Evolving Network Model Research Articles

A novel local-world evolving network model for power grid

Many systems in the real world can be described by evolving models of complex network, one of which is power system. Classical network models show great inconsistence with practical power grids. Based on the essential evolution characteristics of power grid, we propose a new local-world evolving network model for complex power grid, which exists in many physical complex networks. The statistical properties and dynamics of the proposed model are analytically studied. It is found that the degree distribution of this new model has a power-law tail, and its scaling exponent is between 3 and ∞. Numerical simulations for West American power grid and North China power grid and their comparison with random network and scale-free network indicate that the new evolving network model captures the essential features of real-world power grid. Furthermore, it is proved that power grid is neither a random network nor a scale-free network.

Growth Dynamics of Cell Assemblies

We explore an evolutionary network model of pulse-coupled neurons in which the changes of evolutionary coupling strengths are based on Hebbian synaptic plasticity. We show that the ongoing changes of the evolutionary network's nodal-and-coupling dynamics will eventually result in group synchrony and sync-dependent circuits. We also tackle the problem of the stability of neural synchrony and the problem of determining the size of synchronously firing neural groups. This leads to describing a phenomenon underlying synchrony and stability of synchrony that neural synchrony allows positive feedback from which a monotonically increasing sequence of coupling strengths and a monotonically increasing region of states for initializing the stability process arise.

Age-Dependent Evolution of the Yeast Protein Interaction Network Suggests a Limited Role of Gene Duplication and Divergence

Proteins interact in complex protein–protein interaction (PPI) networks whose topological properties—such as scale-free topology, hierarchical modularity, and dissortativity—have suggested models of network evolution. Currently preferred models invoke preferential attachment or gene duplication and divergence to produce networks whose topology matches that observed for real PPIs, thus supporting these as likely models for network evolution. Here, we show that the interaction density and homodimeric frequency are highly protein age–dependent in real PPI networks in a manner which does not agree with these canonical models. In light of these results, we propose an alternative stochastic model, which adds each protein sequentially to a growing network in a manner analogous to protein crystal growth (CG) in solution. The key ideas are (1) interaction probability increases with availability of unoccupied interaction surface, thus following an anti-preferential attachment rule, (2) as a network grows, highly connected sub-networks emerge into protein modules or complexes, and (3) once a new protein is committed to a module, further connections tend to be localized within that module. The CG model produces PPI networks consistent in both topology and age distributions with real PPI networks and is well supported by the spatial arrangement of protein complexes of known 3-D structure, suggesting a plausible physical mechanism for network evolution.

Cascading failures in local-world evolving networks

The local-world (LW) evolving network model shows a transition for the degree distribution between the exponential and power-law distributions, depending on the LW size. Cascading failures under intentional attacks in LW network models with different LW sizes were investigated using the cascading failures load model. We found that the LW size has a significant impact on the network’s robustness against deliberate attacks. It is much easier to trigger cascading failures in LW evolving networks with a larger LW size. Therefore, to avoid cascading failures in real networks with local preferential attachment such as the Internet, the World Trade Web and the multi-agent system, the LW size should be as small as possible.

Controlling load distribution in the queuing systems

A method to control load distribution in the closed exponential queuing networks with one class of customers was proposed. It is based on simultaneous use of routing control and control of servicing intensities. Consideration was given to an evolution model of the queuing network with control of load distribution and an approximate method of analysis of the queuing networks of this type. A technique to calculate the stationary distribution and formulas to calculate other stationary characteristics of the queuing networks with control of load distribution were described. Examples of analysis of the queuing networks of this type were presented.

A self-organized model for network evolution

Here we provide a detailed analysis, along with some extensions and additonal investigations, of a recently proposed [1] self-organized model for the evolution of complex networks. Vertices of the network are characterized by a fitness variable evolving through an extremal dynamics process, as in the Bak-Sneppen [2] model representing a prototype of Self-Organized Criticality. The network topology is in turn shaped by the fitness variable itself, as in the fitness network model [3]. The system self-organizes to a nontrivial state, characterized by a power-law decay of dynamical and topological quantities above a critical threshold. The interplay between topology and dynamics in the system is the key ingredient leading to an unexpected behaviour of these quantities.

