Neural networks and chaos: construction, evaluation of chaotic networks, and prediction of chaos with multilayer feedforward networks

Many research works deal with chaotic neural networks for various fields of application. Unfortunately, up to now, these networks are usually claimed to be chaotic without any mathematical proof. The purpose of this paper is to establish, based on a rigorous theoretical framework, an equivalence between chaotic iterations according to Devaney and a particular class of neural networks. On the one hand, we show how to build such a network, on the other hand, we provide a method to check if a neural network is a chaotic one. Finally, the ability of classical feedforward multilayer perceptrons to learn sets of data obtained from a dynamical system is regarded. Various boolean functions are iterated on finite states. Iterations of some of them are proven to be chaotic as it is defined by Devaney. In that context, important differences occur in the training process, establishing with various neural networks that chaotic behaviors are far more difficult to learn.     

Emergent criticality through adaptive information processing in boolean networks

We study information processing in populations of boolean networks with evolving connectivity and systematically explore the interplay between the learning capability, robustness, the network topology, and the task complexity. We solve a long-standing open question and find computationally that, for large system sizes N, adaptive information processing drives the networks to a critical connectivity K(c)=2. For finite size networks, the connectivity approaches the critical value with a power law of the system size N. We show that network learning and generalization are optimized near criticality, given that the task complexity and the amount of information provided surpass threshold values. Both random and evolved networks exhibit maximal topological diversity near K(c). We hypothesize that this diversity supports efficient exploration and robustness of solutions. Also reflected in our observation is that the variance of the fitness values is maximal in critical network populations. Finally, we discuss implications of our results for determining the optimal topology of adaptive dynamical networks that solve computational tasks.     

Stability analysis of peer-to-peer networks against churn

Users of the peer-to-peer system join and leave the network randomly, which makes the overlay network dynamic and unstable in nature. In this paper, we propose an analytical framework to assess the robustness of p2p networks in the face of user churn. We model the peer churn through degree-independent as well as degree-dependent node failure. Lately, superpeer networks are becoming the most widely used topology among the p2p networks. Therefore, we perform the stability analysis of superpeer networks as a case study. We validate the analytically derived results with the help of simulation.      

Investigating the topology of interacting networks

Network theory provides various tools for investigating the structural or functional topology of many complex systems found in nature, technology and society. Nevertheless, it has recently been realised that a considerable number of systems of interest should be treated, more appropriately, as interacting networks or networks of networks. Here we introduce a novel graph-theoretical framework for studying the interaction structure between subnetworks embedded within a complex network of networks. This framework allows us to quantify the structural role of single vertices or whole subnetworks with respect to the interaction of a pair of subnetworks on local, mesoscopic and global topological scales. Climate networks have recently been shown to be a powerful tool for the analysis of climatological data. Applying the general framework for studying interacting networks, we introduce coupled climate subnetworks to represent and investigate the topology of statistical relationships between the fields of distinct climatological variables. Using coupled climate subnetworks to investigate the terrestrial atmosphere’s three-dimensional geopotential height field uncovers known as well as interesting novel features of the atmosphere’s vertical stratification and general circulation. Specifically, the new measure “cross-betweenness” identifies regions which are particularly important for mediating vertical wind field interactions. The promising results obtained by following the coupled climate subnetwork approach present a first step towards an improved understanding of the Earth system and its complex interacting components from a network perspective.     

Synchronization in complex networks with a modular structure

Networks with a community (or modular) structure arise in social and biological sciences. In such a network individuals tend to form local communities, each having dense internal connections. The linkage among the communities is, however, much more sparse. The dynamics on modular networks, for instance synchronization, may be of great social or biological interest. (Here by synchronization we mean some synchronous behavior among the nodes in the network, not, for example, partially synchronous behavior in the network or the synchronizability of the network with some external dynamics.) By using a recent theoretical framework, the master-stability approach originally introduced by Pecora and Carroll in the context of synchronization in coupled nonlinear oscillators, we address synchronization in complex modular networks. We use a prototype model and develop scaling relations for the network synchronizability with respect to variations of some key network structural parameters. Our results indicate that random, long-range links among distant modules is the key to synchronization. As an application we suggest a viable strategy to achieve synchronous behavior in social networks.     

