Hypernetworks induce stable hyperlocking.

Code ↔ Paper

3 matches between paragraphs of the paper and lines of its authors' code, computed by the harvester (lexical-v1). Click a colored paragraph or line to see its counterpart.

The 3 matches

[1] § Results › Hyperlocking for other nonlinear oscillators and larger networks ↔ Fig6.jl, lines 47–95 · score 0.69 · Stuart Landau oscillators, coupling strength, triplet coupling, pairwise coupling, networks
[2] § Results › Hyperlocking for other nonlinear oscillators and larger networks ↔ Fig6.jl, lines 254–294 · score 0.65 · triadic coupling, Triplet phase, pairwise links, coupling strength, pairwise locking, arrows
[3] § Results › Hyperlocking for other nonlinear oscillators and larger networks ↔ Fig6.jl, lines 1–45 · score 0.56 · directed pairwise links, directed graph, chosen, nodes, triadic, triplet

Paper

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The authors' code

Julia · 3,108 lines · 125 KB · MIT · 3 matches

### A Pluto.jl notebook ###
# v0.20.24
using Markdown
using InteractiveUtils
# ╔═╡ e6e2b124-f53c-11f0-1dd6-e74cdd0384f9
using DifferentialEquations,Plots, LaTeXStrings, Graphs,GraphRecipes, Random, NetworkLayout
# ╔═╡ 52b3f226-d907-4cae-bf86-bb671e244b26
function generatetriadicnetw(n,probp,probt;frange=0.5)#generates a triadic network represented by three directed graphs. 'n' is the number of nodes. The generated network 'g' is a directed pairwise network, where nodes are chosen with probability 'probp'. Among all triplets of nodes which have no connections in g we chose with probability 'probt' a number of triplets and fix randomly the role of the nodes in the triplet: 'b' and 'c' are the connected pair and 'a' modulates their interaction. The network 'tp' contains the pairwise links (b<->c). The network 'tm' has directed links (b->a) and (c->a) whenever 'a' modulates the interaction of 'b' and 'c'.
rng=MersenneTwister(123)
#tr=rand(rng)
w = (1. -frange) .+ 2. *frange*rand(rng,n)
g = erdos_renyi(n, probp, is_directed=true, seed=123)
tm = DiGraph(n,0)
tp = DiGraph(n,0)
emptytriags = []
triplets = []
for j1 in 1:n
for j2 in j1+1:n
for j3 in j2+1:n
if !(j1 in neighbors(g,j2)) && !(j2 in neighbors(g,j1)) &&
!(j2 in neighbors(g,j3)) && !(j3 in neighbors(g,j2)) &&
!(j3 in neighbors(g,j1)) && !(j1 in neighbors(g,j3))
push!(emptytriags,(j1,j2,j3))
end
end
end
end
for jj in 1:length(emptytriags)
if rand(rng)<probt
a=rand(rng,1:3)
b=mod(a,3)+1
c=mod(b,3)+1
push!(triplets,[emptytriags[jj][a],emptytriags[jj][b],emptytriags[jj][c]])
add_edge!(tp,(emptytriags[jj][b],emptytriags[jj][c]))
add_edge!(tp,(emptytriags[jj][c],emptytriags[jj][b]))
add_edge!(tm,(emptytriags[jj][b],emptytriags[jj][a]))
add_edge!(tm,(emptytriags[jj][c],emptytriags[jj][a]))
end
end
(g,tm,tp,w, emptytriags,triplets)
end
# ╔═╡ 3eaa505d-83c2-462f-92ac-4e223e25babf
function largetriadicnetwork!(du,u,p,t) # right hand side for Stuart-Landau network with multiple triadic motifs. Vector of natural frequencies ω, pairwise network g, triplets are represented by two pairwise networks tp and tm, triplet coupling strength α, pairwise coupling strength β, SL parameter λ.
