Scale-free Power-law Distribution Research Articles

A Network Clustering Algorithm for Protein Complex Detection Fused with Power-Law Distribution Characteristic

Network clustering for mining protein complexes from protein–protein interaction (PPI) networks has emerged as a prominent research area in data mining and bioinformatics. Accurately identifying complexes plays a crucial role in comprehending cellular organization and functionality. Network characteristics are often useful in enhancing the performance of protein complex detection methods. Many protein complex detection algorithms have been proposed, primarily focusing on local micro-topological structure metrics while overlooking the potential power-law distribution characteristic of community sizes at the macro global level. The effective use of this distribution characteristic information may be beneficial for mining protein complexes. This paper proposes a network clustering algorithm for protein complex detection fused with power-law distribution characteristic. The clustering algorithm constructs a cluster generation model based on scale-free power-law distribution to generate a cluster with a dense center and relatively sparse periphery. Following the cluster generation model, a candidate cluster is obtained. From a global perspective, the number distribution of clusters of varying sizes is taken into account. If the candidate cluster aligns with the constraints defined by the power-law distribution function of community sizes, it is designated as the final cluster; otherwise, it is discarded. To assess the prediction performance of the proposed algorithm, the gold standard complex sets CYC2008 and MIPS are employed as benchmarks. The algorithm is compared to DPClus, IPCA, SEGC, Core, SR-MCL, and ELF-DPC in terms of F-measure and Accuracy on several widely used protein–protein interaction networks. The experimental results show that the algorithm can effectively detect protein complexes and is superior to other comparative algorithms. This study further enriches the connection between analyzing complex network topology features and mining network function modules, thereby significantly contributing to the improvement of protein complex detection performance.

History effects in the creep of a disordered brittle material

We study the creep behavior of a disordered brittle material (concrete) under successive loading steps, using acoustic emission and ultrasonic sensing to track internal damage. The primary creep rate is observed to follow a (Omori-type) power-law decay in the strain rate, the number of acoustic emission events, as well as the amplitudes of the ultrasonic beams, supporting a brittle-creep mechanism. The distribution of acoustic emission event energies is observed to have a scale-free power-law distribution instead of a truncated one expected for a system approaching a critical point at failure. The main outcome is, however, the discovery of unexpected history effects that make the material less prone to creep when it has been previously deformed and damaged under primary creep. With the help of a progressive damage model implementing thermal activation, we interpret this as an aging-under-stress phenomenon: During an initial creep step at relatively low applied stress, the easy-to-damage sites are exhausted first, depleting the excitation spectrum at low stress gap values. Consequently on reloading, although previously damaged, the primary creep restarts but the material creeps (and damages) less than it would under the same stress but without precreeping. Besides shedding a new light on the fundamental physics of creep of disordered brittle materials, this has important practical consequences in the interpretation of some experimental procedures, such as stress-stepping experiments.

The Role of Local Influential Users in Spread of Situational Crisis Information

Abstract Extensive spread of situational information is important for communities in response to crises/disasters. Among various mechanisms affecting the spread of information on social media, influential users play a critical role in enhancing information spread. This study examines the attributes and activities of local influential users as well as their interactions with ordinary users on Twitter during 2017’s Hurricane Harvey. The results show that the influence across local influential users has a scale-free power law distribution and also indicates a major limitation in spreading information caused by insufficient interaction among influential users themselves. The findings suggest that influential users should play a boundary-spanning and brokerage role in addition to their information hub role in order to be more effective in enhancing the spread of situational information.

