No abstract

Systems composed of many interacting dynamical networks—such as the human body with its biological networks or the global economic network consisting of regional clusters—often exhibit complicated collective dynamics. Three fundamental processes that are typically present are failure, damage spread and recovery. Here we develop a model for such systems and find a very rich phase diagram that becomes increasingly more complex as the number of interacting networks increases. In the simplest example of two interacting networks we find two critical points, four triple points, ten allowed transitions and two ‘forbidden' transitions, as well as complex hysteresis loops. Remarkably, we find that triple points play the dominant role in constructing the optimal repairing strategy in damaged interacting systems. To test our model, we analyse an example of real interacting financial networks and find evidence of rapid dynamical transitions between well-defined states, in agreement with the predictions of our model.

In order to model volatile real-world network behavior, we analyze phase-flipping dynamical scale-free network in which nodes and links fail and recover. We investigate how stochasticity in a parameter governing the recovery process affects phase-flipping dynamics, and find the probability that no more than q% of nodes and links fail. We derive higher moments of the fractions of active nodes and active links, fn(t) and f ℓ (t), and define two estimators to quantify the level of risk in a network. We find hysteresis in the correlations of fn(t) due to failures at the node level, and derive conditional probabilities for phase-flipping in networks. We apply our model to economic and traffic networks. Across a broad range of human activities-from medicine, weather, and traffic management to intelligence services and military operations-forecasting theories help us estimate the probability of future outcomes. In general, the greater the uncertainty of outcome, the more crucial that we be able to forcast future behavior. Since the nodes of many dynamic systems , such as traffic patterns and physiological networks, periodically fail and then recover-a disease spreads through an organism and then after a finite period of time the organism recovers-the forecasting power [22][23][24] is highly relevant-it allows us to estimate the probability of future node and link failure and recovery and to quantify the level of risk in any given dynamic network.A recent paper details how the nodes in dynamic regular networks and Erdős Renyi networks (i) inherently fail, (ii) contiguously fail due to the failure of neighboring nodes, and (iii) recover [25]. These networks exhibit phase-flipping between "active" and "inactive" collective network modes. Here we analyze networks with highly heterogeneous degree distributions and we describe scale-free phase-flipping networks in which nodes and links fail and recover. We describe the collective behavior of these networks using two time-dependent network variables: the fraction of active nodes f n (t) and the fraction of active links f ℓ (t). We place an emphasis on forecasting in dynamic networks-we want to calculate how many nodes will fail at any future time t-and quantify how risky networks are. (i) At each time t any node in the system can independently fail, breaking its links with all other nodes, with a probability p. The internal failure state of node i we denote by spin |s i (if i is active, |s i = |1 ).(ii) The external failure states we denote by spin |S i , where |S i is |1 if node i has more than T h % active neighbors, and |0 (for a subsequent time τ ′ = 1) with probability p 2 if ≤ T h % of i's neighbors are active. For scale-free networks, a percentage threshold T h is the more appropriate choice than the constant T h used in Ref. [1,25]. Node i-described by the two-spin state |s i , S i -is active only if both spins are up (1), i.e., if |s i , S i = |1, 1 . (iii) After a time period τ , the nodes recover from internal failure. Usually τ is random, but we also analyze the c...

Deep neural networks have demonstrated improved performance at predicting the sequence specificities of DNA- and RNA-binding proteins compared to previous methods that rely on k-mers and position weight matrices. To gain insights into why a DNN makes a given prediction, model interpretability methods, such as attribution methods, can be employed to identify motif-like representations along a given sequence. Because explanations are given on an individual sequence basis and can vary substantially across sequences, deducing generalizable trends across the dataset and quantifying their effect size remains a challenge. Here we introduce global importance analysis (GIA), a model interpretability method that quantifies the population-level effect size that putative patterns have on model predictions. GIA provides an avenue to quantitatively test hypotheses of putative patterns and their interactions with other patterns, as well as map out specific functions the network has learned. As a case study, we demonstrate the utility of GIA on the computational task of predicting RNA-protein interactions from sequence. We first introduce a convolutional network, we call ResidualBind, and benchmark its performance against previous methods on RNAcompete data. Using GIA, we then demonstrate that in addition to sequence motifs, ResidualBind learns a model that considers the number of motifs, their spacing, and sequence context, such as RNA secondary structure and GC-bias.

Deep neural networks have demonstrated improved performance at predicting the sequence specificities of DNA- and RNA-binding proteins compared to previous methods that rely on k-mers and position weight matrices. For model interpretability, attribution methods have been employed to reveal learned patterns that resemble sequence motifs. First-order attribution methods only quantify the independent importance of single nucleotide variants in a given sequence – it does not provide the effect size of motifs (or their interactions with other patterns) on model predictions. Here we introduce global importance analysis (GIA), a new model interpretability method that quantifies the population-level effect size that putative patterns have on model predictions. GIA provides an avenue to quantitatively test hypotheses of putative patterns and their interactions with other patterns, as well as map out specific functions the network has learned. As a case study, we demonstrate the utility of GIA on the computational task of predicting RNA-protein interactions from sequence. We first introduce a new convolutional network, we call ResidualBind, and benchmark its performance against previous methods on RNAcompete data. Using GIA, we then demonstrate that in addition to sequence motifs, ResidualBind learns a model that considers the number of motifs, their spacing, and sequence context, such as RNA secondary structure and GC-bias.

Gradients of a deep neural network’s predictions with respect to the inputs are used in a variety of downstream analyses, notably in post hoc explanations with feature attribution methods. For data with input features that live on a lower-dimensional manifold, we observe that the learned function can exhibit arbitrary behaviors off the manifold, where no data exists to anchor the function during training. This leads to a random component in the gradients which manifests as noise. We introduce a simple correction for this off-manifold gradient noise for the case of categorical input features, where input values are subject to a probabilistic simplex constraint, and demonstrate its effectiveness on regulatory genomics data. We find that our correction consistently leads to a significant improvement in gradient-based attribution scores.

Estimating the critical points at which complex systems abruptly flip from one state to another is one of the remaining challenges in network science. Due to lack of knowledge about the underlying stochastic processes controlling critical transitions, it is widely considered difficult to determine the location of critical points for real-world networks, and it is even more difficult to predict the time at which these potentially catastrophic failures occur. We analyse a class of decaying dynamic networks experiencing persistent failures in which the magnitude of the overall failure is quantified by the probability that a potentially permanent internal failure will occur. When the fraction of active neighbours is reduced to a critical threshold, cascading failures can trigger a total network failure. For this class of network we find that the time to network failure, which is equivalent to network lifetime, is inversely dependent upon the magnitude of the failure and logarithmically dependent on the threshold. We analyse how permanent failures affect network robustness using network lifetime as a measure. These findings provide new methodological insight into system dynamics and, in particular, of the dynamic processes of networks. We illustrate the network model by selected examples from biology, and social science.

Post hoc attribution methods can provide insights into the learned patterns from deep neural networks (DNNs) trained on high-throughput functional genomics data. However, in practice, their resultant attribution maps can be challenging to interpret due to spurious importance scores for seemingly arbitrary nucleotides. Here, we identify a previously overlooked attribution noise source that arises from how DNNs handle one-hot encoded DNA. We demonstrate this noise is pervasive across various genomic DNNs and introduce a statistical correction that effectively reduces it, leading to more reliable attribution maps. Our approach represents a promising step towards gaining meaningful insights from DNNs in regulatory genomics.