The complexity and robustness of metro networks

Transportation systems, being real-life examples of networks, are particularly interesting to analyze from the viewpoint of the new and rapidly emerging field of network science. Two particular concepts seem to be particularly relevant: scale-free patterns and small-worlds. By looking at 33 metro systems in the world, this paper adapts network science methodologies to the transportation literature, and offers one application to the robustness of metros; here, metro refers to urban rail transit with exclusive right-of-way, whether it is underground, at grade or elevated. We find that most metros are indeed scale-free (with scaling factors ranging from 2.10 to 5.52) and small-worlds; they show atypical behaviors, however, with increasing size. In particular, the presence of transfer-hubs (stations hosting more than three lines) results in relatively large scaling factors. The analysis provides insights/recommendations for increasing the robustness of metro networks. Smaller networks should focus on creating transfer stations, thus generating cycles to offer alternative routes. For larger networks, few stations seem to detain a certain monopole on transferring, it is therefore important to create additional transfers, possibly at the periphery of city centers; the Tokyo system seems to remarkably incorporate these properties.     

Disorder and decision cost in spatial networks

We introduce the concept of decision cost of a spatial graph, which measures the disorder of a given network taking into account not only the connections between nodes but their position in a two-dimensional map. The influence of the network size is evaluated and we show that normalization of the decision cost allows us to compare the degree of disorder of networks of different sizes. Under this framework, we measure the disorder of the connections between airports of two different countries and obtain some conclusions about which of them is more disordered. The introduced concepts (decision cost and disorder of spatial networks) can easily be extended to Euclidean networks of higher dimensions, and also to networks whose nodes have a certain fitness property (i.e., one-dimensional).     

The impact of sampling density upon cortical network analysis: regions or points

The choice of representation has a fundamental influence on the network analysis results of an empirical data set. The answers to two basic questions - how to define a node and how to define an edge between a pair of nodes - are not obvious in the network analysis of brain imaging data. We considered the first question in the case of magnetic resonance imaging (MRI)-based cortical thickness networks. We selected network nodes to represent vertices of a cortical surface mesh or cortical brain regions. The first network represents the maximal level of detail available in the analysis of cortical thickness networks, while the latter network represents the typical level of detail in the current network analysis studies. We compared the network analysis results between these two representations. The basic network measures behaved approximately as expected when the level of detail increased. However, the overall connectivity of nodes was greater in the vertex level, degree of clustering was smaller in the vertex level, and the node centralities were different between the levels. Further, many parameters of vertex-level network were more robust to the selection of the correlation threshold used to define the edges of network. We conclude that albeit many qualitative network properties were consistent between the two resolution levels, the vertex-level resolution revealed details that were not visible at the regional-level networks, and this additional detail could be useful for some applications. Finally, a similar methodology as the one used here could be used to study effects of the sampling density in other brain-imaging-based networks, for example, in resting-state functional MRI.     

Complexity in human transportation networks: a comparative analysis of worldwide air transportation and global cargo-ship movements

We present a comparative network-theoretic analysis of the two largest global transportation networks: the worldwide air-transportation network (WAN) and the global cargo-ship network (GCSN). We show that both networks exhibit surprising statistical similarities despite significant differences in topology and connectivity. Both networks exhibit a discontinuity in node and link betweenness distributions which implies that these networks naturally segregate into two different classes of nodes and links. We introduce a technique based on effective distances, shortest paths and shortest path trees for strongly weighted symmetric networks and show that in a shortest path tree representation the most significant features of both networks can be readily seen. We show that effective shortest path distance, unlike conventional geographic distance measures, strongly correlates with node centrality measures. Using the new technique we show that network resilience can be investigated more precisely than with contemporary techniques that are based on percolation theory. We extract a functional relationship between node characteristics and resilience to network disruption. Finally we discuss the results, their implications and conclude that dynamic processes that evolve on both networks are expected to share universal dynamic characteristics.     

