Small World Networks in Education Sciences

Dynamical properties of spiking neural networks with small world topologies

In recent years, small-world graphs have gained considerable interest as models of real-world systems, which often display features residing between regularity and randomness. The most notable of these models is the Watts-Strogatz graph [1], though alternatives have been proposed [2]. The unifying characteristics of these models are that any two nodes are joined with a small number of links between them (i.e. short path length), while at the same time connected node pairs exhibit an abundance of triangular relations resulting in a high degree of local redundancy (i.e. high clustering). Theoretical investigations of small-world graph models have generally applied asymptotic evaluations in the limit of large system size [3] or the continuum approximation [4] to the algorithmic definition of the graph, in the absence of an analytic representation. In this study, we introduce a generative model of directed small-world graphs, a canonical model of Watts-Strogatz digraphs, and propose an approach that yields the graph’s defining adjacency matrix in algebraic terms, with the goal to provide mathematically rigorous access to the study of finite-size small-world graphs [5]. The proposed approach makes use of random annihilation operators whose algebraic properties can be utilized to assess algebraically well-defined graph-theoretic measures in an analytically exact framework, valid nonasymptotically for all graph sizes. We demonstrated the application of our approach by calculating, for the first time, the asymmetry index and total clustering coefficient of small worlds in an exact fashion. We then utilize the exact nonasymptotic expression for the clustering coefficient in order to assess the small-worldness of structural brain networks in an analytic setting. Using the number of nodes and edges of the given brain networks to construct the equivalent small-world network, we observe that a significant edge rewiring of at least 20% up to 60% is required to produce the small-worldness indices observed in these networks. Importantly, the maximum of the small-worldness index however occurs in all cases at one order of magnitude lower than the required rewiring found. This result suggests that neural graphs reside far away from the small-world regime of the Watts-Strogatz model.

Small worlds are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is required. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman--Watts small world, the mixing time is of order log2n. This confirms a prediction of Richard Durrett, who proved a lower bound of order log2n and an upper bound of order log3n.

Content location in unstructured peer-to-peer (P2P) networks is a challenging problem. In this paper, we present a novel Interest-based Small World (ISW) network to address the problem, by constructing a cluster overlay in the unstructured P2P network based on the small world paradigm and user interest. There are many attractive properties of a small world network, such as low average hop distance and high clustering coefficient. Interest locality can improve the awareness of user's indeed intentions. Our scheme combines their advantage to create a better solution. The simulation results show that our scheme outperforms other schemes significantly.

PSO may easily get trapped in a local optimum, when solving complex multimodal problems. In this paper, an improved PSO based on small world network and comprehensive is proposed. The learning exemplar of each particle includes three parts: the global best particle, its own best particle (pbest) and the pbest of its neighborhood. And a random position around itself is needed to increase a probability to jump to that promising region. These strategies enable the diversity of the swarm to be preserved to discourage premature convergence. In benchmark function test, the SCPSO algorithm achieves better solutions than other PSOs.

Small-world networks are characterized by large clustering coefficients and small characteristic path lengths. These properties are induced by replacing a small fraction of short-range local scale links of a geometric/Euclidean graph with long-range global scale links. Advances in wireless networks allow for cost-efficient addition of secondary long-range wireless interfaces in devices. We derive analytical mean-field solutions for 1) clustering coefficient and 2) characteristic path length of peer-to-peer D2D wireless networks with topologies mimicking small-world networks. In graph-theoretic terms these topologies correspond to geometric graphs of randomly deployed nodes in two-dimensions with range limited shortcuts. The models show that in spite of the fact that links used to create shortcuts are range limited, the network still retains the essential phase difference between characteristic path length and clustering coefficient that is the hallmark of small-world networks when a small fraction of all nodes, as little as 1%-5%, have range limited shortcut links. We also demonstrate the utility of these models as design tools for determining deployment parameters of small-world wireless sensor networks.

Diffusion of Innovations on Community Based Small Worlds: The Role of Correlation between Social Spheres

The small-world phenomena exhibits highly localized clustering and short-cut paths between vertices in a graph that reflect observed properties in social networks, epidemiological models and other real-world networks. The small-world models rely on the application of constraint-based randomness or the derivation of constraints on randomness to simulate the desired network complexities and their associated network connection properties. In this paper, rather than exploring the random properties of small-world networks, we employ deterministic strategies in the design of a computationally efficient distributed neuronal-axon network simulator that results in a small world network. These strategies are derived by addressing the parallel complexities of the proposed neuronal-axon network simulator, and also from physical constraints imposed by resource limitations of the distributed simulation architecture. The outcome of this study is the realization of a neuronal-axon network simulator that exhibits small-world characteristics of clustering with a logarithmic degree of separation between nodes without the need for long-range communication edges. The importance of this result is the deterministic application of reasoned optimization rules from which the small-world network emerges.

Small-world phenomenon can occur in coupled dynamical systems which are highly clustered at a local level and yet strongly coupled at the global level. We show that cellular neural networks (CNN's) can exhibit "small-world phenomenon". We generalize the "characteristic path length" from previous works on "small-world phenomenon" into a "characteristic coupling strength" for measuring the average coupling strength of the outputs of CNN's. We also provide a simplified algorithm for calculating the "characteristic coupling strength" with a reasonable amount of computing time. We define a "clustering coefficient" and show how it can be calculated by a horizontal "hole detection" CNN, followed by a vertical "hole detection" CNN. Evolutions of the game-of-life CNN with different initial conditions are used to illustrate the emergence of a "small-world phenomenon". Our results show that the well-known game-of-life CNN is not a small-world network. However, generalized CNN life games whose individuals have strong mobility and high survival rate can exhibit small-world phenomenon in a robust way. Our simulations confirm the conjecture that a population with a strong mobility is more likely to qualify as a small world. CNN games whose individuals have weak mobility can also exhibit a small-world phenomenon under a proper choice of initial conditions. However, the resulting small worlds depend strongly on the initial conditions, and are therefore not robust.

