Deciphering the generating rules and functionalities of complex networks

Neural Correlation Integrated Adaptive Point Process Filtering on Population Spike Trains

Brain encodes information through neural spiking activities that modulate external environmental stimuli and underlying internal states. Population of neurons coordinate through functional connectivity to plan movement trajectories and accurately activate neuromuscular activities. Motor Brain-machine interface (BMI) is a platform to study the relationship between behaviors and neural ensemble activities. In BMI, point process filters model directly on spike timings to extract underlying states such as motion intents from observed multi-neuron spike trains. However, these methods assume the encoded information from individual neurons is conditionally independent, which leads to less precise estimation. It is necessary to incorporate functional neural connectivity into a point process filter to improve the state estimation. In this paper, we propose a neural correlation integrated adaptive point process filter (CIPPF) that can incorporate the information from functional neural connectivity from population spike trains in a recursive Bayesian framework. Functional neural connectivity information is approximated by an artificial neural network to provide extra updating information for the posterior estimation. Gaussian approximation is applied on the probability distribution to obtain a closed-form solution. Our proposed method is validated on both simulation and real data collected from the rat two-lever discrimination task. Due to the simultaneous modeling of functional neural connectivity and single neuronal tuning properties, the proposed method shows better decoding performance. This suggests the possibility to improve BMI performance by processing the coordinated neural population activities.

Autocatalytic Nucleation and Self-Assembly of Inorganic Nanoparticles into Complex Biosimilar Networks

Self-replication of bioorganic molecules and oil microdroplets have been explored as models in prebiotic chemistry. An analogous process for inorganic nanomaterials would involve the autocatalytic nucleation of metal, semiconductor, or ceramic nanoparticles-an area that remains largely uncharted. Demonstrating such systems would be both fundamentally intriguing and practically relevant, especially if the resulting particles self-assemble into complex structures beyond the capabilities of molecules or droplets. Here, we show that autocatalytic nucleation occurs with silver nanoparticles, which subsequently self-assemble into chains through spatially restricted attachment. In dispersions containing "hedgehog" particles, these reactions produce complex colloids with hierarchical spike organization. On solid surfaces, autocatalytic nucleation of nanoparticles yields conformal networks with hierarchical organization, including nanoparticle "colonies." We analyzed the complexity of both types of solid-stabilized particle assemblies via graph theory (GT). The complexity index of idealized spiky colloids is comparable to that of complex algal skeletons. The GT analysis of the percolating nanoparticle networks revealed their similarities to the bacterial, but not fungal, biofilms. We conclude that coupling autocatalytic nucleation with self-assembly enables the generation of complex, biosimilar particles and films. This work establishes mathematical and structural parallels between biotic and abiotic matter, integrating self-organization, autocatalytic nucleation, and theoretical description of complex systems. Utilization of quantitative descriptors of connectivity patterns opens possibility to GT-based biomimetic engineering of conductive coatings and other complex nanostructures.

Machine learning-based identification of general transcriptional predictors for plant disease

This study investigated the generalizability of Arabidopsis thaliana immune responses across diverse pathogens, including Botrytis cinerea, Sclerotinia sclerotiorum, and Pseudomonas syringae, using a data-driven, machine learning approach. Machine learning models were trained to predict disease development from early transcriptional responses. Feature selection techniques based on network science and topology were used to train models employing only a fraction of the transcriptome. Machine learning models trained on one pathosystem where then validated by predicting disease development in new pathosystems. The identified feature selection gene sets were enriched for pathways related to biotic, abiotic, and stress responses, though the specific genes involved differed between feature sets. This suggests common immune responses to diverse pathogens that operate via different gene sets. The study demonstrates that machine learning can uncover both established and novel components of the plant's immune response, offering insights into disease resistance mechanisms. These predictive models highlight the potential to advance our understanding of multigenic outcomes in plant immunity and can be further refined for applications in disease prediction.

Graph Theoretical Description of Phase Transitions in Complex Multiscale Phases with Supramolecular Assemblies

Phase transitions are typically quantified using order parameters, such as crystal lattice distances and radial distribution functions, which can identify subtle changes in crystalline materials or high-contrast phases with large structural differences. However, the identification of phases with high complexity, multiscale organization and of complex patterns during the structural fluctuations preceding phase transitions, which are essential for understanding the system pathways between phases, is challenging for those traditional analyses. Here, it is shown that for two model systems- thermotropic liquid crystals and a lyotropic water/surfactant mixtures-graph theoretical (GT) descriptors can successfully identify complex phases combining molecular and nanoscale levels of organization that are hard to characterize with traditional methodologies. Furthermore, the GT descriptors also reveal the pathways between the different phases. Specifically, centrality parameters and node-based fractal dimension quantify the system behavior preceding the transitions, capturing fluctuation-induced breakup of aggregates and their long-range cooperative interactions. GT parameterization can be generalized for a wide range of chemical systems and be instrumental for the growth mechanisms of complex nanostructures.

