threshold pi*_1 and the consensus-equilibrium characterization have no proven closed form.

Editorial extensions

If this is right

If the central claim holds, communities described by this model are genuinely history-dependent for a whole range of social effort, not just at a single threshold.
The explicit consensus equilibria mean that, under the paper's assumptions, the emergent collective decision is a unanimous opinion whose size is computable from the social effort and the triadic-to-pairwise ratio alpha.
The threshold pi*_1 provides a lower bound on how much social effort is needed to make deadlock impossible to maintain, and it is strictly smaller than the pairwise threshold pi_1.
The saddle-node at pi*_1 gives a mechanistic explanation of sudden 'jumps' in opinion: a small increase in social effort can abruptly move a community from deadlock to a nontrivial decision.
In the limit of no triadic interactions (alpha=0), the model reduces to the known pairwise pitchfork and pi*_1=pi_1, recovering the established baseline.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the proportional-influence assumption (Assumption 2(iii)) is stronger than the qualitative phenomenon; the unfolding picture likely survives for generic small triadic tensors, but pi*_1 would no longer have a closed-form expression.
Editorial inference: the model suggests a hysteresis interpretation—if social effort is ramped up above pi_1 and then lowered, a committed community may remain on the nontrivial branch until effort falls below pi*_1, making the decision robust to moderate loss of engagement.
Editorial inference: a natural testable extension is to measure pairwise and triadic participation separately in a real collective (e.g., a resident group or animal colony); the model predicts the width of the bistable window grows with alpha, the ratio of triadic to pairwise engagement.
Editorial inference: for h>2 the same unfolding mechanism should appear, but the scalar reduction g would acquire higher-degree terms, so the number and stability of coexisting decisions could differ from the h=2 case.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this is a clean and useful extension of the h=1 saturated decision-making model to h=2 hypernetworks, and the stability results that are actually proven are solid. Second, the headline claim—a bistable interval (π*_1, π1)—is not established by the included proofs. Section IV-C explicitly defers Lemmas 3 and 4, and Lemma 3 is exactly the result that would show the nontrivial equilibrium is locally stable throughout that interval. Theorem 4's Lyapunov–Schmidt normal form only gives a stable branch close to π1; nothing rules out a secondary bifurcation or loss of stability further down. So the abstract overstates what is proven.

What is genuinely new: Theorem 2's explicit threshold π*_1 and the consensus equilibrium construction in Lemma 2, both under Assumption 2(iii). That assumption is clearly ad hoc—'proportional influence' is a tractability condition, not an empirical one—but it is stated plainly, and it gives a sharp result. The singularity-theory part is textbook recognition (pitchfork and its unfolding), but the derivation of κ2>0 is handled carefully. The numerical example is a single random realization, but it faithfully illustrates the mechanism.

The soft spots are real. The missing proofs are load-bearing, not cosmetic. Assumption 2(iii) is also load-bearing for the explicit threshold: without it, you lose consensus equilibria and the global bound. The local analysis assumes A^(3) is small, and the paper says so, but never quantifies 'small.' These are addressable rather than fatal.

Who is this for? Researchers in higher-order opinion dynamics and networked bifurcation analysis. They'll get a well-structured model and a plausible mechanism, with the main gap honestly flagged. I'd accept it for peer review: the problem is relevant and the partial results are rigorous. I would not cite it as a finished result until Lemmas 3 and 4 appear with proofs, and the smallness condition gets an explicit bound.

Referee Report

3 major / 5 minor

Summary. The paper studies a cooperative decision-making dynamics model over hypernetworks with up to 2-interactions, given by Eq. (5). Under sigmoidal odd nonlinearities, the authors analyze equilibria as a function of the social-effort parameter pi. They prove local stability of the origin below pi_1 (Lemma 1), a conservative global stability threshold (Theorem 1), an explicit global threshold pi*_1 under identical nonlinearities and a proportional-influence condition (Theorem 2), a necessary condition for nontrivial equilibria (Theorem 3), existence of consensus equilibria for pi > pi*_1 (Lemma 2), and a local Lyapunov-Schmidt normal form showing an unfolding of a pitchfork (Theorem 4). The paper claims a bistability interval (pi*_1, pi_1) in which both the deadlock state and a nontrivial consensus equilibrium are locally stable, and illustrates this with a numerical example.

