Centrality Measures for Graphons: Accounting for Uncertainty in Networks

To achieve control objectives for extremely largescale complex networks using standard methods is essentially intractable. In this work, a theory of the approximate control of complex network systems is proposed and developed by the use of graphon theory and the theory of infinite dimensional systems. First, the graphon dynamical system models are formulated in an appropriate infinite dimensional space in order to represent arbitrary-size networks of linear dynamical systems, and to define the convergence of sequences of network systems with limits in the space. The exact controllability and the approximate controllability of graphon dynamical systems are then investigated. Second, the minimum energy state-to-state control problem and the linear quadratic regulator problem for systems on complex networks are considered. The control problem for the graphon limit systems is solved in each case and an approximation is defined which yields control laws for the original control problems. Furthermore, the convergence properties of the approximation schemes are established. A systematic control design methodology is developed within this framework. Finally, numerical examples of networks with randomly sampled weightings are presented to illustrate the effectiveness of the graphon control methodology.

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“…Hence the convergence is point-wise in time and uniform in t over [0, T ] and hence (19) holds. Furthermore,…”

Section: B Proof Of Theoremmentioning

confidence: 92%

Graphon Control of Large-Scale Networks of Linear Systems

Gao

Caines

2020

IEEE Trans. Automat. Contr.

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“…In the study of kernels, various norms are relevant to consider [29], [31], [4]. For 1 ≤ p < ∞, we define the L p norm of a kernel as For W ∈ W 1 , we have the following inequalities between L p norms and the cut norm:…”

Section: B Normsmentioning

confidence: 99%

“…Definition 1 (Sampled Graph [4]): Given a graphon W and a size N ∈ N, we say that the graph G is sampled from W if it is obtained through:…”

Section: Sampling and Approximationmentioning

confidence: 99%

Graphon-Based Sensitivity Analysis of SIS Epidemics

Vizuete

Frasca

Garin

2020

IEEE Control Syst. Lett.

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In this work, we use the spectral properties of graphons to study stability and sensitivity to noise of deterministic SIS epidemics over large networks. We consider the presence of additive noise in a linearized SIS model and we derive a noise index to quantify the deviation from the diseasefree state due to noise. For finite networks, we show that the index depends on the adjacency eigenvalues of its graph. We then assume that the graph is a random sample from a piecewise Lipschitz graphon with finite rank and, using the eigenvalues of the associated graphon operator, we find an approximation of the index that is tight when the network size goes to infinity. A numerical example is included to illustrate the results.

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“…[27,28,29,30,31,32]. Notably in the case of complements, the strategy of each agent at equilibrium is proportional to its Bonacich centrality in the underlying graphon, as recently defined in [33]. For such linear quadratic network games we also study the effect of different targeted intervention policies.…”

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confidence: 99%

“…The concurrent work [40] suggests the use of graphons to extend the setup of mean-field games (which differently from network games are dynamic and stochastic games) to heterogeneous settings. Finally, we remark that the idea of interpreting observed graphs as random realizations from an underlying random graph model has recently been used in the study of centrality measures in [41] for stochastic block models and in [33] for graphon models. The authors of these papers study among others Bonacich centrality, which is known to coincide with the equilibrium of a specific type of network games (with scalar nonnegative strategies, quadratic payoff functions and strategic complements).…”

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confidence: 99%

Graphon Games: A Statistical Framework for Network Games and Interventions

Parise

Ozdaglar

2019

SSRN Journal

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In this paper, we introduce a new class of infinite population games, which we term graphon games. As a first contribution, we show that graphon games can be used to describe strategic behavior in heterogeneous populations of infinite size. We establish existence and uniqueness of graphon equilibria and derive general comparative statics results. As a second contribution, we study the equilibria of an ensemble of finite network games sampled from a stochastic network formation process (represented by the graphon). We provide explicit bounds on the distance of the equilibrium of any finite sampled network game and the corresponding graphon equilibrium in terms of the population size, and we characterize optimal interventions in sampled network games by a planner who knows the graphon but not the realization of the sampled network. Finally, as a third contribution, we relax the assumption that agents know the sampled network and establish a tight link between the graphon equilibrium and the Bayesian Nash equilibrium of an incomplete information network game sampled from the same graphon.

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Centrality Measures for Graphons: Accounting for Uncertainty in Networks

Cited by 63 publications

References 61 publications

Graphon Control of Large-Scale Networks of Linear Systems

Graphon Control of Large-Scale Networks of Linear Systems

Graphon-Based Sensitivity Analysis of SIS Epidemics

Graphon Games: A Statistical Framework for Network Games and Interventions

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