Infinite Network Research Articles

Transitions in growing networks using a structural complexity approach

Structure changes or transitions are common in growing networks (complex networks, graphs, etc.) and must be precisely determined. The introduced quantitative measure of the structural complexity of the network based on a procedure similar to the renormalization process allows one to reveal such changes. The proposed concept of the network structural complexity accounts for the difference between the actual and averaged network structures on different scales and corresponds to the qualitative comprehension of complexity. The structural complexity can be found for the weighted networks also. The structural complexities for various types of growing networks exhibiting transitions similar to phase transitions were found—the deterministic infinite and finite size artificial networks of different natures including percolation structures, and the time series of various types of cardiac rhythms mapped to complex networks using the parametric visibility graph algorithm. In all the cases the structural complexity of the growing network reaches a maximum near the transition point: the formation of a giant component in the graph or at the percolation threshold for two-dimensional and three-dimensional square lattices when a giant cluster having a fractal structure has emerged. Therefore, the structural complexity of the network allows us to detect and study processes similar to a second-order phase transition in complex networks. The structural complexity of a network node can serve as a kind of centrality index, auxiliary, or generalization to the local clustering coefficient. Such an index provides another new ranking manner for the network nodes. Being an easily computable measure, the network structural complexity might help to reveal different features of complex systems and processes of the real world. Published by the American Physical Society 2024

Operator Theory and Analysis of Infinite Networks: Theory and Applications

Operator Theory and Analysis of Infinite Networks: Theory and Applications

Confinement in the Transverse Field Ising Model on the Heavy Hex Lattice.

Inspired by a recent quantum computing experiment [Y. Kim etal., Nature (London), 618, 500-5 (2023)NATUAS0028-083610.1038/s41586-023-06096-3], we study the emergence of confinement in the transverse field Ising model on a decorated hexagonal lattice. Using an infinite tensor network state optimized with belief propagation we show how a quench from a broken symmetry state leads to striking nonthermal behavior underpinned by persistent oscillations and saturation of the entanglement entropy. We explain this phenomenon by constructing a minimal model based on the confinement of elementary excitations. Our model is in excellent agreement with our numerical results. For quenches to larger values of the transverse field and/or from nonsymmetry broken states, our numerical results display the expected signatures of thermalization: a linear growth of entanglement entropy in time, propagation of correlations, and the saturation of observables to their thermal averages. These results provide a physical explanation for the unexpected classical simulability of the quantum dynamics.

Dynamics of finite width Kernel and prediction fluctuations in mean field neural networks*

Abstract We analyze the dynamics of finite width effects in wide but finite feature learning neural networks. Starting from a dynamical mean field theory description of infinite width deep neural network kernel and prediction dynamics, we provide a characterization of the O ( 1 / width ) fluctuations of the dynamical mean field theory order parameters over random initializations of the network weights. Our results, while perturbative in width, unlike prior analyses, are non-perturbative in the strength of feature learning. We find that once the mean field/µP parameterization is adopted, the leading finite size effect on the dynamics is to introduce initialization variance in the predictions and feature kernels of the networks. In the lazy limit of network training, all kernels are random but static in time and the prediction variance has a universal form. However, in the rich, feature learning regime, the fluctuations of the kernels and predictions are dynamically coupled with a variance that can be computed self-consistently. In two layer networks, we show how feature learning can dynamically reduce the variance of the final tangent kernel and final network predictions. We also show how initialization variance can slow down online learning in wide but finite networks. In deeper networks, kernel variance can dramatically accumulate through subsequent layers at large feature learning strengths, but feature learning continues to improve the signal-to-noise ratio of the feature kernels. In discrete time, we demonstrate that large learning rate phenomena such as edge of stability effects can be well captured by infinite width dynamics and that initialization variance can decrease dynamically. For convolutional neural networks trained on CIFAR-10, we empirically find significant corrections to both the bias and variance of network dynamics due to finite width.

Emerging Roles of Gossypol in Therapy: Innovations in Prodrug Design and Nanoformulation Frontiers.

Stimuli activatable systems have the potential to deliver drugs to targeted areas by releasing therapeutic agents in response to diseased specific microenvironments such as the acidic environment commonly found in diseased tissues. This review article focuses on gossypol, a bioactive compound with inherent toxicity attributed to various factors, including the presence of its formyl groups. It highlights the potential of imine-linked gossypol-based prodrugs and nanoparticle formulations for targeted delivery and controlled release. The unique presence of polyphenolic cores on gossypol can be utilized to prepare nanoparticles. This review offers valuable insights into designing safer and more effective drug delivery systems by elucidating the masking effect and stimuli-responsive release mechanisms. Numerous examples demonstrate the conversion of formyl groups to imines, creating prodrugs that mask reactive functionalities and offer pH-responsive release. This insight can guide the design of combination therapeutics, where a second drug with an amine terminal group can form imine-linked prodrugs. Additionally, the second part discusses the use of polyphenolic moieties to create stable nanoparticles from infinite polymeric networks. Through a comprehensive examination of gossypol's properties and applications, this review emphasizes the broader implications of such a masking strategy for optimizing the therapeutic benefits of many similar bioactive compounds while minimizing adverse effects.

