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Register a new userTopology of correlation based minimal spanning trees in real and model markets
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Authors
Giovanni Bonanno
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We present here a topological characterization of the minimal spanning tree that can be obtained by considering the price return correlations of stocks traded in a financial market. We compare the minimal spanning tree obtained from a large group of stocks traded at the New York Stock Exchange during a 12-year trading period with the one obtained from surrogated data simulated by using simple market models. We find that the empirical tree has features of a complex network that cannot be reproduced, even as a first approximation, by a random market model and by the one-factor model.
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An important problem in statistical physics concerns the fascinating connections between partition functions of lattice models studied in equilibrium statistical mechanics on the one hand and graph theoretical enumeration problems on the other hand. We investigate the nature of the relationship between the number of spanning trees and the partition function of the Ising model on the square lattice. The spanning tree generating function $T(z)$ gives the spanning tree constant when evaluated at $z=1$, while giving he lattice green function when differentiated. It is known that for the infinite square lattice the partition function $Z(K)$ of the Ising model evaluated at the critical temperature $K=K_c$ is related to $T(1)$. Here we show that this idea in fact generalizes to all real temperatures. We prove that $ ( Z(K) {rm sech~} 2K ~!)^2 = k expbig[ T(k) big] $, where $k= 2 tanh(2K) {rm sech}(2K)$. The identical Mahler measure connects the two seemingly disparate quantities $T(z)$ and $Z(K)$. In turn, the Mahler measure is determined by the random walk structure function. Finally, we show that the the above correspondence does not generalize in a straightforward manner to non-planar lattices.
The minimum spanning tree (MST) is a combinatorial optimization problem: given a connected graph with a real weight (cost) on each edge, find the spanning tree that minimizes the sum of the total cost of the occupied edges. We consider the random MST, in which the edge costs are (quenched) independent random variables. There is a strongly-disordered spin-glass model due to Newman and Stein [Phys. Rev. Lett. 72, 2286 (1994)], which maps precisely onto the random MST. We study scaling properties of random MSTs using a relation between Kruskals greedy algorithm for finding the MST, and bond percolation. We solve the random MST problem on the Bethe lattice (BL) with appropriate wired boundary conditions and calculate the fractal dimension D=6 of the connected components. Viewed as a mean-field theory, the result implies that on a lattice in Euclidean space of dimension d, there are of order W^{d-D} large connected components of the random MST inside a window of size W, and that d = d_c = D = 6 is a critical dimension. This differs from the value 8 suggested by Newman and Stein. We also critique the original argument for 8, and provide an improved scaling argument that again yields d_c=6. The result implies that the strongly-disordered spin-glass model has many ground states for d>6, and only of order one below six. The results for MSTs also apply on the Poisson-weighted infinite tree, which is a mean-field approach to the continuum model of MSTs in Euclidean space, and is a limit of the BL. In a companion paper we develop an epsilon=6-d expansion for the random MST on critical percolation clusters.
A new approach to the understanding of complex behavior of financial markets index using tools from thermodynamics and statistical physics is developed. Physical complexity, a magnitude rooted in Kolmogorov-Chaitin theory is applied to binary sequences built up from real time series of financial markets indexes. The study is based on NASDAQ and Mexican IPC data. Different behaviors of this magnitude are shown when applied to the intervals of series placed before crashes and to intervals when no financial turbulence is observed. The connection between our results and The Efficient Market Hypothesis is discussed.
We review results on the scaling of the optimal path length in random networks with weighted links or nodes. In strong disorder we find that the length of the optimal path increases dramatically compared to the known small world result for the minimum distance. For ErdH{o}s-Renyi (ER) and scale free networks (SF), with parameter $lambda$ ($lambda >3$), we find that the small-world nature is destroyed. We also find numerically that for weak disorder the length of the optimal path scales logaritmically with the size of the networks studied. We also review the transition between the strong and weak disorder regimes in the scaling properties of the length of the optimal path for ER and SF networks and for a general distribution of weights, and suggest that for any distribution of weigths, the distribution of optimal path lengths has a universal form which is controlled by the scaling parameter $Z=ell_{infty}/A$ where $A$ plays the role of the disorder strength, and $ell_{infty}$ is the length of the optimal path in strong disorder. The relation for $A$ is derived analytically and supported by numerical simulations. We then study the minimum spanning tree (MST) and show that it is composed of percolation clusters, which we regard as super-nodes, connected by a scale-free tree. We furthermore show that the MST can be partitioned into two distinct components. One component the {it superhighways}, for which the nodes with high centrality dominate, corresponds to the largest cluster at the percolation threshold which is a subset of the MST. In the other component, {it roads}, low centrality nodes dominate. We demonstrate the significance identifying the superhighways by showing that one can improve significantly the global transport by improving a very small fraction of the network.
Given a compact $E subset mathbb{R}^n$ and $s > 0$, the maximum distance problem seeks a compact and connected subset of $mathbb{R}^n$ of smallest one dimensional Hausdorff measure whose $s$-neighborhood covers $E$. For $Esubset mathbb{R}^2$, we prove that minimizing over minimum spanning trees that connect the centers of balls of radius $s$, which cover $E$, solves the maximum distance problem. The main difficulty in proving this result is overcome by the proof of Lemma 3.5, which states that one is able to cover the $s$-neighborhood of a Lipschitz curve $Gamma$ in $mathbb{R}^2$ with a finite number of balls of radius $s$, and connect their centers with another Lipschitz curve $Gamma_ast$, where $mathcal{H}^1(Gamma_ast)$ is arbitrarily close to $mathcal{H}^1(Gamma)$. We also present an open source package for computational exploration of the maximum distance problem using minimum spanning trees, available at https://github.com/mtdaydream/MDP_MST.
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