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.

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 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

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.

Social Community Classification Based on Laplacian Spectrum Feature

2011 ◽  
Vol 63-64 ◽  
pp. 142-146
Author(s):  
Bo Wang ◽  
Yi Qiong Xu ◽  
Yao Ming Zhou
Keyword(s):  
Community Structure ◽  
Common Property ◽  
Distance Measure ◽  
Network Models ◽  
Small World ◽  
Laplacian Spectrum ◽  
Free Graph ◽  
Scale Free ◽  
Spectrum Feature ◽  
Complex Network Models

Community structure is a common property that exists in social networks. Community structure analysis is important for understanding network structure and analyzing the network characteristics. Recently community detecting methods are reported continually, but within community the structure is still complex. This paper proposed a method applied to classify community by using Laplacian spectrum feature, and defined the distance measure between the features extracted from difference community. As experiments, this paper studied three complex network models: the random graph of Erdös-Rényi, the small world of Watts and Strogatz and the scale-free graph, and classified them based on Laplacian spectrum feature successfully. The result shows the Laplacian spectrum feature and similarity measure are effective for classification.

scholarly journals Optimisation of the coalescent hyperbolic embedding of complex networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bianka Kovács ◽  
Gergely Palla
Keyword(s):  
Small World ◽  
Hyperbolic Distance ◽  
Greedy Routing ◽  
Scale Free ◽  
Recent Approach ◽  
Strong Motivation ◽  
Angular Coordinates ◽  
Small World Property ◽  
Logarithmic Loss ◽  
Hyperbolic Embedding

AbstractSeveral observations indicate the existence of a latent hyperbolic space behind real networks that makes their structure very intuitive in the sense that the probability for a connection is decreasing with the hyperbolic distance between the nodes. A remarkable network model generating random graphs along this line is the popularity-similarity optimisation (PSO) model, offering a scale-free degree distribution, high clustering and the small-world property at the same time. These results provide a strong motivation for the development of hyperbolic embedding algorithms, that tackle the problem of finding the optimal hyperbolic coordinates of the nodes based on the network structure. A very promising recent approach for hyperbolic embedding is provided by the noncentered minimum curvilinear embedding (ncMCE) method, belonging to the family of coalescent embedding algorithms. This approach offers a high-quality embedding at a low running time. In the present work we propose a further optimisation of the angular coordinates in this framework that seems to reduce the logarithmic loss and increase the greedy routing score of the embedding compared to the original version, thereby adding an extra improvement to the quality of the inferred hyperbolic coordinates.

COMPARING EFFECTS OF CLUSTER-COUPLED PATTERNS ON OPINION DYNAMICS

2012 ◽  
Vol 23 (07) ◽  
pp. 1250050 ◽  
Author(s):  
YUN LIU ◽  
XIA-MENG SI ◽  
YAN-CHAO ZHANG
Keyword(s):  
Community Structure ◽  
Social Cohesion ◽  
Small World ◽  
Opinion Dynamics ◽  
Scale Free ◽  
Update Rules ◽  
Opinion Evolution ◽  
Multiple Cluster ◽  
Better Than ◽  
Continuous Opinions

Community structure is another important feature besides small-world and scale-free property of complex networks. Communities can be coupled through specific fixed links between nodes, or occasional encounter behavior. We introduce a model for opinion evolution with multiple cluster-coupled patterns, in which the interconnectivity denotes the coupled degree of communities by fixed links, and encounter frequency controls the coupled degree of communities by encounter behaviors. Considering the complicated cognitive system of people, the CODA (continuous opinions and discrete actions) update rules are used to mimic how people update their decisions after interacting with someone. It is shown that, large interconnectivity and encounter frequency both can promote consensus, reduce competition between communities and propagate some opinion successfully across the whole population. Encounter frequency is better than interconnectivity at facilitating the consensus of decisions. When the degree of social cohesion is same, small interconnectivity has better effects on lessening the competence between communities than small encounter frequency does, while large encounter frequency can make the greater degree of agreement across the whole populations than large interconnectivity can.

SYNCHRONIZABILITY OF COMPLEX NETWORKS WITH COMMUNITY STRUCTURE

2012 ◽  
Vol 23 (04) ◽  
pp. 1250029 ◽  
Author(s):  
MAHDI JALILI
Keyword(s):  
Community Structure ◽  
Real World ◽  
Order Parameter ◽  
Small World ◽  
Laplacian Matrix ◽  
Small World Networks ◽  
Scale Free ◽  
Community Connections ◽  
Scale Free Networks ◽  
Kuramoto Oscillators

Many real-world networks show community structure characterized by dense intra-community connections and sparse inter-community links. In this paper we investigated the synchronization properties of such networks. In this work we constructed such networks in a way that they consist of a number of communities with scale-free or small-world structure. Furthermore, with a probability, the intra-community connections are rewired to inter-community links. Two synchronizability measures were considered as the eigenratio of the Laplacian matrix and the phase order parameter obtained for coupled nonidentical Kuramoto oscillators. We found a power-law relation between the eigenratio and the inter-community rewiring probability in which as the rewiring probability increased, the eigenratio decreased, and hence, the synchronizability enhanced. The phase order parameter also increased by increasing the rewiring probability. Also, small-world networks with community structure showed better synchronization properties as compared to scale-free networks with community structure.

scholarly journals On equilibrium Metropolis simulations on self-organized urban street networks

2021 ◽  
Vol 6 (1) ◽  
Author(s):  
Jérôme G. M. Benoit ◽  
Saif Eddin G. Jabari
Keyword(s):  
State Of The Art ◽  
Small World ◽  
Metropolis Algorithm ◽  
The State ◽  
Scale Free ◽  
Urban Street ◽  
Amount Of Information ◽  
Street Networks ◽  
Self Organized ◽  
Wide Range

AbstractUrban street networks of unplanned or self-organized cities typically exhibit astonishing scale-free patterns. This scale-freeness can be shown, within the maximum entropy formalism (MaxEnt), as the manifestation of a fluctuating system that preserves on average some amount of information. Monte Carlo methods that can further this perspective are cruelly missing. Here we adapt to self-organized urban street networks the Metropolis algorithm. The “coming to equilibrium” distribution is established with MaxEnt by taking scale-freeness as prior hypothesis along with symmetry-conservation arguments. The equilibrium parameter is the scaling; its concomitant extensive quantity is, assuming our lack of knowledge, an amount of information. To design an ergodic dynamics, we disentangle the state-of-the-art street generating paradigms based on non-overlapping walks into layout-at-junction dynamics. Our adaptation reminisces the single-spin-flip Metropolis algorithm for Ising models. We thus expect Metropolis simulations to reveal that self-organized urban street networks, besides sustaining scale-freeness over a wide range of scalings, undergo a crossover as scaling varies—literature argues for a small-world crossover. Simulations for Central London are consistent against the state-of-the-art outputs over a realistic range of scaling exponents. Our illustrative Watts–Strogatz phase diagram with scaling as rewiring parameter demonstrates a small-world crossover curving within the realistic window 2–3; it also shows that the state-of-the-art outputs underlie relatively large worlds. Our Metropolis adaptation to self-organized urban street networks thusly appears as a scaling variant of the Watts–Strogatz model. Such insights may ultimately allow the urban profession to anticipate self-organization or unplanned evolution of urban street networks.

THREE-DIMENSIONAL APOLLONIAN NETWORKS

2006 ◽  
Vol 17 (08) ◽  
pp. 1219-1226 ◽  
Author(s):  
DANYEL J. B. SOARES ◽  
JOSÉ S. ANDRADE ◽  
HANS J. HERRMANN ◽  
LUCIANO R. da SILVA
Keyword(s):  
Porous Media ◽  
Three Dimensional ◽  
Small World ◽  
Clustering Coefficient ◽  
Force Chains ◽  
Two Dimensional ◽  
Space Filling ◽  
Scale Free ◽  
Wide Range ◽  
Granular Packings

We discuss the three-dimensional Apollonian network introduced by Andrade et al.1 for the two-dimensional case. These networks are simultaneously scale-free, small world, Euclidean, space-filling and matching graphs and have a wide range of applications going from the description of force chains in polydisperse granular packings to the geometry of fully fragmented porous media. Some of the properties of these networks, namely, the connectivity exponent, the clustering coefficient, the shortest path, and vertex betweenness are calculated and found to be particularly rich.

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