Recurrence Networks

Recurrence matrices can be reinterpreted as adjacency matrices of complex networks embedded in state space, such that each node or vertex of the network corresponds to a point of the timeseries, and the links of the network connect pairs of points that are mutually close the phase space. The relationship between a recurrence matrix $R$ and its corresponding adjacency matrix $A$ is:

\[R[i,j] = A[i,j] - \delta[i,j]\]

i.e. there is an edge in the associated network between every two neighboring points in the phase space, excluding self-connections (points in the Line Of Identity or main diagonal of $R$).

This definition assumes that $A$ represents an undirected graph, so $R$ must be a symmetric matrix as corresponding to a RecurrenceMatrix or a JointRecurrenceMatrix.

While RQA characterizes the properties of line structures in the recurrence plots, which consider dynamical aspects (e.g. continuity of recurrences, length of sequences, etc.), the analysis of recurrence networks does not take into account time information, since network properties are independent of the ordering of vertices. On the other hand, recurrence network analysis (RNA) provides information about geometric characteristics of the state space, like homogeneity of the connections, clustering of points, etc. More details about the theoretical framework of RNA can be found in the following papers:

1. R.V. Donner et al. "Recurrence networks — a novel paradigm for nonlinear time series analysis", New Journal of Physics 12, 033025 (2010)
2. R.V. Donner et al. "Complex Network Analysis of Recurrences", in: Webber, C.L. & Marwan N. (eds.) Recurrence Quantification Analysis. Theory and Best Practices, Springer, pp. 101-165 (2015).

Creation and visualization of Recurrence Networks

The JuliaGraphs organization provides multiple packages for Julia to create, visualize and analyze complex networks. In particular, the package LightGraphs defines the type SimpleGraph that can be used to represent undirected networks. Such graphs can be created from symmetric recurrence matrices, as in the following example with a Hénon map:

using RecurrenceAnalysis, DynamicalSystemsBase
using Graphs: SimpleGraph

# make trajectory of Henon map
henon_rule(x, p, n) = SVector(1.0 - p[1]*x[1]^2 + x[2], p[2]*x[1])
u0 = zeros(2)
p0 = [1.4, 0.3]
henon = DeterministicIteratedMap(henon_rule, u0, p0)
X, t = trajectory(henon, 200)
# Cast it into a recurrence network
R = RecurrenceMatrix(X, 0.25; metric = Chebyshev())
network = SimpleGraph(R)

{201, 2945} undirected simple Int64 graph

There are various plotting tools that can be used to visualize such graphs. For instance, the following plot made with the package GraphMakie.jl.

using GraphMakie, CairoMakie
graphplot(network)

Recurrence Network measures

LightGraphs has a large set of functions to extract local measures (associated to particular vertices or edges) and global coefficients associated to the whole network. For SimpleGraphs created from recurrence matrices, as the variable network in the previous example, the vertices are labelled with numeric indices following the same ordering as the rows or columns of the given matrix. So for instance degree(network, i) would give the degree of the i-th point of the timeseries (number of connections with other points), whereas degree(network) would give a vector of such measures ordered as the original timeseries.

As in RQA, we provide a function that computes a selection of commonly used global RNA measures, directly from the recurrence matrix:

rna(R::AbstractRecurrenceMatrix)
rna(args...; kwargs...)