Small-world Connections Research Articles

Network Origins of the Global Economy: East vs. West in a Complex Systems Perspective

Network Origins of the Global Economy: East vs. West in a Complex Systems Perspective

State Transitions During Discrimination Learning in the Gerbil Auditory Cortex Analyzed by Network Causality Metrics.

This work studies the evolution of cortical networks during the transition from escape strategy to avoidance strategy in auditory discrimination learning in Mongolian gerbils trained by the well-established two-way active avoidance learning paradigm. The animals were implanted with electrode arrays centered on the surface of the primary auditory cortex and electrocorticogram (ECoG) recordings were made during performance of an auditory Go/NoGo discrimination task. Our experiments confirm previous results on a sudden behavioral change from the initial naïve state to an avoidance strategy as learning progresses. We employed two causality metrics using Granger Causality (GC) and New Causality (NC) to quantify changes in the causality flow between ECoG channels as the animals switched to avoidance strategy. We found that the number of channel pairs with inverse causal interaction significantly increased after the animal acquired successful discrimination, which indicates structural changes in the cortical networks as a result of learning. A suitable graph-theoretical model is developed to interpret the findings in terms of cortical networks evolving during cognitive state transitions. Structural changes lead to changes in the dynamics of neural populations, which are described as phase transitions in the network graph model with small-world connections. Overall, our findings underscore the importance of functional reorganization in sensory cortical areas as a possible neural contributor to behavioral changes.

Synchronizability of directed networks: The power of non-existent ties.

The understanding of how synchronization in directed networks is influenced by structural changes in network topology is far from complete. While the addition of an edge always promotes synchronization in a wide class of undirected networks, this addition may impede synchronization in directed networks. In this paper, we develop the augmented graph stability method, which allows for explicitly connecting the stability of synchronization to changes in network topology. The transformation of a directed network into a symmetrized-and-augmented undirected network is the central component of this new method. This transformation is executed by symmetrizing and weighting the underlying connection graph and adding new undirected edges with consideration made for the mean degree imbalance of each pair of nodes. These new edges represent "non-existent ties" in the original directed network and often control the location of critical nodes whose directed connections can be altered to manipulate the stability of synchronization in a desired way. In particular, we show that the addition of small-world shortcuts to directed networks, which makes "non-existent ties" disappear, can worsen the synchronizability, thereby revealing a destructive role of small-world connections in directed networks. An extension of our method may open the door to studying synchronization in directed multilayer networks, which cannot be effectively handled by the eigenvalue-based methods.

Stability and instability of a neuron network with excitatory and inhibitory small-world connections

This study considers a delayed neural network with excitatory and inhibitory shortcuts. The global stability of the trivial equilibrium is investigated based on Lyapunov’s direct method and the delay-dependent criteria are obtained. It is shown that both the excitatory and inhibitory shortcuts decrease the stability interval, but a time delay can be employed as a global stabilizer. In addition, we analyze the bounds of the eigenvalues of the adjacent matrix using matrix perturbation theory and then obtain the generalized sufficient conditions for local stability. The possibility of small inhibitory shortcuts is helpful for maintaining stability. The mechanisms of instability, bifurcation modes, and chaos are also investigated. Compared with methods based on mean-field theory, the proposed method can guarantee the stability of the system in most cases with random events. The proposed method is effective for cases where excitatory and inhibitory shortcuts exist simultaneously in the network.

Griffiths phase on hierarchical modular networks with small-world edges.

The Griffiths phase has been proposed to induce a stretched critical regime that facilitates self-organizing of brain networks for optimal function. This phase stems from the intrinsic structural heterogeneity of brain networks, i.e., the hierarchical modular structure. In this work, the Griffiths phase is studied in modified hierarchical networks with small-world connections based on the 3-regular Hanoi network. Through extensive simulations, the hierarchical level-dependent inter-module wiring probabilities are identified to determine the emergence of the Griffiths phase. Numerical results and the complementary spectral analysis of the relevant networks can be helpful for a deeper understanding of the essential structural characteristics of finite-dimensional networks to support the Griffiths phase.

