In adaptive dynamical networks, the dynamics of the nodes and the edges influence each other. We show that we can treat such systems as a closed feedback loop between edge and node dynamics. Using recent advances on the stability of feedback systems from control theory, we derive local, sufficient conditions for steady states of such systems to be linearly stable. These conditions are local in the sense that they are written entirely in terms of the (linearized) behavior of the edges and nodes. We apply these conditions to the Kuramoto model with inertia written in an adaptive form and the adaptive Kuramoto model. For the former, we recover a classic result, and for the latter, we show that our sufficient conditions match necessary conditions where the latter are available, thus completely settling the question of linear stability in this setting. The method we introduce can be readily applied to a vast class of systems. It enables straightforward evaluation of stability in highly heterogeneous systems.
REFERENCES
1.
T.
Gross
and B.
Blasius
, “Adaptive coevolutionary networks: A review
,” J. R. Soc. Interface
5
, 259
–271
(2008
). 2.
T.
Gross
, C. J.
Dommar D’Lima
, and B.
Blasius
, “Epidemic dynamics on an adaptive network
,” Phys. Rev. Lett.
96
, 208701
(2006
). 3.
V.
Marceau
, P.-A.
Noël
, L.
Hébert-Dufresne
, A.
Allard
, and L. J.
Dubé
, “Adaptive networks: Coevolution of disease and topology
,” Phys. Rev. E
82
, 036116
(2010
). 4.
S. V.
Scarpino
, A.
Allard
, and L.
Hébert-Dufresne
, “The effect of a prudent adaptive behaviour on disease transmission
,” Nat. Phys.
12
, 1042
–1046
(2016
). 5.
B.
Kozma
and A.
Barrat
, “Consensus formation on adaptive networks
,” Phys. Rev. E
77
, 016102
(2008
). 6.
H.
Rainer
and U.
Krause
, “Opinion dynamics and bounded confidence: Models, analysis and simulation
,” J. Artif. Soc. Soc. Simul.
5
, 1
–12
(2002
).7.
F.
Vazquez
, V. M.
Eguíluz
, and M. S.
Miguel
, “Generic absorbing transition in coevolution dynamics
,” Phys. Rev. Lett.
100
, 108702
(2008
). 8.
I. D.
Couzin
, C. C.
Ioannou
, G.
Demirel
, T.
Gross
, C. J.
Torney
, A.
Hartnett
, L.
Conradt
, S. A.
Levin
, and N. E.
Leonard
, “Uninformed individuals promote democratic consensus in animal groups
,” Science
334
, 1578
–1580
(2011
). 9.
B.
Skyrms
and R.
Pemantle
, “A dynamic model of social network formation,” in Adaptive Networks: Theory, Models and Applications (Springer, 2009), pp. 231–251.10.
A.-L.
Do
, L.
Rudolf
, and T.
Gross
, “Patterns of cooperation: Fairness and coordination in networks of interacting agents
,” New J. Phys.
12
, 063023
(2010
). 11.
S.
Bornholdt
and T.
Rohlf
, “Topological evolution of dynamical networks: Global criticality from local dynamics
,” Phys. Rev. Lett.
84
, 6114
(2000
). 12.
C.
Meisel
and T.
Gross
, “Adaptive self-organization in a realistic neural network model
,” Phys. Rev. E
80
, 061917
(2009
). 13.
C.
Kuehn
, “Time-scale and noise optimality in self-organized critical adaptive networks
,” Phys. Rev. E
85
, 026103
(2012
). 14.
R. L. G.
Raimundo
, P. R.
Guimaraes
, and D. M.
Evans
, “Adaptive networks for restoration ecology
,” Trends Ecol. Evol.
33
, 664
–675
(2018
). 15.
R.
Berner
, T.
Gross
, C.
Kuehn
, J.
Kurths
, and S.
Yanchuk
, “Adaptive dynamical networks
,” Phys. Rep.
1031
, 1
–59
(2023
). 16.
G.
Demirel
, F.
Vazquez
, G. A.
Böhme
, and T.
Gross
, “Moment-closure approximations for discrete adaptive networks
,” Phys. D
267
, 68
–80
(2014
). 17.
L. A.
Segel
and S. A.
Levin
, “Application of nonlinear stability theory to the study of the effects of diffusion on predator-prey interactions
,” AIP Conf. Proc.
27
, 123
–152
(1976
). “Application of nonlinear stability theory to the study of the effects of diffusion on predator-prey interactions,” in AIP Conference Proceedings (American Institute of Physics, 1976), Vol. 27, pp. 123–152, see https://www.jasss.org/5/3/2.html18.
L. M.
Pecora
and T. L.