Topological change of soil microbiota networks for forest resilience under global warming

Forest management by thinning can mitigate the detrimental impact of increasing drought caused by global warming. Growing evidence shows that the soil microbiota can coordinate the dynamic relationship between forest functions and drought intensity, but how they function as a cohesive whole remains elusive. We outline a statistical topology model to chart the roadmap of how each microbe acts and interacts with every other microbe to shape the dynamic changes of microbial communities under forest management. To demonstrate its utility, we analyze a soil microbiota data collected from a two-way longitudinal factorial experiment involving three stand densities and three levels of rainfall over a growing season in artificial plantations of a forest tree - larix (Larix kaempferi). We reconstruct the most sophisticated soil microbiota networks that code maximally informative microbial interactions and trace their dynamic trajectories across time, space, and environmental signals. By integrating GLMY homology theory, we dissect the topological architecture of these so-called omnidirectional networks and identify key microbial interaction pathways that play a pivotal role in mediating the structure and function of soil microbial communities. The statistical topological model described provides a systems tool for studying how microbial community assembly alters its structure, function and evolution under climate change.

Multifractal perturbations to multiplicative cascades promote multifractal nonlinearity with asymmetric spectra

Biological and psychological processes have been conceptualized as emerging from intricate multiplicative interactions among component processes across various spatial and temporal scales. Among the statistical models employed to approximate these intricate nonlinear interactions across scales, one prominent framework is that of cascades. Despite decades of empirical work using multifractal formalisms, several fundamental questions persist concerning the proper interpretations of multifractal evidence of nonlinear cross-scale interactivity. Does multifractal spectrum width depend on multiplicative interactions, constituent noise processes participating in those interactions, or both? We conducted numerical simulations of cascade time series featuring component noise processes characterizing a range of nonlinear temporal correlations: nonlinearly multifractal, linearly multifractal (obtained via the iterative amplitude adjusted wavelet transform of nonlinearly multifractal), phase-randomized linearity (obtained via the iterative amplitude adjustment Fourier transform of nonlinearly multifractal), and phase and amplitude randomized (obtained via shuffling of nonlinearly multifractal). Our findings show that the multiplicative interactions coordinate with the nonlinear temporal correlations of noise components to dictate emergent multifractal properties. Multiplicative cascades with stronger nonlinear temporal correlations make multifractal spectra more asymmetric with wider left sides. However, when considering multifractal spectral differences between the original and surrogate time series, even multiplicative cascades produce multifractality greater than in surrogate time series, even with linearized multifractal noise components. In contrast, additivity among component processes leads to a linear outcome. These findings provide a robust framework for generating multifractal expectations for biological and psychological models in which cascade dynamics flow from one part of an organism to another.

Complexity synchronization in living matter: a mini review

Fractal time series have been argued to be ubiquitous in human physiology and some of the implications of that ubiquity are quite remarkable. One consequence of the omnipresent fractality is complexity synchronization (CS) observed in the interactions among simultaneously recorded physiologic time series discussed herein. This new kind of synchronization has been revealed in the interaction triad of organ-networks (ONs) consisting of the mutually interacting time series generated by the brain (electroencephalograms, EEGs), heart (electrocardiograms, ECGs), and lungs (Respiration). The scaled time series from each member of the triad look nothing like one another and yet they bear a deeply recorded synchronization invisible to the naked eye. The theory of scaling statistics is used to explain the source of the CS observed in the information exchange among these multifractal time series. The multifractal dimension (MFD) of each time series is a measure of the time-dependent complexity of that time series, and it is the matching of the MFD time series that provides the synchronization referred to as CS. The CS is one manifestation of the hypothesis given by a "Law of Multifractal Dimension Synchronization" (LMFDS) which is supported by data. Therefore, the review aspects of this paper are chosen to make the extended range of the LMFDS hypothesis sufficiently reasonable to warrant further empirical testing.

