Modular networks with hierarchical organization: The dynamical implications of complex structure

Several networks occurring in real life have modular structures that are arranged in a hierarchical fashion. In this paper, we have proposed a model for such networks, using a stochastic generation method. Using this model we show that, the scaling relation between the clustering and degree of the nodes is not a necessary property of hierarchical modular networks, as had previously been suggested on the basis of a deterministically constructed model. We also look at dynamics on such networks, in particular, the stability of equilibria of network dynamics and of synchronized activity in the network. For both of these, we find that, increasing modularity or the number of hierarchical levels tends to increase the probability of instability. As both hierarchy and modularity are seen in natural systems, which necessarily have to be robust against environmental fluctuations, we conclude that additional constraints are necessary for the emergence of hierarchical structure, similar to the occurrence of modularity through multi-constraint optimization as shown by us previously.      

The structure and resilience of financial market networks

Financial markets can be viewed as a highly complex evolving system that is very sensitive to economic instabilities. The complex organization of the market can be represented in a suitable fashion in terms of complex networks, which can be constructed from stock prices such that each pair of stocks is connected by a weighted edge that encodes the distance between them. In this work, we propose an approach to analyze the topological and dynamic evolution of financial networks based on the stock correlation matrices. An entropy-related measurement is adopted to quantify the robustness of the evolving financial market organization. It is verified that the network topological organization suffers strong variation during financial instabilities and the networks in such periods become less robust. A statistical robust regression model is proposed to quantity the relationship between the network structure and resilience. The obtained coefficients of such model indicate that the average shortest path length is the measurement most related to network resilience coefficient. This result indicates that a collective behavior is observed between stocks during financial crisis. More specifically, stocks tend to synchronize their price evolution, leading to a high correlation between pair of stock prices, which contributes to the increase in distance between them and, consequently, decrease the network resilience.     

Multiple dynamical time-scales in networks with hierarchically nested modular organization

Many natural and engineered complex networks have intricate mesoscopic organization, e.g., the clustering of the constituent nodes into several communities or modules. Often, such modularity is manifested at several different hierarchical levels, where the clusters defined at one level appear as elementary entities at the next higher level. Using a simple model of a hierarchical modular network, we show that such a topological structure gives rise to characteristic time-scale separation between dynamics occurring at different levels of the hierarchy. This generalizes our earlier result for simple modular networks, where fast intramodular and slow intermodular processes were clearly distinguished. Investigating the process of synchronization of oscillators in a hierarchical modular network, we show the existence of as many distinct time-scales as there are hierarchical levels in the system. This suggests a possible functional role of such mesoscopic organization principle in natural systems, viz., in the dynamical separation of events occurring at different spatial scales.     

The effect of topology on organization of synchronous behavior in dynamical networks with adaptive couplings

We study the influence of the initial topology of connections on the organization of synchronous behavior in networks of phase oscillators with adaptive couplings. We found that networks with a random sparse structure of connections predominantly demonstrate the scenario as a result of which chimera states are formed. The formation of chimera states retains the features of the hierarchical organization observed in networks with global connections [D.V. Kasatkin, S. Yanchuk, E. Schöll, V.I. Nekorkin, Phys. Rev. E 96, 062211 (2017)], and also demonstrates a number of new properties due to the presence of a random structure of network topology. In this case, the formation of coherent groups takes a much longer time interval, and the sets of elements that form these groups can be significantly rearranged during the evolution of the network. We also found chimera states, in which along with the coherent and incoherent groups, there are subsets, whose different elements can be synchronized with each other for sufficiently long periods of time.     

Deterministic factors of stock networks based on cross-correlation in financial market

The stock market has been known to form homogeneous stock groups with a higher correlation among different stocks according to common economic factors that influence individual stocks. We investigate the role of common economic factors in the market in the formation of stock networks, using the arbitrage pricing model reflecting essential properties of common economic factors. We find that the degree of consistency between real and model stock networks increases as additional common economic factors are incorporated into our model. Furthermore, we find that individual stocks with a large number of links to other stocks in a network are more highly correlated with common economic factors than those with a small number of links. This suggests that common economic factors in the stock market can be understood in terms of deterministic factors.     

The hierarchical backbone of complex networks

Given any complex directed network, a set of acyclic subgraphs can be extracted that will provide valuable information about its hierarchical structure. This Letter presents how the interpretation of the network weight matrix as a transition matrix allows the hierarchical backbone to be identified and characterized in terms of the concepts of hierarchical degree, which expresses the total weights of virtual edges established along successive transitions. The potential of the proposed approach is illustrated with respect to simulated random and preferential-attachment networks as well as real data related to word associations and gene sequencing.     