(ω,g,tp,tm,α,β,λ) = p
n = length(ω)
function sl(u,omega) #Stuart-Landau oscillator
(λ + omega * im - abs(u)^2)*u
end
function h(u1,u2)
(u2 -conj(u1))
end
for jj in 1:n
du[jj] = sl(u[jj],ω[jj]) - β/ne(g) * sum(u[jj] .- u[neighbors(g,jj)]) +
2*α/ne(tm) * sum(u[neighbors(tm,jj)] .* h.(u[neighbors(tp,jj)] ,u[jj]))
end
end
# ╔═╡ cdb2b1f3-32c1-49db-89b0-eaef69938816
function randcomp(n) #gives a vector of 'n' random complex numbers uniformly distributed in the unit square.
u=[]
for j in 1:n
push!(u,rand()+ im*rand())
end
u
end
# ╔═╡ 4911ccf3-3645-4e18-9372-dadda7597922
function runsim(ttrans,trec,p) #Runs a simulation with right hand side 'largetriadicnetwork!' and parameters 'p'. After a start from random initial conditions a trajectory of length 'ttrans' is discarded and then a trajectory of length 'trec' is recorded.
probtrans = ODEProblem(largetriadicnetwork!, 0.3*randcomp(length(p[1])),ttrans,p)
soltrans = solve(probtrans,save_everystep=false,abstol=1e-4 )
probrec = ODEProblem(largetriadicnetwork!, soltrans.u[end],trec,p)
solrec = solve(probrec,alg=Tsit5(),abstol=1e-8 )
end
# ╔═╡ 1b12c3d3-88d7-4c94-849a-692ea4735161
function phase(sol,i;ref=1.,step=0.1) #extracts the unwraped phase of oscillator 'i' in a network solution 'sol'.
trange = 0.:step:sol.t[end]
ph(t)=mod(angle(sol(t)[i]),2*pi)
uwp=[ph(0.)]
shift = 0.
for ii in trange[2:end]
shift = shift + div(ph(ii) - ph(ii-step),2*pi,RoundNearest)
push!(uwp,ph(ii) -shift*2*pi)
end
trange,uwp
end
# ╔═╡ cd6bc01f-c797-413e-b378-cbafbe5caeac
function alleffr(gg,sol) #extracts all effective frequencies from a given solution 'sol' with underlying network 'gg' (the network is just used for indexing the nodes)
alfr=[]
for jj in vertices(gg)
ph = phase(sol,jj)[2]
push!(alfr,(ph[end]-ph[1])/sol.t[end])
end
alfr
end
# ╔═╡ 4cd1363a-58a3-48f1-988c-a8db7e899156
function effreqtr(tnw,sol) # displays for a given hypernetwork the triplets together with their natural and effective frequencies
effr=[]
natfr=[]
for jj in 1: length(tnw[6])
a=tnw[6][jj][1]
b=tnw[6][jj][2]
c=tnw[6][jj][3]
pha, phb, phc = phase(sol,a)[2], phase(sol,b)[2], phase(sol,c)[2]
push!(effr,[(pha[end]-pha[1])/sol.t[end],(phb[end]-phb[1])/sol.t[end], (phc[end]-phc[1])/sol.t[end]])
push!(natfr,[tnw[4][a],tnw[4][b],tnw[4][c]])
end
tnw[6],effr,natfr
end
# ╔═╡ 1b516697-9ad7-4996-8363-92e1b6c9d3fe
function pwphd(sol,j,k;step=0.1)#extracts the unwraped pairwise phase difference of oscillators 'j-k' in a network solution 'sol'.
trange = 0.:step:sol.t[end]
pdiff = [mod(angle(sol(0.)[j])-angle(sol(0.)[k]),2*pi)]
shift = 0.
for ii in trange[2:end]
shift=shift+div(mod(angle(sol(ii)[j])-angle(sol(ii)[k]),2*pi)-
mod(angle(sol(ii-step)[j]) -angle(sol(ii-step)[k]),2*pi),2*pi,RoundNearest)
push!(pdiff,mod(angle(sol(ii)[j])-angle(sol(ii)[k]),2*pi) - shift*2*pi)
end
trange,pdiff
end
# ╔═╡ 4633d5d0-76d9-4ae0-aef0-34ab65cba09e
function trphd(sol,i,j,k;step=0.1)#extracts the unwraped triplet phase of oscillators 'i+j-2k' in a network solution 'sol'.
trange = 0.:step:sol.t[end]
tr(t)=mod(angle(sol(t)[i])+angle(sol(t)[j])-2*angle(sol(t)[k]),2*pi)
pdiff=[tr(0.)]