From pixels to connections: exploring in vitro neuron reconstruction software for network graph generation

Digital reconstruction has been instrumental in deciphering how in vitro neuron architecture shapes information flow. Emerging approaches reconstruct neural systems as networks with the aim of understanding their organization through graph theory. Computational tools dedicated to this objective build models of nodes and edges based on key cellular features such as somata, axons, and dendrites. Fully automatic implementations of these tools are readily available, but they may also be purpose-built from specialized algorithms in the form of multi-step pipelines. Here we review software tools informing the construction of network models, spanning from noise reduction and segmentation to full network reconstruction. The scope and core specifications of each tool are explicitly defined to assist bench scientists in selecting the most suitable option for their microscopy dataset. Existing tools provide a foundation for complete network reconstruction, however more progress is needed in establishing morphological bases for directed/weighted connectivity and in software validation.

Postural control in gymnasts: anisotropic fractal scaling reveals proprioceptive reintegration in vestibular perturbation

Dexterous postural control subtly complements movement variability with sensory correlations at many scales. The expressive poise of gymnasts exemplifies this lyrical punctuation of release with constraint, from coarse grain to fine scales. Dexterous postural control upon a 2D support surface might collapse the variation of center of pressure (CoP) to a relatively 1D orientation-a direction often oriented towards the focal point of a visual task. Sensory corrections in dexterous postural control might manifest in temporal correlations, specifically as fractional Brownian motions whose differences are more and less correlated with fractional Gaussian noises (fGns) with progressively larger and smaller Hurst exponent H. Traditional empirical work examines this arrangement of lower-dimensional compression of CoP along two orthogonal axes, anteroposterior (AP) and mediolateral (ML). Eyes-open and face-forward orientations cultivate greater variability along AP than ML axes, and the orthogonal distribution of spatial variability has so far gone hand in hand with an orthogonal distribution of H, for example, larger in AP and lower in ML. However, perturbing the orientation of task focus might destabilize the postural synergy away from its 1D distribution and homogenize the temporal correlations across the 2D support surface, resulting in narrower angles between the directions of the largest and smallest H. We used oriented fractal scaling component analysis (OFSCA) to investigate whether sensory corrections in postural control might thus become suborthogonal. OFSCA models raw 2D CoP trajectory by decomposing it in all directions along the 2D support surface and fits the directions with the largest and smallest H. We studied a sample of gymnasts in eyes-open and face-forward quiet posture, and results from OFSCA confirm that such posture exhibits the classic orthogonal distribution of temporal correlations. Head-turning resulted in a simultaneous decrease in this angle Δθ, which promptly reversed once gymnasts reoriented their heads forward. However, when vision was absent, there was only a discernible negative trend in Δθ, indicating a shift in the angle's direction but not a statistically significant one. Thus, the narrowing of Δθ may signify an adaptive strategy in postural control. The swift recovery of Δθ upon returning to a forward-facing posture suggests that the temporary reduction is specific to head-turning and does not impose a lasting burden on postural control. Turning the head reduced the angle between these two orientations, facilitating the release of postural degrees of freedom towards a more uniform spread of the CoP across both dimensions of the support surface. The innovative aspect of this work is that it shows how fractality might serve as a control parameter of adaptive mechanisms of dexterous postural control.

Beyond multilayer perceptrons: Investigating complex topologies in neural networks

This study delves into the crucial aspect of network topology in artificial neural networks (NNs) and its impact on model performance. Addressing the need to comprehend how network structures influence learning capabilities, the research contrasts traditional multilayer perceptrons (MLPs) with models built on various complex topologies using novel network generation techniques. Drawing insights from synthetic datasets, the study reveals the remarkable accuracy of complex NNs, particularly in high-difficulty scenarios, outperforming MLPs. Our exploration extends to real-world datasets, highlighting the task-specific nature of optimal network topologies and unveiling trade-offs, including increased computational demands and reduced robustness to graph damage in complex NNs compared to MLPs. This research underscores the pivotal role of complex topologies in addressing challenging learning tasks. However, it also signals the necessity for deeper insights into the complex interplay among topological attributes influencing NN performance. By shedding light on the advantages and limitations of complex topologies, this study provides valuable guidance for practitioners and paves the way for future endeavors to design more efficient and adaptable neural architectures across various applications.

