Spectral Properties of the Hata Tree

The Hata tree is the unique self-similar set in the complex plane determined by the contractions φ0(z) = cz and φ1(z) = (1-|c|2)z+|c|2, where c is a complex number such that |c| and |1-c| are in (0,1). There are four main results in the paper. First, by applying linear algebra and spectral theory it is possible to construct a dynamical system that can compute the eigenvalues of the probabilistic Laplacian on graph approximations to the Hata tree. Conclusions are made about the spectrum of the Laplacian on the limiting graphs. Second, the Sabot theory (c.f. [29]) is applied to construct a simpler dynamical system to compute the eigenvalues of a class of normalized graph Laplacians (including the probabilistic Laplacian) on these approximating graphs. Third, it is possible to reconstruct the Hata tree as the union of two copies of a mixed affine nested fractal identified at a point. Using techniques from [13], some results are stated on the spectral asymptotics of the eigenvalue counting function of a certain class of Laplacians (not including the probabilistic Laplacian) on this mixed affine nested fractal. In the final part, a spectral analysis is performed on graph approximations to the Basilica Julia set of the polynomial z2-1. In [5], the authors give a dynamical system that can be used to construct finite approximations and classify the different possible infinite blow-ups. In this paper, the techniques from the first part are used to construct a dynamical system that can compute the eigenvalues of Laplacian operators on these finite graph approximations. In addition, it is shown that the spectrum of the Laplacian on blow-ups satisfying certain conditions is pure point

Analyzing Self-similar and Fractal Properties of the C. Elegans Neural Network

The brain is one of the most studied and highly complex systems in the biological world. While much research has concentrated on studying the brain directly, our focus is the structure of the brain itself: at its core an interconnected network of nodes (neurons). A better understanding of the structural connectivity of the brain should elucidate some of its functional properties. In this paper we analyze the connectome of the nematode Caenorhabditis elegans. Consisting of only 302 neurons, it is one of the better-understood neural networks. Using a Laplacian Matrix of the 279-neuron “giant component” of the network, we use an eigenvalue counting function to look for fractal-like self similarity. This matrix representation is also used to plot visualizations of the neural network in eigenfunction coordinates. Small-world properties of the system are examined, including average path length and clustering coefficient. We test for localization of eigenfunctions, using graph energy and spacial variance on these functions. To better understand results, all calculations are also performed on random networks, branching trees, and known fractals, as well as fractals which have been “rewired” to have small-world properties. We propose algorithms for generating Laplacian matrices of each of these graphs

Analyzing self-similar and fractal properties of the C. elegans neural network.

The brain is one of the most studied and highly complex systems in the biological world. While much research has concentrated on studying the brain directly, our focus is the structure of the brain itself: at its core an interconnected network of nodes (neurons). A better understanding of the structural connectivity of the brain should elucidate some of its functional properties. In this paper we analyze the connectome of the nematode Caenorhabditis elegans. Consisting of only 302 neurons, it is one of the better-understood neural networks. Using a Laplacian Matrix of the 279-neuron "giant component" of the network, we use an eigenvalue counting function to look for fractal-like self similarity. This matrix representation is also used to plot visualizations of the neural network in eigenfunction coordinates. Small-world properties of the system are examined, including average path length and clustering coefficient. We test for localization of eigenfunctions, using graph energy and spacial variance on these functions. To better understand results, all calculations are also performed on random networks, branching trees, and known fractals, as well as fractals which have been "rewired" to have small-world properties. We propose algorithms for generating Laplacian matrices of each of these graphs

Spacial Variance

<p>(a) <i>C. elegans</i> neural network (b) Random Graph () (c) Sierpinski Gasket, Level 5 (d) Sierpinski Gasket Rewiring ().</p