Search

Network control theory can be used to model how one should steer the brain between different states by driving a specific region with an input. The needed energy to control a network is often used to quantify its controllability, and controlling brain networks requires diverse energy depending on the selected input region. We use the theory of how input node placement affects the longest control chain (LCC) in the controllability of brain networks to study the role of the architecture of white matter fibers in the required control energy. We show that the energy needed to control human brain networks is related to the LCC, i.e., the longest distance between the input region and other regions in the network. We indicate that regions that control brain networks with lower energy have small LCCs. These regions align with areas that can steer the brain around the state space smoothly. By contrast, regions that need higher energy to move the brain toward different target states have larger LCCs. We also investigate the role of the number of paths between regions in the control energy. Our results show that the more paths between regions, the lower cost needed to control brain networks. We evaluate the number of paths by counting specific motifs in brain networks since determining all paths in graphs is a difficult problem., Comment: 14 pages, 8 figures

The minimum number of inputs needed to control a network is frequently used to quantify its controllability. Control of linear dynamics through a minimum set of inputs, however, often has prohibitively large energy requirements and there is an inherent trade-off between minimizing the number of inputs and control energy. To better understand this trade-off, we study the problem of identifying a minimum set of input nodes such that controllabililty is ensured while restricting the length of the longest control chain. The longest control chain is the maximum distance from input nodes to any network node, and recent work found that reducing its length significantly reduces control energy. We map the longest control chain-constraint minimum input problem to finding a joint maximum matching and minimum dominating set. We show that this graph combinatorial problem is NP-complete, and we introduce and validate a heuristic approximation. Applying this algorithm to a collection of real and model networks, we investigate how network structure affects the minimum number of inputs, revealing, for example, that for many real networks reducing the longest control chain requires only few or no additional inputs, only the rearrangement of the input nodes., Comment: 16 pages, 9 figures, supplementary

We use machine learning tools to model the line interaction of failure cascading in power grid networks. We first collect data sets of simulated trajectories of possible consecutive line failure following an initial random failure and considering actual constraints in a model power network until the system settles at a steady state. We use weighted $l_1$-regularized logistic regression-based models to find static and dynamic models that capture pairwise and latent higher-order lines' failure interactions using pairwise statistical data. The static model captures the failures' interactions near the steady states of the network, and the dynamic model captures the failure unfolding in a time series of consecutive network states. We test models over independent trajectories of failure unfolding in the network to evaluate their failure predictive power. We observe asymmetric, strongly positive, and negative interactions between different lines' states in the network. We use the static interaction model to estimate the distribution of cascade size and identify groups of lines that tend to fail together, and compare against the data. The dynamic interaction model successfully predicts the network state for long-lasting failure propagation trajectories after an initial failure., Comment: 10 pages, 7 figures

We investigate to what extent the degree sequence of a directed network constrains the number of driver nodes. We develop a pair of algorithms that take a directed degree sequence as input and aim to output a network with the maximum or minimum number of driver nodes. We find an upper bound for the maximum and a lower bound for the minimum, and show that the algorithms achieve these bounds for all real and model networks, with few exceptions characterized by tiny system size and heterogeneous degree distributions. Applying these algorithms to a broad range of real networks, we show the gap between the upper and lower bounds can vary dramatically across different degree sequences. Thus, we introduce the notion of \bl{structural control} complexity to capture how much more difficult it is to control a specific network beyond what is required given its degree sequence, suggesting additional structure is present. Using model networks, we numerically and analytically investigate how typical features of the degree distribution affect the range of required driver nodes. We find that the minimum is determined by the number of sources or sinks, while the maximum is strongly affected by the presence of hubs., Comment: 12 pages, 8 figures

Predicting missing links in real networks is an important problem in network science to which considerable efforts have been devoted, giving as a result a vast plethora of link prediction methods in the literature. In this work, we take a different point of view on the problem and study the theoretical limitations to the predictability of missing links. In particular, we hypothesise that there is an irreducible uncertainty in link prediction on real networks as a consequence of the random nature of their formation process. By considering ensembles defined by well-known network models, we prove analytically that even the best possible link prediction method for an ensemble, given by the ranking of the ensemble connection probabilities, yields a limited precision. This result suggests a theoretical limitation to the predictability of links in real complex networks. Finally, we show that connection probabilities inferred by fitting network models to real networks allow to estimate an upper-bound to the predictability of missing links, and we further propose a method to approximate such bound from incomplete instances of real-world networks.