Significance. If fully established, the result would provide a clean mechanism by which higher-order interactions break the symmetry of the classical 1-interaction pitchfork bifurcation and create a bistable decision-making interval with an explicit threshold. The Lyapunov computations are explicit, the assumptions are stated, and the local singularity analysis follows a standard route. The numerical example is useful. However, the central interval-level bistability claim is not proven by the included material: the only stability statement for nontrivial equilibria is deferred, and the local normal form does not cover the whole interval. The significance is therefore conditional on completing that proof or appropriately weakening the claim.

major comments (3)

[Sec. IV-C / Lemma 3; Sec. V] The abstract and Sec. V claim bistability for every pi in (pi*_1, pi_1). This requires local asymptotic stability of the upper consensus equilibrium on the whole interval. Lemma 3 is the only stability statement for nontrivial equilibria, but its proof is explicitly deferred ('Due to the length of the proofs...'), and its statement covers only pi in (pi_1, pi_2), where the origin is already unstable. It does not address pi < pi_1. Theorem 4's normal form is derived locally at (0, pi_1, 0) by Lyapunov-Schmidt reduction and is local in pi and in A^(3); it cannot exclude a secondary bifurcation or loss of stability as pi decreases from pi_1 to pi*_1. Thus the claimed bistable interval is not established by the included proofs.
[Assumption 2; Sec. IV-B, Theorem 2 and Lemma 2] The explicit threshold pi*_1 and the consensus equilibria rely on Assumption 2(iii) and identical nonlinearities. However, Assumption 2 as stated only requires symmetry and no self-loops for A^(3)_i; nonnegativity of A^(3)_i is not assumed, although it is used in proofs, e.g., 'A^(3)_i >= 0' appears in the proof of Theorem 3 and in the H-bound. The proportional-influence condition is also a strong algebraic restriction with no structural or empirical justification; all quantitative results depend on it. The authors should add nonnegativity to the formal assumptions and either derive the proportional-influence condition from a more primitive model or discuss robustness of the bistability interval under generic perturbations of A^(3).
[Theorem 4; Sec. IV-B] Theorem 4 is presented as a proof that higher-order interactions unfold the pitchfork, but the proof is essentially a citation to [17] and [31] with no derivation of the stated Lyapunov-Schmidt coefficients. The smallness of A^(3), which is mentioned informally before the theorem, is not stated among the theorem hypotheses. Since the theorem is used to justify the stable upper branch, the hypotheses and the normal-form derivation should be made precise, or the theorem should be clearly labeled as a local formal result with the proof based on the cited references.

minor comments (5)

[Sec. IV intro] The notation pi_2 = lambda_{n-1}(Delta^{-1} A) should use A^(2) consistently; in Lemma 3 the correct matrix is used.
[Proof of Theorem 2] The existence and uniqueness of the tangent point pi* is asserted informally ('This implies...'). A short argument that pi* is well-defined and that g(epsilon, pi) < 0 for all epsilon > 0 when pi < pi* would improve rigor.
[Assumption 2(iii)] The assumption 1_n^T A_i^(3) 1_n = alpha [A^(2)1_n]_i is purely algebraic. It would help to state explicitly where each part of Assumptions 1 and 2 is used (A.5 in Theorem 2, nonnegativity in Theorem 3, etc.).
[Sec. V, Fig. 2] The caption for Fig. 2(b) refers to 'system (3)' in the text; the hypernetwork system is Eq. (5). Also, the sentence 'x*_2 is unstable for any pi < pi_1' is asserted without a proof or reference; this is part of the missing stability analysis.
[Theorem 4] Calling the unfolding an 'n^3-parameter unfolding' is imprecise: the normal form (13) has only pi and the imperfection coefficient kappa_2 as effective unfolding parameters, while the A^(3) entries are constrained by the proportional-influence and symmetry assumptions. A sentence clarifying the counting would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bistability interval is derived from model assumptions via standard singularity theory, not from fitting or self-referential definitions.

