The Analysis of Complex Structure for China Education Network

The previous research on the controllability of complex networks is qualitative issue which can only identify whether the network is controllable or not, and it is difficult to assess the controlled ability of network. In this paper, a quantitative measurement index of network controllability is introduced. And the index is used to study the controllability of the IEEE power network system. The relationship between controllability and network parameters (including the average degree, the clustering coefficient, the average path length) of power network is analyzed by comparing with the ER random network, WS small world network, BA scale-free network and CM network. The results show that the power networks have a small average path length and a higher clustering coefficient, which belong to the small world network. And, the controllability of power networks is larger than the corresponding model networks, indicating that power networks are relatively easier to be controlled.

Based on the Koch network constructed using Koch fractals, we proposed a class of expanded Koch networks in this paper. The original triangle is replaced by r-polygon, and each node generates m sub r-polygons by every step, which makes the Koch network more general. We studied the structure and properties of the networks. The exact analytical result of the degree distribution, clustering coefficient and average path length were obtained. When parameters m and r satisfy some certain conditions, the networks follow a power-law distribution and have a small average path length. Finally, we introduced the random walk on the network. Our discussions focused on the trapping problem, particularly the calculation and derivation of mean first passage time (MFPT) and global mean first passage time (GMFPT). In addition, we also gave the relationship between the above results and the network size.

The propagation of various opinions in social networks, which influences human inter-relationships and even social structure, and hence is a most important part of social life. We have incorporated social balance into opinion propagation in social networks are influenced by social balance. The edges in networks can represent both friendly or hostile relations, and change with the opinions of individual nodes. We introduce a model to characterize the coevolutionary dynamics of these two dynamical processes on Watts-Strogatz (WS) small-world network. We employ two distinct evolution rules (i) opinion renewal; and (ii) relation adjustment. By changing the rewiring probability, and thus the small-worldness of the WS network, we found that the time for the system to reach balanced states depends critically on both the average path length and clustering coefficient of the network, which is different than other networked process like epidemic spreading. In particular, the system equilibrates most quickly when the underlying network demonstrates strong small-worldness, i.e., small average path lengths and large clustering coefficient. We also find that opinion clusters emerge in the process of the network approaching the global equilibrium, and a measure of global contrariety is proposed to quantify the balanced state of a social network.

It is highly desirable and challenging for a wireless ad hoc network to have self-organization properties in order to achieve wide network characteristics. Studies have shown that Small-World properties, primarily low average path length (APL) and high clustering coefficient, are desired properties for networks in general. However, due to the spatial nature of the wireless networks, achieving small-world properties remains highly challenging. Studies also show that, wireless ad hoc networks with small-world properties show a degree of distribution that lies between geometric and power law. In this paper, we show that in a wireless ad hoc network with non-uniform node density with only local information, we can significantly reduce the APL and retain the clustering coefficient. To achieve our goal, our algorithm first identifies logical regions using the Lateral Inhibition technique, then identifies the nodes that beamform and finally the beam properties using Flocking. We use Lateral Inhibition and Flocking because they enable us to use local state information as opposed to other techniques. We support our work with simulation results and analysis, which show that a reduction of up to 40% can be achieved for a high-density network. We also show the effect of hopcount used to create regions on APL, clustering coefficient and connectivity.

A novel edge rewiring strategy for tuning structural properties in networks

Research shows the degree distribution of firms' innovation network in accordance with power-law distribution as continuous evolution of firms' innovation network, which has larger clustering coefficient and smaller average path length. Therefore, the network structure and simulation results of two stages are compared with the data of Taiyuan stainless steel firms' innovation network. The size of Taiyuan stainless steel firms' innovation network is constantly expanding, so is the clustering coefficient, while the average path length is constantly shrinking.

Functional restoration of the cerebral cortex relies on activity-dependent neuroplasticity following stroke. However, optimizing robotic rehabilitation for hand recovery remains a significant challenge. This proof-of-concept study investigates the feasibility of a configurable soft pneumatic robot in modulating the brain network among ten participants with hand motor impairments after stroke. The programmable robotic intervention was administered in a randomized sequence for resting-state assessment, slow-mode, and fast-mode robotic therapy by adjusting working mode, action time, and interaction duration. Functional near-infrared spectroscopy (fNIRS) was employed to measure cortical activity and functional connectivity. Additionally, graph theory metrics, including clustering coefficient, average path length, small-world index, global efficiency, degree centrality, and eigenvector centrality, were derived from fNIRS-based functional connectivity matrices. The results demonstrated that robotic intervention significantly improved clustering coefficient (P = 0.034), average path length (P = 0.007), and global efficiency (P = 0.001). While small-world index, degree centrality, and eigenvector centrality showed an increasing trend, these differences did not reach statistical significance. Moreover, fast-mode therapy induced more substantial changes in clustering coefficient compared to slow-mode therapy, suggesting a potentially stronger effect on neural reorganization. These findings preliminarily support the use of soft robotics with adjustable paradigms to enhance brain network connectivity and facilitate neuroplasticity following stroke. The observed improvements in global efficiency and small-world properties indicate that robotic therapy optimizes cortical organization, promoting functional recovery. Further studies with larger sample sizes and personalized intervention protocols are needed to confirm these results and explore their long-term effects.

