On controlling dynamic complex networks

A general framework applicable for characterizing dynamic complex networks is presented. The framework 1) incorporates a revised Kuramoto model to define constituent dynamics, 2) explores information entropy for the description of global ensemble behaviors, 3) defines the variation of the state of connected constituents using energy, and 4) introduces two new time-dependent parameters, i.e., degrees of coupling, to delineate the extent to which the state of one constituent impacts the other. Information entropy which defines the randomness of constituent energy at the microscopic level provides a definitive measure for the ensemble dynamics at the macroscopic level. Whether a dynamic complex network is evolving toward synchronization or deteriorating and collapsing can be determined by tracking ensemble entropy in time. Two popular topological network structures are examined under the framework for their respective network responses. It is found that, in addition to misrepresenting the true network dynamics, static network structures do not differentiate themselves in resolving network properties such as average path length and degree distribution, thus rendering similar interpretations for the underlying network.

A betweenness structural entropy of complex networks

One of the most remarkable features in “Life” or “Living Systems” is the coexistence between evolvability and robustness. Diversity of cell types in organisms is such a typical example. Indeed, Embryo-Stem (ES) cells differentiate into various types of cells with different phenotypes depending on the multiple environmental factors, and the cells can repeat the stable cell division. It is an irreversible transition in evolution as pointed out by Smith and Szathonany.1) The process of cell differentiation obeys a dynamical rule of the gene-gene interaction. Dynamics of Boolean networks has been studied as a simple model for variety of cell type in genetic networks since a pioneering work by Kauffman.2),3) A Boolean network consists of N nodes, each of which receives ki inputs such that the degree is ki. In the so-called Kauffman model — random Boolean network (RBN) model, each node receives a certain fixed number of inputs such that the degree is ki ≡ K. However, as a more realistic modeling of biological systems we expect that the fluctuation of the number of input-degree is treated as a random variable with a probability distribution function such as the inverse power-law distribution or the exponential distribution or the Poisson distribution. Actually it is demonstrated that the in-degree distribution appears to be exponential in E.coil and to be inverse power-law in yeast.4)–13) The relationship between the stability of network structure and the robustness of the attractors to the perturbation is very important for understanding the realistic gene regulatory networks (GRN) must be mutually correlated with each others. However, we deal with simple models in a sense that we simulate the Boolean dynamics in the fixed network structure and it does not influence the network growth. In the present paper, we investigate some intrinsic properties of attractor states in Boolean dynamics in complex networks with the fluctuation of the number of

This study presents a proposition for describing the dynamics of real-world networks under the general framework of complex networks. Outward behaviors of complex networks are the manifestation of the coupled dynamics at the macroscopic level and the individual dynamics at the microscopic level. At the macroscopic level a law of coupling governs the interactions of network constituents. At the microscopic level, the dynamics of individual constituent is defined by energy that follows normal distribution. Constituent dynamics are bounded by physical constraints. Consequently, network dynamics can be quantified using information entropy which is a function of constituent energy. In real-world networks, differences between individual constituents exist due to differing mechanical properties and dynamics. Consequently, network dynamics are of different layers and hierarchies. Construct of network governing equations formulated under the general framework of complex networks are demonstrated using two real-world networks — a brain network and a lymph node network. Brain network is constructed by the neurons that each connected by the synapse. Brain network dynamics is composed by the law of coupling defined by the synaptic dynamics through the transmitting of neurotransmitters that couples the individual neuron dynamics. Since different classifications exist among neurotransmitters and neurons, the post synaptic neuron can present either inhibitory or excitatory action. The inhibitory and excitatory behavior of the neurons changes the mechanical properties of each neuron and further alters the brain network dynamics. Consequently, the brain network emerges dynamics with different layers. Lymph node network drains fluid from blood vessels, filter the lymph (the interstitial fluid lymphatic system collects from the blood circulation) through lymph nodes, and transport the lymph back to the blood circulation. Lymph node dynamics is composed by the dynamics of lymph transportation along the lymph node network and the individual lymph node dynamics that involves lymphocytes-pathogens interactions (adaptive immune response). In each lymph node, lymphocytes fight off the pathogens which also emerges a network dynamics such as the interaction between T cells and HIV viruses. Finally, the lymph is collected from each lymph nodes and drained back to the blood circulation. As a result, the lymph node network has the dynamics of different hierarchies where the lymphocytes-pathogens dynamics exists within each lymph node at the lower hierarchy is further under the influence of the lymph transportation dynamics among the whole lymph node network on the higher hierarchy. Since the constituent dynamics of the brain network and lymph node network can be defined by energy that follows normal distribution and both are bounded by physical constraints, the network dynamics of both cases can be quantified through information entropy. Features pertaining to the global as well as individual constituent dynamics of the networks are identified that are insightful to the control of such complex networks.