full rationale

The paper's derivation chain is self-contained with respect to its main modeling steps. The thresholds pi_1 and pi*_1 are defined from the Jacobian and from the scalar function g(eps,pi) in (9)-(10), respectively, not fitted to the target bistability. Lemma 2 derives consensus equilibria directly from Assumption 2(iii) and (A.5); Theorem 4 obtains the pitchfork unfolding (13) by a Lyapunov-Schmidt reduction and cites external singularity theory [31], with the h=1 base case from [17]/[20] used only at A^(3)=0. The interval (pi*_1, pi_1) is a consequence of Theorem 2, Lemma 1 and Lemma 2, not an input. The only concern is a rigor gap: Lemma 3 (local stability of the nontrivial equilibria on the interval) is stated but its proof is deferred to future work and justified by 'similarity to [20]'. This is a load-bearing missing proof and should be addressed by the authors, but it is a completeness/correctness issue, not circularity: the claimed stability is not assumed in the model or defined into the thresholds.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. Its theoretical edifice rests on the model assumptions (especially the ad hoc proportional-influence condition), prior results by the same authors or their close collaborators, and the local small-A^(3) assumption. The free parameter alpha controls the mixing of 1- and 2-interactions and is chosen by hand in the numerical example.

free parameters (1)

alpha (proportional influence ratio) = 1 in the numerical example; arbitrary nonnegative scalar in Assumption 2(iii)
Introduced to make the consensus-equilibria analysis tractable. It ties each agent's total 2-interaction weight to its 1-interaction degree, and the explicit thresholds pi*_1, pi_1=1+alpha, and the consensus equilibria in Lemma 2 all depend on alpha.

assumptions (6)

domain assumption Assumption 1 (A.1)-(A.5): odd, strictly increasing, saturated, sigmoidal, and identical nonlinearities psi_i, with psi_i'(0)=1.
Defines the admissible opinion-transmission functions. Assumption (A.5) (identical nonlinearities) and the sigmoidal shape are used in Theorems 2 and Lemma 2 to derive the scalar reduction g(epsilon, pi).
domain assumption Assumption 2(i)-(ii): each A_i^(3) is symmetric and has zero diagonal (no self-loops in 2-interactions).
Structural assumptions on the hypergraph that ensure the Jacobian at the origin is symmetric and that the Lyapunov analysis in Theorem 1 goes through.
ad hoc to paper Assumption 2(iii): 1_n^T A_i^(3) 1_n = alpha [A^(2) 1_n]_i for all i.
This proportional-influence condition is load-bearing for the explicit global-stability threshold in Theorem 2 and for the existence of consensus equilibria in Lemma 2. It implies delta_i = (1+alpha) [A^(2)1_n]_i and makes the problem reduce to the scalar equation g(epsilon, pi)=0. Without it, the claimed interval and thresholds have no proven explicit form.
domain assumption Prior h=1 results from [17] and [20] on the pitchfork bifurcation and stability of nontrivial equilibria.
Theorem 4(i) and Lemma 3 rely directly on the Lyapunov-Schmidt reduction and stability theorems of [17] and [20]. These are published external results, but they are assumed correct and are not re-derived here.
ad hoc to paper Implicit smallness of A^(3) for the local Lyapunov-Schmidt analysis in Theorem 4.
Stated in Section IV-B: 'we implicitly assume that the parameters describing the higher-order interactions, A_i^(3), are small.' This smallness is never quantified, yet it is what legitimizes the local normal form (13) and the unfolding description.
standard math Standard matrix theory (Perron-Frobenius, Ger'sgorin) and singularity/unfolding theory from [31].
Used in Lemma 1, Theorem 1, Theorem 3, and Theorem 4 to connect the matrix spectra with the bifurcation structure. These are accepted mathematical tools, not specific to this paper.

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Cite this review

Pith. "Pith review of Collective decision-making dynamics in hypernetworks." pith.science (2026). https://pith.science/paper/XSZVZEES

@misc{pith2026250905182,
  author       = {Pith},
  title        = {Pith review of: Collective decision-making dynamics in hypernetworks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XSZVZEES}},
  note         = {Machine review of arXiv:2509.05182}
}

read the original abstract

This work describes a collective decision-making dynamical process in a multiagent system under the assumption of cooperative higher-order interactions within the community, modeled as a hypernetwork. The nonlinear interconnected system is characterized by saturated nonlinearities that describe how agents transmit their opinion state to their neighbors in the hypernetwork, and by a bifurcation parameter representing the community's social effort. We show that the presence of higher-order interactions leads to the unfolding of a pitchfork bifurcation, introducing an interval for the social effort parameter in which the system exhibits bistability. With equilibrium points representing collective decisions, this implies that, depending on the initial conditions, the community will either remain in a deadlock state (with the origin as the equilibrium point) or reach a nontrivial decision. A numerical example is given to illustrate the results.