GAHLS: an optimized graph analytics based high level synthesis framework

The urgent need for low latency, high-compute and low power on-board intelligence in autonomous systems, cyber-physical systems, robotics, edge computing, evolvable computing, and complex data science calls for determining the optimal amount and type of specialized hardware together with reconfigurability capabilities. With these goals in mind, we propose a novel comprehensive graph analytics based high level synthesis (GAHLS) framework that efficiently analyzes complex high level programs through a combined compiler-based approach and graph theoretic optimization and synthesizes them into message passing domain-specific accelerators. This GAHLS framework first constructs a compiler-assisted dependency graph (CaDG) from low level virtual machine (LLVM) intermediate representation (IR) of high level programs and converts it into a hardware friendly description representation. Next, the GAHLS framework performs a memory design space exploration while account for the identified computational properties from the CaDG and optimizing the system performance for higher bandwidth. The GAHLS framework also performs a robust optimization to identify the CaDG subgraphs with similar computational structures and aggregate them into intelligent processing clusters in order to optimize the usage of underlying hardware resources. Finally, the GAHLS framework synthesizes this compressed specialized CaDG into processing elements while optimizing the system performance and area metrics. Evaluations of the GAHLS framework on several real-life applications (e.g., deep learning, brain machine interfaces) demonstrate that it provides 14.27× performance improvements compared to state-of-the-art approaches such as LegUp 6.2.

A unified approach of detecting phase transition in time-varying complex networks

Deciphering the non-trivial interactions and mechanisms driving the evolution of time-varying complex networks (TVCNs) plays a crucial role in designing optimal control strategies for such networks or enhancing their causal predictive capabilities. In this paper, we advance the science of TVCNs by providing a mathematical framework through which we can gauge how local changes within a complex weighted network affect its global properties. More precisely, we focus on unraveling unknown geometric properties of a network and determine its implications on detecting phase transitions within the dynamics of a TVCN. In this vein, we aim at elaborating a novel and unified approach that can be used to depict the relationship between local interactions in a complex network and its global kinetics. We propose a geometric-inspired framework to characterize the network's state and detect a phase transition between different states, to infer the TVCN's dynamics. A phase of a TVCN is determined by its Forman-Ricci curvature property. Numerical experiments show the usefulness of the proposed curvature formalism to detect the transition between phases within artificially generated networks. Furthermore, we demonstrate the effectiveness of the proposed framework in identifying the phase transition phenomena governing the training and learning processes of artificial neural networks. Moreover, we exploit this approach to investigate the phase transition phenomena in cellular re-programming by interpreting the dynamics of Hi-C matrices as TVCNs and observing singularity trends in the curvature network entropy. Finally, we demonstrate that this curvature formalism can detect a political change. Specifically, our framework can be applied to the US Senate data to detect a political change in the United States of America after the 1994 election, as discussed by political scientists.

Hierarchical growth in neural networks structure: Organizing inputs by Order of Hierarchical Complexity

Several studies demonstrate that the structure of the brain increases in hierarchical complexity throughout development. We tested if the structure of artificial neural networks also increases in hierarchical complexity while learning a developing task, called the balance beam problem. Previous simulations of this developmental task do not reflect a necessary premise underlying development: a more complex structure can be built out of less complex ones, while ensuring that the more complex structure does not replace the less complex one. In order to address this necessity, we segregated the input set by subsets of increasing Orders of Hierarchical Complexity. This is a complexity measure that has been extensively shown to underlie the complexity behavior and hypothesized to underlie the complexity of the neural structure of the brain. After segregating the input set, minimal neural network models were trained separately for each input subset, and adjacent complexity models were analyzed sequentially to observe whether there was a structural progression. Results show that three different network structural progressions were found, performing with similar accuracy, pointing towards self-organization. Also, more complex structures could be built out of less complex ones without substituting them, successfully addressing catastrophic forgetting and leveraging performance of previous models in the literature. Furthermore, the model structures trained on the two highest complexity subsets performed better than simulations of the balance beam present in the literature. As a major contribution, this work was successful in addressing hierarchical complexity structural growth in neural networks, and is the first that segregates inputs by Order of Hierarchical Complexity. Since this measure can be applied to all domains of data, the present method can be applied to future simulations, systematizing the simulation of developmental and evolutionary structural growth in neural networks.