Spectrum,intensity and coherence in weighted networks of a financial market

We construct a correlation matrix based financial network for a set of New York Stock Exchange (NYSE) traded stocks with stocks corresponding to nodes and the links between them added one after the other, according to the strength of the correlation between the nodes. The eigenvalue spectrum of the correlation matrix reflects the structure of the market, which also shows in the cluster structure of the emergent network. The stronger and more compact a cluster is, the earlier the eigenvalue representing the corresponding business sector occurs in the spectrum. On the other hand, if groups of stocks belonging to a given business sector are considered as a fully connected subgraph of the final network, their intensity and coherence can be monitored as a function of time. This approach indicates to what extent the business sector classifications are visible in market prices, which in turn enables us to gauge the extent of group-behaviour exhibited by stocks belonging to a given business sector.     

Self-contained algorithms to detect communities in networks

The investigation of community structures in networks is an important issue in many domains and disciplines. In this paper we present a new class of local and fast algorithms which incorporate a quantitative definition of community. In this way the algorithms for the identification of the community structure become fully self-contained and one does not need additional non-topological information in order to evaluate the accuracy of the results. The new algorithms are tested on artificial and real-world graphs. In particular we show how the new algorithms apply to a network of scientific collaborations both in the unweighted and in the weighted version. Moreover we discuss the applicability of these algorithms to other non-social networks and we present preliminary results about the detection of community structures in networks of interacting proteins.Received: 7 November 2003, Published online: 14 May 2004PACS: 89.75.Hc Networks and genealogical trees - 87.23.Ge Dynamics of social systems - 87.90. + y Other topics in biological and medical physics     

The volatility in a multi-share financial market model

Single index financial market models cannot account for the empirically observed complex interactions between shares in a market. We describe a multi-share financial market model and compare characteristics of the volatility, that is the variance of the price fluctuations, with empirical characteristics. In particular we find its probability distribution is similar to a log normal distribution but with a long power-law tail for the large fluctuations, and that the time development shows superdiffusion. Both these results are in good quantitative agreement with observations.     

Multifractal properties of the Indian financial market

We investigate the multifractal properties of the logarithmic returns of the Indian financial indices (BSE & NSE) by applying the multifractal detrended fluctuation analysis. The results are compared with that of the US S&P 500 index. Numerically we find that qth-order generalized Hurst exponents h(q) and τ(q) change with the moments q. The nonlinear dependence of these scaling exponents and the singularity spectrum f(α) show that the returns possess multifractality. By comparing the MF-DFA results of the original series to those for the shuffled series, we find that the multifractality is due to the contributions of long-range correlations as well as the broad probability density function. The financial markets studied here are compared with the Binomial Multifractal Model (BMFM) and have a smaller multifractal strength than the BMFM.     

Synchronizability of dynamical scale-free networks subject to random errors

In this paper the robustness of network synchronizability against random deletion of nodes, i.e. errors, in dynamical scale-free networks was studied. To this end, two measures of network synchronizability, namely, the eigenratio of the Laplacian and the order parameter quantifying the degree of phase synchrony were adopted, and the synchronizability robustness on preferential attachment scale-free graphs was investigated. The findings revealed that as the network size decreases, the robustness of its synchronizability against random removal of nodes declines, i.e. the more the number of randomly removed nodes from the network, the worse its synchronizability. We also showed that this dependence of the synchronizability on the network size is different with that in the growing scale-free networks. The profile of a number of network properties such as clustering coefficient, efficiency, assortativity, and eccentricity, as a function of the network size was investigated in these two cases, growing scale-free networks and those with randomly removed nodes. The results showed that these processes are also different in terms of these metrics.     

On the use of tryptophan to detect slow orientational motions

For our investigations on orientational motions of membrane proteins with the method of time-resolved fluorescence anisotropy decay (TRFA), we first wanted to test the use of tryptophan fluorescence in detecting slow motions on the time scale of tens of nanoseconds and to get more insight into the possible motions of membrane proteins by investigating a simple system. We performed TRFA measurements on a short α-helical 21-amino acid peptide in different environments (Vogel, H.,et al. Proc. Natl. Acad. Sci. USA 85, 5067–5071, 1988). In each case, we got three relexation time constants. The longest of these depends strongly on changes in the environment, whereas the two shorter times show only weak dependencies. So we conclude that the longest time belongs to the rotational diffusion of the entire peptide and the other to internal motions.     