Social organisms often show collective behaviors such as group foraging or movement. Collective behaviors can emerge from interactions between group members and may depend on the behavior of key individuals. When social interactions change over time, collective behaviors may change because these behaviors emerge from interactions among individuals. Despite the importance of, and growing interest in, the temporal dynamics of social interactions, it is not clear how to quantify changes in interactions over time or measure their stability. Furthermore, the temporal scale at which we should observe changes in social networks to detect biologically meaningful changes is not always apparent. Here we use multilayer network analysis to quantify temporal dynamics of social networks of the social spider Stegodyphus dumicola and determine how these dynamics relate to individual and group behaviors. We found that social interactions changed over time at a constant rate. Variation in both network structure and the identity of a keystone individual was not related to the mean or variance of the collective prey attack speed. Individuals that maintained a large and stable number of connections, despite changes in network structure, were the boldest individuals in the group. Therefore, social interactions and boldness are linked across time, but group collective behavior is not influenced by the stability of the social network. Our work demonstrates that dynamic social networks can be modeled in a multilayer framework. This approach may reveal biologically important temporal changes to social structure in other systems.

Exploiting local and repeated structure in Dynamic Bayesian Networks

The discovery of communities in complex networks is a challenging problem with various applications in the real world. Classic examples of networks include transport networks, the immune system, human brain and social networks. Given a certain grouping of nodes into communities, a good measure is needed to evaluate the quality of the community structure based on the definition that a strong community has dense intra-connections and sparse outside community links. This paper investigates several fitness functions in an evolutionary approach to community detection in complex networks. Moreover, these fitness functions are used to study dynamic networks using an extended evolutionary algorithm designed to handle changes in the network structure. Computational experiments are performed for several real-world networks which have a known community structure and thus can be evaluated. The obtained results confirm the ability of the proposed method to efficiently detect communities for both static and dynamic complex networks.

Different modelling approaches have been used to relate the structure of mutualistic interactions with the stability of communities. However, inconsistencies arise when we compare modelling outcomes with the patterns of interactions observed in empirical studies. To shed light on these inconsistencies, we explored the network structure–stability relationship by incorporating the cost of mutualistic interactions, a long ignored feature of mutualisms, into population dynamics models. We assessed the changes in the relationship between network structure (species richness, connectance, modularity) and community stability (species persistence, resilience), and between network structure and community structural attributes (average abundance), using models with increasing levels of cost for mutualistic communities. We found that adding the potential cost of mutualistic interactions affected the strength of the network structure–stability relationship. Our results revive the question of whether the structure of mutualistic networks determines community stability.

Research on inter-organizational conflicts among the participants of construction projects typically concentrate on the binary perspective of the project owner and contractor, disregarding the correlation of the complex relationship networks on conflict incidents. This research will propose a systematic set of strategies for reducing inter-organizational conflicts and enhancing project performance from the perspective of trust networks governance. Specifically, the research aims to uncover the interactive mechanisms among the trust network of all participants in construction projects, inter-organizational conflicts, and project performance, thereby filling the existing research gaps. The PLS-SEM approach was employed to analyze 207 valid questionnaires from the participants of construction projects. Data analysis revealed that both the density and stability of the trust networks can restrain various forms of inter-organizational conflicts, and inter-organizational conflicts can impede the attainment of project performance. The density of trust networks can enhance project performance, yet the extent to which the stability of network structure is related to this aspect remains unproven. These findings are conducive to optimizing the performance level of engineering projects and reducing the probability of inter-organizational conflicts.

