The impact of input node placement in the controllability of structural brain networks.

Network controllability refers to the ability to steer the state of a network towards a target state by driving certain nodes, known as input nodes. This concept can be applied to brain networks for studying brain function and its relation to the structure, which has numerous practical applications. Brain network controllability involves using external signals such as electrical stimulation to drive specific brain regions and navigate the neurophysiological activity level of the brain around the state space. Although controllability is mainly theoretical, the energy required for control is critical in real-world implementations. With a focus on the structural brain networks, this study explores the impact of white matter fiber architecture on the control energy in brain networks using the theory of how input node placement affects the LCC (the longest distance between inputs and other network nodes). Initially, we use a single input node as it is theoretically possible to control brain networks with just one input. We show that highly connected brain regions that lead to lower LCCs are more energy-efficient as a single input node. However, there may still be a need for a significant amount of control energy with one input, and achieving controllability with less energy could be of interest. We identify the minimum number of input nodes required to control brain networks with smaller LCCs, demonstrating that reducing the LCC can significantly decrease the control energy in brain networks. Our results show that relying solely on highly connected nodes is not effective in controlling brain networks with lower energy by using multiple inputs because of densely interconnected brain network hubs. Instead, a combination of low and high-degree nodes is necessary.

Input node placement restricting the longest control chain in controllability of complex networks.

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.

Higher-order interaction learning of line failure cascading in power networks.

Line failure cascading in power networks is a complex process that involves direct and indirect interactions between lines' states. We consider the inverse problem of learning statistical models to find the sparse interaction graph from the pairwise statistics collected from line failures data in the steady states and over time. We show that the weighted l-regularized pairwise maximum entropy models successfully capture pairwise and indirect higher-order interactions undistinguished by observing the pairwise statistics. The learned models reveal asymmetric, strongly positive, and negative interactions between the network's different lines' states. We evaluate the predictive performance of models over independent trajectories of failure unfolding in the network. The static model captures the failures' interactions by maximizing the log-likelihood of observing each link state conditioned to other links' states near the steady states. We use the learned interactions to reconstruct the network's steady states using the Glauber dynamics, predicting the cascade size distribution, inferring the co-susceptible line groups, and comparing the results against the data. The dynamic interaction model is learned by maximizing the log-likelihood of the network's state in state trajectories and can successfully predict the network state for failure propagation trajectories after an initial failure.

Eigenvalue sensitivity-based analysis for evaluation of biological network stability versus disturbances.

Network modeling is an effective tool for understanding the properties of complex systems. Networks are widely used to help us gain insight into biological systems. In this way, the cell, gene, and protein are denoted as nodes, and the connection elements are regarded as links or edges. In this paper, a novel stochastic strategy is developed for identifying the most influential edges on the stability of biological networks. Regarding the principles of networks and control-theory basics like Jacobian and eigenvalue sensitivity-based analysis, a new criterion is proposed, called "random sensitivity index matrix" (RSIM). RSIM evaluates the eigenvalue sensitivity of all edges in a network in the presents of stochastic disturbances based on the Monte Carlo algorithm. Through the values of RSIM elements, the sensitive edges are identifiable. In addition, the contribution of each edge in network instability has been compared through different percentages of disturbances. Different percentages of disturbances did not change the results. The performance of the proposed method was verified by simulation results for Lac (lactose) operon and MAPK (Mitogen-activated protein kinases) as two sample biological networks.

Erratum: Precision as a measure of predictability of missing links in real networks [Phys. Rev. E 101, 052318 (2020)].

This corrects the article DOI: 10.1103/PhysRevE.101.052318.

Precision as a measure of predictability of missing links in real networks.

Predicting missing links in real networks is an important open 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 focus on predictability instead of prediction. By considering ensembles defined by well-known network models, we prove analytically that even the best possible link prediction method, given by the ensemble connection probabilities, yields a limited precision that depends quantitatively on the topological properties-such as degree heterogeneity, clustering, and community structure-of the ensemble. This suggests an absolute limitation to the predictability of missing links in real networks, due to the irreducible uncertainty arising from the random nature of link formation processes. We show that a predictability limit can be estimated in real networks, and we propose a method to approximate such a bound from real-world networks with missing links. The predictability limit gives a benchmark to gauge the quality of link prediction methods in real networks.