Figures

Figures reproduced from arXiv: 2509.05182 by the authors.

**Figure 1.** Hypergraph G 2 and associated adjacency tensors A(2) and A (3) i , i = 1, . . . , n. The colormap represents the edge weights for both 1- and 2-interactions. consumption. Therefore, similar to the co-authorship hypernetworks example [33], the social hypernetwork of the KTH Live-In Lab tenants captures (multi-)tenant collaborations across various sustainable activities (mobility, food, water consumption, and electri… view at source ↗

**Figure 2.** Bifurcation diagram of system (5), depicted for a component xi . (a): Without 2-interactions, that is, A(3) = 0. (b): With 2-interactions. The adjacency tensor A(3) satisfies Assumption 2(iii) with α = 1. [6] A. Sarker, N. Northrup, and A. Jadbabaie, Generalizing Homophily to Simplicial Complexes, vol. 1078. Springer, 2023. [7] A. Hickok, Y. Kureh, H. Z. Brooks, M. Feng, and M. A. Porter, “A bounded-confidence model… view at source ↗

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Works this paper leans on

33 extracted references · 29 canonical work pages

[17]

Multiequilibria analysis for a class of collective decision-making networked systems,

A. Fontan and C. Altafini, “Multiequilibria analysis for a class of collective decision-making networked systems,”IEEE Transactions on Control of Network Systems, vol. 5, pp. 1931–1940, 12 2018

work page 1931
[31]

Golubitsky and D

M. Golubitsky and D. G. Schaeffer,Singularities and Groups in Bi- furcation Theory. Volume I, vol. 51 ofApplied Mathematical Sciences. Springer New York, 1985

work page 1985
[1]

Networks beyond pairwise interactions: Structure and dynamics,

F. Battiston, G. Cencetti, I. Iacopini, V . Latora, M. Lucas, A. Patania, J. G. Young, and G. Petri, “Networks beyond pairwise interactions: Structure and dynamics,”Physics Reports, vol. 874, pp. 1–92, 2020

work page 2020
[2]

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, 2021

work page 2021
[3]

Centralities in simplicial complexes. Applications to protein interaction networks,

E. Estrada and G. J. Ross, “Centralities in simplicial complexes. Applications to protein interaction networks,”Journal of Theoretical Biology, vol. 438, pp. 46–60, 2018

work page 2018
[4]

Higher-order interaction between species inhibits bacterial invasion of a phototroph-predator microbial commu- nity,

H. Mickalide and S. Kuehn, “Higher-order interaction between species inhibits bacterial invasion of a phototroph-predator microbial commu- nity,”Cell Systems, vol. 9, pp. 521–533, 12 2019

work page 2019
[5]

Modelling non- linear consensus dynamics on hypergraphs,

R. Sahasrabuddhe, L. Neuh ¨auser, and R. Lambiotte, “Modelling non- linear consensus dynamics on hypergraphs,”Journal of Physics: Complexity, vol. 2, no. 2, 2021. (a) (b) Fig. 2: Bifurcation diagram of system (5), depicted for a componentx i. (a): Without 2-interactions, that is,A (3) = 0. (b): With 2-interactions. The adjacency tensorA (3) satisfies Assu...

work page 2021
[6]

Sarker, N

A. Sarker, N. Northrup, and A. Jadbabaie,Generalizing Homophily to Simplicial Complexes, vol. 1078. Springer, 2023

work page 2023

Show all 33 references

[7]

A bounded-confidence model of opinion dynamics on hypergraphs,

A. Hickok, Y . Kureh, H. Z. Brooks, M. Feng, and M. A. Porter, “A bounded-confidence model of opinion dynamics on hypergraphs,” SIAM Journal on Applied Dynamical Systems, vol. 21, pp. 1–32, 3 2022

2022
[8]

Consensus Dynamics and Opinion Formation on Hypergraphs,

L. Neuh ¨auser, R. Lambiotte, and M. T. Schaub, “Consensus Dynamics and Opinion Formation on Hypergraphs,” inHigher-Order Systems. Understanding Complex Systems(F. Battiston and G. Petri, eds.), pp. 347–376, Springer, 2022