Diversity of Structures and Bonding in Alkali Metal Ureaphosphanes

AbstractWhile organoelement compounds of lithium, sodium and potassium have been much studied for decades and consequently have found forests of applications, those of the heavier alkali metals, rubidium and caesium would barely manage to fill a tree. However, recently the literature has seen some little growth spurts with these metals, hinting at a possible fertile future in areas such as homogeneous catalysis provided more work is put into their fundamental development. Here we report the synthesis and crystal structures of lithium, rubidium and caesium derivatives of the ureaphosphane Ph2PCH2CH2NHC(=O)NHPh, chosen because it offers O, N, P, and π‐coordination sites. Though one may expect such alkali metal compounds to be essentially similar, the caesium complex has novel features where Cs+ engages in a side‐on coordination to the C=O bond and in a weak bond to the P centre, both of which are absent in the Rb structure. Less surprisingly, the lithium derivative is tetrameric in contrast to the infinite networks of the rubidium and caesium structures. All alkali metal derivatives were made with deprotonating the ureaphosphane by a suitable base, including the sodium and potassium complexes though these two complexes could not be obtained in a crystalline form.

Universal bounds for spreading on networks.

Spreading (diffusion) of innovations is a stochastic process on social networks. When the key driving mechanism is the peer effect (word of mouth), the rate at which the aggregate adoption level increases with time depends strongly on the network structure. In many applications, however, the network structure is unknown. To estimate the aggregate adoption level as a function of time for such innovations, we show that the minimal and maximal adoption levels are attained on a homogeneous two-node network and on a homogeneous infinite complete network, respectively. Solving the Bass model on these two networks yields explicit lower and upper bounds for the expected adoption level on any network. These bounds are tight, and they also hold for the individual adoption probabilities of nodes. The gap between the lower and upper bounds increases monotonically with the ratio of the rates of internal and external influences.

Percolating cosmic string loops from evaporating primordial black holes

The Pulsar timing data from NANOGrav Collaboration has regenerated interest in the possibility of observing stochastic gravitational wave background arising from cosmic strings. Standard theory of formation of cosmic strings is based on a spontaneous symmetry breaking (SSB) phase transition in the early universe, with a string network forming via the so called Kibble mechanism. This string network rapidly evolves and reaches a scaling solution with a given spectrum of string loops and long strings at any given time. This scenario necessarily requires that the entire observable Universe goes through the SSB phase transition. This would not be possible, e.g., in models of low energy inflation, where the reheat temperature is much lower than the energy scale of cosmic strings. We point out a very different possibility, where a network of even high energy scale cosmic strings can form when the temperature of the Universe is much lower. We consider local heating of plasma in the early universe by evaporating primordial black holes (PBHs). It is known that for suitable masses of PBHs, Hawking radiation of evaporating primordial black holes may re-heat the surrounding plasma to high temperatures, restoring certain symmetries locally which are broken at the ambient temperature of the Universe at that stage. Expansion of the hot plasma cools it so that the locally restored symmetry is spontaneously broken again. If this SSB supports formation of cosmic strings, then string loops will form in this region around the PBH. Further, resulting temperature gradients lead to large pressure gradients such that plasma will develop radial flow with the string loops getting stretched as they get dragged by the flow. For a finite density of PBHs of suitable masses, one will get local hot spots, each one contributing to expanding cosmic string loops. For suitable PBH density, the loops from different regions may intersect. If that happens, then intercommutation of strings can lead to percolation, leading to the possibility of formation of infinite string network, even when the entire universe never goes through the respective SSB phase transition.

A new reduced geometric model for simulation and design of NETmix reactors: LoopNUB

NETmix is a mesosized static mixer comprising a network of mixing chambers and channels, which has experienced a fast industrial uptake. The flow inside this mixer has been previously simulated using the NETmix Unit Block (NUB) - a vertical subdomain of the network that uses lateral Periodic Boundary Conditions (PBCs). A new model is introduced, enabling further reduction of the simulated space – the LoopNUB – that considers PBCs in the sidewalls and between the inlets and outlets, representing an infinite network. The LoopNUB with four rows proved suitable for simulating flow dynamics in NETmix for distances larger than 8 unit cells from inlet channels. LoopNUB was verified, and the results were validated from NUB simulations, which had been validated from experiments. LoopNUB enables the accurate simulation of flow dynamics and heat transfer while decreasing simulation time and computational resources such as disk and memory usage.