The Polynomial Volume Law of Complex Networks in the Context of Local and Global Optimization

Many complex networks expose global hub structures: for some nodes, the number of incident edges far exceeds the average, leading to a small average shortest path length. Such ‘small-world properties’ are often guided by a scale-free power-law distribution of the node degrees, and self-organization inside the network has been identified as a reason driving the emergence of this structure. Small-world networks have recently raised lots of interest, because they capture the global topology of the World-Wide Web, metabolic, and social networks. While small-world networks reflect global structures, little attention is paid to the local structure of complex networks. In this article neighbourhoods are demonstrated to share a common local structure in many real complex networks, manifested by a polynomial volume law. This law can, in case of networks that are embedded in space, be explained in terms of the embedding and the properties of Euclidean space. A model of hierarchical spatial networks is introduced to examine the effect of global structures, in particular of hierarchies, on the polynomial volume law. It turns out that the law is robust against the coexistence of such global structures. The local structure of space and global optimization can both be found in transport, brain, and communication networks, which suggests the polynomial volume law, often in combination with hierarchies or other global optimization principles, to be a generic property inherent to many networks.

Scale-free networks in metabolomics.

Metabolomics is an expanding discipline in biology. It is the process of portraying the phenotype of a cell, tissue or species organism using a comprehensive set of metabolites. Therefore, it is of interest to understand complex systems such as metabolomics using a scale-free topology. Genetic networks and the World Wide Web (WWW) are described as networks with complex topology. Several large networks have vertex connectivity that goes beyond a scale-free power-law distribution. It is observed that (a) networks expand constantly by the addition of recent vertices, and (b) recent vertices attach preferentially to sites that are already well connected. Scalefree networks are determined with precision using vital features such as a structure, a disease and a patient. This is pertinent to the understanding of complex systems such as metabolomics. Hence, we describe the relevance of scale-free networks in the understanding of metabolomics in this article.

Global energy flows embodied in international trade: A combination of environmentally extended input–output analysis and complex network analysis

Significant energy flows embodied in international trade have evolved into a complex interconnected flow network. Therefore, this study applies a variety of complex network analysis tools to uncover the structure of embodied energy flow network (EEFN) at global, regional and national level, based on environmentally extended input–output analysis (EEIOA). At global level, small-world nature has been found, implying the economies are highly connected through embodied energy transfer. EEFN is proved to be a heterogeneous network due to the scale-free power-law distribution of degree/strength. At regional level, 4 communities are detected and members in the same regional cooperative organizations, such as EU, ASEAN, NAFTA and AU, tend to be classified into the same community, indicating that EEFN embodies the characteristics of regionalization and multi-polarization. At national level, some key economies, such as USA, China and Germany, are always at the forefront of network-based centrality measures and EEIOA-based accountings. Furthermore, the security of embodied energy supply is evaluated for each economy. Consequently, policy implications of the results are discussed, which could provide additional insights for policy formulation to enhance energy security.

Taming active turbulence with patterned soft interfaces

Active matter embraces systems that self-organize at different length and time scales, often exhibiting turbulent flows apparently deprived of spatiotemporal coherence. Here, we use a layer of a tubulin-based active gel to demonstrate that the geometry of active flows is determined by a single length scale, which we reveal in the exponential distribution of vortex sizes of active turbulence. Our experiments demonstrate that the same length scale reemerges as a cutoff for a scale-free power law distribution of swirling laminar flows when the material evolves in contact with a lattice of circular domains. The observed prevalence of this active length scale can be understood by considering the role of the topological defects that form during the spontaneous folding of microtubule bundles. These results demonstrate an unexpected strategy for active systems to adapt to external stimuli, and provide with a handle to probe the existence of intrinsic length and time scales.

Investigating the Influence Relationship Models for Stocks in Indian Equity Market: A Weighted Network Modelling Study.

The socio-economic systems today possess high levels of both interconnectedness and interdependencies, and such system-level relationships behave very dynamically. In such situations, it is all around perceived that influence is a perplexing power that has an overseeing part in affecting the dynamics and behaviours of involved ones. As a result of the force & direction of influence, the transformative change of one entity has a cogent aftereffect on the other entities in the system. The current study employs directed weighted networks for investigating the influential relationship patterns existent in a typical equity market as an outcome of inter-stock interactions happening at the market level, the sectorial level and the industrial level. The study dataset is derived from 335 constituent stocks of ‘Standard & Poor Bombay Stock Exchange 500 index’ and study period is 1st June 2005 to 30th June 2015. The study identifies the set of most dynamically influential stocks & their respective temporal pattern at three hierarchical levels: the complete equity market, different sectors, and constituting industry segments of those sectors. A detailed influence relationship analysis is performed for the sectorial level network of the construction sector, and it was found that stocks belonging to the cement industry possessed high influence within this sector. Also, the detailed network analysis of construction sector revealed that it follows scale-free characteristics and power law distribution. In the industry specific influence relationship analysis for cement industry, methods based on threshold filtering and minimum spanning tree were employed to derive a set of sub-graphs having temporally stable high-correlation structure over this ten years period.

