scholarly journals Graph fractal dimension and the structure of fractal networks

2020 ◽  
Vol 8 (4) ◽  
Author(s):  
Pavel Skums ◽  
Leonid Bunimovich
Keyword(s):  
Complex Networks ◽  
Evolutionary Biology ◽  
Real Life ◽  
Fractal Dimensions ◽  
Network Models ◽  
Self Similarity ◽  
Descriptive Complexity ◽  
Network Communities ◽  
Fractal Properties ◽  
Continuous Objects

Abstract Fractals are geometric objects that are self-similar at different scales and whose geometric dimensions differ from so-called fractal dimensions. Fractals describe complex continuous structures in nature. Although indications of self-similarity and fractality of complex networks has been previously observed, it is challenging to adapt the machinery from the theory of fractality of continuous objects to discrete objects such as networks. In this article, we identify and study fractal networks using the innate methods of graph theory and combinatorics. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colourings and graph descriptive complexity, and analyse the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework in evolutionary biology and virology by analysing networks of viral strains sampled at different stages of evolution inside their hosts. Our methodology revealed gradual self-organization of intra-host viral populations over the course of infection and their adaptation to the host environment. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.

scholarly journals An Extended Correlation Dimension of Complex Networks

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 710
Author(s):  
Sheng Zhang ◽  
Wenxiang Lan ◽  
Weikai Dai ◽  
Feng Wu ◽  
Caisen Chen
Keyword(s):  
Complex Networks ◽  
Correlation Dimension ◽  
Fractal Dimensions ◽  
Small World ◽  
Scaling Analysis ◽  
Fractal Nature ◽  
Small World Networks ◽  
Self Similarity ◽  
Weighted Networks ◽  
Fractal Properties

Fractal and self-similarity are important characteristics of complex networks. The correlation dimension is one of the measures implemented to characterize the fractal nature of unweighted structures, but it has not been extended to weighted networks. In this paper, the correlation dimension is extended to the weighted networks. The proposed method uses edge-weights accumulation to obtain scale distances. It can be used not only for weighted networks but also for unweighted networks. We selected six weighted networks, including two synthetic fractal networks and four real-world networks, to validate it. The results show that the proposed method was effective for the fractal scaling analysis of weighted complex networks. Meanwhile, this method was used to analyze the fractal properties of the Newman–Watts (NW) unweighted small-world networks. Compared with other fractal dimensions, the correlation dimension is more suitable for the quantitative analysis of small-world effects.

scholarly journals Modelling community structure and temporal spreading on complex networks

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Vesa Kuikka
Keyword(s):  
Community Structure ◽  
Complex Networks ◽  
Community Detection ◽  
Network Structure ◽  
Network Connectivity ◽  
Network Models ◽  
Building Blocks ◽  
Detection Algorithm ◽  
Research Area ◽  
Detection Methods

AbstractWe present methods for analysing hierarchical and overlapping community structure and spreading phenomena on complex networks. Different models can be developed for describing static connectivity or dynamical processes on a network topology. In this study, classical network connectivity and influence spreading models are used as examples for network models. Analysis of results is based on a probability matrix describing interactions between all pairs of nodes in the network. One popular research area has been detecting communities and their structure in complex networks. The community detection method of this study is based on optimising a quality function calculated from the probability matrix. The same method is proposed for detecting underlying groups of nodes that are building blocks of different sub-communities in the network structure. We present different quantitative measures for comparing and ranking solutions of the community detection algorithm. These measures describe properties of sub-communities: strength of a community, probability of formation and robustness of composition. The main contribution of this study is proposing a common methodology for analysing network structure and dynamics on complex networks. We illustrate the community detection methods with two small network topologies. In the case of network spreading models, time development of spreading in the network can be studied. Two different temporal spreading distributions demonstrate the methods with three real-world social networks of different sizes. The Poisson distribution describes a random response time and the e-mail forwarding distribution describes a process of receiving and forwarding messages.

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

scholarly journals Optimizing community detection in social networks using antlion and K-median

2019 ◽  
Vol 8 (4) ◽  
Author(s):  
Amany A. Naem ◽  
Neveen I. Ghali
Keyword(s):  
Social Networks ◽  
Objective Function ◽  
Community Detection ◽  
Optimization Problem ◽  
Real Life ◽  
Optimization Methods ◽  
Population Based ◽  
Detection Problem ◽  
Network Communities ◽  
Antlion Optimization

Antlion Optimization (ALO) is one of the latest population based optimization methods that proved its good performance in a variety of applications. The ALO algorithm copies the hunting mechanism of antlions to ants in nature. Community detection in social networks is conclusive to understanding the concepts of the networks. Identifying network communities can be viewed as a problem of clustering a set of nodes into communities. k-median clustering is one of the popular techniques that has been applied in clustering. The problem of clustering network can be formalized as an optimization problem where a qualitatively objective function that captures the intuition of a cluster as a set of nodes with better in ternal connectivity than external connectivity is selected to be optimized. In this paper, a mixture antlion optimization and k-median for solving the community detection problem is proposed and named as K-median Modularity ALO. Experimental results which are applied on real life networks show the ability of the mixture antlion optimization and k-median to detect successfully an optimized community structure based on putting the modularity as an objective function.

scholarly journals FRACTALS, NATURAL DISASTERS AND ECOLOGICAL PROBLEMS OF MALDIVES

2018 ◽  
pp. 18-25
Author(s):  
Boyko Ranguelov ◽  
Fathimath Shadiya
Keyword(s):  
Local Population ◽  
Fractal Dimensions ◽  
Point Of View ◽  
Negative Consequences ◽  
Fractal Nature ◽  
Fractal Properties ◽  
Level Increase ◽  
Ocean Level ◽  
Ecological Problems ◽  
Negative Phenomena

A new idea about the fractal nature of Maldives archipelago is under investigation. The origin of this famous Maldivian islands’ country is still questionable from geodynamic point of view. The present study is focused to the assessment of the fractal properties and the coefficients of the nonlinearity (fractal dimensions) of the areal spatial distribution of the major atolls of the Maldives. This is the most vulnerable area in the world from point of view of the global warming and the possible negative consequences to the country and population from the ocean level increase. From another side the natural hazards (tsunamis, storms, etc.) are common negative phenomena attacking the country. The strongly developed tourism – more than 30% of the GDP and the increased urbanization is another factor creating ecological problems to the local population. The relationships between the fractal nature and the possible ways to avoid the pollution are also in the focus of this research.

A conservation law, entropy principle and quantization of fractal dimensions in hadron interactions

2018 ◽  
Vol 33 (10) ◽  
pp. 1850057 ◽  
Author(s):  
I. Zborovský
Keyword(s):  
Conservation Law ◽  
Fractal Structure ