Network Dynamics

2019 ◽  
pp. 207-242
Author(s):  
David D. Nolte
Keyword(s):  
Random Graphs ◽  
Small World ◽  
Distance Matrix ◽  
Kuramoto Model ◽  
Global Synchronization ◽  
Scale Free ◽  
Wide Range ◽  
Network Topologies ◽  
The Kuramoto Model ◽  
Static Connectivity

A language of nodes and links, degree and moments, and adjacency matrix and distance matrix, among others, is defined and used to capture the wide range of different types and properties of network topologies. Regular graphs and random graphs have fundamentally different connectivities that play a role in dynamic processes such as diffusion and synchronization on a network. Three common random graphs are the Erdös–Rényi (ER) graph, the small-world (SW) graph, and the scale-free (SF) graph. Random graphs give rise to critical phenomena based on static connectivity properties, such as the percolation threshold, but also exhibit dynamical thresholds for the diffusion of states across networks and the synchronization of oscillators. The vaccination threshold for diseases propagating on networks and the global synchronization transition in the Kuramoto model are examples of dynamical processes that can be used to probe network topologies.

scholarly journals The inherent community structure of hyperbolic networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bianka Kovács ◽  
Gergely Palla
Keyword(s):  
Community Structure ◽  
Random Graphs ◽  
Small World ◽  
Label Propagation ◽  
Hyperbolic Distance ◽  
Scale Free ◽  
Wide Range ◽  
Community Finding ◽  
Optimisation Model ◽  
Low Dimensional

AbstractA remarkable approach for grasping the relevant statistical features of real networks with the help of random graphs is offered by hyperbolic models, centred around the idea of placing nodes in a low-dimensional hyperbolic space, and connecting node pairs with a probability depending on the hyperbolic distance. It is widely appreciated that these models can generate random graphs that are small-world, highly clustered and scale-free at the same time; thus, reproducing the most fundamental common features of real networks. In the present work, we focus on a less well-known property of the popularity-similarity optimisation model and the $${\mathbb {S}}^1/{\mathbb {H}}^2$$ S 1 / H 2 model from this model family, namely that the networks generated by these approaches also contain communities for a wide range of the parameters, which was certainly not an intention at the design of the models. We extracted the communities from the studied networks using well-established community finding methods such as Louvain, Infomap and label propagation. The observed high modularity values indicate that the community structure can become very pronounced under certain conditions. In addition, the modules found by the different algorithms show good consistency, implying that these are indeed relevant and apparent structural units. Since the appearance of communities is rather common in networks representing real systems as well, this feature of hyperbolic models makes them even more suitable for describing real networks than thought before.

scholarly journals Analysis on Invulnerability of Wireless Sensor Network towards Cascading Failures Based on Coupled Map Lattice

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Xiuwen Fu ◽  
Yongsheng Yang ◽  
Haiqing Yao
Keyword(s):  
Random Network ◽  
Small World ◽  
Coupled Map Lattice ◽  
Wireless Sensor ◽  
Cascading Failures ◽  
Scale Free Network ◽  
Scale Free ◽  
Network Topologies ◽  
Free Network ◽  
Entire Network

Previous research of wireless sensor networks (WSNs) invulnerability mainly focuses on the static topology, while ignoring the cascading process of the network caused by the dynamic changes of load. Therefore, given the realistic features of WSNs, in this paper we research the invulnerability of WSNs with respect to cascading failures based on the coupled map lattice (CML). The invulnerability and the cascading process of four types of network topologies (i.e., random network, small-world network, homogenous scale-free network, and heterogeneous scale-free network) under various attack schemes (i.e., random attack, max-degree attack, and max-status attack) are investigated, respectively. The simulation results demonstrate that the rise of interference R and coupling coefficient ε will increase the risks of cascading failures. Cascading threshold values Rc and εc exist, where cascading failures will spread to the entire network when R>Rc or ε>εc. When facing a random attack or max-status attack, the network with higher heterogeneity tends to have a stronger invulnerability towards cascading failures. Conversely, when facing a max-degree attack, the network with higher uniformity tends to have a better performance. Besides that, we have also proved that the spreading speed of cascading failures is inversely proportional to the average path length of the network and the increase of average degree k can improve the network invulnerability.

scholarly journals Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Géza Ódor ◽  
Jeffrey Kelling
Keyword(s):  
Control Parameter ◽  
Mean Field ◽  
Small World ◽  
Kuramoto Model ◽  
Crossover Point ◽  
Small World Graphs ◽  
Synchronization Phase ◽  
Experimental Values ◽  
The Kuramoto Model ◽  
Parameter Dependent

AbstractThe hypothesis, that cortical dynamics operates near criticality also suggests, that it exhibits universal critical exponents which marks the Kuramoto equation, a fundamental model for synchronization, as a prime candidate for an underlying universal model. Here, we determined the synchronization behavior of this model by solving it numerically on a large, weighted human connectome network, containing 836733 nodes, in an assumed homeostatic state. Since this graph has a topological dimension d < 4, a real synchronization phase transition is not possible in the thermodynamic limit, still we could locate a transition between partially synchronized and desynchronized states. At this crossover point we observe power-law–tailed synchronization durations, with τt ≃ 1.2(1), away from experimental values for the brain. For comparison, on a large two-dimensional lattice, having additional random, long-range links, we obtain a mean-field value: τt ≃ 1.6(1). However, below the transition of the connectome we found global coupling control-parameter dependent exponents 1 < τt ≤ 2, overlapping with the range of human brain experiments. We also studied the effects of random flipping of a small portion of link weights, mimicking a network with inhibitory interactions, and found similar results. The control-parameter dependent exponent suggests extended dynamical criticality below the transition point.