A local-world node deleting evolving network model

A new type network growth rule which comprises node addition with the concept of local-world connectivity and node deleting is studied. A series of theoretical analysis and numerical simulation to the LWD network are conducted in this Letter. Firstly, the degree distribution p ( k ) of this network changes no longer pure scale free but truncates by an exponential tail and the truncation in p ( k ) increases as p a decreases. Secondly, the connectivity is tighter, as the local-world size M increases. Thirdly, the average path length L increases and the clustering coefficient 〈 C 〉 decreases as generally node deleting increases. Finally, 〈 C 〉 trends up when the local-world size M increases, so as to k max . Hence, the expanding local-world can compensate the infection of the node deleting.

Self-organized criticality in quantum gravity

We study a simple model of spin network evolution motivated by the hypothesis that the emergence of classical spacetime from a discrete microscopic dynamics may be a self-organized critical process. Self-organized critical systems are statistical systems that naturally evolve without fine tuning to critical states in which correlation functions are scale invariant. We study several rules for evolution of frozen spin networks in which the spins labeling the edges evolve on a fixed graph. We find evidence for a set of rules which behaves analogously to sand pile models in which a critical state emerges without fine tuning, in which some correlation functions become scale invariant.

Epidemic spreading behavior in local-world evolving networks

The susceptible–infected–susceptible (SIS) epidemic model is presented which can be used to study the epidemic spreading behavior in the local-world evolving network model. Local-world evolving model displays a transition from the exponential network to the scale-free network with respect to the connectivity distribution. From theoretical analyses and numerical simulations, we find that the epidemic spreading behavior on local-world networks also takes on some kind of transitional behaviors. The transitional behavior is further verified by comparing the spreading behavior of local-world network with that of random and scale-free networks. Some feasible control strategies are also proposed to keep from the epidemic spreading on networks.

Epidemic spreading behavior with time delay on local-world evolving networks

An improved susceptible-infected-susceptible (SIS) model in the local-world evolving network model is presented to study the epidemic spreading behavior with time delay, which is added into the infected phase. The local-world evolving model displays a transition from the exponential network to the scale-free network with respect to the degree distribution. Two typical delay regimes, i.e., uniform and degree-dependent delays are incorporated into the SIS epidemic model to investigate the epidemic infection processes in the local-world network model. The results indicate that the infection delay will promote the epidemic outbreaks, increase the prevalence and reduce the critical threshold of epidemic spreading. It is also found that local-world size M will considerably influence the epidemic spreading behavior with time delay in the local-world network through large-scale numerical simulations. Simulation results are also of relevance to fight epidemic outbreaks.

Reconstruction of ancestral protein interaction networks for the bZIP transcription factors

As whole-genome protein-protein interaction datasets become available for a wide range of species, evolutionary biologists have the opportunity to address some of the unanswered questions surrounding the evolution of these complex systems. Protein interaction networks from divergent organisms may be compared to investigate how gene duplication, deletion, and rewiring processes have shaped the evolution of their contemporary structures. However, current approaches for comparing observed networks from multiple species lack the phylogenetic context necessary to reconstruct the evolutionary history of a network. Here we show how probabilistic modeling can provide a platform for the quantitative analysis of multiple protein interaction networks. We apply this technique to the reconstruction of ancestral networks for the bZIP family of transcription factors and find that excellent agreement is obtained with an alternative sequence-based method for the prediction of leucine zipper interactions. Further analysis shows our probabilistic method to be significantly more robust to the presence of noise in the observed network data than a simple parsimony-based approach. In addition, the integration of evidence over multiple species means that the same method may be used to improve the quality of noisy interaction data for extant species. The ancestral states of a protein interaction network have been reconstructed here by using an explicit probabilistic model of network evolution. We anticipate that this model will form the basis of more general methods for probing the evolutionary history of biochemical networks.