Pinning control of scale-free dynamical networks

Recently, it has been demonstrated that many large complex networks display a scale-free feature, that is, their connectivity distributions have the power-law form. In the present work, control of a scale-free dynamical network by applying local feedback injections to a fraction of network nodes is investigated. The specifically and randomly pinning schemes are considered. The specifically pinning of the most highly connected nodes is shown to require a significantly smaller number of local controllers as compared to the randomly pinning scheme. The method is applied to an array of Chua's oscillators as an example.     

Optimization-based structure identification of dynamical networks

The topological structure of a dynamical network plays a pivotal part in its properties, dynamics and control. Thus, understanding and modeling the structure of a network will lead to a better knowledge of its evolutionary mechanisms and to a better cottoning on its dynamical and functional behaviors. However, in many practical situations, the topological structure of a dynamical network is usually unknown or uncertain. Thus, exploring the underlying topological structure of a dynamical network is of great value. In recent years, there has been a growing interest in structure identification of dynamical networks. As a result, various methods for identifying the network structure have been proposed. However, in most of the previous work, few of them were discussed in the perspective of optimization. In this paper, an optimization algorithm based on the projected conjugate gradient method is proposed to identify a network structure. It is straightforward and applicable to networks with or without observation noise. Furthermore, the proposed algorithm is applicable to dynamical networks with partially observed component variables for each multidimensional node, as well as small-scale networks with time-varying structures. Numerical experiments are conducted to illustrate the good performance and universality of the new algorithm.     

The mean complexities in the regimes of dynamical networks with full oscillations binding

We continue to apply the notion of mean complexities to study dynamical networks. We show that the mean complexities can help to single out the nodes with similar features (and dynamical behavior) and to reveal some properties of the topology of the networks. We found that the nodes with the same degree (number of connections) have equal values of the mean complexities in the regime of full binding. At the same time, the mean complexities of nodes with different degree follow a descending order with respect to the degree.     

k-Core organization of complex networks

We analytically describe the architecture of randomly damaged uncorrelated networks as a set of successively enclosed substructures--k-cores. The k-core is the largest subgraph where vertices have at least k interconnections. We find the structure of k-cores, their sizes, and their birthpoints--the bootstrap percolation thresholds. We show that in networks with a finite mean number zeta2 of the second-nearest neighbors, the emergence of a k-core is a hybrid phase transition. In contrast, if zeta2 diverges, the networks contain an infinite sequence of k-cores which are ultrarobust against random damage.     

The risks and returns of stock investment in a financial market

The risks and returns of stock investment are discussed via numerically simulating the mean escape time and the probability density function of stock price returns in the modified Heston model with time delay. Through analyzing the effects of delay time and initial position on the risks and returns of stock investment, the results indicate that: (i) There is an optimal delay time matching minimal risks of stock investment, maximal average stock price returns and strongest stability of stock price returns for strong elasticity of demand of stocks (EDS), but the opposite results for weak EDS; (ii) The increment of initial position recedes the risks of stock investment, strengthens the average stock price returns and enhances stability of stock price returns. Finally, the probability density function of stock price returns and the probability density function of volatility and the correlation function of stock price returns are compared with other literatures. In addition, good agreements are found between them.     

Stabilization of avalanche processes on dynamical networks

The stabilization of avalanches on dynamical networks has been studied. Dynamical networks are networks where the structure of links varies in time owing to the presence of the individual “activity” of each site, which determines the probability of establishing links with other sites per unit time. An interesting case where the times of existence of links in a network are equal to the avalanche development times has been examined. A new mathematical model of a system with the avalanche dynamics has been constructed including changes in the network on which avalanches are developed. A square lattice with a variable structure of links has been considered as a dynamical network within this model. Avalanche processes on it have been simulated using the modified Abelian sandpile model and fixed-energy sandpile model. It has been shown that avalanche processes on the dynamical lattice under study are more stable than a static lattice with respect to the appearance of catastrophic events. In particular, this is manifested in a decrease in the maximum size of an avalanche in the Abelian sandpile model on the dynamical lattice as compared to that on the static lattice. For the fixed-energy sandpile model, it has been shown that, in contrast to the static lattice, where an avalanche process becomes infinite in time, the existence of avalanches finite in time is always possible.     

Topology identification of weighted complex dynamical networks

Recently, various papers investigated the geometry features, synchronization and control of complex network provided with certain topology. While, the exact topology of a network is sometimes unknown or uncertain. Using Lyapunov theory, we propose an adaptive feedback controlling method to identify the exact topology of a rather general weighted complex dynamical network model. By receiving the network nodes evolution, the topology of such kind of network with identical or different nodes, or even with switching topology can be monitored. Experiments show that the methods presented in this paper are of high accuracy with good performance.