Self-similarity of complex networks

A simplex path integral and a simplex renormalization group for high-order interactions. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2024;87:087601. [PMID: 39077989 DOI: 10.1088/1361-6633/ad5c99] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

Modern theories of phase transitions and scale invariance are rooted in path integral formulation and renormalization groups (RGs). Despite the applicability of these approaches in simple systems with only pairwise interactions, they are less effective in complex systems with undecomposable high-order interactions (i.e. interactions among arbitrary sets of units). To precisely characterize the universality of high-order interacting systems, we propose a simplex path integral and a simplex RG (SRG) as the generalizations of classic approaches to arbitrary high-order and heterogeneous interactions. We first formalize the trajectories of units governed by high-order interactions to define path integrals on corresponding simplices based on a high-order propagator. Then, we develop a method to integrate out short-range high-order interactions in the momentum space, accompanied by a coarse graining procedure functioning on the simplex structure generated by high-order interactions. The proposed SRG, equipped with a divide-and-conquer framework, can deal with the absence of ergodicity arising from the sparse distribution of high-order interactions and can renormalize a system with intertwined high-order interactions at thep-order according to its properties at theq-order (p⩽q). The associated scaling relation and its corollaries provide support to differentiate among scale-invariant, weakly scale-invariant, and scale-dependent systems across different orders. We validate our theory in multi-order scale-invariance verification, topological invariance discovery, organizational structure identification, and information bottleneck analysis. These experiments demonstrate the capability of our theory to identify intrinsic statistical and topological properties of high-order interacting systems during system reduction.

Unraveling the mesoscale organization induced by network-driven processes

Complex systems are characterized by emergent patterns created by the nontrivial interplay between dynamical processes and the networks of interactions on which these processes unfold. Topological or dynamical descriptors alone are not enough to fully embrace this interplay in all its complexity, and many times one has to resort to dynamics-specific approaches that limit a comprehension of general principles. To address this challenge, we employ a metric-that we name Jacobian distance-which captures the spatiotemporal spreading of perturbations, enabling us to uncover the latent geometry inherent in network-driven processes. We compute the Jacobian distance for a broad set of nonlinear dynamical models on synthetic and real-world networks of high interest for applications from biological to ecological and social contexts. We show, analytically and computationally, that the process-driven latent geometry of a complex network is sensitive to both the specific features of the dynamics and the topological properties of the network. This translates into potential mismatches between the functional and the topological mesoscale organization, which we explain by means of the spectrum of the Jacobian matrix. Finally, we demonstrate that the Jacobian distance offers a clear advantage with respect to traditional methods when studying human brain networks. In particular, we show that it outperforms classical network communication models in explaining functional communities from structural data, therefore highlighting its potential in linking structure and function in the brain.

Direct computations of viscoelastic moduli of biomolecular condensates

In vitro facsimiles of biomolecular condensates are formed by different types of intrinsically disordered proteins including prion-like low complexity domains (PLCDs). PLCD condensates are viscoelastic materials defined by time-dependent, sequence-specific complex shear moduli. Here, we show that viscoelastic moduli can be computed directly using a generalization of the Rouse model and information regarding intra- and inter-chain contacts that is extracted from equilibrium configurations of lattice-based Metropolis Monte Carlo (MMC) simulations. The key ingredient of the generalized Rouse model is the Zimm matrix that we compute from equilibrium MMC simulations. We compute two flavors of Zimm matrices, one referred to as the single-chain model that accounts only for intra-chain contacts, and the other referred to as a collective model, that accounts for inter-chain interactions. The single-chain model systematically overestimates the storage and loss moduli, whereas the collective model reproduces the measured moduli with greater fidelity. However, in the long time, low-frequency domain, a mixture of the two models proves to be most accurate. In line with the theory of Rouse, we find that a continuous distribution of relaxation times exists in condensates. The single crossover frequency between dominantly elastic versus dominantly viscous behaviors is influenced by the totality of the relaxation modes. Hence, our analysis suggests that viscoelastic fluid-like condensates are best described as generalized Maxwell fluids. Finally, we show that the complex shear moduli can be used to solve an inverse problem to obtain distributions of relaxation times that underlie the dynamics within condensates.

Network Spreading from Network Dimension

Continuous-state network spreading models provide critical numerical and analytic insights into transmission processes in epidemiology, rumor propagation, knowledge dissemination, and many other areas. Most of these models reflect only local features such as adjacency, degree, and transitivity, so can exhibit substantial error in the presence of global correlations typical of empirical networks. Here, we propose mitigating this limitation via a network property ideally suited to capturing spreading. This is the network correlation dimension, which characterizes how the number of nodes within range of a source typically scales with distance. Applying the approach to susceptible-infected-recovered processes leads to a spreading model which, for a wide range of networks and epidemic parameters, can provide more accurate predictions of the early stages of a spreading process than important established models of substantially higher complexity. In addition, the proposed model leads to a basic reproduction number that provides information about the final state not available from popular established models.

Research on proactive defense and dynamic repair of complex networks considering cascading effects

Cascading effects can result in the nonlinear propagation of failures in complex networks, ultimately leading to network collapse. Research on the fault propagation principles, defense strategies, and repair strategies can help mitigate the effects of cascading failures. Especially, proactive defense and dynamic repair are flexible and effective methods to ensure network security. Most studies on the cascade of complex networks are based on the unprocessed initial information of the network. However, marginal nodes are a type of node that cloaks the initial information of the network. In this study, we rank the importance of nodes according to the intensity of network energy confusion after the removal of this node, clarify the meaning of marginal nodes and proposed two methods to screen marginal nodes. The results indicated that the proactive removal of marginal nodes can effectively reduce the effect of cascading failures without causing any negative disturbance to the energy flow of the network. In addition, network repair according to the proposed strategy can minimize the cascade effect in the repair process and improve repair efficiency.