Bifractality of fractal scale-free networks

The presence of large-scale real-world networks with various architectures has motivated active research towards a unified understanding of diverse topologies of networks. Such studies have revealed that many networks with scale-free and fractal properties exhibit the structural multifractality, some of which are actually bifractal. Bifractality is a particular case of the multifractal property, where only two local fractal dimensions d_{f}^{min} and d_{f}^{max}(>d_{f}^{min}) suffice to explain the structural inhomogeneity of a network. In this work we investigate analytically and numerically the multifractal property of a wide range of fractal scale-free networks (FSFNs) including deterministic hierarchical, stochastic hierarchical, nonhierarchical, and real-world FSFNs. Then we demonstrate how commonly FSFNs exhibit the bifractal property. The results show that all these networks possess the bifractal nature. We conjecture from our findings that any FSFN is bifractal. Furthermore, we find that in the thermodynamic limit the lower local fractal dimension d_{f}^{min} describes substructures around infinitely high-degree hub nodes and finite-degree nodes at finite distances from these hub nodes, whereas d_{f}^{max} characterizes local fractality around finite-degree nodes infinitely far from the infinite-degree hub nodes. Since the bifractal nature of FSFNs may strongly influence time-dependent phenomena on FSFNs, our results will be useful for understanding dynamics such as information diffusion and synchronization on FSFNs from a unified perspective.

Network bypasses sustain complexity

Real-world networks are neither regular nor random, a fact elegantly explained by mechanisms such as the Watts-Strogatz or the Barabási-Albert models, among others. Both mechanisms naturally create shortcuts and hubs, which while enhancing the network's connectivity, also might yield several undesired navigational effects: They tend to be overused during geodesic navigational processes-making the networks fragile-and provide suboptimal routes for diffusive-like navigation. Why, then, networks with complex topologies are ubiquitous? Here, we unveil that these models also entropically generate network bypasses: alternative routes to shortest paths which are topologically longer but easier to navigate. We develop a mathematical theory that elucidates the emergence and consolidation of network bypasses and measure their navigability gain. We apply our theory to a wide range of real-world networks and find that they sustain complexity by different amounts of network bypasses. At the top of this complexity ranking we found the human brain, which points out the importance of these results to understand the plasticity of complex systems.

Topological energy of networks

Energy is an important network indicator defined by the eigenvalues of an adjacency matrix that includes the neighbor information for each node. This article expands the definition of network energy to include higher-order information between nodes. We use resistance distances to characterize the distances between nodes and order complexes to extract higher-order information. Topological energy ( T E), defined by the resistance distance and order complex, reveals the characteristics of the network structure from multiple scales. In particular, calculations show that the topological energy can be used to distinguish graphs with the same spectrum well. In addition, topological energy is robust, and small random perturbations of edges do not significantly affect the T E values. Finally, we find that the energy curve of the real network is significantly different from that of the random graph, thus showing that T E can be used to distinguish the network structure well. This study shows that T E is an indicator that distinguishes the structure of a network and has some potential applications for real-world problems.

Case report: Cortico-ocular interaction networks in NBA2K

The sport industry has never seen growth such as eSports'. Using synchronized monitoring of two biological processes on a 25-year-old gamer, we investigated how his brain (via EEG) and eyes (via pupil dilation) interacted dynamically over time as an integrated network during NBA2K playing time. After the spectral decomposition of the different Brain and Eye signals into seven frequency bands, we calculated the bivariate equal-time Pearson's cross-correlation between each pair of EEG/Eye spectral power time series. On average, our results show a reorganization of the cortico-muscular network across three sessions (e.g., new interactions, hemispheric asymmetry). These preliminary findings highlight the potential need for individualized, specific, adaptive, and periodized interventions and encourage the continuation of this line of research for the creation of general theories of networks in eSports gaming. Future studies should recruit larger samples, investigate different games, and explore cross-frequency coordination among other key organ systems.

Exploring robust architectures for deep artificial neural networks

The architectures of deep artificial neural networks (DANNs) are routinely studied to improve their predictive performance. However, the relationship between the architecture of a DANN and its robustness to noise and adversarial attacks is less explored, especially in computer vision applications. Here we investigate the relationship between the robustness of DANNs in a vision task and their underlying graph architectures or structures. First we explored the design space of architectures of DANNs using graph-theoretic robustness measures and transformed the graphs to DANN architectures using various image classification tasks. Then we explored the relationship between the robustness of trained DANNs against noise and adversarial attacks and their underlying architectures. We show that robustness performance of DANNs can be quantified before training using graph structural properties such as topological entropy and Olivier-Ricci curvature, with the greatest reliability for complex tasks and large DANNs. Our results can also be applied for tasks other than computer vision such as natural language processing and recommender systems.

Asim Waqas and colleagues investigated the relationship between the architectures of deep artificial neural networks (DANNs) and their robustness to noise and adversarial attacks in computer vision. The researchers found that the robustness of DANNs was highly correlated with graph-theoretic measures of entropy and curvature. This finding could help design more robust DANN architectures.