Patent citation network in nanotechnology (1976–2004)

The patent citation networks are described using critical node, core network, and network topological analysis. The main objective is understanding of the knowledge transfer processes between technical fields, institutions and countries. This includes identifying key influential players and subfields, the knowledge transfer patterns among them, and the overall knowledge transfer efficiency. The proposed framework is applied to the field of nanoscale science and engineering (NSE), including the citation networks of patent documents, submitting institutions, technology fields, and countries. The NSE patents were identified by keywords “full-text” searching of patents at the United States Patent and Trademark Office (USPTO). The analysis shows that the United States is the most important citation center in NSE research. The institution citation network illustrates a more efficient knowledge transfer between institutions than a random network. The country citation network displays a knowledge transfer capability as efficient as a random network. The technology field citation network and the patent document citation network exhibit a␣less efficient knowledge diffusion capability than a random network. All four citation networks show a tendency to form local citation clusters.     

Percolation on shopping and cashback electronic commerce networks

Many realistic networks live in the form of multiple networks, including interacting networks and interdependent networks. Here we study percolation properties of a special kind of interacting networks, namely Shopping and Cashback Electronic Commerce Networks (SCECNs). We investigate two actual SCECNs to extract their structural properties, and develop a mathematical framework based on generating functions for analyzing directed interacting networks. Then we derive the necessary and sufficient condition for the absence of the system-wide giant in- and out- component, and propose arithmetic to calculate the corresponding structural measures in the sub-critical and supercritical regimes. We apply our mathematical framework and arithmetic to those two actual SCECNs to observe its accuracy, and give some explanations on the discrepancies. We show those structural measures based on our mathematical framework and arithmetic are useful to appraise the status of SCECNs. We also find that the supercritical regime of the whole network is maintained mainly by hyperlinks between different kinds of websites, while those hyperlinks between the same kinds of websites can only enlarge the sizes of in-components and out-components.     

Measurement of the Spectral Efficiency of a Heterogeneous Network Architecture of the NG-PON Type for a Quasilinear Propagation Regime

The objective of future optical fiber networks is to provide an efficient infrastructure capable of supporting an increasing and variable number of data traffic generated by the diversification of applications with different speed requirements that the current legacy Line Speed networks Single Line Rate (SLR), with predefined modulation formats, cannot supply, because they do not offer enough flexibility to meet the requirements of the demands with such a wide range of granularities. Therefore, next-generation optical networks will be highly heterogeneous in nature, incorporating mixed modulation formats and Mixed Line Rates (MLR). In this work, an analysis of the measurement of the spectral efficiency of a heterogeneous network architecture of the next-generation passive optical network (NG-PON) type is reported for a quasilinear propagation regime through the use of the equation adapted from Shannon’s information theory and developed by the group from the GNTT Research of the University of Cauca, where it was found that it is better to transmit channels of 10 Gbps and 40 Gbps with robust modulations in MLR networks to make an improvement in the spectral efficiency of the network, achieving the same amount of information in a smaller bandwidth or more information in the same bandwidth.     

Combinatorial explosion in model gene networks

The explosive growth in knowledge of the genome of humans and other organisms leaves open the question of how the functioning of genes in interacting networks is coordinated for orderly activity. One approach to this problem is to study mathematical properties of abstract network models that capture the logical structures of gene networks. The principal issue is to understand how particular patterns of activity can result from particular network structures, and what types of behavior are possible. We study idealized models in which the logical structure of the network is explicitly represented by Boolean functions that can be represented by directed graphs on n-cubes, but which are continuous in time and described by differential equations, rather than being updated synchronously via a discrete clock. The equations are piecewise linear, which allows significant analysis and facilitates rapid integration along trajectories. We first give a combinatorial solution to the question of how many distinct logical structures exist for n-dimensional networks, showing that the number increases very rapidly with n. We then outline analytic methods that can be used to establish the existence, stability and periods of periodic orbits corresponding to particular cycles on the n-cube. We use these methods to confirm the existence of limit cycles discovered in a sample of a million randomly generated structures of networks of 4 genes. Even with only 4 genes, at least several hundred different patterns of stable periodic behavior are possible, many of them surprisingly complex. We discuss ways of further classifying these periodic behaviors, showing that small mutations (reversal of one or a few edges on the n-cube) need not destroy the stability of a limit cycle. Although these networks are very simple as models of gene networks, their mathematical transparency reveals relationships between structure and behavior, they suggest that the possibilities for orderly dynamics in such networks are extremely rich and they offer novel ways to think about how mutations can alter dynamics. (c) 2000 American Institute of Physics.     