Scaling theory of fractal complex networks

We show that fractality in complex networks arises from the geometric self-similarity of their built-in hierarchical community-like structure, which is mathematically described by the scale-invariant equation for the masses of the boxes with which we cover the network when determining its box dimension. This approach-grounded in both scaling theory of phase transitions and renormalization group theory-leads to the consistent scaling theory of fractal complex networks, which complements the collection of scaling exponents with several new ones and reveals various relationships between them. We propose the introduction of two classes of exponents: microscopic and macroscopic, characterizing the local structure of fractal complex networks and their global properties, respectively. Interestingly, exponents from both classes are related to each other and only a few of them (three out of seven) are independent, thus bridging the local self-similarity and global scale-invariance in fractal networks. We successfully verify our findings in real networks situated in various fields (information-the World Wide Web, biological-the human brain, and social-scientific collaboration networks) and in several fractal network models.

Fractal networks: Topology, dimension, and complexity

Over the past two decades, the study of self-similarity and fractality in discrete structures, particularly complex networks, has gained momentum. This surge of interest is fueled by the theoretical developments within the theory of complex networks and the practical demands of real-world applications. Nonetheless, translating the principles of fractal geometry from the domain of general topology, dealing with continuous or infinite objects, to finite structures in a mathematically rigorous way poses a formidable challenge. In this paper, we overview such a theory that allows to identify and analyze fractal networks through the innate methodologies of graph theory and combinatorics. It establishes the direct graph-theoretical analogs of topological (Lebesgue) and fractal (Hausdorff) dimensions in a way that naturally links them to combinatorial parameters that have been studied within the realm of graph theory for decades. This allows to demonstrate that the self-similarity in networks is defined by the patterns of intersection among densely connected network communities. Moreover, the theory bridges discrete and continuous definitions by demonstrating how the combinatorial characterization of Lebesgue dimension via graph representation by its subsets (subgraphs/communities) extends to general topological spaces. Using this framework, we rigorously define fractal networks and connect their properties with established combinatorial concepts, such as graph colorings and descriptive complexity. The theoretical framework surveyed here sets a foundation for applications to real-life networks and future studies of fractal characteristics of complex networks using combinatorial methods and algorithms.

Fractal and first-passage properties of a class of self-similar networks

A class of self-similar networks, obtained by recursively replacing each edge of the current network with a well-designed structure (generator) and known as edge-iteration networks, has garnered considerable attention owing to its role in presenting rich network models to mimic real objects with self-similar structures. The generator dominates the structural and dynamic properties of edge-iteration networks. However, the general relationships between these networks' structural and dynamic properties and their generators remain unclear. We study the fractal and first-passage properties, such as the fractal dimension, walk dimension, resistance exponent, spectral dimension, and global mean first-passage time, which is the mean time for a walker, starting from a randomly selected node and reaching the fixed target node for the first time. We disclose the properties of the generators that dominate the fractal and first-passage properties of general edge-iteration networks. A clear relationship between the fractal and first-passage properties of the edge-iteration networks and the related properties of the generators are presented. The upper and lower bounds of these quantities are also discussed. Thus, networks can be customized to meet the requirements of fractal and dynamic properties by selecting an appropriate generator and tuning their structural parameters. The results obtained here shed light on the design and optimization of network structures.

Chimera-like states in neural networks and power systems

Partial, frustrated synchronization, and chimera-like states are expected to occur in Kuramoto-like models if the spectral dimension of the underlying graph is low: ds<4. We provide numerical evidence that this really happens in the case of the high-voltage power grid of Europe (ds<2), a large human connectome (KKI113) and in the case of the largest, exactly known brain network corresponding to the fruit-fly (FF) connectome (ds<4), even though their graph dimensions are much higher, i.e., dgEU≃2.6(1) and dgFF≃5.4(1), dgKKI113≃3.4(1). We provide local synchronization results of the first- and second-order (Shinomoto) Kuramoto models by numerical solutions on the FF and the European power-grid graphs, respectively, and show the emergence of chimera-like patterns on the graph community level as well as by the local order parameters.

The Risk Function of Breast and Ovarian Cancers in the Avrami-Dobrzyński Cellular Phase-Transition Model

Specifying the role of genetic mutations in cancer development is crucial for effective screening or targeted treatments for people with hereditary cancer predispositions. Our goal here is to find the relationship between a number of cancerogenic mutations and the probability of cancer induction over the lifetime of cancer patients. We believe that the Avrami-Dobrzyński biophysical model can be used to describe this mechanism. Therefore, clinical data from breast and ovarian cancer patients were used to validate this model of cancer induction, which is based on a purely physical concept of the phase-transition process with an analogy to the neoplastic transformation. The obtained values of model parameters established using clinical data confirm the hypothesis that the carcinogenic process strongly follows fractal dynamics. We found that the model's theoretical prediction and population clinical data slightly differed for patients with the age below 30 years old, and that might point to the existence of an ancillary protection mechanism against cancer development. Additionally, we reveal that the existing clinical data predict breast or ovarian cancers onset two years earlier for patients with BRCA1/2 mutations.

A Self-Similarity Logic May Shape the Organization of the Nervous System

From the morphological point of view, the nervous system exhibits a fractal, self-similar geometry at various levels of observations, from single cells up to cell networks. From the functional point of view, it is characterized by a hierarchical organization in which self-similar structures (networks) of different miniaturizations are nested within each other. In particular, neuronal networks, interconnected to form neuronal systems, are formed by neurons, which operate thanks to their molecular networks, mainly having proteins as components that via protein-protein interactions can be assembled in multimeric complexes working as micro-devices. On this basis, the term "self-similarity logic" was introduced to describe a nested organization where, at the various levels, almost the same rules (logic) to perform operations are used. Self-similarity and self-similarity logic both appear to be intimately linked to the biophysical evidence for the nervous system being a pattern-forming system that can flexibly switch from one coherent state to another. Thus, they can represent the key concepts to describe its complexity and its concerted, holistic behavior.

