Maths Project Repository Database

The so called visibility algorithms [1,2] are a family of simple algorithms that map a time series $\{x_1,x_2,\ldots,x_n\}$ into a graph of n vertices where each two vertices share an edge if a concrete "visibility criterion" holds. One can then associate a graph/network to a given time series, i.e. one can link graph theory/network theory with dynamics. A graph can be represented by its adjacency matrix A, whose entries are $A_{ij}=1$ if nodes i and j share an edge, and 0 otherwise. This project is about studying the spectral properties (that is, the eigenvalues) of these matrices.

Note: this is mainly a research project (i.e. exciting but hard). The student is expected to implement numerical routines in the computer, to learn some theory and to develop analytical calculations.

1. L. Lacasa, B. Luque, F. Ballesteros, J. Luque and J. C. Nuño, From time series to complex networks: the visibility graph, Proc. Natl. Acad. Sci. USA 105 13 (2008) 4972–4975. 
2. B. Luque, L. Lacasa, J. Luque and F. J. Ballesteros, Horizontal visibility graphs: exact results for random time series, Physical Review E 80 046103 (2009).