Directed networks’ different link formation mechanisms causing degree distribution distinction

Within undirected networks, scientists have shown much interest in presenting power-law features. For instance, Barabási and Albert (1999) claimed that a common property of many large networks is that vertex connectivity follows scale-free power-law distribution, and in another study Barabási et al. (2002) showed power law evolution in the social network of scientific collaboration. At the same time, Jiang et al. (2011) discussed deviation from power-law distribution; others indicated that size effect (Bagrow et al., 2008), information filtering mechanism (Mossa et al., 2002), and birth and death process (Shi et al., 2005) could account for this deviation. Within directed networks, many authors have considered that outlinks follow a similar mechanism of creation as inlinks’ (Faloutsos et al., 1999; Krapivsky et al., 2001; Tanimoto, 2009) with link creation rate being the linear function of node degree, resulting in a power-law shape for both indegree and outdegree distribution. Some other authors have made an assumption that directed networks, such as scientific collaboration or citation, behave as undirected, resulting in a power-law degree distribution accordingly (Barabási et al., 2002). At the same time, we claim (1) Outlinks feature different degree distributions than inlinks; where different link formation mechanisms cause the distribution distinctions, (2) in/outdegree distribution distinction holds for different levels of system decomposition; therefore this distribution distinction is a property of directed networks. First, we emphasize in/outlink formation mechanisms as causal factors for distinction between indegree and outdegree distributions (where this distinction has already been noticed in Barker et al. (2010) and Baxter et al. (2006)) within a sample network of OSS projects as well as Java software corpus as a network. Second, we analyze whether this distribution distinction holds for different levels of system decomposition: open-source-software (OSS) project–project dependency within a cluster, package–package dependency within a project and class–class dependency within a package. We conclude that indegree and outdegree dependencies do not lead to similar type of degree distributions, implying that indegree dependencies follow overall power-law distribution (or power-law with flat-top or exponential cut-off in some cases), while outdegree dependencies do not follow heavy-tailed distribution.

Evolution of the statistical distribution in a topological defect network.

The complex networks of numerous topological defects in hexagonal manganites are highly relevant to vastly different phenomena from the birth of our cosmos to superfluidity transition. The topological defects in hexagonal manganites form two types of domain networks: type-I without and type-II with electric self-poling. A combined phase-field simulations and experimental study shows that the frequencies of domains with N-sides, i.e. of N-gons, in a type-I network are fitted by a lognormal distribution, whereas those in type-II display a scale-free power-law distribution with exponent ∼2. A preferential attachment process that N-gons with a larger N have higher probability of coalescence is responsible for the emergence of the scale-free networks. Since the domain networks can be observed, analyzed, and manipulated at room temperature, hexagonal manganites provide a unique opportunity to explore how the statistical distribution of a topological defect network evolves with an external electric field.

Conservation of Information: Software’sHidden Clockwork?

In this paper it is proposed that the Conservation of Hartley-Shannon Information (hereafter contracted to H-S Information) plays the same role in discrete systems as the Conservation of Energy does in physical systems. In particular, using a variational approach, it is shown that the symmetry of scale-invariance, power-laws and the Conservation of H-S Information are intimately related and lead to the prediction that the component sizes of any software system assembled from components made from discrete tokens always asymptote to a scale-free power-law distribution in the unique alphabet of tokens used to construct each component. This is then validated to a very high degree of significance on some 100 million lines of software in seven different programming languages independently of how the software was produced, what it does, who produced it or what stage of maturity it has reached. A further implication of the theory presented here is that the average size of components depends only on their unique alphabet, independently of the package they appear in. This too is demonstrated on the main data set and also on 24 additional Fortran 90 packages.