scholarly journals Unimodular lattice triangulations as small-world and scale-free random graphs

2015 ◽  
Vol 17 (2) ◽  
pp. 023013 ◽  
Author(s):  
B Krüger ◽  
E M Schmidt ◽  
K Mecke
Keyword(s):  
Random Graphs ◽  
Small World ◽  
Unimodular Lattice ◽  
Scale Free

TRAFFIC CONGESTION ANALYSIS IN COMPLEX NETWORKS BASED ON VARIOUS ROUTING STRATEGIES

2007 ◽  
Vol 21 (15) ◽  
pp. 929-939 ◽  
Author(s):  
XUAN GUO ◽  
HONGTAO LU
Keyword(s):  
Complex Networks ◽  
Traffic Congestion ◽  
Small World ◽  
Modern Society ◽  
Information Communication ◽  
Shortest Path Routing ◽  
Scale Free ◽  
Deterministic Routing ◽  
Routing Strategy ◽  
Network Topologies

Networks, acting as infrastructure for information communication, play an important role in modern society, therefore, the elements affecting the efficiency of network traffic are worthy of deep research. In this paper, we investigate numerically the problem of traffic congestion in complex networks through the use of various routing strategies. Three types of complex networks structures, namely Poisson random networks, small-world networks and scale-free networks, are considered. Three different routing strategies are used on networks: deterministic routing strategy, preferential routing strategy and shortest path routing strategy. We evaluate the efficiency of different routing strategies on different network topologies and show how the network structures and routing strategies influence the traffic congestion status.

scholarly journals The network structure affects the fixation probability when it couples to the birth-death dynamics in finite population

2021 ◽  
Vol 17 (10) ◽  
pp. e1009537
Author(s):  
Mohammad Ali Dehghani ◽  
Amir Hossein Darooneh ◽  
Mohammad Kohandel
Keyword(s):  
Network Structure ◽  
Evolutionary Dynamics ◽  
Finite Population ◽  
Small World ◽  
Fixation Time ◽  
Fixation Probability ◽  
Neutral Drift ◽  
Scale Free ◽  
Wide Range ◽  
And Mathematics

The study of evolutionary dynamics on graphs is an interesting topic for researchers in various fields of science and mathematics. In systems with finite population, different model dynamics are distinguished by their effects on two important quantities: fixation probability and fixation time. The isothermal theorem declares that the fixation probability is the same for a wide range of graphs and it only depends on the population size. This has also been proved for more complex graphs that are called complex networks. In this work, we propose a model that couples the population dynamics to the network structure and show that in this case, the isothermal theorem is being violated. In our model the death rate of a mutant depends on its number of neighbors, and neutral drift holds only in the average. We investigate the fixation probability behavior in terms of the complexity parameter, such as the scale-free exponent for the scale-free network and the rewiring probability for the small-world network.

Synchronization of Chaotic Circuits with Stochastically-Coupled Network Topology

2021 ◽  
Vol 31 (01) ◽  
pp. 2150015
Author(s):  
Yoko Uwate ◽  
Yoshifumi Nishio ◽  
Thomas Ott
Keyword(s):  
Secure Communication ◽  
Communication Systems ◽  
Small World ◽  
Chaotic Circuit ◽  
Scale Free ◽  
Wide Range ◽  
Scale Free Networks ◽  
Dynamic Topology ◽  
Chaotic Circuits ◽  
Fully Coupled

In recent years, research on synchronization between coupled chaotic circuits has attracted interest in a wide range of fields. This is because the synchronization of coupled chaotic circuits is a multidisciplinary phenomenon that occurs in various applications, such as broadband communication systems or secure communication. In this study, we propose a coupled chaotic circuit network model with stochastic couplings. We investigate the synchronization phenomena observed for the proposed network using different network structures such as fully-coupled, random, small world and scale-free networks. We find that the same synchronization characteristics can be obtained for these networks with a dynamic topology as when the coupling strength is changed in static networks.

scholarly journals Quantitative Controllability Index of Complex Networks

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Lifu Wang ◽  
Yali Zhang ◽  
Jingxiao Han ◽  
Zhi Kong
Keyword(s):  
Complex Networks ◽  
Condition Number ◽  
Random Network ◽  
Small World ◽  
Scale Free Network ◽  
Scale Free ◽  
Network Topologies ◽  
Free Network ◽  
The Relationship ◽  
Controllability Index

In this paper, the controllability issue of complex network is discussed. A new quantitative index using knowledge of control centrality and condition number is constructed to measure the controllability of given networks. For complex networks with different controllable subspace dimensions, their controllability is mainly determined by the control centrality factor. For the complex networks that have the equal controllable subspace dimension, their different controllability is mostly determined by the condition number of subnetworks’ controllability matrix. Then the effect of this index is analyzed based on simulations on various types of network topologies, such as ER random network, WS small-world network, and BA scale-free network. The results show that the presented index could reflect the holistic controllability of complex networks. Such an endeavour could help us better understand the relationship between controllability and network topology.

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