Stability, Instability and Bifurcation Modes of a Delayed Small World Network with Excitatory or Inhibitory Short-Cuts

This paper presents a detailed analysis on the stability and instability of a coupled oscillator network with small world connections. This network consists of regular connections, excitatory short-cuts or inhibitory short-cuts. By using the perturbation theory of matrix, we give the upper and lower bounds of maximum and minimum eigenvalues of the coupling strength matrix, and then give the generalized sufficient conditions that guarantee the system complete stability or complete instability. In addition, we analyze the effects of the short-cut possibility, excitatory or inhibitory short-cut strength and time delay on the system stability. We also analyze the instability mechanism and bifurcation modes. In addition, the studies on the robustness stability show that the stability of this network is more robust to change of short-cut connections than the regular network. Compared to the mean-field theory, the stability conditions from the proposed method are more conservational. However, the proposed method can guarantee the complete stability even if the randomness is in the system. They are more useful and adaptive than mean-field theory especially when the excitatory and inhibitory connections exist simultaneously.

Damage spreading in spatial and small-world random Boolean networks.

The study of the response of complex dynamical social, biological, or technological networks to external perturbations has numerous applications. Random Boolean networks (RBNs) are commonly used as a simple generic model for certain dynamics of complex systems. Traditionally, RBNs are interconnected randomly and without considering any spatial extension and arrangement of the links and nodes. However, most real-world networks are spatially extended and arranged with regular, power-law, small-world, or other nonrandom connections. Here we explore the RBN network topology between extreme local connections, random small-world, and pure random networks, and study the damage spreading with small perturbations. We find that spatially local connections change the scaling of the Hamming distance at very low connectivities (K ≪ 1) and that the critical connectivity of stability K(s) changes compared to random networks. At higher K, this scaling remains unchanged. We also show that the Hamming distance of spatially local networks scales with a power law as the system size N increases, but with a different exponent for local and small-world networks. The scaling arguments for small-world networks are obtained with respect to the system sizes and strength of spatially local connections. We further investigate the wiring cost of the networks. From an engineering perspective, our new findings provide the key design trade-offs between damage spreading (robustness), the network's wiring cost, and the network's communication characteristics.

Computer Simulations of Chua’s Circuits with Small-World Connection

Based on the two-dimensional array of Chua’s circuit, the complex dynamic phenomena of Chua’s circuits with small world connections was simulated. Such networks can be thought of as a model of nonlinear phenomena in spatially extended systems. And this means for understanding complex interactions existing in real systems where separate cells can communicate in various ways. In such small-world networks, the Chua’s circuit is taken as a standard chaotic cell. The results of computer simulation show that the spiral wave patterns could form in certain initial conditions in the presence of the small-world connection. In addition, it is found that the small-world connection could affect the pattern formation greatly.

Chaos in complex motor networks induced by Newman—Watts small-world connections

We investigate how dynamical behaviours of complex motor networks depend on the Newman—Watts small-world (NWSW) connections. Network elements are described by the permanent magnet synchronous motor (PMSM) with the values of parameters at which each individual PMSM is stable. It is found that with the increase of connection probability p, the motor in networks becomes periodic and falls into chaotic motion as p further increases. These phenomena imply that NWSW connections can induce and enhance chaos in motor networks. The possible mechanism behind the action of NWSW connections is addressed based on stability theory.

Channel noise-induced phase transition of spiral wave in networks of Hodgkin-Huxley neurons

The phase transition of spiral waves in networks of Hodgkin-Huxley neurons induced by channel noise is investigated in detail. All neurons in the networks are coupled with small-world connections, and the results are compared with the case for regular networks, in which all neurons are completely coupled with nearest-neighbor connections. A statistical variable is defined to study the collective behavior and phase transition of the spiral wave due to the channel noise and topology of the network. The effect of small-world connection networks is described by local regular networks and long-range connection with certain probability p. The numerical results confirm that (1) a stable rotating spiral wave can be developed and maintain robust with low p, where the breakup of the spiral wave and turbulence result from increasing the probability p to a certain threshold; (2) appropriate intensity of the optimized channel noise can develop a spiral wave among turbulent states in small-world connection networks of H-H neurons; and (3) regular connection networks are more robust to channel noise than small-world connection networks. A spiral wave in a small-world network encounters instability more easily as the membrane temperature is increased to a certain high threshold.

