This thesis is devoted to the study of complex systems as dynamical networks. Complex systems typically have many interacting components with non-trivial dynamics, and often have a natural network representation. Examples of such networks are abundant, from the world wide web and power plants, to gene regulation and the human brain.
However, often we can not observe these networks themselves, but can only passively observe the dynamics of their nodes. For example, neural activity is often available but not the connections between neurons. This motivated researchers to attempt to infer networks from observations of systems’ dynamics. These efforts resulted in the application of network methods to systems whose underlying structure is not intuitively described by a network, e.g., Earth’s climate. Therefore, it is important that the appropriateness and robustness of these analyses are critically analyzed — the focus of the bulk of this thesis.
Recent work has outlined a way to assess how much information about a system can be captured by pairwise relationships — from which these networks are generally constructed. As most commonly formulated however, the method is based on cross-correlations, and therefore only sensitive to linear relationships. We extend this by using mutual informations, which are sensitive to a wide range of nonlinear relationships undetectable by the cross-correlation. We illustrate the advantages of our method using phase oscillators and fMRI data. Our method also allows a novel method for network inference, letting us estimate the conditional mutual informations between all pairs of variables when given their mutual informations. When applied to phase oscillator networks, this has resulted in superior results than using the mutual information alone.
I also address the robustness of dynamical networks. We show that contrary to previous results, networks formed during El Niño periods have more connectivity than during “normal” periods, where the discrepancy is a result of the original analysis mixing different climate regimes. Finally, we address the question of inferring time delays for constructing dynamical networks. We show that a common technique is biased towards large time lags, and suggest an alternative estimator that does not suffer from this bias.