Homological landscape of human brain functional sub-circuits

Human whole-brain functional connectivity networks have been shown to exhibit both local/quasilocal (e.g., set of functional sub-circuits induced by node or edge attributes) and non-local (e.g., higher-order functional coordination patterns) properties. Nonetheless, the non-local properties of topological strata induced by local/quasilocal functional sub-circuits have yet to be addressed. To that end, we proposed a homological formalism that enables the quantification of higher-order characteristics of human brain functional sub-circuits. Our results indicated that each homological order uniquely unravels diverse, complementary properties of human brain functional sub-circuits. Noticeably, the H 1 homological distance between rest and motor task were observed at both whole-brain and sub-circuit consolidated level which suggested the self-similarity property of human brain functional connectivity unraveled by homological kernel. Furthermore, at the whole-brain level, the rest-task differentiation was found to be most prominent between rest and different tasks at different homological orders: i) Emotion task H 0 , ii) Motor task H 1 , and iii) Working memory task H 2 . At the functional sub-circuit level, the rest-task functional dichotomy of default mode network is found to be mostly prominent at the first and second homological scaffolds. Also at such scale, we found that the limbic network plays a significant role in homological reconfiguration across both task- and subject- domain which sheds light to subsequent Investigations on the complex neuro-physiological role of such network. From a wider perspective, our formalism can be applied, beyond brain connectomics, to study non-localized coordination patterns of localized structures stretching across complex network fibers.

The nature and nurture of network evolution

Although the origin of the fat-tail characteristic of the degree distribution in complex networks has been extensively researched, the underlying cause of the degree distribution characteristic across the complete range of degrees remains obscure. Here, we propose an evolution model that incorporates only two factors: the node's weight, reflecting its innate attractiveness (nature), and the node's degree, reflecting the external influences (nurture). The proposed model provides a good fit for degree distributions and degree ratio distributions of numerous real-world networks and reproduces their evolution processes. Our results indicate that the nurture factor plays a dominant role in the evolution of social networks. In contrast, the nature factor plays a dominant role in the evolution of non-social networks, suggesting that whether nodes are people determines the dominant factor influencing the evolution of real-world networks.

Hierarchical network structure as the source of hierarchical dynamics (power-law frequency spectra) in living and non-living systems: How state-trait continua (body plans, personalities) emerge from first principles in biophysics

Living systems are hierarchical control systems that display a small world network structure. In such structures, many smaller clusters are nested within fewer larger ones, producing a fractal-like structure with a 'power-law' cluster size distribution (a mereology). Just like their structure, the dynamics of living systems shows fractal-like qualities: the timeseries of inner message passing and overt behavior contain high frequencies or 'states' (treble) that are nested within lower frequencies or 'traits' (bass), producing a power-law frequency spectrum that is known as a 'state-trait continuum' in the behavioral sciences. Here, we argue that the power-law dynamics of living systems results from their power-law network structure: organisms 'vertically encode' the deep spatiotemporal structure of their (anticipated) environments, to the effect that many small clusters near the base of the hierarchy produce high frequency signal changes and fewer larger clusters at its top produce ultra-low frequencies. Such ultra-low frequencies exert a tonic regulatory pressure that produces morphological as well as behavioral traits (i.e., body plans and personalities). Nested-modular structure causes higher frequencies to be embedded within lower frequencies, producing a power-law state-trait continuum. At the heart of such dynamics lies the need for efficient energy dissipation through networks of coupled oscillators, which also governs the dynamics of non-living systems (e.q., earthquakes, stock market fluctuations). Since hierarchical structure produces hierarchical dynamics, the development and collapse of hierarchical structure (e.g., during maturation and disease) should leave specific traces in system dynamics (shifts in lower frequencies, i.e. morphological and behavioral traits) that may serve as early warning signs to system failure. The applications of this idea range from (bio)physics and phylogenesis to ontogenesis and clinical medicine.

Multifractality of Complex Networks Is Also Due to Geometry: A Geometric Sandbox Algorithm

Over the past three decades, describing the reality surrounding us using the language of complex networks has become very useful and therefore popular. One of the most important features, especially of real networks, is their complexity, which often manifests itself in a fractal or even multifractal structure. As a generalization of fractal analysis, the multifractal analysis of complex networks is a useful tool for identifying and quantitatively describing the spatial hierarchy of both theoretical and numerical fractal patterns. Nowadays, there are many methods of multifractal analysis. However, all these methods take into account only the fact of connection between nodes (and eventually the weight of edges) and do not take into account the real positions (coordinates) of nodes in space. However, intuition suggests that the geometry of network nodes' position should have a significant impact on the true fractal structure. Many networks identified in nature (e.g., air connection networks, energy networks, social networks, mountain ridge networks, networks of neurones in the brain, and street networks) have their own often unique and characteristic geometry, which is not taken into account in the identification process of multifractality in commonly used methods. In this paper, we propose a multifractal network analysis method that takes into account both connections between nodes and the location coordinates of nodes (network geometry). We show the results for different geometrical variants of the same network and reveal that this method, contrary to the commonly used method, is sensitive to changes in network geometry. We also carry out tests for synthetic as well as real-world networks.

Scale-free dynamics of COVID-19 in a Brazilian city

A common basis to address the dynamics of directly transmitted infectious diseases, such as COVID-19, are compartmental (or SIR) models. SIR models typically assume homogenous population mixing, a simplification that is convenient but unrealistic. Here we validate an existing model of a scale-free fractal infection process using high-resolution data on COVID-19 spread in São Caetano, Brazil. We find that transmission can be described by a network in which each infectious individual has a small number of susceptible contacts, of the order of 2-5. This model parameter correlated tightly with physical distancing measured by mobile phone data, such that in periods of greater distancing the model recovered a lower average number of contacts, and vice versa. We show that the SIR model is a special case of our scale-free fractal process model in which the parameter that reflects population structure is set at unity, indicating homogeneous mixing. Our more general framework better explained the dynamics of COVID-19 in São Caetano, used fewer parameters than a standard SIR model and accounted for geographically localized clusters of disease. Our model requires further validation in other locations and with other directly transmitted infectious agents.

