Visibility Graph

A visibility graph is a method for converting a time series into a complex network using a geometric criterion: two time points are connected if the line segment joining their values does not intersect any intermediate data point. Introduced by Lacasa, Luque, Ballesteros, Luque, and Nuño in 2008, the method maps ordinal and structural properties of the time series onto topological properties of the resulting graph.

Unlike recurrence networks, which require phase space reconstruction and a distance threshold, visibility graphs are constructed directly from the time series values. The resulting networks inherit properties of the original series: periodic series produce regular graphs with homogeneous degree distributions; random series produce graphs whose degree distributions follow exponential laws; and fractal series produce scale-free networks with power-law degree distributions. This means the network topology encodes the temporal structure without requiring prior knowledge of the underlying dynamics.

Visibility graphs have been used to analyze seismic data, financial time series, and physiological signals. Their principal advantage is simplicity: they require no parameter tuning (no embedding dimension, no recurrence threshold) and are computationally efficient. Their principal limitation is that they discard metric information — the actual values of the time series — preserving only relative heights and visibility relations.

The visibility graph demonstrates that network structure can emerge from almost any sequential data, not just from systems with well-defined phase spaces. This is both its strength and its danger: the method is so general that it risks finding network structure where none is meaningful. The crucial question is not whether a time series can be turned into a graph, but whether the graph's properties say something true about the system that the time series alone does not.