Complex dynamical networks are ubiquitous in many fields of science from engineering to biology, physics, and sociology. Collective behavior, and in particular synchronization,) is one of the most interesting consequences of interaction of dynamical systems over complex networks. In this thesis we study some aspects of synchronization in dynamical networks. The first section of the study discuses the problem of synchronizability in dynamical networks. Although synchronizability, i.e. the ease by which interacting dynamical systems can synchronize their activity, has been frequently used in research studies, there is no single interpretation for that. Here we give some possible interpretations of synchronizability and investigate to what extent they coincide. We show that in unweighted dynamical networks different interpretations of synchronizability do not lie in the same line, in general. However, in networks with high degrees of synchronization properties, the networks with properly assigned weights for the links or the ones with well-performed link rewirings, the different interpretations of synchronizability go hand in hand. We also show that networks with nonidentical diffusive connections whose weights are assigned using the connection-graph-stability method are better synchronizable compared to networks with identical diffusive couplings. Furthermore, we give an algorithm based on node and edge betweenness centrality measures to enhance the synchronizability of dynamical networks. The algorithm is tested on some artificially constructed dynamical networks as well as on some real-world networks from different disciplines. In the second section we study the synchronization phenomenon in networks of Hindmarsh-Rose neurons. First, the complete synchronization of Hindmarsh-Rose neurons over Newman-Watts networks is investigated. By numerically solving the differential equations of the dynamical network as well as using the master-stability-function method we determine the synchronizing coupling strength for diffusively coupled Hindmarsh-Rose neurons. We also consider clustered networks with dense intra-cluster connections and sparse inter-cluster links. In such networks, the synchronizability is more influenced by the inter-cluster links than intra-cluster connections. We also consider the case where the neurons are coupled through both electrical and chemical connections and obtain the synchronizing coupling strength using numerical calculations. We investigate the behavior of interacting locally synchronized gamma oscillations. We construct a network of minimal number of neurons producing synchronized gamma oscillations. By simulating giant networks of this minimal module we study the dependence of the spike synchrony on some parameters of the network such as the probability and strength of excitatory/inhibitory couplings, parameter mismatch, correlation of thalamic input and transmission time-delay. In the third section of the thesis we study the interdependencies within the time series obtained through electroencephalography (EEG) and give the EEG specific maps for patients suffering from schizophrenia or Alzheimer's disease. Capturing the collective coherent spatiotemporal activity of neuronal populations measured by high density EEG is addressed using measures estimating the synchronization within multivariate time series. Our EEG power analysis on schizophrenic patients, which is based on a new parametrization of the multichannel EEG, shows a relative increase of power in alpha rhythm over the anterior brain regions against its reduction over posterior regions. The correlations of these patterns with the clinical picture of schizophrenia as well as discriminating of the schizophrenia patients from normal control subjects supports the concept of hypofrontality in schizophrenia and renders the alpha rhythm as a sensitive marker of it. By applying a multivariate synchronization estimator, called S-estimator, we reveal the whole-head synchronization topography in schizophrenia. Our finding shows bilaterally increased synchronization over temporal brain regions and decreased synchronization over the postcentral/parietal brain regions. The topography is stable over the course of several months as well as over all conventional EEG frequency bands. Moreover, it correlates with the severity of the illness characterized by positive and negative syndrome scales. We also reveal the EEG features specific to early Alzheimer's disease by applying multivariate phase synchronization method. Our analyses result in a specific map characterized by a decrease in the values of phase synchronization over the fronto-temporal and an increase over temporo-parieto-occipital region predominantly of the left hemisphere. These abnormalities in the synchronization maps correlate with the clinical scores associated to the patients and are able to discriminate patients from normal control subjects with high precision.

The Potts model is a powerful tool to uncover community structure in complex networks. Here, we propose a framework to reveal the optimal number of communities and stability of network structure by quantitatively analyzing the dynamics of the Potts model. Specifically we model the community structure detection Potts procedure by a Markov process, which has a clear mathematical explanation. Then we show that the local uniform behavior of spin values across multiple timescales in the representation of the Markov variables could naturally reveal the network's hierarchical community structure. In addition, critical topological information regarding multivariate spin configuration could also be inferred from the spectral signatures of the Markov process. Finally an algorithm is developed to determine fuzzy communities based on the optimal number of communities and the stability across multiple timescales. The effectiveness and efficiency of our algorithm are theoretically analyzed as well as experimentally validated.