Scaling relationships and evolution of distributary networks on wave‐influenced deltas

No genetic model can explain the variability in distributary network pattern on modern deltas. Here we derive scaling relationships for two processes known to create distributary channels and, with these laws, construct a simple model for distributary network evolution. The first process is mouth‐bar deposition at the shoreline and subsequent channel bifurcation; the second is avulsion—the wholesale abandonment of a channel in favor of a new path. The former creates relatively small networks with power‐law distributions of channel length; the latter generates relatively few, backwater‐scale distributaries. Frequency‐magnitude plots of channel length on natural deltas agree with theoretical predictions and show a clear separation in scale that reflects these processes: Mouth‐bar distributary lengths scale with the width of the parent channel, and avulsive distributary lengths scale with the backwater length; intermediate channel lengths are relatively rare. Wave energy controls network topology by suppressing mouth‐bar development, thereby preferentially eliminating smaller‐scale distributaries.

Using Likelihood-Free Inference to Compare Evolutionary Dynamics of the Protein Networks of H. pylori and P. falciparum

Gene duplication with subsequent interaction divergence is one of the primary driving forces in the evolution of genetic systems. Yet little is known about the precise mechanisms and the role of duplication divergence in the evolution of protein networks from the prokaryote and eukaryote domains. We developed a novel, model-based approach for Bayesian inference on biological network data that centres on approximate Bayesian computation, or likelihood-free inference. Instead of computing the intractable likelihood of the protein network topology, our method summarizes key features of the network and, based on these, uses a MCMC algorithm to approximate the posterior distribution of the model parameters. This allowed us to reliably fit a flexible mixture model that captures hallmarks of evolution by gene duplication and subfunctionalization to protein interaction network data of Helicobacter pylori and Plasmodium falciparum. The 80% credible intervals for the duplication–divergence component are [0.64, 0.98] for H. pylori and [0.87, 0.99] for P. falciparum. The remaining parameter estimates are not inconsistent with sequence data. An extensive sensitivity analysis showed that incompleteness of PIN data does not largely affect the analysis of models of protein network evolution, and that the degree sequence alone barely captures the evolutionary footprints of protein networks relative to other statistics. Our likelihood-free inference approach enables a fully Bayesian analysis of a complex and highly stochastic system that is otherwise intractable at present. Modelling the evolutionary history of PIN data, it transpires that only the simultaneous analysis of several global aspects of protein networks enables credible and consistent inference to be made from available datasets. Our results indicate that gene duplication has played a larger part in the network evolution of the eukaryote than in the prokaryote, and suggests that single gene duplications with immediate divergence alone may explain more than 60% of biological network data in both domains.

Evolving network with different edges

We propose a scale-free network similar to Barabasi-Albert networks but with two different types of edges. This model is based on the idea that in many cases there are more than one kind of link in a network and when a new node enters the network both old nodes and different kinds of links compete to obtain it. The degree distribution of both the total degree and the degree of each type of edge is analyzed and found to be scale-free. Simulations are shown to confirm these results.

The physical position neighbourhood evolving network model

Many social, technological, biological and economical systems are properly described by evolved network models. In this paper, a new evolving network model with the concept of physical position neighbourhood connectivity is proposed and studied. This concept exists in many real complex networks such as communication networks. The simulation results for network parameters such as the first nonzero eigenvalue and maximal eigenvalue of the graph Laplacian, clustering coefficients, average distances and degree distributions for different evolving parameters of this model are presented. The dynamical behaviour of each node on the consensus problem is also studied. It is found that the degree distribution of this new model represents a transition between power-law and exponential scaling, while the Barábasi–Albert scale-free model is only one of its special (limiting) cases. It is also found that the time to reach a consensus becomes shorter sharply with increasing of neighbourhood scale of the nodes.