Structure-based prediction of nucleic acid binding residues by merging deep learning- and template-based approaches

Accurate prediction of nucleic binding residues is essential for the understanding of transcription and translation processes. Integration of feature- and template-based strategies could improve the prediction of these key residues in proteins. Nevertheless, traditional hybrid algorithms have been surpassed by recently developed deep learning-based methods, and the possibility of integrating deep learning- and template-based approaches to improve performance remains to be explored. To address these issues, we developed a novel structure-based integrative algorithm called NABind that can accurately predict DNA- and RNA-binding residues. A deep learning module was built based on the diversified sequence and structural descriptors and edge aggregated graph attention networks, while a template module was constructed by transforming the alignments between the query and its multiple templates into features for supervised learning. Furthermore, the stacking strategy was adopted to integrate the above two modules for improving prediction performance. Finally, a post-processing module dependent on the random walk algorithm was proposed to further correct the integrative predictions. Extensive evaluations indicated that our approach could not only achieve excellent performance on both native and predicted structures but also outperformed existing hybrid algorithms and recent deep learning methods. The NABind server is available at http://liulab.hzau.edu.cn/NABind/.

Fractal Design of Hierarchical PtPd with Enhanced Exposed Surface Atoms for Highly Catalytic Activity and Stability

Hierarchical assembly of arc-like fractal nanostructures not only has its unique self-similarity feature for stability enhancement but also possesses the structural advantages of highly exposed surface-active sites for activity enhancement, remaining a great challenge for high-performance metallic nanocatalyst design. Herein, we report a facile strategy to synthesize a novel arc-like hierarchical fractal structure of PtPd bimetallic nanoparticles (h-PtPd) by using pyridinium-type ionic liquids as the structure-directing agent. Growth mechanisms of the arc-like nanostructured PtPd nanoparticles have been fully studied, and precise control of the particle sizes and pore sizes has been achieved. Due to the structural features, such as size control by self-similarity growth of subunits, structural stability by nanofusion of subunits, and increased numbers of exposed active atoms by the curved homoepitaxial growth, h-PtPd displays outstanding electrocatalytic activity toward oxygen reduction reaction and excellent stability during hydrothermal treatment and catalytic process.

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.

Emergent information dynamics in many-body interconnected systems

The information implicitly represented in the state of physical systems allows for their analysis using analytical techniques from statistical mechanics and information theory. This approach has been successfully applied to complex networks, including biophysical systems such as virus-host protein-protein interactions and whole-brain models in health and disease, drawing inspiration from quantum statistical physics. Here we propose a general mathematical framework for modeling information dynamics on complex networks, where the internal node states are vector valued, allowing each node to carry multiple types of information. This setup is relevant for various biophysical and sociotechnological models of complex systems, ranging from viral dynamics on networks to models of opinion dynamics and social contagion. Instead of focusing on node-node interactions, we shift our attention to the flow of information between network configurations. We uncover fundamental differences between widely used spin models on networks, such as voter and kinetic dynamics, which cannot be detected through classical node-based analysis. We illustrate the mathematical framework further through an exemplary application to epidemic spreading on a low-dimensional network. Our model provides an opportunity to adapt powerful analytical methods from quantum many-body systems to study the interplay between structure and dynamics in interconnected systems.

Chaotic renormalization group flow and entropy gradients over Haros graphs

Haros graphs have been recently introduced as a set of graphs bijectively related to real numbers in the unit interval. Here we consider the iterated dynamics of a graph operator R over the set of Haros graphs. This operator was previously defined in the realm of graph-theoretical characterization of low-dimensional nonlinear dynamics and has a renormalization group (RG) structure. We find that the dynamics of R over Haros graphs is complex and includes unstable periodic orbits of arbitrary period and nonmixing aperiodic orbits, overall portraiting a chaotic RG flow. We identify a single RG stable fixed point whose basin of attraction is associated with the set of rational numbers, and find periodic RG orbits that relate to (pure) quadratic irrationals and aperiodic RG orbits, related with (nonmixing) families of nonquadratic algebraic irrationals and transcendental numbers. Finally, we show that the graph entropy of Haros graphs is globally decreasing as the RG flows towards its stable fixed point, albeit in a strictly nonmonotonic way, and that such graph entropy remains constant inside the periodic RG orbit associated to a subset of irrationals, the so-called metallic ratios. We discuss the possible physical interpretation of such chaotic RG flow and put results regarding entropy gradients along RG flow in the context of c-theorems.

Predictive algorithm for the regional spread of coronavirus infection across the Russian Federation

BACKGROUND

Outbreaks of infectious diseases are a complex phenomenon with many interacting factors. Regional health authorities need prognostic modeling of the epidemic process.

METHODS

For these purposes, various mathematical algorithms can be used, which are a useful tool for studying the infections spread dynamics. Epidemiological models act as evaluation and prognosis models. The authors outlined the experience of developing a short-term predictive algorithm for the spread of the COVID-19 in the region of the Russian Federation based on the SIR model: Susceptible (vulnerable), Infected (infected), Recovered (recovered). The article describes in detail the methodology of a short-term predictive algorithm, including an assessment of the possibility of building a predictive model and the mathematical aspects of creating such forecast algorithms.

RESULTS

Findings show that the predicted results (the mean square of the relative error of the number of infected and those who had recovered) were in agreement with the real-life situation: σ(I) = 0.0129 and σ(R) = 0.0058, respectively.

CONCLUSIONS

The present study shows that despite a large number of sophisticated modifications, each of which finds its scope, it is advisable to use a simple SIR model to quickly predict the spread of coronavirus infection. Its lower accuracy is fully compensated by the adaptive calibration of parameters based on monitoring the current situation with updating indicators in real-time.

Correlation dimension in empirical networks

Network correlation dimension governs the distribution of network distance in terms of a power-law model and profoundly impacts both structural properties and dynamical processes. We develop new maximum likelihood methods which allow us robustly and objectively to identify network correlation dimension and a bounded interval of distances over which the model faithfully represents structure. We also compare the traditional practice of estimating correlation dimension by modeling as a power law the fraction of nodes within a distance to a proposed alternative of modeling as a power law the fraction of nodes at a distance. In addition, we illustrate a likelihood ratio technique for comparing the correlation dimension and small-world descriptions of network structure. Improvements from our innovations are demonstrated on a diverse selection of synthetic and empirical networks. We show that the network correlation dimension model accurately captures empirical network structure over neighborhoods of substantial size and span and outperforms the alternative small-world network scaling model. Our improved methods tend to lead to higher estimates of network correlation dimension, implying that prior studies could have produced or utilized systematic underestimates of dimension.

