Near-Minimal Spanning Trees: a Scaling Exponent in Probability Models

We study the relation between the minimal spanning tree (MST) on many random points and the "near-minimal" tree which is optimal subject to the constraint that a proportion $\delta$ of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model, the ratio of lengths should scale as $1 + \Theta(\delta^2)$. We prove this scaling result in the model of the lattice with random edge-lengths and in the Euclidean model.Comment: 24 pages, 3 figure

Optimal Path and Minimal Spanning Trees in Random Weighted Networks

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 Erd\H{o}s-R\'enyi (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.Comment: review, accepted at IJB

Correlation Networks Among Currencies

By analyzing the foreign exchange market data of various currencies, we derive a hierarchical taxonomy of currencies constructing minimal-spanning trees. Clustered structure of the currencies and the key currency in each cluster are found. The clusters match nicely with the geographical regions of corresponding countries in the world such as Asia or East Europe, the key currencies are generally given by major economic countries as expected.Comment: 12 pages, 3 figures, 1 tabl

STORAGE REDUCTION THROUGH MINIMAL SPANNING TREES

In this paper, we shall show that a minimal spanning tree for a set of data can be used to reduce the amount of memory space required to store the data. Intuitively, the more points we have, the more likely our method will be better than the straightforward method where the data is stored in the form of a matrix. In Section 3, we shall show that once the number of samples exceeds a certain threshold, it is guaranteed that our method is better. Experiments were conducted on a set of randomly generated artificial data and a set of patient data. In the arttficial data experiment, we saved 23% for the worst case and 45% for the best case. In the patient data experiment, we saved 73% of the memory space