Universal distribution of component frequencies in biological and technological systems

Bacterial genomes and large-scale computer software projects both consist of a large number of components (genes or software packages) connected via a network of mutual dependencies. Components can be easily added or removed from individual systems, and their use frequencies vary over many orders of magnitude. We study this frequency distribution in genomes of ∼500 bacterial species and in over 2 million Linux computers and find that in both cases it is described by the same scale-free power-law distribution with an additional peak near the tail of the distribution corresponding to nearly universal components. We argue that the existence of a power law distribution of frequencies of components is a general property of any modular system with a multilayered dependency network. We demonstrate that the frequency of a component is positively correlated with its dependency degree given by the total number of upstream components whose operation directly or indirectly depends on the selected component. The observed frequency/dependency degree distributions are reproduced in a simple mathematically tractable model introduced and analyzed in this study.

The Impact of Sampling and Network Topology on the Estimation of Social Intercorrelations

With the growing popularity of online social networks, it is becoming more important for marketing researchers to understand and measure social intercorrelations among consumers. The authors show that the estimation of consumers' social intercorrelations can be significantly affected by the sampling method used in the study and the topology of the social network. Through a series of simulation studies using a spatial model, the authors find that the magnitude of social intercorrelations in consumer networks tends to be underestimated if samples of the networks are used (rather than using the entire population of the network). The authors further demonstrate that sampling methods that better preserve the network structure perform best in recovering the social intercorrelations. However, this advantage decreases in networks characterized by the scale-free power-law distribution for the number of connections of each member. The authors discuss the insights they glean from these findings and propose a method to obtain unbiased estimation of the magnitude of social intercorrelations.

Scale-free behavior of displacement bursts: Lower limit and scaling exponent

Plastic flow of micro-samples proceeds in an intermittent displacement bursts, showing a scale-free power-law distribution. Using maximum likelihood estimator and the Kolmogorov-Smirnov statistic, we investigate the statistics of displacement bursts such as lower limit and scaling exponent, for selected micro-samples from previous researches. The results of lower limit indicate that all the crystals considered share a common lower limit, the magnitude of which is found to be comparable to the largest burst size governed by short-range interactions (i.e., junctions). Additionally, the calculated results of the scaling exponents challenge those observed in many previous studies where other statistical techniques are employed. We demonstrate that our results are consistently manifested in the experimental data.

Weighted BA Scale-Free Random Graph Model

Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of three generic mechanisms: (i) networks expand continuously by the addition of new vertices, (ii) new vertex with different number edges of weighted selected that connected to different vertices in the system, and (iii) new vertices attach preferentially to sites that are already well connected. A model based on these three ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.

Self-Organized Criticality Underlies Arrhythmogenic Calcium Waves in Cardiac Myocytes

In cardiac myocytes, calcium (Ca)-induced Ca release can give rise to propagating Ca waves that promote cardiac arrhythmias. To understand the mechanism underlying the transition from localized Ca sparks to global Ca waves, we developed a mathematical model of a spatially-distributed network of diffusively coupled Ca release units (CRUs) comprised of junctional/longitudinal sarcoplasmic reticulum and dyadic/cytoplasmic spaces with physiologically realistic dimensions. As total Ca was increased in the model, Ca spark cluster sizes initially exhibited an exponential distribution, which transitioned to a scale-free power-law (f-β) distribution near the threshold at which random Ca sparks transitioned to Ca waves. The power-law relationship indicates that Ca release patterns in the CRU network are governed by a dynamical mechanism called self-organized criticality, the same process underlying many natural world phenomena such as avalanches and earthquakes. We tested this prediction experimentally in saponin-permeabilized cardiac myocytes. As free Ca, buffered with 0.5 mM EGTA, was increased, Ca spark cluster size transitioned from an exponential distribution to a power-law distribution at a free Ca concentration of 400 nM. Consistent with self-organized criticality, this Ca concentration was near the threshold for spark-to-wave transition since fully propagating waves were observed above 400 nM Ca. Below this concentration, Ca was released mostly as individual sparks, and spark cluster sizes followed an exponential rather than power-law distribution. In conclusion, our findings provide both theoretical and experimental evidence that the transition from Ca sparks to arrhythmogenic Ca waves in cardiac myocytes is mediated by the dynamical process of self-organized criticality, common to many natural phenomena. This provides a theoretical framework for developing interventions which modulate the Ca spark-to-wave transition threshold as a potential therapeutic strategy for preventing arrhythmias.