Topological probability and connection strength induced activity in complex neural networks

Recent experimental evidence suggests that some brain activities can be assigned to small-world networks. In this work, we investigate how the topological probability p and connection strength C affect the activities of discrete neural networks with small-world (SW) connections. Network elements are described by two-dimensional map neurons (2DMNs) with the values of parameters at which no activity occurs. It is found that when the value of p is smaller or larger, there are no active neurons in the network, no matter what the value of connection strength is; for a given appropriate connection strength, there is an intermediate range of topological probability where the activity of 2DMN network is induced and enhanced. On the other hand, for a given intermediate topological probability level, there exists an optimal value of connection strength such that the frequency of activity reaches its maximum. The possible mechanism behind the action of topological probability and connection strength is addressed based on the bifurcation method. Furthermore, the effects of noise and transmission delay on the activity of neural network are also studied.

SMALL PURE CARBON MOLECULES WITH SMALL-WORLD NETWORKS USING DENSITY FUNCTIONAL THEORY SIMULATIONS

The possible existence of small, pure carbon molecules based on small-world networks is addressed using density functional theory simulations. A ring of atoms with one or more small-world connections between pairs of non-nearest-neighbor sites was chosen for the network topology. The small-world connections are made with and without additional carbon atoms placed along the link. The energy per atom of these small-world carbon systems is compared with benchmark molecules such as the C 20 ring, bowl, and cage isomers, the C 60 Buckyball, monocyclic pure carbon rings ranging from C 4 to C 60, bare linear carbon chains ranging from C 2 to C 36, and various graphitic fragments without hydrogens. The results of the energy per atom for some of these small-world clusters provide an indication that such pure carbon molecules are reasonable for real world synthesis.

Small-World Connections to Induce Firing Activity and Phase Synchronization in Neural Networks

We investigate how the firing activity and the subsequent phase synchronization of neural networks with small-world topological connections depend on the probability p of adding-links. Network elements are described by two-dimensional map neurons (2DMNs) in a quiescent original state. Neurons burst for a given coupling strength when the topological randomness p increases, which is absent in a regular-lattice neural network. The bursting activity becomes frequent and synchronization of neurons emerges as topological randomness further increases. The maximal firing frequency and phase synchronization appear at a particular value of p. However, if the randomness p further increases, the firing frequency decreases and synchronization is apparently destroyed.

Predictive protocol of flocks with small-world connection pattern

By introducing a predictive mechanism with small-world connections, we propose a new motion protocol for self-driven flocks. The small-world connections are implemented by randomly adding long-range interactions from the leader to a few distant agents, namely, pseudoleaders. The leader can directly affect the pseudoleaders, thereby influencing all the other agents through them efficiently. Moreover, these pseudoleaders are able to predict the leader's motion several steps ahead and use this information in decision making towards coherent flocking with more stable formation. It is shown that drastic improvement can be achieved in terms of both the consensus performance and the communication cost. From the engineering point of view, the current protocol allows for a significant improvement in the cohesion and rigidity of the formation at a fairly low cost of adding a few long-range links embedded with predictive capabilities. Significantly, this work uncovers an important feature of flocks that predictive capability and long-range links can compensate for the insufficiency of each other. These conclusions are valid for both the attractive and repulsive swarm model and the Vicsek model.

STOCHASTIC RESONANCE IN HINDMARSH–ROSE NEURAL NETWORK WITH SMALL-WORLD CONNECTIONS

We study stochastic resonance (SR) in Hindmarsh–Rose (HR) neural network with small-world (SW) connections driven by external periodic stimulus, focusing on the dependence of properties of SR on the network structure parameters. It is found that, the SW neural network enhances SR compared with single neuron. By turning coupling strength c, two categories of SR were gained. With the connection-rewiring probability p increasing, the resonance curve becomes more and more sharp and the peak value increases gradually and then reaches saturation. The SW network enhances the SR peak value compared with regular network and widens resonance in ascending range compared with random network. When decreasing node degree k, the resonance range is enlarged, and the signal noise ratio (SNR) curve becomes a two peak one from a classic single peak SR curve, and then the stochastic resonance phenomenon almost disappears.