Hydantoin-D,L-valine: Synthesis, characterization, and non-covalent interaction analysis from crystallographic studies, DFTB calculations, and Hirshfeld surface analysis

The title compound, hydantoin-D,L-daline or (RS)-5-isopropyl-imidazolidine-2,4‑dione, a new amino acid hydantoin derivative with formula C6H10N2O2 has been synthesized and structurally characterized by FT-IR, NMR, and X-ray diffraction techniques. Spectroscopy results are consistent with the skeleton structure. The powder X-ray diffraction data confirms the phase purity of the crystalline sample. Single-crystal X-ray diffraction analysis indicated that the title compound crystallizes in the monoclinic space group P21/c (N°14), Z = 4, and unit cell parameters a = 5.493(3) Å, b = 23.53(2) Å, c = 6.254(3) Å, β= 115.09(4)°, V = 732.1(9) Å3. The molecular structure and crystal packing are stabilized by intermolecular N–H···O, C–H···O, and C–H···π hydrogen bonds forming an infinite two-dimensional network with chains described by the graph-set notation C(5), and centrosymmetric rings with graphs R22(8) and R44(16). The non-covalent interactions were analyzed employing DFTB calculations, which reproduce the main features of the stereochemistry and geometries exhibited by X-ray diffraction. Additionally, these intermolecular interactions were analyzed using Hirshfeld surfaces analysis and 2D fingerprint plots.

Interplay of bipyridine and 4,4′-diaminodiphenylmethane ligands in crystal design of cadmium-based coordination polymers: Structures and unusual photoluminescence quenching

Starting from Cd(ClO4)2·2H2O, four positively charged coordination polymers were prepared and characterized by IR spectroscopic and single-crystal X-ray diffraction methods. The obtained crystalline solids included one-dimensional coordination polymers, {[Cd(dadpm)(2,2′-bpy)2](ClO4)2}n (1), and {[Cd(dadpm)(2,2′-bpy)2](ClO4)2·0.5(H2O)}n (2), and two-dimensional coordination polymers, {[Cd(4,4′-bpy)2(H2O)2](ClO4)2(dadpm)2}n (3), and {[Cd(4,4′-bpy)2(H2O)2](ClO4)2(4,4′-bpy)0.5(dadpm)·EtOH}n (4) (dadpm = 4,4′-diaminodiphenylmethane, 2,2′-bpy = 2,2′-bipyridine; 4,4′-bpy = 4,4′-bipyridine). The similar one-dimensional coordination backbones in 1 and 2 were originated from the mononuclear Cd(II) coordination cores with the N6 distorted octahedral geometries going from two terminal bidentate-chelate 2,2′-bpy and two bidentate-bridging dadpm ligands that mediated the successive metal atoms in coordination chains. The charge-balanced perchlorate anions were located in the interchain space complemented by water molecules in 2. The coordination chains were interconnected into H-bonded networks via NH(NH2)⋯O hydrogen bonds. The 2D coordination polymers 3 and 4 comprised the same positively charged square-grid infinite networks {[Cd(4,4′-bpy)2(H2O)2]2+}n with sql topology and with cadmium atom in the N4O2 tetragonally distorted octahedral coordination geometry as a single node. The charge-compensated ClO4− anions and the dadpm molecules were located in the interlayer space in 3 being complemented by the 4,4′-bpy and EtOH molecules in 4. The solid-state photoluminescence in all reported coordination polymers was significantly attenuated and redistributed along the spectrum in comparison with the pure dadpm which revealed the dual emission in the form of a strong narrow signal in the ultraviolet and a weak signal in the yellow regions of spectrum.

Expected Policy Gradient for Network Aggregative Markov Games in Continuous Space.

In this article, we investigate the Nash-seeking problem of a set of agents, playing an infinite network aggregative Markov game. In particular, we focus on a noncooperative framework where each agent selfishly aims at maximizing its long-term average reward without having explicit information on the model of the environment dynamics and its own reward function. The main contribution of this article is to develop a continuous multiagent reinforcement learning (MARL) algorithm for the Nash-seeking problem in infinite dynamic games with convergence guarantee. To this end, we propose an actor-critic MARL algorithm based on expected policy gradient (EPG) with two general function approximators to estimate the value function and the Nash policy of the agents. We consider continuous state and action spaces and adopt a newly proposed EPG to alleviate the variance of the gradient approximation. Based on such formulation and under some conventional assumptions (e.g., using linear function approximators), we prove that the policies of the agents converge to the unique Nash equilibrium (NE) of the game. Furthermore, an estimation error analysis is conducted to investigate the effects of the error arising from function approximation. As a case study, the framework is applied on a cloud radio access network (C-RAN) by modeling the remote radio heads (RRHs) as the agents and the congestion of baseband units (BBUs) as the dynamics of the environment.