Scaling law: A global topological property of genetic sequences

Here, we study the topological properties of genetic sequence by viewing the entire sequence as a whole. First, a systematic way of coding the gene-combinations in a genetic sequence is developed. Next, we apply the coding method to real genetic sequences, and find a scale-free power-law distribution for some particular kinds of gene-combinations. Furthermore, we also present a model to reproduce the observed scale-free feature, which is based on three generic mechanisms: 1) Growth mechanism - genetic sequences expands continuously by the addition of new codon; 2) Preferential replication mechanism - the newly added codon at every time step is a replication of one of existed condons, and the probability that a codon is replicated is proportional to its emerging times in existed sequence; and 3) Mutation mechanism - following (2) the newly added codon has a small probability to mutate. To our knowledge, no report has been published to study the genetic sequences in such way.

On necessary conditions for scale-freedom in complex networks, with applications to computer communication systems

Many complex networks exhibit a scale-free, power-law distribution of vertex degrees. This common feature is a consequence of two generic mechanisms relating to the formation of real networks: (i) networks tend to expand over time through the addition of new vertices and (ii) new vertices attach preferentially to those that are already well connected. We show that for many natural or man-made complex networks possessing a scale-free power-law distribution with the exponent γ ≥ 2, the number of degree-1 vertices, when nonzero, is of the same order as the network size N and that the average degree is of order at most log N. Our results expose another necessary characteristic of such networks. Furthermore, our method has the benefit of relying only on conditions that are static and easily verified for arbitrary networks. We use the preceding results to derive a closed-form formula approximating the distance distribution in scale-free networks. Such distributions are applied extensively in the fields of computer communication and software architecture, among other domains.

A Small-World Model of Scale-Free Networks: Features and Verifications

It is now well known that many large-sized complex networks obey a scale-free power-law vertex-degree distribution. Here, we show that when the vertex degrees of a large-sized network follow a scale-free power-law distribution with exponent  2, the number of degree-1 vertices, if nonzero, is of order N and the average degree is of order lower than log N, where N is the size of the network. Furthermore, we show that the number of degree-1 vertices is divisible by the least common multiple of , , . . ., , and l is less than log N, where l = < is the vertex-degree sequence of the network. The method we developed here relies only on a static condition, which can be easily verified, and we have verified it by a large number of real complex networks.

A novel weighted evolving network model based on clique overlapping growth

A novel weighted evolving network model based on the clique overlapping growth was proposed. The model shows different network characteristics under two different selection mechanisms that are preferential selection and random selection. On the basis of mean-field theory, this model under the two different selection mechanisms was analyzed. The analytic equations of distributions of the number of cliques that a vertex joins and the vertex strength of the model were given. It is proved that both distributions follow the scale-free power-law distribution in preferential selection mechanism and the exponential distribution in random selection mechanism, respectively. The analytic expressions of exponents of corresponding distributions were obtained. The agreement between the simulations and analytical results indicates the validity of the theoretical analysis. Finally, three real transport bus networks (BTNs) of Beijing, Shanghai and Hangzhou in China were studied. By analyzing their network properties, it is discovered that these real BTNs belong to a kind of weighted evolving network model with clique overlapping growth and random selection mechanism that was proposed in this context.