Pitman-Yor process mixture model for community structure exploration considering latent interaction patterns

The statistical model for community detection is a promising research area in network analysis. Most existing statistical models of community detection are designed for networks with a known type of community structure, but in many practical situations, the types of community structures are unknown. To cope with unknown community structures, diverse types should be considered in one model. We propose a model that incorporates the latent interaction pattern, which is regarded as the basis of constructions of diverse community structures by us. The interaction pattern can parameterize various types of community structures in one model. A collapsed Gibbs sampling inference is proposed to estimate the community assignments and other hyper-parameters. With the Pitman-Yor process as a prior, our model can automatically detect the numbers and sizes of communities without a known type of community structure beforehand. Via Bayesian inference, our model can detect some hidden interaction patterns that offer extra information for network analysis. Experiments on networks with diverse community structures demonstrate that our model outperforms four state-of-the-art models.     

Social networks: Ideas for future research from the vantage point of political science

In this short piece, Bunce and Csanadi draw upon their expertise in political science and political economy to offer some observations about the analysis of social networks. Using both examples and questions they highlight the importance of structural variations in networks, including differences in the motivations behind network formation; the subsequent development of networks, including extension, contraction and duration; and the effects of individual decision-makers on network dynamics and, at the same time, the effects of network structure and dynamics on individual decision-makers.     

Networks, dynamics, and modularity

The identification of general principles relating structure to dynamics has been a major goal in the study of complex networks. We propose that the special case of linear network dynamics provides a natural framework within which a number of interesting yet tractable problems can be defined. We report the emergence of modularity and hierarchical organization in evolved networks supporting asymptotically stable linear dynamics. Numerical experiments demonstrate that linear stability benefits from the presence of a hierarchy of modules and that this architecture improves the robustness of network stability to random perturbations in network structure. This work illustrates an approach to network science which is simultaneously structural and dynamical in nature.     

Spectral coarse graining and synchronization in oscillator networks

Coarse graining techniques offer a promising alternative to large-scale simulations of complex dynamical systems, as long as the coarse-grained system is truly representative of the initial one. Here, we investigate how the dynamical properties of oscillator networks are affected when some nodes are merged together to form a coarse-grained network. Moreover, we show that there exists a way of grouping nodes preserving as much as possible some crucial aspects of the network dynamics. This coarse graining approach provides a useful method to simplify complex oscillator networks, and more generally, networks whose dynamics involves a Laplacian matrix.     

Invariants of Spin Networks Embedded in Three-Manifolds

We study the invariants of spin networks embedded in a three-dimensional manifold which are based on the path integral for SU(2) BF-Theory. These invariants appear naturally in Loop Quantum Gravity, and have been defined as spin-foam state sums. By using the Chain-Mail technique, we give a more general definition of these invariants, and show that the state-sum definition is a special case. This provides a rigorous proof that the state-sum invariants of spin networks are topological invariants. We derive various results about the BF-Theory spin network invariants, and we find a relation with the corresponding invariants defined from Chern-Simons Theory, i.e. the Witten-Reshetikhin-Turaev invariants. We also prove that the BF-Theory spin network invariants coincide with V. Turaev’s definition of invariants of coloured graphs embedded in 3-manifolds and thick surfaces, constructed by using shadow-world evaluations. Our framework therefore provides a unified view of these invariants.     

Compressed sensing in random access networks with applications to underwater monitoring

For networks that are deployed for long-term monitoring of environmental phenomena, it is of crucial importance to design an efficient data gathering scheme that prolongs the life-time of the network. To this end, we consider a Random Access Compressed Sensing (RACS) scheme that considerably reduces the power and bandwidth usage of a large network. Motivated by underwater applications, we propose a continuous-time RACS that eliminates the need for synchronization and scheduling which are difficult to achieve in a distributed acoustic network. We provide an analytical framework for system design that ensures fast recovery and power-efficiency. Through analysis and examples, we demonstrate that recovery of the field can be attained using only a fraction of the resources used by a conventional TDMA network, while employing a scheme which is simple to implement.