shift = 0.
for ii in trange[2:end]
shift = shift + div(tr(ii) - tr(ii-step),2*pi,RoundNearest)
push!(pdiff,tr(ii) -shift*2*pi)
end
trange,pdiff
end
# ╔═╡ a5430ebc-31b4-48fc-8913-5dfc063c77c7
function plottrnw(tnw) #plots a triadic network structure produced by 'generatetriadicnetw'.
function extractxy(pos)
x=[]
y=[]
for jj in 1: length(pos)
push!(x,pos[jj][1])
push!(y,pos[jj][2])
end
x,y
end
pos = stress(tnw[1])
x,y=extractxy(pos)
m=length(tnw[6])
mg=DiGraph(2*m,0)
xmod=[]
xmodt=[]
ymod=[]
ymodt=[]
for jj in 1:m
add_edge!(mg,(jj,m+jj))
push!(xmod, x[tnw[6][jj][1]])
push!(ymod, y[tnw[6][jj][1]])
push!(xmodt, (x[tnw[6][jj][2]]+x[tnw[6][jj][3]])/2.)
push!(ymodt, (y[tnw[6][jj][2]]+y[tnw[6][jj][3]])/2.)
end
p1=graphplot(mg,x=append!(xmod,xmodt), y=append!(ymod,ymodt), edgecolor=:blue,linewidth=2, curvature=0.0,markersize = 0.001,style=:dash,size=(200,200),annotation=(-1.2,1.5,"(a)"),annotationfontsizes=12)
graphplot!(p1,tnw[3],x=x,y=y,edgecolor=:blue,linewidth=2, curvature=0.0,markersize = 0.001,arrow=:none)
graphplot!(p1,tnw[1], x=x, y=y,curves=true, nodeshape=:circle, markersize = 0.35, fontsize=10,nodecolor=[:red,:blue,:blue,:blue,:red,:blue,:blue,:blue,:blue,:blue, :blue,:blue,:red,:blue,:blue,:blue,:blue,:blue,:blue,:blue],
names=["a"," "," "," ","b"," "," "," "," "," "," "," ","c"," "," "," "," "," "," "," "])
end
# ╔═╡ 0d0a0938-67f9-4d70-a17c-479d522bde77
md"### Test system with $n=3$
Just for testing we recover the results for the triadic motif."
# ╔═╡ 68ec7863-abd5-4b40-933b-29101875939f
gg=complete_graph(3) #network of pairwise links
# ╔═╡ 4c0c9c36-12e3-41ff-a1ba-c2b38e2dea4c
trnwtest=generatetriadicnetw(3,0.,1.) #network defining the single higher order link
# ╔═╡ 03ac5976-dc02-4f2a-bb68-fbc36a816ab1
graphplot(trnwtest[2],curves=false, names = 1:3 )
# ╔═╡ c4f4a4df-1cbe-4116-a312-52b1ac2a9e3e
graphplot(trnwtest[3],curves=false, names = 1:3 )
# ╔═╡ f2dd9a65-8228-4493-a2a0-60861be05000
ww=[0.98,0.95,1.05]; #natural frequencies
# ╔═╡ 358b33bd-d522-4198-86bd-a1d4ddc0072c
soltest = runsim(500,100,(ww,gg,trnwtest[3],trnwtest[2],0.3,0.0,0.2));
# ╔═╡ e9f32600-1488-47b3-b70e-5464d3e29423
function plotp(sol;step=0.1) #plots pairwise phase differences of solution 'sol'
trange=0.:step:sol.t[end]
pdiff12=[mod(angle(sol(0.)[1])-angle(sol(0.)[2]),2*pi)]
pdiff23=[mod(angle(sol(0.)[2])-angle(sol(0.)[3]),2*pi)]
pdiff31=[mod(angle(sol(0.)[3])-angle(sol(0.)[1]),2*pi)]
shift12=0.
shift23=0.
shift31=0.
for ii in trange[2:end]
shift12=shift12+div(mod(angle(sol(ii)[1])-angle(sol(ii)[2]),2*pi)-
mod(angle(sol(ii-step)[1]) -angle(sol(ii-step)[2]),2*pi),2*pi,RoundNearest)
shift23=shift23+div(mod(angle(sol(ii)[2])-angle(sol(ii)[3]),2*pi)-
mod(angle(sol(ii-step)[2]) -angle(sol(ii-step)[3]),2*pi),2*pi,RoundNearest)
shift31=shift31+div(mod(angle(sol(ii)[3])-angle(sol(ii)[1]),2*pi)-
mod(angle(sol(ii-step)[3]) -angle(sol(ii-step)[1]),2*pi),2*pi,RoundNearest)
push!(pdiff12,mod(angle(sol(ii)[1])-angle(sol(ii)[2]),2*pi) -shift12*2*pi)
push!(pdiff23,mod(angle(sol(ii)[2])-angle(sol(ii)[3]),2*pi) -shift23*2*pi)
push!(pdiff31,mod(angle(sol(ii)[3])-angle(sol(ii)[1]),2*pi) -shift31*2*pi)
end
