Interacting diffusions on sparse graphs: hydrodynamics from local weak limits

Oliveira, Roberto I.; Reis, Guilherme H.; Stolerman, Lucas M.

doi:10.1214/20-ejp505

Cited by 19 publications

(16 citation statements)

References 24 publications

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“…initial conditions (respectively, Gibbs measure with the same interaction potential). The only other empirical measure convergence result in the sparse graph regime that we are aware of is [32], which obtains similar convergence results for a slightly different model of Markovian interacting diffusions with identity diffusion coefficient, weighted pairwise interactions, an i.i.d. random environment and i.i.d.…”

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confidence: 53%

“…Discussion of our results. Both works [24] and [32] can be viewed as implementing, for sparse graph sequences, the first step of the two-step approach of [18] mentioned above for complete graphs, namely showing that the limit of {X Gn } n∈N exists and can be characterized as the unique solution to a countably infinite coupled system of SDEs. However, both these works leave open the important question of providing an autonomous characterization of the marginal dynamics of these infinite system of SDEs.…”

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confidence: 99%

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Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees

Lacker¹,

Ramanan²,

Wu³

2020

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Consider a system of homogeneous interacting diffusive particles labeled by the nodes of a unimodular Galton-Watson tree, where the state of each node evolves infinitesimally like a d-dimensional diffusion whose drift coefficient depends on (the histories of) its own state and the states of neighboring nodes, and whose diffusion coefficient depends only on its own state. Under suitable regularity assumptions on the coefficients, an autonomous characterization is obtained for the marginal distribution of the dynamics of the neighborhood of a typical node in terms of a certain local equation, which is a new kind of stochastic differential equation that is nonlinear in the sense of McKean. This equation describes a finite-dimensional non-Markovian stochastic process whose infinitesimal evolution at any time depends not only on the structure and current state of the neighborhood, but also on the conditional law of the current state given the past of the neighborhood process until that time. Such marginal distributions are of interest because they arise as weak limits of both marginal distributions and empirical measures of interacting diffusions on many sequences of sparse random graphs, including the configuration model and Erdös-Rényi graphs whose average degrees converge to a finite non-zero limit. The results obtained complement classical results in the mean-field regime, which characterize the limiting dynamics of homogeneous interacting diffusions on complete graphs, as the number of nodes goes to infinity, in terms of a corresponding nonlinear Markov process. However, in the sparse graph setting, the topology of the graph matters, and the analysis requires a completely different approach. Important ingredients of the proofs include conditional independence and symmetry properties for particle trajectories on unimodular Galton-Watson trees, a stochastic analytic result on projections of Itô processes, and a recursive construction for establishing well-posedness of the local equation.

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confidence: 53%

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confidence: 99%

Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees

Lacker¹,

Ramanan²,

Wu³

2020

Preprint

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“…We review the results for different graph families below. The following literature is based on a wide range of population models, including stochastic networks (Kar and Moura (2011)), epidemics Xavier (2013), Santos, Kar, Moura, andXavier (2016)), Kuramoto models Medvedev (2019), Medvedev (2019), Chiba, Medvedev, and Mizuhara (2018)) and interacting diffusions (Delattre, Giacomin, and Luçon (2016), Coppini, Dietert, and Giacomin (2020), Olivera and Reis (2019), Oliveira, Reis, andStolerman (2020), Lacker, Ramanan, andWu (2019)). The details and applications of each of these models are different, but the fundamental hurdles of the distributed information setting is present in all of them.…”

Section: Related Workmentioning

confidence: 99%

“…Recently, Oliveira, Reis, and Stolerman (2020) as well as Lacker, Ramanan, and Wu (2019) have examined the sparse case where the average degree G is bounded. The techniques in this regime are much more complicated and are outside the purview of our results.…”

Section: Related Workmentioning

confidence: 99%

Mean-field Approximations for Stochastic Population Processes with Heterogenous Interactions

Sridhar¹,

Kar²

2021

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This paper studies a general class of stochastic population processes in which agents interact with one another over a network. Agents update their behaviors in a random and decentralized manner based only on their current state and the states of their neighbors. It is well known that when the number of agents is large and the network is a complete graph (has all-to-all information access), the macroscopic behavior of the population converges to a differential equation called a mean-field approximation. When the network is not complete, it is unclear in general whether there exists a suitable mean-field approximation for the macroscopic behavior of the population. This paper provides general conditions on the network and policy dynamics for which a suitable mean-field approximation exists. First, we show that as long as the network is well-connected, the macroscopic behavior of the population concentrates around the same mean-field system as the complete-graph case. Next, we show that as long as the network is sufficiently dense, the macroscopic behavior of the population concentrates around a mean-field system that is, in general, different from the mean-field system obtained in the complete-graph case. Finally, we provide conditions under which the mean-field approximation is equivalent to the one obtained in the complete-graph case.

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“…Note also that all the present works address the case where the graph of interaction has diverging degrees. The case with sparse interaction (see [56,42] in the diffusive case) remains open for Hawkes processes and will be the object of future works.…”

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confidence: 99%

Multivariate Hawkes processes on inhomogeneous random graphs

Agathe-Nerine¹

2021

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We consider a population of N interacting neurons, represented by a multivariate Hawkes process : the firing rate of each neuron depends on the history of the connected neurons. Contrary to the mean-field framework where the interaction occurs on the complete graph, the connectivity between particles is given by a random possibly diluted and inhomogeneous graphs where the probability of presence of each edge depends on the spatial position of its vertices. We address the well-posedness of this system and the behaviour as N → ∞. A crucial issue will be to understand how spatial inhomogeneity influences the large time behaviour of the system.

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Interacting diffusions on sparse graphs: hydrodynamics from local weak limits

Cited by 19 publications

References 24 publications

Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees

Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees

Mean-field Approximations for Stochastic Population Processes with Heterogenous Interactions

Multivariate Hawkes processes on inhomogeneous random graphs

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