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We present a rigorous and precise analysis of the maximum degree and the average degree in a dynamic duplication-divergence graph model introduced by Sol\'e, Pastor-Satorras et al. in which the graph grows according to a duplication-divergence mechanism, i.e. by iteratively creating a copy of some node and then randomly alternating the neighborhood of a new node with probability $p$. This model captures the growth of some real-world processes e.g. biological or social networks. In this paper, we prove that for some $0 < p < 1$ the maximum degree and the average degree of a duplication-divergence graph on $t$ vertices are asymptotically concentrated with high probability around $t^p$ and $\max\{t^{2 p - 1}, 1\}$, respectively, i.e. they are within at most a polylogarithmic factor from these values with probability at least $1 - t^{-A}$ for any constant $A > 0$.

Physics informed neural networks (PINNs) have recently been very successfully applied for efficiently approximating inverse problems for PDEs. We focus on a particular class of inverse problems, the so-called data assimilation or unique continuation problems, and prove rigorous estimates on the generalization error of PINNs approximating them. An abstract framework is presented and conditional stability estimates for the underlying inverse problem are employed to derive the estimate on the PINN generalization error, providing rigorous justification for the use of PINNs in this context. The abstract framework is illustrated with examples of four prototypical linear PDEs. Numerical experiments, validating the proposed theory, are also presented.

Hesitant fuzzy sets are widely used in the instances of uncertainty and hesitation. The inclusion relationship is an important and foundational definition for sets. Hesitant fuzzy set, as a kind of set, needs explicit definition of inclusion relationship. Base on the hesitant fuzzy membership degree of discrete form, several kinds of inclusion relationships for hesitant fuzzy sets are proposed. And then some foundational propositions of hesitant fuzzy sets and the families of hesitant fuzzy sets are presented. Finally, some foundational propositions of hesitant fuzzy information systems with respect to parameter reductions are put forward, and an example and an algorithm are given to illustrate the processes of parameter reductions.

The identification of essential proteins in protein-protein interaction networks (PINs) can help to discover drug targets and prevent disease. In order to improve the accuracy of the identification of essential proteins, researchers attempted to obtain a refined PIN by combining multiple biological information to filter out some unreliable interactions in the PIN. Unfortunately, such approaches drastically reduce the number of nodes in the PIN after multiple refinements and result in a sparser PIN. It makes a considerable portion of essential proteins unidentifiable. In this paper, we propose a multi-layer refined network (MR-PIN) that addresses this problem. Firstly, four refined networks are constructed by respectively integrating different biological information into the static PIN to form a multi-layer heterogeneous network. Then scores of proteins in each network layer are calculated by the existing node ranking method, and the importance score of a protein in the MR-PIN is evaluated in terms of the geometric mean of its scores in all layers. Finally, all nodes are sorted by their importance scores to determine their essentiality. To evaluate the effectiveness of the multi-layer refined network model, we apply 16 node ranking methods on the MR-PIN, and compare the results with those on the SPIN, DPIN and RDPIN. Then the predictive performances of these ranking methods are validated in terms of the identification number of essential protein at top100 - top600, sensitivity, specificity, positive predictive value, negative predictive value, F-measure, accuracy, Jackknife, ROCAUC and PRAUC. The experimental results show that the MR-PIN is superior to the existing refined PINs in the identification accuracy of essential proteins.

Many real-world networks exhibit the phenomenon of edge clustering, which is typically measured by the average clustering coefficient. Recently, an alternative measure, the average closure coefficient, is proposed to quantify local clustering. It is shown that the average closure coefficient possesses a number of useful properties and can capture complementary information missed by the classical average clustering coefficient. In this paper, we study the asymptotic distribution of the average closure coefficient of a heterogeneous Erd\"{o}s-R\'{e}nyi random graph. We prove that the standardized average closure coefficient converges in distribution to the standard normal distribution. In the Erd\"{o}s-R\'{e}nyi random graph, the variance of the average closure coefficient exhibits the same phase transition phenomenon as the average clustering coefficient.

For safe and effective operation of humanoid robots in human-populated environments, the problem of commanding a large number of Degrees of Freedom (DoF) while simultaneously considering dynamic obstacles and human proximity has still not been solved. We present a new reactive motion controller that commands two arms of a humanoid robot and three torso joints (17 DoF in total). We formulate a quadratic program that seeks joint velocity commands respecting multiple constraints while minimizing the magnitude of the velocities. We introduce a new unified treatment of obstacles that dynamically maps visual and proximity (pre-collision) and tactile (post-collision) obstacles as additional constraints to the motion controller, in a distributed fashion over surface of the upper-body of the iCub robot (with 2000 pressure-sensitive receptors). The bio-inspired controller: (i) produces human-like minimum jerk movement profiles; (ii) gives rise to a robot with whole-body visuo-tactile awareness, resembling peripersonal space representations. The controller was extensively experimentally validated, including a physical human-robot interaction scenario.

We present a demonstration of image classification using an echo-state network (ESN) relying on a single simulated spintronic nanostructure known as the vortex-based spin-torque oscillator (STVO) delayed in time. We employ an ultrafast data-driven simulation framework called the data-driven Thiele equation approach (DD-TEA) to simulate the STVO dynamics. This allows us to avoid the challenges associated with repeated experimental manipulation of such a nanostructured system. We showcase the versatility of our solution by successfully applying it to solve classification challenges with the MNIST, EMNIST-letters and Fashion MNIST datasets. Through our simulations, we determine that within a large ESN the results obtained using the STVO dynamics as an activation function are comparable to the ones obtained with other conventional nonlinear activation functions like the reLU and the sigmoid. While achieving state-of-the-art accuracy levels on the MNIST dataset, our model's performance on EMNIST-letters and Fashion MNIST is lower due to the relative simplicity of the system architecture and the increased complexity of the tasks. We expect that the DD-TEA framework will enable the exploration of deeper architectures, ultimately leading to improved classification accuracy.

An important question in statistical network analysis is how to estimate models of discrete and dependent network data with intractable likelihood functions, without sacrificing computational scalability and statistical guarantees. We demonstrate that scalable estimation of random graph models with dependent edges is possible, by establishing convergence rates of pseudo-likelihood-based $M$-estimators for discrete undirected graphical models with exponential parameterizations and parameter vectors of increasing dimension in single-observation scenarios. We highlight the impact of two complex phenomena on the convergence rate: phase transitions and model near-degeneracy. The main results have possible applications to discrete and dependent network, spatial, and temporal data. To showcase convergence rates, we introduce a novel class of generalized $\beta$-models with dependent edges and parameter vectors of increasing dimension, which leverage additional structure in the form of overlapping subpopulations to control dependence. We establish convergence rates of pseudo-likelihood-based $M$-estimators for generalized $\beta$-models in dense- and sparse-graph settings.

We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.

Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.