plot(trange,[pdiff12,pdiff23,pdiff31],color=[:purple :orange :cyan],framestyle=:box,labels=[L"\hat\phi_1-\hat\phi_2" L"\hat\phi_2-\hat\phi_3" L"\hat\phi_3-\hat\phi_1"], yrange=[-7,7], size=(600,200),xticks=false,grid=false,legend_position=:bottomright, yticks=[-5.,0.,5.],annotation=(1,5.4,"(b)"),annotationfontsizes=12,legendfontsize=12,linewidth=2)
end
# ╔═╡ d151ef53-846c-4f82-8300-fd4be387fd9e
function plott(sol;step=0.1) #plots unwrapped triple phase differences of solution 'sol'
tr1(t)=mod(angle(sol(t)[1])+angle(sol(t)[2])-2*angle(sol(t)[3]),2*pi)
tr2(t)=mod(angle(sol(t)[2])+angle(sol(t)[3])-2*angle(sol(t)[1]),2*pi)
tr3(t)=mod(angle(sol(t)[3])+angle(sol(t)[1])-2*angle(sol(t)[2]),2*pi)
trange=0.:step:sol.t[end]
pdiff1=[tr1(0.)]
pdiff2=[tr2(0.)]
pdiff3=[tr3(0.)]
shift1=0.
shift2=0.
shift3=0.
for ii in trange[2:end]
shift1 = shift1 + div(tr1(ii) - tr1(ii-step),2*pi,RoundNearest)
shift2 = shift2 + div(tr2(ii) - tr2(ii-step),2*pi,RoundNearest)
shift3 = shift3 + div(tr3(ii) - tr3(ii-step),2*pi,RoundNearest)
push!(pdiff1,tr1(ii) -shift1*2*pi)
push!(pdiff2,tr2(ii) -shift2*2*pi)
push!(pdiff3,tr3(ii) -shift3*2*pi)
end
plot(trange,[pdiff1,pdiff2,pdiff3],framestyle=:box,labels=[L"\hat\phi_1+\hat\phi_2-2\hat\phi_3" L"\hat\phi_2+\hat\phi_3-2\hat\phi_1" L"\hat\phi_3+\hat\phi_1-2\hat\phi_2"], yrange=[-12,12] ,size=(600,200),grid=false,legend_position=:bottomright,color=[:deeppink :darkgreen :navy], xlabel=L"t", yticks=[-10,0,10],annotation=(1,10.,"(c)"),annotationfontsizes=12,legendfontsize=12, linewidth=2,xguidefontsize=12)
end
# ╔═╡ 1d85b4b1-8822-4328-9f7d-3aac5299328c
rp(t,u)=(t,real(u)) #function for plotting (extracts real part of a component vs time)
# ╔═╡ 01471628-7089-41a2-b6d8-a684ca3e88d9
begin
p1=plot(soltest, idxs = [(rp, 0, 1),(rp, 0, 2),(rp, 0, 3)], xticks=false,
xlabel="", framestyle=:box, grid=false,size=(600,200), yrange=(-1.5,1.5), yticks=[-1.,0.,1.], labels=[L"\mathrm{Re} (z_1)" L"\mathrm{Re}(z_2)" L"\mathrm{Re}(z_3)"], annotation=(4,1.2,"(a)"), annotationfontsizes=12, legendfontsize=12,linewidth=2,guidefontsize=14); #plots time traces of real parts
p2=plotp(soltest); #plots pairwise phase differences
p3=plott(soltest); #plots triplet phases
fig1=plot(p1,p2,p3,layout=(3,1),size=(600,600)) # combined plot
end
# ╔═╡ 810d442b-1b30-44ed-b109-fe4ab0c0dd93
md" ### Network with pairwise and triadic coupling
We simulate now a hypernetwork with randomly chosen pairwise directed links and triadic hyperlinks. The triadic links a chosen from triplets of nodes that have no pairwise connections among them, but are linked to the other nodes by pairwise and, possibly, also by one of the other triadic links.
"
# ╔═╡ 9a976ce0-a318-43b8-b934-274a2790be89
tnw=generatetriadicnetw(20,0.2,0.05,frange=0.3)
# ╔═╡ 2f73156a-a8b2-4d81-8afe-5d0561b8a4f1
md"
This hypernetwork has $n=20$ nodes with directed pairwise links indicated by the black arrows and triadic hyperlinks indicated in blue. The dashed arrows pointing not to a node but to another blue solid link indicate, where a third node modulates a pairwise connection, generating together a triplet hyperlink. Within the triplet of red nodes will adapt the natural frequencies such that together they are sufficiently close to full resonance. All other natural frequencies are chosen with equal probability from the interval $[0.7,1.3]$.