2022
[9]

Multibody interactions and nonlinear consensus dynamics on networked systems,

L. Neuh ¨auser, A. Mellor, and R. Lambiotte, “Multibody interactions and nonlinear consensus dynamics on networked systems,”Physical Review E, vol. 101, no. 3, pp. 1–11, 2020

2020
[10]

Synchronization induced by directed higher-order interactions,

L. Gallo, R. Muolo, L. V . Gambuzza, V . Latora, M. Frasca, and T. Car- letti, “Synchronization induced by directed higher-order interactions,” Communications Physics, vol. 5, no. 1, 2022

2022
[11]

A tutorial on modeling and analysis of dynamic social networks. Part I,

A. V . Proskurnikov and R. Tempo, “A tutorial on modeling and analysis of dynamic social networks. Part I,”Annual Reviews in Control, vol. 43, pp. 65–79, 2017

2017
[12]

A tutorial on modeling and analysis of dynamic social networks. Part II,

A. V . Proskurnikov and R. Tempo, “A tutorial on modeling and analysis of dynamic social networks. Part II,”Annual Reviews in Control, vol. 45, pp. 166–190, 2018

2018
[13]

Dynamics over signed networks,

G. Shi, C. Altafini, and J. S. Baras, “Dynamics over signed networks,” SIAM Review, vol. 61, pp. 229–257, 1 2019

2019
[14]

Reaching a consensus,

M. H. DeGroot, “Reaching a consensus,”Journal of the American Statistical Association, vol. 69, no. 345, pp. 118–121, 1974

1974
[15]

Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes,

Y . Zhang, M. Lucas, and F. Battiston, “Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes,”Nature Communications, vol. 14, p. 1605, 3 2023

2023
[16]

Spectral conditions for stability and stabilization of positive equilibria for a class of nonlin- ear cooperative systems,

P. U. Abara, F. Ticozzi, and C. Altafini, “Spectral conditions for stability and stabilization of positive equilibria for a class of nonlin- ear cooperative systems,”IEEE Transactions on Automatic Control, vol. 63, no. 2, pp. 402–417, 2018

2018
[18]

Multiagent Decision-Making Dynamics Inspired by Honeybees,

R. Gray, A. Franci, V . Srivastava, and N. E. Leonard, “Multiagent Decision-Making Dynamics Inspired by Honeybees,”IEEE Transac- tions on Control of Network Systems, vol. 5, pp. 793–806, 6 2018

2018
[19]

Neurons with graded response have collective compu- tational properties like those of two-state neurons.,

J. J. Hopfield, “Neurons with graded response have collective compu- tational properties like those of two-state neurons.,”Proc. Natl. Acad. Sci. USA, vol. 81, no. 10, pp. 3088–3092, 1984

1984
[20]

The role of frustration in collective decision-making dynamical processes on multiagent signed networks,

A. Fontan and C. Altafini, “The role of frustration in collective decision-making dynamical processes on multiagent signed networks,” IEEE Transactions on Automatic Control, vol. 67, pp. 5191–5206, 10 2022

2022
[21]

Nonlinear opinion dynamics with tunable sensitivity,

A. Bizyaeva, A. Franci, and N. E. Leonard, “Nonlinear opinion dynamics with tunable sensitivity,”IEEE Transactions on Automatic Control, vol. 68, pp. 1415–1430, 3 2023

2023
[22]

Multi-topic belief formation through bifurcations over signed social networks,

A. Bizyaeva, A. Franci, and N. E. Leonard, “Multi-topic belief formation through bifurcations over signed social networks,”IEEE Transactions on Automatic Control, no. to appear, pp. 1–16, 2025

2025
[23]

A signed network perspective on the gov- ernment formation process in parliamentary democracies,

A. Fontan and C. Altafini, “A signed network perspective on the gov- ernment formation process in parliamentary democracies,”Scientific Reports, vol. 11, 12 2021

2021
[24]

Online learning for nonlinear dynamical systems without the i.i.d. condition,

L. Zhang and S. Zhang, “Online learning for nonlinear dynamical systems without the i.i.d. condition,”arXiv:2504.02995, 2025

2025 arXiv
[25]