Identification of Relevant Protein Interactions with Partial Knowledge: A Complex Network and Deep Learning Approach

Protein-protein interactions (PPIs) are the basis for understanding most cellular events in biological systems. Several experimental methods, e.g., biochemical, molecular, and genetic methods, have been used to identify protein-protein associations. However, some of them, such as mass spectrometry, are time-consuming and expensive. Machine learning (ML) techniques have been widely used to characterize PPIs, increasing the number of proteins analyzed simultaneously and optimizing time and resources for identifying and predicting protein-protein functional linkages. Previous ML approaches have focused on well-known networks or specific targets but not on identifying relevant proteins with partial or null knowledge of the interaction networks. The proposed approach aims to generate a relevant protein sequence based on bidirectional Long-Short Term Memory (LSTM) with partial knowledge of interactions. The general framework comprises conducting a scale-free and fractal complex network analysis. The outcome of these analyses is then used to fine-tune the fractal method for the vital protein extraction of PPI networks. The results show that several PPI networks are self-similar or fractal, but that both features cannot coexist. The generated protein sequences (by the bidirectional LSTM) also contain an average of 39.5% of proteins in the original sequence. The average length of the generated sequences was 17% of the original one. Finally, 95% of the generated sequences were true.

Hub-collision avoidance and leaf-node options algorithm for fractal dimension and renormalization of complex networks

The box-covering method plays a fundamental role in the fractal property recognition and renormalization analysis of complex networks. This study proposes the hub-collision avoidance and leaf-node options (HALO) algorithm. In the box sampling process, a forward sampling rule (for avoiding hub collisions) and a reverse sampling rule (for preferentially selecting leaf nodes) are determined for bidirectional network traversal to reduce the randomness of sampling. In the box selection process, the larger necessary boxes are preferentially selected to join the solution by continuously removing small boxes. The compact-box-burning (CBB) algorithm, the maximum-excluded-mass-burning (MEMB) algorithm, the overlapping-box-covering (OBCA) algorithm, and the algorithm for combining small-box-removal strategy and maximum box sampling with a sampling density of 30 (SM30) are compared with HALO in experiments. Results on nine real networks show that HALO achieves the highest performance score and obtains 11.40%, 7.67%, 2.18%, and 8.19% fewer boxes than the compared algorithms, respectively. The algorithm determinism is significantly improved. The fractal dimensions estimated by covering four standard networks are more accurate. Moreover, different from MEMB or OBCA, HALO is not affected by the tightness of the hubs and exhibits a stable performance in different networks. Finally, the time complexities of HALO and the compared algorithms are all O(N²), which is reasonable and acceptable.

The effect of self-organizing map architecture based on the value migration network centrality measures on stock return. Evidence from the US market

Complex financial systems are the subject of current research interest. The notion of complex network is used for understanding the value migration process. Based on the stock data of 498 companies listed in the S&P500, the value migration network has been constructed using the MST-Pathfinder filtering network approach. The analysis covered 471 companies included in the largest component of VMN. Three methods: (i) complex networks; (ii) artificial neural networks and (iii) MARS regression, are developed to determine the effect of network centrality measures and rate of return on shares. A network-based data mining analysis has revealed that the topological position in the value migration network has a pronounced impact on the stock's returns.

The Analysis of the Power Law Feature in Complex Networks

Consensus about the universality of the power law feature in complex networks is experiencing widespread challenges. In this paper, we propose a generic theoretical framework in order to examine the power law property. First, we study a class of birth-and-death networks that are more common than BA networks in the real world, and then we calculate their degree distributions; the results show that the tails of their degree distributions exhibit a distinct power law feature. Second, we suggest that in the real world two important factors-network size and node disappearance probability-will affect the analysis of power law characteristics in observation networks. Finally, we suggest that an effective way of detecting the power law property is to observe the asymptotic (limiting) behavior of the degree distribution within its effective intervals.

Detecting the ultra low dimensionality of real networks

Reducing dimension redundancy to find simplifying patterns in high-dimensional datasets and complex networks has become a major endeavor in many scientific fields. However, detecting the dimensionality of their latent space is challenging but necessary to generate efficient embeddings to be used in a multitude of downstream tasks. Here, we propose a method to infer the dimensionality of networks without the need for any a priori spatial embedding. Due to the ability of hyperbolic geometry to capture the complex connectivity of real networks, we detect ultra low dimensionality far below values reported using other approaches. We applied our method to real networks from different domains and found unexpected regularities, including: tissue-specific biomolecular networks being extremely low dimensional; brain connectomes being close to the three dimensions of their anatomical embedding; and social networks and the Internet requiring slightly higher dimensionality. Beyond paving the way towards an ultra efficient dimensional reduction, our findings help address fundamental issues that hinge on dimensionality, such as universality in critical behavior.

An empirical study on network conversion of stock time series based on STL method

A complex network has been widely used to reveal the rule of a complex system. How to convert the stock data into a network is an open issue since the stock data are so large and their random volatility is strong. In this paper, a seasonal trend decomposition procedure based on the loess ( S T L) method is applied to convert the stock time series into a directed and weighted symbolic network. Three empirical stock datasets, including the closing price of Shanghai Securities Composite Index, S&P 500 Index, and Nikkei 225 Index, are considered. The properties of these stock time series are revealed from the topological characteristics of corresponding symbolic networks. The results show that: (1) both the weighted indegree and outdegree distributions obey the power-law distribution well; (2) fluctuations of stock closing price are revealed by related network topological properties, such as weighting degree, betweenness, pageranks, and clustering coefficient; and (3) stock closing price, in particular, periods such as financial crises, can be identified by modularity class of the symbolic networks. Moreover, the comparison between the S T L method and the visibility graph further highlights the advantages of the S T L method in terms of the time complexity of the algorithm. Our method offers a new idea to study the network conversion of stock time series.

