Continuum limits for adaptive network dynamics

Gkogkas, Marios Antonios; Kuehn, Christian; Xu, Chuang

doi:10.4310/cms.2023.v21.n1.a4

Cited by 11 publications

(3 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Opinion formation on evolving networks is also studied in [22]. We mention the work [26], where the authors study a Kuramoto-type model in a weighted network, whose weights are allowed to depend on the phase of the oscillators. In the previous works the graph is a way of keeping track of the identity or label of different particles and does not bring further structural properties.…”

Section: Introductionmentioning

confidence: 99%

On evolution PDEs on co-evolving graphs

Esposito,

Mikolás

2024

DCDS

View full text Add to dashboard Cite

We provide a well-posedness theory for a class of nonlocal continuity equations on co-evolving graphs. We describe the connection among vertices through an edge weight function and we let it evolve in time, coupling its dynamics with the dynamics on the graph. This is relevant in applications to opinion dynamics and transportation networks. Existence and uniqueness of suitably defined solutions is obtained by exploiting the Banach fixed-point theorem. We consider different time scales for the evolution of the weight function: faster and slower than the flow defined on the graph. The former leads to graphs whose weight functions depend nonlocally on the density configuration at the vertices, while the latter induces static graphs. Furthermore, we prove a discrete-to-continuum limit for the PDEs under study as the number of vertices converges to infinity.

show abstract

Section: Introductionmentioning

confidence: 99%

On evolution PDEs on co-evolving graphs

Esposito,

Mikolás

2024

DCDS

View full text Add to dashboard Cite

show abstract

“…Even if node dynamics are given by a simple Kuramoto-type model, it is a challenge to understand the dynamics of large networks with adaptivity. While the continuum limit of Kuramoto-type networks with STDP-like adaptivity can be described by integro-differential equations from a theoretical perspective (Gkogkas, Kuehn, & Xu, 2021), these do not necessarily elucidate the resulting network dynamics or yield a computational advantage. Approaches like the Ott-Antonsen reduction (Ott & Antonsen, 2008), which have been instrumental to derive low-dimensional descriptions of phase oscillators (see (Bick, Goodfellow, Laing, & Martens, 2020) and references therein) are not directly applicable to adaptive networks where all connection weights evolve independently of one another.…”

Section: Introductionmentioning

confidence: 99%

Mean-field approximations with adaptive coupling for networks with spike-timing-dependent plasticity

Duchet

Bick

Byrne

2022

Preprint

View full text Add to dashboard Cite

Understanding the effect of spike-timing-dependent plasticity (STDP) is key to elucidate how neural networks change over long timescales and to design interventions aimed at modulating such networks in neurological disorders. However, progress is restricted by the significant computational cost associated with simulating neural network models with STDP, and by the lack of low-dimensional description that could provide analytical insights. To approximate STDP in phase oscillator networks, simpler phase-difference-dependent plasticity (PDDP) rules have been developed, which prescribe synaptic changes based on phase differences of neuron pairs rather than differences in spike timing. Here we construct mean-field approximations for phase oscillator networks with STDP to describe the dynamics of large networks of adaptive oscillators. We first show that single-harmonic PDDP rules can approximate a simple form of symmetric STDP, while multi-harmonic rules are required to accurately approximate causal STDP. We then derive exact expressions for the evolution of the average PDDP coupling weight in terms of network synchrony. For adaptive networks of Kuramoto oscillators that form clusters, we formulate a low-dimensional description based on mean field dynamics and average coupling weight. Finally, we show that such a two-cluster mean-field model can be fitted to provide a low-dimensional approximation of a full adaptive network with symmetric STDP. Our framework represents a step towards a low-dimensional description of adaptive networks with STDP, and could for example inform the development of new therapies aimed at maximizing the long-lasting effects of brain stimulation.

show abstract

“…Graphs are indeed a suitable mathematical structure to classify and represent data, as done in [7,34,35,59,47,64] and the references therein. Furthermore, it is worth to mention recent advances in the use of graphs in the context of social dynamics or opinion formation, [55,4,63], kinetic networks, [10], and synchronization, [37].…”

mentioning

confidence: 99%

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

Esposito

Patacchini

Schlichting

et al. 2021

Arch Rational Mech Anal

122

View full text Add to dashboard Cite

We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

show abstract

Continuum limits for adaptive network dynamics

Cited by 11 publications

References 21 publications

On evolution PDEs on co-evolving graphs

On evolution PDEs on co-evolving graphs

Mean-field approximations with adaptive coupling for networks with spike-timing-dependent plasticity

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

Contact Info

Product

Resources

About