Modeling collective be- haviors: A moment-based approach,

S. Zhang, A. Ringh, X. Hu, and J. Karlsson, “Modeling collective be- haviors: A moment-based approach,”IEEE Transactions on Automatic Control, vol. 66, no. 1, pp. 33–48, 2020

2020
[26]

Global exponential stability of high-order Hopfield neural networks with state-dependent impulses,

Z. He, C. Li, H. Li, and Q. Zhang, “Global exponential stability of high-order Hopfield neural networks with state-dependent impulses,” Physica A: Statistical Mechanics and its Applications, vol. 542, p. 123434, 2020

2020
[27]

Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes,

O. T. Courtney and G. Bianconi, “Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes,”Physical Review E, vol. 93, p. 062311, 6 2016

2016
[28]

Node and edge nonlinear eigenvector centrality for hypergraphs,

F. Tudisco and D. J. Higham, “Node and edge nonlinear eigenvector centrality for hypergraphs,”Communications Physics, vol. 4, p. 201, 9 2021

2021
[29]

R. A. Horn and C. R. Johnson,Matrix analysis. Cambridge University Press, second ed., 2013

2013
[30]

Nonquadratic Lyapunov functions for robust control,

F. Blanchini, “Nonquadratic Lyapunov functions for robust control,” Automatica, vol. 31, pp. 451–461, 3 1995

1995
[32]

Social interactions for a sustain- able lifestyle: The design of an experimental case study,

A. Fontan, M. Farjadnia, J. Llewellyn, C. Katzeff, M. Molinari, V . Cvetkovic, and K. Johansson, “Social interactions for a sustain- able lifestyle: The design of an experimental case study,”IFAC- PapersOnLine, vol. 56, no. 2, pp. 657–663, 2023

2023
[33]

Random walks on hypergraphs,

T. Carletti, F. Battiston, G. Cencetti, and D. Fanelli, “Random walks on hypergraphs,”Physical Review E, vol. 101, p. 022308, 2 2020

2020

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [17]

Multiequilibria analysis for a class of collective decision-making networked systems,

A. Fontan and C. Altafini, “Multiequilibria analysis for a class of collective decision-making networked systems,”IEEE Transactions on Control of Network Systems, vol. 5, pp. 1931–1940, 12 2018

work page 1931

[2] [31]

Golubitsky and D

M. Golubitsky and D. G. Schaeffer,Singularities and Groups in Bi- furcation Theory. Volume I, vol. 51 ofApplied Mathematical Sciences. Springer New York, 1985

work page 1985

[3] [1]

Networks beyond pairwise interactions: Structure and dynamics,

F. Battiston, G. Cencetti, I. Iacopini, V . Latora, M. Lucas, A. Patania, J. G. Young, and G. Petri, “Networks beyond pairwise interactions: Structure and dynamics,”Physics Reports, vol. 874, pp. 1–92, 2020

work page 2020

[4] [2]

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, 2021

work page 2021

[5] [3]

Centralities in simplicial complexes. Applications to protein interaction networks,

E. Estrada and G. J. Ross, “Centralities in simplicial complexes. Applications to protein interaction networks,”Journal of Theoretical Biology, vol. 438, pp. 46–60, 2018

work page 2018

[6] [4]

Higher-order interaction between species inhibits bacterial invasion of a phototroph-predator microbial commu- nity,

H. Mickalide and S. Kuehn, “Higher-order interaction between species inhibits bacterial invasion of a phototroph-predator microbial commu- nity,”Cell Systems, vol. 9, pp. 521–533, 12 2019

work page 2019

[7] [5]

Modelling non- linear consensus dynamics on hypergraphs,

R. Sahasrabuddhe, L. Neuh ¨auser, and R. Lambiotte, “Modelling non- linear consensus dynamics on hypergraphs,”Journal of Physics: Complexity, vol. 2, no. 2, 2021. (a) (b) Fig. 2: Bifurcation diagram of system (5), depicted for a componentx i. (a): Without 2-interactions, that is,A (3) = 0. (b): With 2-interactions. The adjacency tensorA (3) satisfies Assu...

work page 2021

[8] [6]

Sarker, N

A. Sarker, N. Northrup, and A. Jadbabaie,Generalizing Homophily to Simplicial Complexes, vol. 1078. Springer, 2023

work page 2023