Intelligent systems cover a wide range of technologies related to hard sciences, such as modeling and control theory, and soft sciences, such as the artificial intelligence (AI). Intelligent systems, including neural networks (NNs), fuzzy logic (FL), and wavelet techniques, utilize the concepts of biological systems and human cognitive capabilities. These three systems have been recognized as a robust and attractive alternative to the some of the classical modeling and control methods. The application of classical NNs, FL, and wavelet technology to dynamic system modeling and control has been constrained by the non-dynamic nature of their popular architectures. The major drawbacks of these architectures are the curse of dimensionality, such as the requirement of too many parameters in NNs, the use of large rule bases in FL, the large number of wavelets, and the long training times, etc. These problems can be overcome with dynamic network structures, referred to as dynamic neural networks (DNNs), dynamic fuzzy networks (DFNs), and dynamic wavelet networks (DWNs), which have unconstrained connectivity and dynamic neural, fuzzy, and wavelet processing units, called “neurons”, “feurons”, and “wavelons”, respectively. The structure of dynamic networks are based on Hopfield networks. Here, we present a comparative study of DNNs, DFNs, and DWNs for non-linear dynamical system modeling. All three dynamic networks have a lag dynamic, an activation function, and interconnection weights. The network weights are adjusted using fast training (optimization) algorithms (quasi-Newton methods). Also, it has been shown that all dynamic networks can be effectively used in non-linear system modeling, and that DWNs result in the best capacity. But all networks have non-linearity properties in non-linear systems. In this study, all dynamic networks are considered as a non-linear optimization with dynamic equality constraints for non-linear system modeling. They encapsulate and generalize the target trajectories. The adjoint theory, whose computational complexity is significantly less than the direct method, has been used in the training of the networks. The updating of weights (identification of network parameters) is based on Broyden–Fletcher–Goldfarb–Shanno method. First, phase portrait examples are given. From this, it has been shown that they have oscillatory and chaotic properties. A dynamical system with discrete events is modeled using the above network structure. There is a localization property at discrete event instants for time and frequency in this example.

Complexity is highly susceptible to variations in the network dynamics, reflected on its underlying architecture where topological organization of cohesive subsets into clusters, system’s modular structure and resulting hierarchical patterns, are cross-linked with functional dynamics of the system. Here we study connection between hierarchical topological scales of the simplicial complexes and the organization of functional clusters – communities in complex networks. The analysis reveals the full dynamics of different combinatorial structures of q-th-dimensional simplicial complexes and their Laplacian spectra, presenting spectral properties of resulting symmetric and positive semidefinite matrices. The emergence of system’s collective behavior from inhomogeneous statistical distribution is induced by hierarchically ordered topological structure, which is mapped to simplicial complex where local interactions between the nodes clustered into subcomplexes generate flow of information that characterizes complexity and dynamics of the full system.

The advent of sophisticated molecular biology techniques allows to deduce the structure of complex biological networks. However, networks tend to be huge and impose computational challenges on traditional mathematical analysis due to their high dimension and lack of reliable kinetic data. To overcome this problem, complex biological networks are decomposed into modules that are assumed to capture essential aspects of the full network's dynamics. The question that begs for an answer is how to identify the core that is representative of a network's dynamics, its function and robustness. One of the powerful methods to probe into the structure of a network is Petri net analysis. Petri nets support network visualization and execution. They are also equipped with sound mathematical and formal reasoning based on which a network can be decomposed into modules. The structural analysis provides insight into the robustness and facilitates the identification of fragile nodes. The application of these techniques to a previously proposed hypoxia control network reveals three functional modules responsible for degrading the hypoxia-inducible factor (HIF). Interestingly, the structural analysis identifies superfluous network parts and suggests that the reversibility of the reactions are not important for the essential functionality. The core network is determined to be the union of the three reduced individual modules. The structural analysis results are confirmed by numerical integration of the differential equations induced by the individual modules as well as their composition. The structural analysis leads also to a coarse network structure highlighting the structural principles inherent in the three functional modules. Importantly, our analysis identifies the fragile node in this robust network without which the switch-like behavior is shown to be completely absent.

The spread and evolution of web hot events is due to the interaction between users. And the behaviour of users is particularly important in micro log. The information spread through the behaviour of users and information exchange among users lead to event evolution. So the behaviour of users plays an important role in information spread and evolution of web hot event. The analysis of behaviour is a key method to obtain the development direction and trend of web hot events. However, it is a challenge to quantify the behaviour of users, especially to properly predict the direction and trend of web hot events according to the behaviour of users. This paper proposes a way to build information spread network according to behaviour of users and have a statistics of degrees of node in network. Then we use information entropy to measure the stability of behaviour network according to node degrees. And we can predict the direction and trend of web hot events by their information entropy. Experimental results show that information entropy can efficiently measure the stability of network, popular events have high entropy and their structure of network is unstable, normal events have low entropy and their structure is stable, which means events with high entropy are more likely to be popular events and may evolution in different directions.