Carroll
, “Master stability functions for synchronized coupled systems
,” Phys. Rev. Lett.
80
, 2109
(1998
). 19.
R.
Berner
, S.
Vock
, E.
Schöll
, and S.
Yanchuk
, “Desynchronization transitions in adaptive networks
,” Phys. Rev. Lett.
126
, 028301
(2021
). 20.
R.
Berner
and S.
Yanchuk
, “Synchronization in networks with heterogeneous adaptation rules and applications to distance-dependent synaptic plasticity
,” Front. Appl. Math. Stat.
7
, 714978
(2021
). 21.
D.
Wang
, W.
Chen
, S. Z.
Khong
, and L.
Qiu
, “On the phases of a complex matrix
,” Linear Algebra Appl.
593
, 152
–179
(2020
). 22.
D.
Wang
, X.
Mao
, W.
Chen
, and L.
Qiu
, “On the phases of a semi-sectorial matrix and the essential phase of a Laplacian
,” Linear Algebra Appl.
676
, 441
–458
(2023
). 23.
W.
Chen
, D.
Wang
, S. Z.
Khong
, and L.
Qiu
, “A phase theory of multi-input multi-output linear time-invariant systems
,” SIAM J. Control Optim.
62
, 1235
–1260
(2024
). 24.
25.
A. L.
Do
, S.
Boccaletti
, J.
Epperlein
, S.
Siegmund
, and T.
Gross
, “Topological stability criteria for networking dynamical systems with Hermitian Jacobian
,” Eur. J. Appl. Math.
27
, 888
–903
(2016
). 26.
J.
Niehues
, R.
Delabays
, and F.
Hellmann
, “Small-signal stability of power systems with voltage droop,” arXiv:2411.10832 (2024).27.
R.
Kogler
, A.
Plietzsch
, P.
Schultz
, and F.
Hellmann
, “Normal form for grid-forming power grid actors
,” PRX Energy
1
, 013008
(2022
). 28.
A.
Büttner
and F.
Hellmann
, “Complex couplings—A universal, adaptive, and bilinear formulation of power grid dynamics
,” PRX Energy
3
, 013005
(2024
). 29.
G.
Zames
, “On the input-output stability of time-varying nonlinear feedback systems Part one: Conditions derived using concepts of loop gain, conicity, and positivity
,” IEEE Trans. Autom. Control
11
, 228
–238
(1966
). 30.
Y.
Kuramoto
, Self-entrainment of a Population of Coupled Non-linear Oscillators
(Springer
, 1975
), pp. 420
–422
.31.
W. S.
Levine
, The Control Systems Handbook: Control System Advanced Methods
(CRC Press
, 2018
).32.
J.
Bechhoefer
, Control Theory for Physicists
, 1st ed. (Cambridge University Press
, 2021
).33.
The term DC means “direct current” and refers to evaluation at zero frequency.
34.
35.
L.
Woolcock
and R.
Schmid
, “Mixed gain/phase robustness criterion for structured perturbations with an application to power system stability
,” IEEE Control Syst. Lett.
7
, 3193
–3198
(2023
). 36.
From here on, we use the same letter for the Laplace transform of a quantity, following convention.
37.
R. A.
Horn
and C. R.
Johnson
, Matrix Analysis
, 2nd ed. (Cambridge University Press
, 2012
).38.
To see this, consider the stacked system of complexified
and the complex conjugate
, which is unitarily equivalent to a real system of stacked
and
.
39.
M.
Thümler
, S. G.
Srinivas
, M.
Schröder
, and M.
Timme
, “Synchrony for weak coupling in the complexified Kuramoto model
,” Phys. Rev. Lett.
130
, 187201
(2023
). 40.
S.
Lee
, L.
Braun
, F.
Bönisch
, M.
Schröder
, M.
Thümler
, and M.
Timme
, “Complexified synchrony
,” Chaos
34
, 053141
(2024
). 41.
A.
Bergen
and D.
Hill
, “A structure preserving model for power system stability analysis
,” IEEE Trans. Power Appar. Syst.
PAS-100
, 25
–35
(1981
). 42.
G.
Filatrella
, A. H.
Nielsen
, and N. F.
Pedersen
, “Analysis of a power grid using a Kuramoto-like model
,” Eur. Phys. J. B
61
, 485
–491
(2008
). 43.
R.
Berner
, S.
Yanchuk
, and E.
Schöll
, “What adaptive neuronal networks teach us about power grids
,” Phys. Rev. E
103
, 042315
(2021
). © 2025 Author(s). Published under an exclusive license by AIP Publishing.
2025
Author(s)
You do not currently have access to this content.