Performance Optimization of Surface Plasmon Resonance Imaging Sensor Network Based on the Multi-Objective Optimization Algorithm

In this work, we report performance optimization of a wireless sensor network (WSN) based on the plain silver surface plasmon resonance imaging (SPRi) sensor. At the sensor node level, we established the refractive index-thickness models for both gold and silver in the sensor and calculated the depth-width ratio (DWR) and penetration depth (PD) values of the sensor of different gold and silver thicknesses by the Jones transfer matrix and Kriging interpolation. We optimized the DWR and PD simultaneously by using the multi-objective optimization genetic algorithm (MOGA). In the following performance optimization of WSN, we simultaneously optimized the transmission success rate and information dimension with the number of nodes and transmission failure rate of the sensor node as variables by the same algorithm. By calculating the information dimension and the transmission success rate of each Pareto optimal solution, we obtained the number of nodes and transmission failure probability of the node available for practical deployment of WSN. The above results indicate that the Pareto optimal solution set obtained from MOGA can help to provide the best solution for the optimization of some certain performance parameters and also assist us in making the trade-off decision in the structure design and network deployment if optimal values of all the performance parameters can be obtained simultaneously.

Introduction to the Class of Prefractal Graphs

Fractals are already firmly rooted in modern science. Research continues on the fractal properties of objects in physics, chemistry, biology and many other scientific fields. Fractal graphs as a discrete representation are used to model and describe the structure of various objects and processes, both natural and artificial. The paper proposes an introduction to prefractal graphs. The main definitions and notation are proposed—the concept of a seed, the operations of processing a seed, the procedure for generating a prefractal graph. Canonical (typical) and non-canonical (special) types of prefractal graphs are considered separately. Important characteristics are proposed and described—the preservation of adjacency of edges for different ranks in the trajectory. The definition of subgraph-seeds of different ranks is given separately. Rules for weighting a prefractal graph by natural numbers and intervals are proposed. Separately, the definition of a fractal graph as infinite is given, and the differences between the concepts of fractal and prefractal graphs are described. At the end of the work, already published works of the authors are proposed, indicating the main backlogs, as well as a list of directions for new research. This work is the beginning of a cycle of works on the study of the properties and characteristics of fractal and prefractal graphs.

Stochastic growth tree networks with an identical fractal dimension: Construction and mean hitting time for random walks

There is little attention paid to stochastic tree networks in comparison with the corresponding deterministic analogs in the current study of fractal trees. In this paper, we propose a principled framework for producing a family of stochastic growth tree networks _T possessing fractal characteristic, where t represents the time step and parameter m is the number of vertices newly created for each existing vertex at generation. To this end, we introduce two types of generative ways, i.e., Edge-Operation and Edge-Vertex-Operation. More interestingly, the resulting stochastic trees turn out to have an identical fractal dimension _d = ln ⁡ 2 ( m + 1 ) / ln ⁡ 2 regardless of the introduction of randomness in the growth process. At the same time, we also study many other structural parameters including diameter and degree distribution. In both extreme cases, our tree networks are deterministic and follow multiple-point degree distribution and power-law degree distribution, respectively. Additionally, we consider random walks on stochastic growth tree networks _T and derive an expectation estimation for mean hitting time ⟨ _H ⟩ in an effective combinatorial manner instead of commonly used spectral methods. The result shows that on average, the scaling of mean hitting time ⟨ _H ⟩ obeys ⟨ _H ⟩ = | _T ^|, where | _T | represents vertex number and exponent λ is equivalent to 1 + ln ⁡ 2 / ln ⁡ 2 ( m + 1 ). In the meantime, we conduct extensive experimental simulations and observe that empirical analysis is in strong agreement with theoretical results.

A New Method of Quantifying the Complexity of Fractal Networks

There is a large body of research devoted to identifying the complexity of structures in networks. In the context of network theory, a complex network is a graph with nontrivial topological features—features that do not occur in simple networks, such as lattices or random graphs, but often occur in graphs modeling real systems. The study of complex networks is a young and active area of scientific research inspired largely by the empirical study of real-world networks, such as computer networks and logistic transport networks. Transport is of great importance for the economic and cultural cooperation of any country with other countries, the strengthening and development of the economic management system, and in solving social and economic problems. Provision of the territory with a well-developed transport system is one of the factors for attracting population and production, serving as an important advantage for locating productive forces and providing an integration effect. In this paper, we introduce a new method for quantifying the complexity of a network based on presenting the nodes of the network in Cartesian coordinates, converting to polar coordinates, and calculating the fractal dimension using the ReScaled ranged (R/S) method. Our results suggest that this approach can be used to determine complexity for any type of network that has fixed nodes, and it presents an application of this method in the public transport system.

Frequency-dependent transition in power-law rheological behavior of living cells

Living cells are active viscoelastic materials exhibiting diverse mechanical behaviors at different time scales. However, dynamical rheological characteristics of cells in frequency range spanning many orders of magnitude, especially in high frequencies, remain poorly understood. Here, we show that a self-similar hierarchical model can capture cell's power-law rheological characteristics in different frequency scales. In low-frequency scales, the storage and loss moduli exhibit a weak power-law dependence on frequency with same exponent. In high-frequency scales, the storage modulus becomes a constant, while the loss modulus shows a power-law dependence on frequency with an exponent of 1.0. The transition between low- and high-frequency scales is defined by a transition frequency based on cell's mechanical parameters. The cytoskeletal differences of different cell types or states can be characterized by changes in mechanical parameters in the model. This study provides valuable insights into potentially using mechanics-based markers for cell classification and cancer diagnosis.

Efficacy and self-similarity of SARS-CoV-2 thermal decontamination

Dry heat decontamination has been shown to effectively inactivate viruses without compromising the integrity of delicate personal protective equipment (PPE), allowing safe reuse and helping to alleviate shortages of PPE that have arisen due to COVID-19. Unfortunately, current thermal decontamination guidelines rely on empirical data which are often sparse, limited to a specific virus, and unable to provide fundamental insight into the underlying inactivation reaction. In this work, we experimentally quantified dry heat decontamination of SARS-CoV-2 on disposable masks and validated a model that treats the inactivation reaction as thermal degradation of macromolecules. Furthermore, upon nondimensionalization, all of the experimental data collapse onto a unified curve, revealing that the thermally driven decontamination process exhibits self-similar behavior. Our results show that heating surgical masks to 70 °C for 5 min inactivates over 99.9% of SARS-CoV-2. We also characterized the chemical and physical properties of disposable masks after heat treatment and did not observe degradation. The model presented in this work enables extrapolation of results beyond specific temperatures to provide guidelines for safe PPE decontamination. The modeling framework and self-similar behavior are expected to extend to most viruses-including yet-unencountered novel viruses-while accounting for a range of environmental conditions.