"
# ╔═╡ fc263a9d-a07e-4fb4-bba5-b43cef24c250
md"First, we simulate the network with only the pairwise links, i.e. $\beta=0$. "
# ╔═╡ c5238c84-a689-49be-8946-beea93c7eb33
sol0=runsim(1000,500,(tnw[4],tnw[1],tnw[3],tnw[2],0.0,1.,0.3));#only paiwise links
# ╔═╡ a103e0e1-a916-49ed-8203-ac3c2303b69b
md" The effective frequencies show that for the given coupling strength there is a pairwise locking among some of the nodes. This depends on their random connectedness and natural frequencies. "
# ╔═╡ 4b7b5165-0e6c-47e2-9cdf-ecf8a22de58d
sort(alleffr(tnw[1],sol0))
# ╔═╡ 7764f4bc-a159-4277-964d-367230a5d8e4
md"Now we inspect the effective frequencies of the oscillators in the triplet hyperlinks. "
# ╔═╡ da41ed56-9094-4f54-88cf-63c1dcc4ce62
effreqtr(tnw,sol0)
# ╔═╡ 91564504-554d-4292-9184-99f53d8f7dca
md" We adapt now two of the frequencies in the first triplet to bring it close to full resonance."
# ╔═╡ e550ba23-adf4-4779-8a35-d17f16e9daff
w=copy(tnw[4])
# ╔═╡ 9847f76f-269f-48ab-b743-c7f99cbf8d22
w[1],w[5]=0.94,1.03
# ╔═╡ 475e441f-d144-41d8-9a21-12d8807c3586
sol1=runsim(1000,500,(w,tnw[1],tnw[3],tnw[2],0.0,1.,0.3));# adapted frequencies 'w', no triplet coupling
# ╔═╡ 7404a469-11d3-434c-83c6-2331b5928a2c
effreqtr(tnw,sol1)
# ╔═╡ 8187154e-4194-4fd5-8252-6aec0887516a
md"Finally, we simulate the full network system with pairwise and triplet links and the adapted natural frequencies."
# ╔═╡ fed71c68-671a-4126-84ca-48d4024a02d5
sol = runsim(1000,500,(w,tnw[1],tnw[3],tnw[2],1.5,1.,0.3)); #adapted frequencies and triplet coupling
# ╔═╡ e4b0d5cc-69c1-4bce-a1b2-b955e470346a
begin
pa=plottrnw(tnw)
pb=plot([pwphd(sol,1,5),pwphd(sol,5,13),pwphd(sol,13,1)], framestyle=:box, grid=false, size=(600,200), labels=[L"\hat\phi_a-\hat\phi_b" L"\hat\phi_b-\hat\phi_c" L"\hat\phi_c-\hat\phi_a"], legend_position=:topright, color=[:purple :orange :cyan], xticks=false, yticks=[-40.,0.,40.], yrange=[-50.,50], annotation=(5,41,"(b)"), annotationfontsizes=12, legendfontsize=12,linewidth=2)
plot!(pb,[pwphd(sol1,1,5),pwphd(sol1,5,13),pwphd(sol1,13,1)],color=[:purple :orange :cyan], opacity=0.5,labels=:none)
pc=plot([trphd(sol,1,5,13),trphd(sol,5,13,1),trphd(sol,13,1,5)], framestyle=:box, grid=false, size=(600,280), labels=[L"\hat\phi_a+\hat\phi_b-2\hat\phi_c" L"\hat\phi_b+\hat\phi_c-2\hat\phi_a" L"\hat\phi_c+\hat\phi_a-2\hat\phi_b"], yrange=[-100,100], legend_position=:bottomright, color=[:darkgreen :deeppink :navy], xlabel=L"t", annotation=(5,87,"(c)"), annotationfontsizes=12, legendfontsize=12,linewidth=2,guidefontsize=12);
plot!(pc,[trphd(sol1,1,5,13),trphd(sol1,5,13,1),trphd(sol1,13,1,5)],
color=[:darkgreen :deeppink :navy], opacity=0.5,labels=:none);
fig6=plot(pa,pb,pc,layout=Plots.grid(3,1,heights=[0.64,0.16,0.20]), size=(600,1000))
end
# ╔═╡ 82d530e5-99ae-466f-bdf2-9929e8a63074
savefig(fig6,"./Figs/Fig6.png")
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Fig6.jl, under MIT · at the source

