Revealing Dynamics, Communities, and Criticality from Data

Complex systems such as ecological communities and neuron networks are essential parts of our everyday lives. These systems are composed of units which interact through intricate networks. The ability to predict sudden changes in the dynamics of these networks, known as critical transitions, from data is important to avert disastrous consequences of major disruptions. Predicting such changes is a major challenge as it requires forecasting the behavior for parameter ranges for which no data on the system are available. We address this issue for networks with weak individual interactions and chaotic local dynamics. We do this by building a model network, termed an effective network, consisting of the underlying local dynamics and a statistical description of their interactions. We show that behavior of such networks can be decomposed in terms of an emergent deterministic component and a fluctuation term. Traditionally, such fluctuations are filtered out. However, as we show, they are key to accessing the interaction structure. We illustrate this approach on synthetic time series of realistic neuronal interaction networks of the cat cerebral cortex and on experimental multivariate data of optoelectronic oscillators. We reconstruct the community structure by analyzing the stochastic fluctuations generated by the network and predict critical transitions for coupling parameters outside the observed range.

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Complex networks, consisting of many interacting units that evolve over time, are found everywhere. One example is a neuronal network, where neurons in the body are wired together in intricate patterns that dictate their function. A small change in the behavior of these neurons can cause a huge change in the network, leading to pathologies such as an epileptic seizure. Researchers would like to know if it is possible to use time-series measurements to build a mathematical model for such a complex system that is good enough to provide predictions of critical transitions before they occur. Here, we describe a new technique that achieves this goal for a large class of complex systems, not just neurons, in a synthetic setting.

Most approaches to analyzing datasets start by filtering out fluctuation terms, assumed to be noise, but it is precisely these terms that contain the key to unlocking the interaction structure in the network. Whether two nodes are predominantly connected to the same nodes can be detected by checking whether the fluctuation at these nodes is correlated. Without prior knowledge about the network, our method allows us to model the dynamics at each node, the impacts of node interactions on the time evolution of the system, and the network structure.

In tests of our approach, we detect with high accuracy which nodes in a cat’s brain are connected to each other (and thus which nodes belong to the auditory region of the brain). Our model is so accurate that it can predict critical transitions that would occur if parameters change, even if no data are available for those parameters.

Figure 1

Effective network of the cat cerebral cortex. We use the local dynamics as a spiking neuron coupled via electric synapses. (a) The cat cerebral cortex network with nodes color coded according to the four functional modules. Rich-club members are indicated by red encircled nodes. (b) The covariance matrix of the data cannot detect communities. (c) The covariance matrix of the fluctuations can distinguish clusters. This matrix has entries color coded (according to the key on the right) with red entries corresponding to pairs of nodes sharing a large numbers of nearest neighbors in the network, while blue nodes correspond to pairs of nodes that share a small number of common neighbors. (d) A model in the cat cortex constructed via the effective network approach. From the matrix in (c) we can recover a representative effective network. The reconstructed network represents the actual network in (a) with good accuracy.

Figure 2

Sparse recovery method on a cat cerebral cortex. Sparse recovery is applied to the data generated by bursting neurons electrically coupled on the cat cerebral cortex. Selecting the threshold parameter $\sigma$ in the method changes the reconstructed network. Here we show the results of sparse recovery method for different enforced sparsity $\sigma$. The nonzero entries of the original network’s adjacency matrix are in blue. The red filled circles represent the nonzero entries in the adjacency matrix of the network reconstructed with the sparse recovery method. As each connection is small in comparison with the isolated dynamics, the sparse recovery tends to neglect them.

Figure 3

Prediction error for misidentification of communities in the reconstructed cat cerebral cortex from synthetic data. For each realization, the chosen parameters are the same as in Fig. 1 and only the overall coupling is changed. Mean and standard deviation of prediction error computed for the network over 50 realizations for each value of $\alpha$. If $\mathrm{\Delta }\alpha >0.42$, the system synchronizes and the procedure cannot reconstruct the community structures.

Figure 4

Prediction of critical transitions in the rich club of the cat cerebral cortex. The level of synchronization $r$ of the rich club is shown for different values of the coupling strength. Insets show time series of neuronal dynamics of four rich-club members, and color of time series matches with the color of nodes in Fig. 1. For values in the gray shaded region, $r$ is increasing toward close to one and the rich club exhibits collective behavior. We can predict the critical coupling ${\alpha }_c$ (standard deviation in shaded region) by studying the effective network obtained from a time series measured at $\mathrm{\Delta }\alpha =0.3$.

Figure 5

Reconstruction of structural power-law exponents $\gamma$ of scale-free networks from data. We estimated $\gamma$ from the multivariate time series obtained from the dynamics random scale-free networks with degree distribution $P(k)\propto k^{-\gamma }$. The plots in (a) compare the functional and effective network approach. We obtain better estimates using the effective network. Panel (b) shows the degree distribution of the original system (in blue) and that estimated from an effective model (in red) for the neural network in the optical lobe of Drosophila melanogaster. We obtained an accuracy of 3% in the structural exponent $\gamma$. Panel (c) shows the true exponent $\gamma$ versus ${\gamma }_{\mathrm{est}}$ obtained with an effective network from data for spiking neuron coupled with chemical synapses. We generated 1000 networks with distinct $\gamma$, from which the ${\gamma }_{\mathrm{est}}$ estimate is within 2% accuracy.

Figure 6

Effective network from experimental data of networks of optoelectronic oscillators. We consider multivariate time series of voltages of a network of 17 weakly coupled optoelectronic oscillators (interaction corresponding to 1% of the oscillator amplitude). In the left-hand panels, we show the actual network used to couple the optoelectronic oscillators in Ref. [52] as adjacency matrix in (a) and graph representation in (d). In the middle panels, we show the reconstruction of the network by a functional network analysis in terms of its adjacency matrix in (b) and graph representation in (e). In the right-hand panels, we show the reconstruction of the network from an analysis of the dynamical fluctuations by applying the effective network approach as adjacency matrix in (c) and graph representation in (f). The effective network provides a striking reconstruction and only two links are misidentified and are indicated in (f) as red links. In the graph representation, the nodes of the network are colored according to the community obtained by a community detection algorithm [38].

Figure 7

Reconstruction scheme with the effective network. From the time series, we build a model for the local evolution $f_i$ at each node. Under the assumption that such rules change from node to node depending on their connectivity, we estimate the coupling function. Using the fluctuations of the time series with respect to the low-dimensional rules, we recover the community structures. Gathering all this information, we obtain an effective network that can be used to predict critical transitions.