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Register a new userSparse estimation of Laplacian eigenvalues in multiagent networks
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We propose a method to efficiently estimate the Laplacian eigenvalues of an arbitrary, unknown network of interacting dynamical agents. The inputs to our estimation algorithm are measurements about the evolution of a collection of agents (potentially one) during a finite time horizon; notably, we do not require knowledge of which agents are contributing to our measurements. We propose a scalable algorithm to exactly recover a subset of the Laplacian eigenvalues from these measurements. These eigenvalues correspond directly to those Laplacian modes that are observable from our measurements. We show how our technique can be applied to networks of multiagent systems with arbitrary dynamics in both continuous- and discrete-time. Finally, we illustrate our results with numerical simulations.
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Let $G$ be a connected undirected graph with $n$, $nge 3$, vertices and $m$ edges. Denote by $rho_1 ge rho_2 ge cdots > rho_n =0$ the normalized Laplacian eigenvalues of $G$. Upper and lower bounds of $rho_i$, $i=1,2,ldots , n-1$, are determined in terms of $n$ and general Randi c index, $R_{-1}$.
In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $pto 1,$ we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.
In this paper we present a decentralized algorithm to estimate the eigenvalues of the Laplacian matrix that encodes the network topology of a multi-agent system. We consider network topologies modeled by undirected graphs. The basic idea is to provide a local interaction rule among agents so that their state trajectory is a linear combination of sinusoids oscillating only at frequencies function of the eigenvalues of the Laplacian matrix. In this way, the problem of decentralized estimation of the eigenvalues is mapped into a standard signal processing problem in which the unknowns are the finite number of frequencies at which the signal oscillates.
The spectral gap of the graph Laplacian with Dirichlet boundary conditions is computed for the graphs of several communication networks at the IP-layer, which are subgraphs of the much larger global IP-layer network. We show that the Dirichlet spectral gap of these networks is substantially larger than the standard spectral gap and is likely to remain non-zero in the infinite graph limit. We first prove this result for finite regular trees, and show that the Dirichlet spectral gap in the infinite tree limit converges to the spectral gap of the infinite tree. We also perform Dirichlet spectral clustering on the IP-layer networks and show that it often yields cuts near the network core that create genuine single-component clusters. This is much better than traditional spectral clustering where several disjoint fragments near the periphery are liable to be misleadingly classified as a single cluster. Spectral clustering is often used to identify bottlenecks or congestion; since congestion in these networks is known to peak at the core, our results suggest that Dirichlet spectral clustering may be better at finding bona-fide bottlenecks.
For $alphain(0,pi)$, let $U_alpha$ denote the infinite planar sector of opening $2alpha$, [ U_alpha=big{ (x_1,x_2)inmathbb R^2: big|arg(x_1+ix_2) big|<alpha big}, ] and $T^gamma_alpha$ be the Laplacian in $L^2(U_alpha)$, $T^gamma_alpha u= -Delta u$, with the Robin boundary condition $partial_ u u=gamma u$, where $partial_ u$ stands for the outer normal derivative and $gamma>0$. The essential spectrum of $T^gamma_alpha$ does not depend on the angle $alpha$ and equals $[-gamma^2,+infty)$, and the discrete spectrum is non-empty iff $alpha<fracpi 2$. In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle $alpha$. In particular, there is just one discrete eigenvalue for $alpha ge frac{pi}{6}$. As $alpha$ approaches $0$, the number of discrete eigenvalues becomes arbitrary large and is minorated by $kappa/alpha$ with a suitable $kappa>0$, and the $n$th eigenvalue $E_n(T^gamma_alpha)$ of $T^gamma_alpha$ behaves as [ E_n(T^gamma_alpha)=-dfrac{gamma^2}{(2n-1)^2 alpha^2}+O(1) ] and admits a full asymptotic expansion in powers of $alpha^2$. The eigenfunctions are exponentially localized near the origin. The results are also applied to $delta$-interactions on star graphs.
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