Metric Dimension of Irregular Convex Triangular Networks

Irregular convex triangular networks consist of the interior of a 6-sided convex polygon drawn on the infinite triangular network. Formal description of these applicable networks is provided. In the main result it is proved that the metric dimension of an irregular convex triangular network is either 2 or 3 and the determination of which value is the right one is specified. Metric dimensions of various graph classes and interconnection networks are also collected.

Gauging tensor networks with belief propagation

Effectively compressing and optimizing tensor networks requires reliable methods for fixing the latent degrees of freedom of the tensors, known as the gauge. Here we introduce a new algorithm for gauging tensor networks using belief propagation, a method that was originally formulated for performing statistical inference on graphical models and has recently found applications in tensor network algorithms. We show that this method is closely related to known tensor network gauging methods. It has the practical advantage, however, that existing belief propagation implementations can be repurposed for tensor network gauging, and that belief propagation is a very simple algorithm based on just tensor contractions so it can be easier to implement, optimize, and generalize. We present numerical evidence and scaling arguments that this algorithm is faster than existing gauging algorithms, demonstrating its usage on structured, unstructured, and infinite tensor networks. Additionally, we apply this method to improve the accuracy of the widely used simple update gate evolution algorithm.

Glycosaminoglycan-mimetic infernan grafted with poly(N-isopropylacrylamide): Toward a thermosensitive polysaccharide

Glycosaminoglycans (GAGs) are essential constituents of the cell surface and extracellular matrix, where they are involved in several cellular processes through their interactions with various proteins. For successful tissue regeneration, developing an appropriate matrix supporting biological activities of cells in a similar manner than GAGs remains still challenging. In this context, this study aims to design a thermosensitive polysaccharide that could further be used as hydrogel for tissue engineering applications. For this purpose, infernan, a marine bacterial exopolysaccharide (EPS) endowed with GAG-mimetic properties was grafted with a thermosensitive polymer, poly(N-isopropylacrylamide) (pNIPAM). Eight grafted polysaccharides were obtained by varying EPS/pNIPAM molar ratio and the molecular weight of pNIPAM. Their physicochemical characteristics and their thermosensitive properties were determined using a multi-technique, experimental approach. In parallel, molecular dynamics and Monte Carlo simulations were applied at two different scales to elucidate, respectively, the molecular conformation of grafted infernan chain and their ability to form an infinite network undergoing a sol-gel transition near the percolation, a necessary condition in hydrogel formation. It comes out from this study that thermosensitive infernan was successfully developed and its potential use in tissue regeneration as a hydrogel scaffold will further be assessed.

Electromagnetic wave absorption properties of porous carbon nanocomposite prepared by explosion method derived from energetic CuFe-MOF

An energetic heteronuclear metal–organic framework (MOF), denoted as {[CuIIFeIII3O(tza)6(H2O)3]·Cl·(NO3)2·4H2O}n (CuFe-MOF), was successfully synthesized and characterized using various techniques, including single-crystal X-ray diffractometry (XRD), powder X-ray diffractometry (PXRD), and scanning/transmission electron microscopy (SEM/TEM). The compound crystallized in the P63m space group, forming an infinite 3D cationic network composed of [Fe3O(tza)6]+ clusters and Cu2+ ions. Subsequently, we transformed CuFe-MOF into porous carbon nanocomposites, C/Cu/Fe3N/Fe4N/Fe3O4/CuFe2O4, using an explosion method. We then investigated the electromagnetic wave (EMW) attenuation properties and mechanisms of the carbon nanocomposite. Notably, the synthesized C/Cu/Fe3N/Fe4N/Fe3O4/CuFe2O4 composite material demonstrated exceptional EMW absorption. At a low frequency of 4.1 GHz with a thickness of 5.0 mm, it achieved a minimum reflection loss (RL) of −47.7 dB. Even at a high frequency of 17.8 GHz and a thickness of 1.5 mm, the minimum RL reached −41.8 dB. Moreover, this material exhibited a wide effective absorption bandwidth (EAB; RL ≤ −10 dB) spanning 14.9 GHz (3.1–18 GHz) for thicknesses between 1.0 and 5.5 mm. These attributes position the carbon composite as a strong candidate for absorption applications within the Ku, X, and C bands, making it suitable for both military and civilian usage. Our mechanistic analysis suggests that the excellent absorption performance of C/Cu/Fe3N/Fe4N/Fe3O4/CuFe2O4 results from a combination of mechanisms, including electrical and magnetic loss. This research introduces an innovative approach for the fabrication of high-performance carbon composite materials via the explosive method, utilizing multinuclear energetic materials.

