Inner Composition Alignment for Inferring Directed Networks from Short Time Series

Identifying causal links (couplings) is a fundamental problem that facilitates the understanding of emerging structures in complex networks. We propose and analyze inner composition alignment—a novel, permutation-based asymmetric association measure to detect regulatory links from very short time series, currently applied to gene expression. The measure can be used to infer the direction of couplings, detect indirect (superfluous) links, and account for autoregulation. Applications to the gene regulatory network of E. coli are presented.

Figure 1

Time series $y^{(1)}$ and $y^{(2)}$ reordered by the permutation ${\pi }^{(1)}$. Horizontal lines are drawn at points of $y^{(2)}$ (${\pi }^{(1)}$). The plot in the inset shows the time series in their original order.

Figure 2

ROC curves for a network of 100 genes of E. coli (upper panels: noise-free case; lower panels: noise level of 0.1). The left and middle panels are obtained for $\iota$ (superfluous removed) including a significance test with 10 000 realizations for different weights—left: slope (black), squared slope (dark gray), maximal excursion (gray); middle: uniform (black), arithmetic mean (dark gray), geometric mean (gray), harmonic mean (light gray). The right panels relate to the correlations: Pearson (black), Spearman (dark gray), Kendall (gray). These results do not incorporate a significance test.

Figure 3

Reconstruction of a regulatory network of 100 genes of E. coli from (a),(b) noise-free time series using threshold 1, (c),(d) time series simulated with noise level 0.1 using threshold 0.95, and (e),(f) with noise level 0.1 using threshold 0.5. Panels (a),(c),(e) show the networks obtained with IOTA, whereas (b),(d),(f) are obtained with Kendall’s $\tau$. The original network (in the lower panels the undirected version) is shown in light gray; correctly identified links are marked in black. False positive links are not shown.

Figure 4

Network reconstruction with IOTA for networks of different sizes [(a)–(i) refer to Table ] and stochastic time series (noise intensity 0.01) of 10 time points. Black links are obtained with threshold 0.75 (dark gray additionally with threshold 0.99); light gray indicates not identified links.