In this paper, we investigate the abnormalities of electroencephalograph (EEG) signals in the Alzheimer's disease (AD) by analyzing 16-scalp electrodes EEG signals and make a comparison with the normal controls. Coherence is introduced to measure the pair-wise normalized linear synchrony and functional correlations between two EEG signals in different frequency domains, and graph analysis is further used to investigate the influence of AD on the functional connectivity of human brain. Data analysis results show that, compared with the control group, the pair-wise coherence of AD group is significantly decreased, especially for the theta and alpha frequency bands in the frontal and parieto-occipital regions. Furthermore, functional connectivity among different brain regions is reconstructed based on EEG, which exhibit obvious small-world properties. Graph analysis demonstrates that the local functional connections between regions for AD decrease. In addition, it is found that small-world properties of AD networks are largely weakened, by calculating its average path lengths, clustering coefficients, global efficiency, local efficiency, and small-worldness. The obtained results show that both pair-wise coherence and functional network can be taken as effective measures to distinguish AD patients from the normal, which may benefit our understanding of the disease.

Dürer’s pentagon is known to the artist Albrecht Dürer. In this paper, we consider directed networks generated by Dürer-type polygons. We aim to present a study on some properties of these networks, such as degree distribution, clustering coefficient and average path length. We show that such networks have the scale-free effect, but do not have the small-world effect.

We have obtained exact results for the Ising model on a hierarchical lattice incorporating three key features characterizing many real-world networks--a scale-free degree distribution, a high clustering coefficient, and the small-world effect. By varying the probability p of long-range bonds, the entire spectrum from an unclustered, non-small-world network to a highly clustered, small-world system is studied. Using the self-similar structure of the network, we obtain analytic expressions for the degree distribution P(k) and clustering coefficient C for all p, as well as the average path length l for p = 0 and 1. The ferromagnetic Ising model on this network is studied through an exact renormalization-group transformation of the quenched bond probability distribution, using up to 562,500 renormalized probability bins to represent the distribution. For p < 0.494, we find power-law critical behavior of the magnetization and susceptibility, with critical exponents continuously varying with p, and exponential decay of correlations away from Tc. For p > or = 0.494, in fact where the network exhibits small-world character, the critical behavior radically changes: We find a highly unusual phase transition, namely an inverted Berezinskii-Kosterlitz-Thouless singularity, between a low-temperature phase with nonzero magnetization and finite correlation length and a high-temperature phase with zero magnetization and infinite correlation length, with power-law decay of correlations throughout the phase. Approaching Tc from below, the magnetization and the susceptibility, respectively, exhibit the singularities of exp(-C/square root of Tc - T) and exp(D/square root of Tc - T), with C and D positive constants. With long-range bond strengths decaying with distance, we see a phase transition with power-law critical singularities for all p, and evaluate an unusually narrow critical region and important corrections to power-law behavior that depend on the exponent characterizing the decay of long-range interactions.

Small-world networks are an important type of complex network structures with small average shortest path lengths and large average clustering coefficients. A variety of generation algorithms for small-world networks have been given in existing studies, but less attention has been paid to how to optimize the small-world property of generative networks. In this paper, considering multiple objective optimization problems, small-worldness is defined to evaluate the small-world properties of generative networks, and Q-learning-based algorithms for small-world network generation are proposed and compared with WS small-world networks. Compared with the WS small-world network, the network generated by the Q-learning algorithm performs better in each metric.

Graph theory has been widely used to study structural and functional brain connectivity changes in healthy aging, and occasionally with clinical samples; in both cases, during task-related and resting-state experiments. Recent studies have focused their interest on dynamic changes during a resting-state fMRI register in order to identify differences in non-stationary patterns associated with the aging process. The objective of this study was to characterize resting-state fMRI network dynamics in order to study the healthy aging process. 114 healthy older adults were measured in a resting-state paradigm using fMRI. A sliding-window approach to graph theory was used to measure the mean degree, average path length, clustering coefficient, and small-worldness of each subnetwork, and the impact of age and time in each graph measure was assessed. A combined effect of age and time was detected in mean degree, average path length, and small-worldness, where participants aged 75 to 79 showed a curvilinear trend with reduced network density and increased small-world coefficient in the middle of the register. An effect of age was observed on average path length, with younger participants showing slightly lower scores.

The increased usage of directional methods of communications to improve medium reuse, network capacity, and bandwidth has prompted research into leveraging directionality in every layer of the network stack. Recently, there has been work on bringing the apparent capacity gains on layer 2 using directional communications methods to layer 3 by using directionality to route packets scalably in unstructured, flat networks. In their protocol, Orthogonal Rendezvous Routing Protocol, Cheng et al. [1] showed that by ”drawing” two lines orthogonal to each other at each node, it is possible to provide over 98% connectivity while maintaining only O(N3/2) evenly distributed states at a cost of only 1.2 path stretch. In this paper, we seek to provide more in-depth performance analysis by tuning additional factors such as the number of directions to transmit, the number of interfaces per node, among others, to understand its affect on varying network densities, topologies, connections, and traffic patterns. We show that by sending packets out in more directions, increased connectivity, smaller average path length, better goodput results only up to a point as compared to other routing protocols. The trade-off, however, is added state information maintained at each node and additional control packets received. We also show that the addition of more interfaces generally yields better packet delivery success, average path length, and goodput.

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In the context of growing networks, we introduce a simple dynamical model that unifies the generic features of real networks: scale-free distribution of degree and the small-world effect. While the average shortest path length increases logarithmically as in random networks, the clustering coefficient assumes a large value independent of system size. We derive analytical expressions for the clustering coefficient in two limiting cases: random [C approximately (ln N)(2)/N] and highly clustered (C=5/6) scale-free networks.