Abstract. In this paper we study the pattern formation of a kinematic aggregation model for biological swarming in two dimensions. The swarm is represented by particles and the dynamics are driven by a gradient flow of a non-local interaction potential which has a local repulsion long range attraction structure. We review and expand upon recent developments of this class of problems as well as present new results. As in previous work, we leverage a co-dimension one formulation of the continuum gradient flow to characterize the stability of ring solutions for general interaction kernels. In the regime of long-wave instability we show that the resulting ground state is a low mode bifurcation away from the ring and use weakly nonlinear analysis to provide conditions for when this bifurcation is a pitchfork. In the regime of short-wave instabilities we show that the rings break up into fully 2D ground states in the large particle limit. We analyze the dependence of the stability of a ring on the number of particles and provide examples of complex multi-ring bifurcation behavior as the number of particles increases. We are also able to provide a solution for the "designer potential" problem in 2D. Finally, we characterize the stability of the rotating rings in the second order kinetic swarming model. IntroductionMathematical models for swarming, schooling, and other aggregative behavior in biology have given us many tools to understand the fundamental behavior of collective motion and pattern formation that occurs in nature [10,6,2,26,25,14,7,13,27,19,33,32,23,11,17,37,38,34,36,9,15,29,21,20,24,8]. One of the key features of many of these models is that the social communication between individuals (sound, chemical detection, sight, etc...) is performed over different scales and are inherently nonlocal [11,22,2]. In the case of swarming, these nonlocal interactions between individuals usually consist of a shorter range repulsion to avoid collisions and medium to long range attraction to keep the swarm cohesive. While some models include anisotropy in this communication (e.g. an organism's eyes may have a limited field of vision) simplified isotropic interactions have been shown to capture many important swarming behaviors including milling [20,10]. More recently it has been shown [17,38,37] that the competition between the desire to avoid collisions and the desire to remain in a cohesive swarm can sometimes result in simple radially symmetric patterns such as rings, annuli and uniform circular patches and other times result in exceedingly complex patterns. Moreover how modelers select the strength and form of the repulsion near the origin
This paper provides both theoretical and algorithmic results for the 1 -relaxation of the Cheeger cut problem. The 2 -relaxation, known as spectral clustering, only loosely relates to the Cheeger cut; however, it is convex and leads to a simple optimization problem. The 1 -relaxation, in contrast, is non-convex but is provably equivalent to the original problem. The 1 -relaxation therefore trades convexity for exactness, yielding improved clustering results at the cost of a more challenging optimization. The first challenge is understanding convergence of algorithms. This paper provides the first complete proof of convergence for algorithms that minimize the 1 -relaxation. The second challenge entails comprehending the 1energy landscape, i.e. the set of possible points to which an algorithm might converge. We show that 1 -algorithms can get trapped in local minima that are not globally optimal and we provide a classification theorem to interpret these local minima. This classification gives meaning to these suboptimal solutions and helps to explain, in terms of graph structure, when the 1 -relaxation provides the solution of the original Cheeger cut problem.
We study the linear stability of flock and mill ring solutions of two individual based models for biological swarming. The individuals interact via a nonlocal interaction potential that is repulsive in the short range and attractive in the long range. We relate the instability of the flock rings with the instability of the ring solution of the first order model. We observe that repulsive-attractive interactions lead to new configurations for the flock rings such as clustering and fattening formation. Finally, we numerically explore mill patterns arising from this kind of interactions together with the asymptotic speed of the system.
Ideas from the image processing literature have recently motivated a new set of clustering algorithms that rely on the concept of total variation. While these algorithms perform well for bi-partitioning tasks, their recursive extensions yield unimpressive results for multiclass clustering tasks. This paper presents a general framework for multiclass total variation clustering that does not rely on recursion. The results greatly outperform previous total variation algorithms and compare well with state-of-the-art NMF approaches.
This work aims at recovering signals that are sparse on graphs. Compressed sensing offers techniques for signal recovery from a few linear measurements and graph Fourier analysis provides a signal representation on graph. In this paper, we leverage these two frameworks to introduce a new Lasso recovery algorithm on graphs. More precisely, we present a non-convex, non-smooth algorithm that outperforms the standard convex Lasso technique. We carry out numerical experiments on three benchmark graph datasets. Sparse Representation on GraphsThe goal of this work is to reconstruct signals on graphs that are supposed to be sparse in the graph Fourier representation. In this context, we will deal here with two main concepts, graph and sparsity, which have gathered a lot of attention in the recent years with the emergence of Compressed Sensing and Big Data. Let us introduce briefly these two concepts in the rest of this section.Graph/network is a powerful tool to represent complex high-dimensional datasets, in the sense that a graph structures data with respect to their similarities. Graphs have become increasingly more considered in applications such as search engines, social networks, airline routes, 3D geometric shapes, human brain connectivity, etc. Mathematics offer strong theoretical tools to analyze graphs with Harmonic Analysis and Spectral Graph Theory. An essential graph analysis tool is the graph Laplacian operator, which is the discrete approximation of the continuum Laplace-Beltrami operator for smooth manifolds. It is known that the eigenvectors of the Laplace-Beltrami operator provide a local parametrization of the manifold [1]. Equivalently, the eigenvectors of the graph Laplacian, also called graph Fourier modes, provides a representation of the graph. Given a graph with (V, E, W ), V , E and W being respectively the set of n nodes, the set of edges and the similarity/adjacency matrix, then the (unnormalized) graph Laplacian operator is defined aswhere D is the diagonal degree matrix s.t. is its Fourier transform. In this paper, we consider three well-known graphs. First, the synthetic LFR graph, which was introduced in [2] to study community graphs. Here, the number of nodes is chosen to be n = 1, 000, the number of communities is 10 and the degree of community overlapping is µ = 0.4. Second, we consider a coarse version (for computational speedup) of the benchmark MNIST dataset of NYU [3] with n = 1, 176 nodes and the number of classes is 10. Last, we use a coarse version of the well-known 20newsgroups dataset of CMU [4] with n = 1, 432 nodes and the number of classes is 20. All three dataset graphs are illustrated on Figure 1 with their graph Laplacian spectrum.
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