Building surrogate temporal network data from observed backbones

Many systems of socioeconomic interests find a convenient representation in the form of temporal networks, i.e., sets of nodes and interactions occurring at specified times. In the corresponding data sets, however, crucial elements coexist with nonessential ones and noise. Several methods have thus been proposed to extract a “network backbone,” i.e., the set of most important links in a network data set. The outcome of such methods can be seen as compressed versions of the original data. However, the question of how to practically use such reduced views of the data has not been tackled: for instance, using them directly in numerical simulations of processes on networks might lead to important biases. Overall, such reduced views of the data might not be actionable without an adequate decompression method. Here, we address this issue by putting forward and exploring several systematic procedures to build surrogate data from various kinds of temporal network backbones. In particular, we explore how much information about the original data needs to be retained alongside the backbone so that the surrogate data can be used in data-driven numerical simulations of spreading processes on a wide range of spreading parameters. We illustrate our results using empirical temporal networks with a broad variety of structures and properties. Our results give hints on how to best summarize complex data sets so that they remain actionable. Moreover, they show how ensembles of surrogate data with similar properties can be obtained from an original single data set, without any modeling assumptions.

Figure 1

Illustration of backbones and surrogates. The backbone construction identifies the most important links, and thereby compresses the original data. The surrogate generation model adds auxiliary links, extracted at random using specific procedures, to create a network of the same size as the original. The more links (information) retained in the backbone construction, the more similar is the surrogate data to the original.

Figure 2

$R_0$ (top row) and $\mathrm{\Omega }$ (bottom row) values obtained from the simulations on the original data and on the backbones, as a function of the SIR parameters, for the Thiers13 data set. Left: original data. Middle: GST backbone with $f=10\%$. Right: GST backbone with $f=5\%$.

Figure 3

Distributions of (aggregated) degrees (left columns) and weights (right column) in the surrogate data obtained by various methods. From top to bottom: (G)ST-OA, (G)ST-RA, (G)ST-RT, and (G)TB-RT. In each case, the blue line shows the distribution for the original data, and the red and green lines denote the surrogate built respectively without and with group information. Using group information yields distributions closer to the original ones. The surrogates were built from backbones with $f=10\%$ of the ties of the original data.

Figure 4

Effect of taking group structure into account when building the surrogate data, for the Thiers13 data set. Each panel shows the relative difference in $R_0$ obtained from the simulations on surrogate data with respect to simulations on the original data. Top: ST-RA. Bottom: GST-RA. Left, $f=40\%$; middle, $f=10\%$; right, $f=5\%$. In each case the backbone timelines are kept, and timelines respecting the statistics of contact and intercontact durations are built for the surrogate ties (BTL-Stats method).

Figure 5

Outcome of SIR processes on surrogate data obtained by various reconstruction methods, for the Thiers13 data set. Relative difference in the values of $R_0$ measured in simulations on the surrogate and on the original data. First row: GST-RA. Second row: GST-OA. Third row: GST-RT. Fourth row: GTB-RT. Left column, $f=40\%$; middle, $f=10\%$; right, $f=5\%$. In each case the backbone timelines are kept, and timelines respecting the statistics of contact and intercontact durations are built for the surrogate ties (BTL-Stats method).