Overview

Authors: Eddie Nijholt¹, Tiago Pereira^1,2, Matthias Wolfrum³, Sagnik Chakraborty⁴, István Z Kiss⁴, Jürgen Kurths⁵

ORCID iDs: Eddie Nijholt, Matthias Wolfrum, Jürgen Kurths

Institutions: Universidade de São Paulo (Brazil); Imperial College London (United Kingdom); Weierstrass Institute for Applied Analysis and Stochastics (Germany); Saint Louis University (United States); Potsdam Institute for Climate Impact Research (Germany)

Journal: Nature communications, volume 17, issue 1, article 7947

Dates: received 10 October 2025; accepted 8 June 2026; published online 24 June 2026

Type: Research article · Language: English

License: CC BY

Identifiers: DOI 10.1038/s41467-026-74556-1 · PMID 42342645 · PMCID PMC13448840 · OpenAlex W7165802382

Open access: gold, a free copy (OpenAlex)

Status: code verified

Methods: Spectral & time-frequency, Connectivity

Keywords: Applied mathematics, Complex networks, Nonlinear phenomena

Topic: Nonlinear Dynamics and Pattern Formation (Computer Networks and Communications, Computer Science), according to OpenAlex

Funding: Ministry of Science, Technology and Innovation | Conselho Nacional de Desenvolvimento Científico e Tecnológico (National Council for Scientific and Technological Development) (312287/2021-6)