RETRACTED ARTICLE: Optimized design and application research of smart interactive screen for wireless networks based on federated learning

The rapid development of infinite networks and information technology has promoted the wide deployment and rapid growth of intelligent interactive devices. However, at the same time, touch interaction technology also faces many challenges such as lack of precision. This study combines federated learning with LayerGesture technology to optimize and design a touch interaction system with higher interaction accuracy and applies it to practice. The analysis results show that with the increase in the number of iterations of the federated model, the accuracy of the human–computer recognition interaction and the amount of information contained in it increases, and the accuracy curve reaches stability at about 2800 times and is at the optimal interaction adaptation. At this point, the loss function also decreases gradually, while the loss factor tends to 0, which verifies the stability of the optimized model. According to the participants’ interaction experience and experimental results, the optimized LayerGesture technique of the federated learning model has an average correctness rate of 90.4% and the lowest average selection time, while the average selection time of LayerGesture in the interaction area at the edge of the screen is 2510 ms and the average correctness rate is 93.60%, which is better than the Shift technique. In addition, the subjective survey results indicated that more participants favored the optimized LayerGesture technique. In summary, this paper’s joint learning algorithm contributes to the recognition effectiveness and efficiency of intelligent interactive systems.

Extended-range percolation in complex networks.

Classical percolation theory underlies many processes of information transfer along the links of a network. In these standard situations, the requirement for two nodes to be able to communicate is the presence of at least one uninterrupted path of nodes between them. In a variety of more recent data transmission protocols, such as the communication of noisy data via error-correcting repeaters, both in classical and quantum networks, the requirement of an uninterrupted path is too strict: two nodes may be able to communicate even if all paths between them have interruptions or gaps consisting of nodes that may corrupt the message. In such a case a different approach is needed. We develop the theoretical framework for extended-range percolation in networks, describing the fundamental connectivity properties relevant to such models of information transfer. We obtain exact results, for any range R, for infinite random uncorrelated networks and we provide a message-passing formulation that works well in sparse real-world networks. The interplay of the extended range and heterogeneity leads to novel critical behavior in scale-free networks.

Piecewise linear functions representable with infinite width shallow ReLU neural networks

This paper analyzes representations of continuous piecewise linear functions with infinite width, finite cost shallow neural networks using the rectified linear unit (ReLU) as an activation function. Through its integral representation, a shallow neural network can be identified by the corresponding signed, finite measure on an appropriate parameter space. We map these measures on the parameter space to measures on the projective n n -sphere cross R \mathbb {R} , allowing points in the parameter space to be bijectively mapped to hyperplanes in the domain of the function. We prove a conjecture of Ongie et al. [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case, arXiv, 2019] that every continuous piecewise linear function expressible with this kind of infinite width neural network is expressible as a finite width shallow ReLU neural network.

Predicting the mass spectrum of polymerizing linoleates using weighted random graph modeling

Biopolymers and biopolymer networks that form via autoxidation, like in drying of oil paint or fat degradation in food components, contain a large variety of monomeric building blocks. While the monomer variety complicates the modeling itself, obtaining experimental validation of infinite polymer networks is inherently difficult as well. A new model is developed, where an automated reaction network generation (ARNG) procedure is used to automatically generate the monomer components, structures and masses, and their reactions. This methodology is combined with random graph (RG) modeling to predict global polymer properties: distributions of numbers of monomer units and molar masses, gel point and gel fraction. This computational framework is applied to two model systems for linseed oil paint binder: the polymerization of ethyl linoleate (EL) and methyl linoleate (ML). A novel method was constructed to deal with the variability of monomer masses that complicates inferring molar mass from monomer number distribution. By modeling the polymer as a weighted random graph where the nodes contain information about the monomer masses in the system, the total weight of the finite connected components is computed. The predicted mass spectrum of finite connected components is used for validation with experimental data. A size exclusion chromatography (SEC) trace of ML is employed, which after calibration using the proposed framework, proves consistency between model and SEC data. The model provides a practical approach to both characterize complex biopolymers as polymers in terms of molar mass distribution and gel point, while preserving the information down to the level of monomeric units.