Correlation, selection and the evolution of species networks

We use a generalised version of the individual-based Tangled Nature model of evolutionary ecology to study the relationship between ecosystem structure and evolutionary history. Our evolved model ecosystems typically exhibit interaction networks with exponential degree distributions and an inverse dependence between connectance and species richness. We use a simplified network evolution model to demonstrate that the observed degree distributions can occur as a consequence of partial correlations in the inheritance process. Further to this, in the limit of low connectance and maximal correlation, distributions of power-law form, P ( k ) ∝ 1 / k , can be achieved. We also show that a hyperbolic relationship between connectance and species richness, C ∼ 1 / D can arise as a consequence of probabilistic constraints on the evolutionary search process.

Identification of functional modules from conserved ancestral protein–protein interactions

The increasing availability of large-scale protein-protein interaction (PPI) data has fueled the efforts to elucidate the building blocks and organization of cellular machinery. Previous studies have shown cross-species comparison to be an effective approach in uncovering functional modules in protein networks. This has in turn driven the research for new network alignment methods with a more solid grounding in network evolution models and better scalability, to allow multiple network comparison. We develop a new framework for protein network alignment, based on reconstruction of an ancestral PPI network. The reconstruction algorithm is built upon a proposed model of protein network evolution, which takes into account phylogenetic history of the proteins and the evolution of their interactions. The application of our methodology to the PPI networks of yeast, worm and fly reveals that the most probable conserved ancestral interactions are often related to known protein complexes. By projecting the conserved ancestral interactions back onto the input networks we are able to identify the corresponding conserved protein modules in the considered species. In contrast to most of the previous methods, our algorithm is able to compare many networks simultaneously. The performed experiments demonstrate the ability of our method to uncover many functional modules with high specificity. Information for obtaining software and supplementary results are available at http://bioputer.mimuw.edu.pl/papers/cappi.

A new community-based evolving network model

In order to describe the community structure upon the dynamical evolution of complex networks, we propose a new community based evolving network (CBEN) model having increasing communities with preferential mechanisms of both community sizes and node degrees, whose cumulative distribution and raw distribution follow scale-invariant power-law distributions P( S⩾ s)∼ s − ν and P( k)∼ k − γ with exponents of ν⩾1 and γ∈[2, + ∞), respectively. Besides, complex networks generated by the CBEN model are hierarchically structured, which cover the range from disassortative networks to assortative networks.

Assortativity and act degree distribution of some collaboration networks

Empirical investigation results on weighted and un-weighted assortativity, act degree distribution, degree distribution and node strength distribution of nine real world collaboration networks have been presented. The investigations propose that act degree distribution, degree distribution and node strength distribution usually show so-called “shifted power law” (SPL) function forms, which can continuously vary from an ideal power law to an ideal exponential decay. Two parameters, α and η , can be used for description of the distribution functions. Another conclusion is that assortativity coefficient and the parameter, α or η , is monotonously dependent on each other. By the collaboration network evolution model introduced in a reference [P. Zhang et al., Physica A 360 (2006) 599], we analytically derived the SPL distributions, which typically appeared in general situations where nodes are selected partially randomly, with a probability p , and partially by linear preferential principle, with the probability 1 - p . The analytic discussion gives an explicit expression on the relationship between the random selection proportion p and the parameters α and η . The numerical simulation results by the model show a monotonic dependence of assortativity on the random selection proportion p . The empirically obtained assortativity versus α or η curve for the four collaboration networks with small maximal act size, T max , shows a good agreement with the model prediction. According to the curves, one can qualitatively judge the random selection proportion of the real world network in its evolution process.

Formation of Circulation in Complex Networks

We study the mechanism that forms the cyclic topology in various scale-free networks. An evolving network model is introduced in which the ratio of the number of edges attaching a new vertex to old vertices and that linking between old vertices is controlled and the edges are attached in three different ways with probability parameters. By controlling the ratio and the probability parameters, we realize model networks showing almost the same cyclic topology as that of various real networks. This indicates that the cyclic property of each network is determined by the ratio of the tree-like structure, triangular structure, and quadrangular structure.