Citations: cited by 1 paper (Europe PMC); 42 references in the paper

Abstract

Hypernetworks capture coupling structures where interactions extend beyond pairs to groups of three or more units, called hyperedges. They are of increasing importance for many systems such as the brain, social groups, ecosystems, and the climate. We describe here a synchronization phenomenon that is distinctive for hypernetworks. We uncover that in a system of three coupled oscillators with resonant frequencies, the coupling by a triadic hypernetwork motif, where a third node modulates the interaction between two others, can induce a stable locking of a phase triplet, while no pairwise locking is observed. Using normal form transformations and phase reduction, we derive analytically how a specific choice of the coupling functions induces this hyperlocking. We confirm our predictions with both numerical simulations and chemical experiments. Our findings uncover a new synchronization mechanism intrinsic to higher-order interactions and open new directions for controlling real-world complex dynamics beyond pairwise frameworks.

Reproduced under the paper's license (CC BY), from the paper cited above.

Repository

Its files are read in the Code ↔ Paper reader above, with 3 matches between paragraphs and lines of code.

Zenodo 19821152

License: MIT

State: the link answers, verified on 27 September 2026

Evidence: files inventoried

Languages: Julia (3)

Size: 4 files, 3 scripts

Software Heritage: not checked

Found in: “Code availability”

Not found: README, license file, CITATION.cff, environment file, tests, continuous integration, documentation

Tools: DifferentialEquations.jl (3 files), Plots.jl (3 files)

Availability: 1 check, the latest on 27 September 2026: the link answers (HTTP 200)

27 September 2026: the link answers (HTTP 200)

3 files

Cone.jl, Julia, 2,977 lines
Fig2-4.jl, Julia, 2,893 lines
Fig6.jl, Julia, 3,108 lines, 3 matches

At the source:

Code availability

The numerical simulations and analysis routines used in this study were implemented in Julia, using the package ’DifferentialEquations.jl’30. The computational notebooks can be downloaded from 10.5281/zenodo.19821152.

Reproduced under the paper's license (CC BY), from the paper cited above.

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zenodo:19928267, at Zenodo; found in “Data availability”

Data availability

Data from numerical simulations are provided at 10.5281/zenodo.19821152. Experimental source data for all figures are deposited at 10.5281/zenodo.19928267.

Reproduced under the paper's license (CC BY), from the paper cited above.

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Recorded: type, language, journal, volume, issue, pages, dates, 6 authors, 3 keywords, 1 funder, 20 references.

Cite

This paper

Nijholt, E., Pereira, T., Wolfrum, M., Chakraborty, S., Kiss, I. Z., & Kurths, J. (2026). Hypernetworks induce stable hyperlocking. Nature communications, 17(1), 7947. https://doi.org/10.1038/s41467-026-74556-1

BibTeX

@article{nijholt2026hypernetworks, author = {Nijholt, Eddie and Pereira, Tiago and Wolfrum, Matthias and Chakraborty, Sagnik and Kiss, István Z and Kurths, Jürgen}, title = {{Hypernetworks induce stable hyperlocking}}, journal = {Nature communications}, year = {2026}, month = jun, volume = {17}, number = {1}, pages = {7947}, publisher = {Nature Publishing Group}, issn = {2041-1723}, doi = {10.1038/s41467-026-74556-1}, url = {https://doi.org/10.1038/s41467-026-74556-1}, pmid = {42342645}, pmcid = {PMC13448840} }

RIS

TY - JOUR AU - Nijholt, Eddie AU - Pereira, Tiago AU - Wolfrum, Matthias AU - Chakraborty, Sagnik AU - Kiss, István Z AU - Kurths, Jürgen TI - Hypernetworks induce stable hyperlocking T2 - Nature communications J2 - Nat Commun PY - 2026 DA - 2026/06/24 VL - 17 IS - 1 SP - 7947 SN - 2041-1723 PB - Nature Publishing Group DO - 10.1038/s41467-026-74556-1 UR - https://doi.org/10.1038/s41467-026-74556-1 LA - en ER -

CSL-JSON

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Hypernetworks induce stable hyperlocking.

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