Coherence resonance in neural networks with small-world connections

The nonlinear response of small-world biological nerve network in the Hodgkin-Huxley model in the absence of external periodic signal is studied. Simulation results show the ordered degree of spike sequence may reach a maximum when the strength of noise is a certain limited value. Namely, coherence resonance phenomenon is produced in small-world biology nerve network. The number of minima of coherence resonance coefficient cv is more than one. This indicates that coherence resonance may occur in different scales of networks.

Patterned Interactions in Complex Systems: Implications for Exploration

Scholars who view organizational, social, and technological systems as sets of interdependent decisions have increasingly used simulation models from the biological and physical sciences to examine system behavior. These models shed light on an enduring managerial question: How much exploration is necessary to discover a good configuration of decisions? The models suggest that, as interactions across decisions intensify and local optima proliferate, broader exploration is required. The models typically assume, however, that the interactions among decisions are distributed randomly. Contrary to this assumption, recent empirical studies of real organizational, social, and technological systems show that interactions among decisions are highly patterned. Patterns such as centralization, small-world connections, power-law distributions, hierarchy, and preferential attachment are common. We embed such patterns into an NK simulation model and obtain dramatic results: Holding fixed the total number of interactions among decisions, a shift in the pattern of interaction can alter the number of local optima by more than an order of magnitude. Thus, the long-run value of broader exploration is significantly greater in the face of some interaction patterns than in the face of others. We develop simple, intuitive rules of thumb that allow a decision maker to examine two interaction patterns and determine which warrants greater investment in broad exploration. We also find that, holding fixed the interaction pattern, an increase in the number of interactions raises the number of local optima regardless of the pattern. This validates prior comparative static results with respect to the number of interactions, but highlights an important implicit assumption in earlier work—that the underlying interaction pattern remains constant as interactions become more numerous.

Stochastic Resonance in Neural Systems with Small-WorldConnections

We study the stochastic resonance (SR) in Hodgkin–Huxley (HH) neuralsystems with small-world (SW) connections under the noise synapticcurrent and periodic stimulus, focusing on the dependence of propertiesof SR on coupling strength c. It is found that there exists a criticalcoupling strength c* such that if c<c*, then the SR can appear onthe SW neural network. Especially, dependence of the critical couplingstrength c* on the number of neurons N shows the monotonic evenalmost linear increase of c* as N increases and c* on the SWnetwork is smaller than that on the random network. For the effect ofthe SW network on the phenomenon of SR, we show that decreasing theconnection-rewiring probability p of the network topology leadsto an enhancement of SR. This indicates that the SR on the SWnetwork is more prominent than that on the random network(p = 1.0). In addition, it is noted that the effect becomes remarkable ascoupling strength increases. Moreover, it is found that the SRweakens but resonance range becomes wider with the increase of c onthe SW neural network.

Traversal times for random walks on small-world networks

We study the mean traversal time for a class of random walks on Newman-Watts small-world networks, in which steps around the edge of the network occur with a transition rate that is different from the rate for steps across small-world connections. When f>>F, the mean time to traverse the network exhibits a transition associated with percolation of the random graph (i.e., small-world) part of the network, and a collapse of the data onto a universal curve. This transition was not observed in earlier studies in which equal transition rates were assumed for all allowed steps. We develop a simple self-consistent effective-medium theory and show that it gives a quantitatively correct description of the traversal time in all parameter regimes except the immediate neighborhood of the transition, as is characteristic of most effective-medium theories.

Magnetic small world nanomaterials: Physical small-world networks

The question addressed is whether magnetic materials based on physical small-world networks are possible. Physical constraints, such as uniform bond length and embedding in three dimensions, are the features added to make small-world networks physical. Results are presented to further determine if physical small-world networks can exist and the effect of the small-world connections on the critical phenomena of Ising models on such networks. Spectra of the Laplacian on randomly collapsed bead-chain networks are studied. The scaling function for the order parameter of an Ising model with physical small-world connections is presented.