Neural networks have dramatically increased our capacity to learn from large, high-dimensional datasets across innumerable disciplines. However, their decisions are not easily interpretable, their computational costs are high, and building and training them are not straightforward processes. To add structure to these efforts, we derive new mathematical results to efficiently measure the changes in entropy as fully-connected and convolutional neural networks process data. By measuring the change in entropy as networks process data effectively, patterns critical to a well-performing network can be visualized and identified. Entropy-based loss terms are developed to improve dense and convolutional model accuracy and efficiency by promoting the ideal entropy patterns. Experiments in image compression, image classification, and image segmentation on benchmark datasets demonstrate these losses guide neural networks to learn rich latent data representations in fewer dimensions, converge in fewer training epochs, and achieve higher accuracy.

Abstract As manipulating the self-assembly of supramolecular and nanoscale constructs at the single-molecule level increasingly becomes the norm, new theoretical scaffolds must be erected to replace the thermodynamic and kinetics based models used to describe traditional bulk phase syntheses. Like the statistical mechanics underpinning these latter theories, the framework we propose uses state probabilities as its fundamental objects; but, contrary to the Gibbsian paradigm, our theory directly models the transition probabilities between the initial and final states of a trajectory, foregoing the need to assume ergodicity. We leverage these probabilities in the context of molecular self-assembly to compute the overall likelihood that a specified experimental condition leads to a desired structural outcome. We demonstrate the application of this framework to a toy model in which N identical molecules can assemble into oligomers of different lengths and conclude with a discussion of how the high computational cost of such a fine-grained model can be overcome through approximation when extending it to larger, more complex systems.

Given a social network, the Influence Maximization (InfMax) problem seeks a seed set of k people that maximizes the expected influence for a viral marketing campaign. However, a solution for a particular seed size k is often not enough to make an informed choice regarding budget and cost-effectiveness. In this article, we propose the computation of Influence Spectrum (InfSpec), the maximum influence at each possible seed set size k within a given range [ k lower , k upper ], thus providing optimal decision making for any availability of budget or influence requirements. As none of the existing methods for InfMax are efficient enough for the task in large networks, we propose LISA (sub-Linear Influence Spectrum Approximation), an efficient approximation algorithm for InfSpec (and also InfMax) with the best-known worst-case guarantees for billion-scale networks. LISA returns an (1-1/e -ϵ)-approximate influence spectrum with high probability (1-δ), where ϵ, δ are precision parameters provided by users. Using statistical decision theory, LISA has an asymptotic optimal running time (in addition to optimal approximation guarantee). In practice, LISA surpasses the state-of-the-art InfMax methods, taking less than 15 minutes to process a network of 41.7 million nodes and 1.5 billions edges.

Given an aftermath of a cascade in the network, i.e. a set VI of "infected" nodes after an epidemic outbreak or a propagation of rumors/worms/viruses, how can we infer the sources of the cascade? Answering this challenging question is critical for computer forensic, vulnerability analysis, and risk management. Despite recent interest towards this problem, most of existing works focus only on single source detection or simple network topologies, e.g. trees or grids. In this paper, we propose a new approach to identify infection sources by searching for a seed set S that minimizes the symmetric difference between the cascade from S and VI, the given set of infected nodes. Our major result is an approximation algorithm, called SISI, to identify infection sources without the prior knowledge on the number of source nodes. SISI, to our best knowledge, is the first algorithm with provable guarantee for the problem in general graphs. It returns a 2/((1-ε)2 Δ-approximate solution with high probability, where Δ denotes the maximum number of nodes in VI that may infect a single node in the network. Our experiments on real-world networks show the superiority of our approach and SISI in detecting true source(s), boosting the F1-measure from few percents, for the state-of-the-art NETSLEUTH, to approximately 50%.

Proteins search along DNA for targets (e.g. transcription initiation sequences) through a combination of sliding, jumping, and intersegment transfers, wherein the sliding process proceeds until the protein-DNA complex dissociates. As such, we propose a diffusion-reaction model of the sliding phase of a total search process, and study the effect of varying reaction rate on the detailed search kinetics. With increasing dissociation rate, only the "fastest" proteins survive to explore downstream sequences, but at the price of increasing rarity, allowing for fluctuations to dominate the behavior at late times and long distances. Predictions of this model for the total search time agree with experimental estimates across an order of magnitude, providing bounds on dissociation rates that suggest a balance is established between maximizing downstream exploration of the DNA while minimizing the number of rare trajectories to guarantee the greatest chance of a successful search.

BACKGROUND: Proteins search along the DNA for targets, such as transcription initiation sequences, according to one-dimensional diffusion, which is interrupted by micro- and macro-hopping events and intersegmental transfers that occur under close packing conditions. RESULTS: A one-dimensional diffusion-reaction model in the form of difference-differential equations is proposed to analyze the nonequilibrium protein sliding kinetics along a segment of bacterial DNA. A renormalization approach is used to derive an expression for the mean first-passage time to arrive at sites downstream of the origin from the occupation probabilities given by the individual transport equations. Monte Carlo simulations are employed to assess the validity of the proposed approach, and all results are interpreted within the context of bacterial transcription. CONCLUSIONS: Mean first-passage times decrease with increasing reaction rates, indicating that, on average, surviving proteins more rapidly locate downstream targets than their reaction-free counterparts, but at the price of increasing rarity. Two qualitatively different screening regimes are identified according to whether the search process operates under "small" or "large" values for the dissociation rate of the protein-DNA complex. Lower bounds are placed on the overall search time for varying reactive conditions. Good agreement with experimental estimates requires the reaction rate reside near the transition between both screening regimes, suggesting that biology balances a need for rapid searches against maximum exploration during each round of the sliding phase.