It is believed that thalamic reticular nucleus (TRN) controls spindles and spike-wave discharges (SWD) in seizure or sleeping processes. The dynamical mechanisms of spatiotemporal evolutions between these two types of activity, however, are not well understood. In light of this, we first use a single-compartment thalamocortical neural field model to investigate the effects of TRN on occurrence of SWD and its transition. Results show that the increasing inhibition from TRN to specific relay nuclei (SRN) can lead to the transition of system from SWD to slow-wave oscillation. Specially, it is shown that stimulations applied in the cortical neuronal populations can also initiate the SWD and slow-wave oscillation from the resting states under the typical inhibitory intensity from TRN to SRN. Then, we expand into a 3-compartment coupled thalamocortical model network in linear and circular structures, respectively, to explore the spatiotemporal evolutions of wave states in different compartments. The main results are: (i) for the open-ended model network, SWD induced by stimulus in the first compartment can be transformed into sleep-like slow UP-DOWN and spindle states as it propagates into the downstream compartments; (ii) for the close-ended model network, weak stimulations performed in the first compartment can result in the consistent experimentally observed spindle oscillations in all three compartments; in contrast, stronger periodic single-pulse stimulations applied in the first compartment can induce periodic transitions between SWD and spindle oscillations. Detailed investigations reveal that multi-attractor coexistence mechanism composed of SWD, spindles and background state underlies these state evolutions. What's more, in order to demonstrate the state evolution stability with respect to the topological structures of neural network, we further expand the 3-compartment coupled network into 10-compartment coupled one, with linear and circular structures, and nearest-neighbor (NN) coupled network as well as its realization of small-world (SW) topology via random rewiring, respectively. Interestingly, for the cases of linear and circular connetivities, qualitatively similar results were obtained in addition to the more irregularity of firings. However, SWD can be eventually transformed into the consistent low-amplitude oscillations for both NN and SW networks. In particular, SWD evolves into the slow spindling oscillations and background tonic oscillations within the NN and SW network, respectively. Our modeling and simulation studies highlight the effect of network topology in the evolutions of SWD and spindling oscillations, which provides new insights into the mechanisms of cortical seizures development.

To address the inherent limitations of traditional Reservoir Computing (RC) in modeling chaotic systems and complex nonlinear dynamics, such as insufficient representational capacity and shallow state memory, this paper proposes a novel RC architecture named SW-MOSIE-RC, which integrates a Small-World (SW) topology with a Multi-Order State Interaction Enhancement (MOSIE) mechanism. The SW structure, constructed via the Watts–Strogatz model, introduces local clustering and short-path characteristics to enhance state coupling efficiency and global dynamical responsiveness within the reservoir. Concurrently, the MOSIE module enables multi-order nonlinear state augmentation, expanding the effective feature dimension and allowing the readout layer to better capture high-dimensional evolutionary patterns. Experiments on chaotic Permanent Magnet Synchronous Motor (PMSM) systems demonstrate that SW-MOSIE-RC achieves high prediction accuracy and strong robustness under noise perturbations, outperforming conventional RC and deep learning baselines.

The World Wide Web has taken the form of a social network and if analyzed carefully, we can extract various communities from this network based on different parameters such as culture, trust, behavior, relationships, etc. The Small World Effect is a kind of behavior that has been discovered in entity networks of many natural phenomena. The set of nodes in a network showing Small World Effect form a local network within the major network highlighting the properties of a Small World Network. This work analyzes three different web networks, i.e. Term-Term similarity network, Document-Document similarity network, and Hyperlinks network, to check whether they show Small World Network behavior or not. We define a criterion and then compare these network graphs against that criterion. The network graph which fulfills that criterion is declared to be a Small World Network.

The topology of the trust network is important to optimize its usage in the trust-aware applications. However, since the users can join trust network ubiquitously, the structure of the highly dynamic trust network is still unknown. This paper contributes to verify that the trust network is the small-world network, and its small-world topology is independent of its dynamics. This is achieved by verifying the scale-freeness of five trust networks extracted from real online sites. Using the small-world nature of the trust network, we optimize the rating prediction mechanism of the conventional trust-aware recommender system. Experimental results clearly show that our proposed mechanism can achieve the maximum accuracy and coverage with the minimum computation complexity for the rating predictions.

In this paper we present and simulate a new self–organising algorithm for P2P unstructured networks inspired by human relationships. This algorithm, called PROSA, tries to simulate the evolution of human relationships from simple acquaintance to friendship and partnership. Our target is to obtain a self–reconfiguring P2P system which possesses some of the desirable features of social communities. The most useful property of many natural social communities is that of beeing “small–worlds”, since in a small–world network queries usually require a small amount of “hops” to walk from a source to a destination peer. We show that PROSA naturally evolves into a small–world network, with an high clustering coefficient and a relly short average path length.

We study the origin of navigability in small-world (SW) networks and propose a general scheme for navigating SW networks. We find that navigability can naturally emerge from self-organization in the absence of prior knowledge about the underlying reference frames of networks. Through a process of information exchange and accumulation on networks, a hidden metric space for navigation on networks is constructed. Navigation based on distances between vertices in the hidden metric space can efficiently deliver messages on SW networks, in which long-range connections play an important role. Numerical simulations further demonstrate that a high cluster coefficient and a low diameter are both necessary for navigability. These interesting results provide profound insights into scalable routing on the Internet due to its distributed and localized requirements.