Mean trapping time for an arbitrary trap site on a class of fractal scale-free trees

Fractals are ubiquitous in nature and random walks on fractals have attracted lots of scientific attention in the past several years. In this work, we consider discrete random walks on a class of fractal scale-free trees (FST), whose topologies are controlled by two integer parameters (i.e., u≥2 and v≥1) and exhibit a wide range of topological properties by suitably varying the parameters u and v. The mean trapping time (MTT), referred to as T_{y}, which is the mean time it takes the walker to be absorbed by the trap fixed at site y of the FST, is addressed analytically on the FST, and the effects of the trap location y on the MTT for the FST and for the general trees are also analyzed. First, a method, which is based on the connection between the MTT and the effective resistances, to derive analytically T_{y} for an arbitrary site y of the FST is presented, and some examples are provided to show the effectiveness of the method. Then, we compare T_{y} for all the possible site y of the trees, and find the sites where T_{y} achieves the minimum (or maximum) on the FST. Finally, we analyze the effects of trap location on the MTT in general trees and find that the average path length (APL) from an arbitrary site to the trap is the decisive factor which dominates the difference in the MTTs for different trap locations on general trees. We find, for any tree, the MTT obtains the minimum (or maximum) at sites where the APL achieves the minimum (or maximum).

Degree correlations in graphs with clique clustering

Correlations among the degrees of vertices in random graphs often occur when clustering is present. In this paper we define a joint-degree correlation function for vertices in the giant component of clustered configuration model networks which are composed of clique subgraphs. We use this model to investigate, in detail, the organization among nearest-neighbor subgraphs for random graphs as a function of subgraph topology as well as clustering. We find an expression for the average joint degree of a neighbor in the giant component at the critical point for these networks. Finally, we introduce a novel edge-disjoint clique decomposition algorithm and investigate the correlations between the subgraphs of empirical networks.

Network medicine in ovarian cancer: topological properties to drug discovery

Network medicine provides network theoretical tools, methods and properties to study underlying laws governing human interactome to identify disease states and disease complexity leading to drug discovery. Within this framework, we investigated the topological properties of ovarian cancer network (OCN) and the roles of hubs to understand OCN organization to address disease states and complexity. The OCN constructed from the experimentally verified genes exhibits fractal nature in the topological properties with deeply rooted functional communities indicating self-organizing behavior. The network properties at all levels of organization obey one parameter scaling law which lacks centrality lethality rule. We showed that $\langle k\rangle $ can be taken as a scaling parameter, where, power law exponent can be estimated from the ratio of network diameters. The betweenness centrality $C_B$ shows two distinct behaviors one shown by high degree hubs and the other by segregated low degree nodes. The $C_B$ power law exponent is found to connect the exponents of distributions of high and low degree nodes. OCN showed the absence of rich-club formation which leads to the missing of a number of attractors in the network causing formation of weakly tied diverse functional modules to keep optimal network efficiency. In OCN, provincial and connector hubs, which includes identified key regulators, take major responsibility to keep the OCN integrity and organization. Further, most of the key regulators are found to be over expressed and positively correlated with immune infiltrates. Finally, few potential drugs are identified related to the key regulators.

A general model of hierarchical fractal scale-free networks

We propose a general model of unweighted and undirected networks having the scale-free property and fractal nature. Unlike the existing models of fractal scale-free networks (FSFNs), the present model can systematically and widely change the network structure. In this model, an FSFN is iteratively formed by replacing each edge in the previous generation network with a small graph called a generator. The choice of generators enables us to control the scale-free property, fractality, and other structural properties of hierarchical FSFNs. We calculate theoretically various characteristic quantities of networks, such as the exponent of the power-law degree distribution, fractal dimension, average clustering coefficient, global clustering coefficient, and joint probability describing the nearest-neighbor degree correlation. As an example of analyses of phenomena occurring on FSFNs, we also present the critical point and critical exponents of the bond-percolation transition on infinite FSFNs, which is related to the robustness of networks against edge removal. By comparing the percolation critical points of FSFNs whose structural properties are the same as each other except for the clustering nature, we clarify the effect of the clustering on the robustness of FSFNs. As demonstrated by this example, the present model makes it possible to elucidate how a specific structural property influences a phenomenon occurring on FSFNs by varying systematically the structures of FSFNs. Finally, we extend our model for deterministic FSFNs to a model of non-deterministic ones by introducing asymmetric generators and reexamine all characteristic quantities and the percolation problem for such non-deterministic FSFNs.

Gaining confidence in inferred networks

Network inference is a notoriously challenging problem. Inferred networks are associated with high uncertainty and likely riddled with false positive and false negative interactions. Especially for biological networks we do not have good ways of judging the performance of inference methods against real networks, and instead we often rely solely on the performance against simulated data. Gaining confidence in networks inferred from real data nevertheless thus requires establishing reliable validation methods. Here, we argue that the expectation of mixing patterns in biological networks such as gene regulatory networks offers a reasonable starting point: interactions are more likely to occur between nodes with similar biological functions. We can quantify this behaviour using the assortativity coefficient, and here we show that the resulting heuristic, functional assortativity, offers a reliable and informative route for comparing different inference algorithms.

Hurst analysis of dynamic networks

The sequence of network snapshots with time stamps is an effective tool for describing system dynamics. First, this article constructs a multifractal analysis of a snapshot network, in which the Hurst integral is used to describe the fractal structure hidden in structural dynamics. Second, we adjusted the network model and conducted comparative analysis to clarify the meaning of the Hurst exponent and found that the snapshot network usually includes multiple fractal structures, such as local and global fractal structures. Finally, we discussed the fractal structure of two real network datasets. We found that the real snapshot network also includes rich dynamics, which can be distinguished by the Hurst exponent. In particular, the dynamics of financial networks includes multifractal structures. This article provides a perspective to study the dynamic networks, thereby indirectly describing the fractal characteristics of complex system dynamics.