Adding links on minimum degree and longest distance strategies for improving network robustness and efficiency

Many real-world networks characterized by power-law degree distributions are extremely vulnerable against malicious attacks. Therefore, it is important to obtain effective methods for strengthening the robustness of the existing networks. Previous studies have been discussed some link addition methods for improving the robustness. In particular, two effective strategies for selecting nodes to add links have been proposed: the minimum degree and longest distance strategies. However, it is unclear whether the effects of these strategies on the robustness are independent or not. In this paper, we investigate the contributions of these strategies to improving the robustness by adding links in distinguishing the effects of degrees and distances as much as possible. Through numerical simulation, we find that the robustness is effectively improved by adding links on the minimum degree strategy for both synthetic trees and real networks. As an exception, only when the number of added links is small, the longest distance strategy is the best. Conversely, the robustness is only slightly improved by adding links on the shortest distance strategy in many cases, even combined with the minimum degree strategy. Therefore, enhancing global loops is essential for improving the robustness rather than local loops.

Objective Supervised Machine Learning-Based Classification and Inference of Biological Neuronal Networks

The classification of biological neuron types and networks poses challenges to the full understanding of the human brain’s organisation and functioning. In this paper, we develop a novel objective classification model of biological neuronal morphology and electrical types and their networks, based on the attributes of neuronal communication using supervised machine learning solutions. This presents advantages compared to the existing approaches in neuroinformatics since the data related to mutual information or delay between neurons obtained from spike trains are more abundant than conventional morphological data. We constructed two open-access computational platforms of various neuronal circuits from the Blue Brain Project realistic models, named Neurpy and Neurgen. Then, we investigated how we could perform network tomography with cortical neuronal circuits for the morphological, topological and electrical classification of neurons. We extracted the simulated data of 10,000 network topology combinations with five layers, 25 morphological type (m-type) cells, and 14 electrical type (e-type) cells. We applied the data to several different classifiers (including Support Vector Machine (SVM), Decision Trees, Random Forest, and Artificial Neural Networks). We achieved accuracies of up to 70%, and the inference of biological network structures using network tomography reached up to 65% of accuracy. Objective classification of biological networks can be achieved with cascaded machine learning methods using neuron communication data. SVM methods seem to perform better amongst used techniques. Our research not only contributes to existing classification efforts but sets the road-map for future usage of brain–machine interfaces towards an in vivo objective classification of neurons as a sensing mechanism of the brain’s structure.

Energy Transport along α-Helix Protein Chains: External Drives and Multifractal Analysis

Energy transport within biological systems is critical for biological functions in living cells and for technological applications in molecular motors. Biological systems have very complex dynamics supporting a large number of biochemical and biophysical processes. In the current work, we study the energy transport along protein chains. We examine the influence of different factors such as temperature, salt concentration, and external mechanical drive on the energy flux through protein chains. We obtain that energy fluctuations around the average value for short chains are greater than for longer chains. In addition, the external mechanical load is the most effective agent on bioenergy transport along the studied protein systems. Our results can help design a functional nano-scaled molecular motor based on energy transport along protein chains.

Unifying structural descriptors for biological and bioinspired nanoscale complexes

Biomimetic nanoparticles are known to serve as nanoscale adjuvants, enzyme mimics and amyloid fibrillation inhibitors. Their further development requires better understanding of their interactions with proteins. The abundant knowledge about protein-protein interactions can serve as a guide for designing protein-nanoparticle assemblies, but the chemical and biological inputs used in computational packages for protein-protein interactions are not applicable to inorganic nanoparticles. Analysing chemical, geometrical and graph-theoretical descriptors for protein complexes, we found that geometrical and graph-theoretical descriptors are uniformly applicable to biological and inorganic nanostructures and can predict interaction sites in protein pairs with accuracy >80% and classification probability ~90%. We extended the machine-learning algorithms trained on protein-protein interactions to inorganic nanoparticles and found a nearly exact match between experimental and predicted interaction sites with proteins. These findings can be extended to other organic and inorganic nanoparticles to predict their assemblies with biomolecules and other chemical structures forming lock-and-key complexes.

The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals

In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of the Hausdorff dimension and the Lebesgue measure, there are aleph-two virtual random fractals with, almost surely, a Hausdorff dimension of a bivariate function of them and the expected Lebesgue measure equal to the latter one. The associated results for three other fractal dimensions are similar to the case given for the Hausdorff dimension. The problem remains unsolved in the case of non-Euclidean abstract fractal spaces.