Predicting Essential Proteins Based on Integration of Local Fuzzy Fractal Dimension and Subcellular Location Information

Essential proteins are indispensable to cells’ survival and development. Prediction and analysis of essential proteins are crucial for uncovering the mechanisms of cells. With the help of computer science and high-throughput technologies, forecasting essential proteins by protein–protein interaction (PPI) networks has become more efficient than traditional approaches (expensive experimental methods are generally used). Many computational algorithms were employed to predict the essential proteins; however, they have various restrictions. To improve the prediction accuracy, by introducing the Local Fuzzy Fractal Dimension (LFFD) of complex networks into the analysis of the PPI network, we propose a novel algorithm named LDS, which combines the LFFD of the PPI network with the protein subcellular location information. By testing the proposed LDS algorithm on three different yeast PPI networks, the experimental results show that LDS outperforms some state-of-the-art essential protein-prediction techniques.

Non-ergodicity of a globular protein extending beyond its functional timescale

Internal motions of folded proteins have been assumed to be ergodic, i.e., that the dynamics of a single protein molecule averaged over a very long time resembles that of an ensemble. Here, by performing single-molecule fluorescence resonance energy transfer (smFRET) experiments and molecular dynamics (MD) simulations of a multi-domain globular protein, cytoplasmic protein-tyrosine phosphatase (SHP2), we demonstrate that the functional inter-domain motion is observationally non-ergodic over the time spans 10⁻¹² to 10⁻⁷ s and 10⁻¹ to 10² s. The difference between observational non-ergodicity and simple non-convergence is discussed. In comparison, a single-strand DNA of similar size behaves ergodically with an energy landscape resembling a one-dimensional linear chain. The observed non-ergodicity results from the hierarchical connectivity of the high-dimensional energy landscape of the protein molecule. As the characteristic time for the protein to conduct its dephosphorylation function is ∼10 s, our findings suggest that, due to the non-ergodicity, individual, seemingly identical protein molecules can be dynamically and functionally different.

Internal motions of folded proteins have been assumed to be ergodic, i.e., that the dynamics of a single protein molecule averaged over a very long time resembles that of an ensemble.

Motif-h: a novel functional backbone extraction for directed networks

Dense networks are very pervasive in social analytics, biometrics, communication, architecture, etc. Analyzing and visualizing such large-scale networks are significant challenges, which are generally met by reducing the redundancy on the level of nodes or edges. Motifs, patterns of the higher order organization compared with nodes and edges, are recently found to be the novel fundamental unit structures of complex networks. In this work, we proposed a novel motif h-backbone (Motif-h) method to extract functional cores of directed networks based on both motif strength and h-bridge. Compared with the state-of-the-art method Motif-DF and Entropy, our method solves two main issues which are often found in existing methods: the Motif-h reconsiders weak ties into our candidate set, and those weak ties often have critical functions of bridges in networks; moreover, our method provides a trade-off between the motif size and the edge strength, which quantifies the core edges accordingly. In the simulations, we compare our method with Motif-DF in four real-world networks and found that Motif-h can streamline the extraction of crucial structures compared with the others with limited edges.

Recent Advances in Multilayer-Structure Dielectrics for Energy Storage Application

An electrostatic capacitor has been widely used in many fields (such as high pulsed power technology, new energy vehicles, etc.) due to its ultrahigh discharge power density. Remarkable progress has been made over the past 10 years by doping ferroelectric ceramics into polymers because the dielectric constant is positively correlated with the energy storage density. However, this method often leads to an increase in dielectric loss and a decrease in energy storage efficiency. Therefore, the way of using a multilayer structure to improve the energy storage density of the dielectric has attracted the attention of researchers. Although research on energy storage properties using multilayer dielectric is just beginning, it shows the excellent effect and huge potential. In this review, the main physical mechanisms of polarization, breakdown and energy storage in multilayer structure dielectric are introduced, the theoretical simulation and experimental results are systematically summarized, and the preparation methods and design ideas of multilayer structure dielectrics are mainly described. This article covers not only an overview of the state-of-the-art advances of multilayer structure energy storage dielectric but also the prospects that may open another window to tune the electrical performance of the electrostatic capacitor via designing a multilayer structure.

Research on Collaborative Efficiency Evaluation of Complex Supplier Network under the Background of Intelligent Manufacturing

Supplier network collaborative efficiency evaluation is important content in the transformation and upgrading of intelligent manufacturing enterprises. Aiming at the shortcomings of existing methods, this paper proposes a new method to evaluate the collaborative efficiency of internal members of a complex supplier network based on complex network theory. Based on the analysis of the characteristics of the complex supplier network, from the perspective of the system, the macro supplier network is divided into multiple multi-level supplier micro subsystems with manufacturing enterprises as the core. In order to reasonably quantify the collaboration relationship of members in the subsystem structure model, the collaboration entropy is introduced as a measurement tool, and combined with the hesitation fuzzy scoring function, and the collaborative evaluation model of the complex supplier network is constructed. By quantifying the collaboration relationship among the members in the subsystem and summarizing it step by step and iteratively, the collaborative efficiency evaluation of the complex supplier network from local to overall is realized. Finally, taking a large battery manufacturing enterprise in China as an example, the proposed method is used to calculate the collaboration entropy, collaborative efficiency, and collaboration ratio of members at different supplier network levels. The results verify the effectiveness of the model.

Deciphering the generating rules and functionalities of complex networks

Network theory helps us understand, analyze, model, and design various complex systems. Complex networks encode the complex topology and structural interactions of various systems in nature. To mine the multiscale coupling, heterogeneity, and complexity of natural and technological systems, we need expressive and rigorous mathematical tools that can help us understand the growth, topology, dynamics, multiscale structures, and functionalities of complex networks and their interrelationships. Towards this end, we construct the node-based fractal dimension (NFD) and the node-based multifractal analysis (NMFA) framework to reveal the generating rules and quantify the scale-dependent topology and multifractal features of a dynamic complex network. We propose novel indicators for measuring the degree of complexity, heterogeneity, and asymmetry of network structures, as well as the structure distance between networks. This formalism provides new insights on learning the energy and phase transitions in the networked systems and can help us understand the multiple generating mechanisms governing the network evolution.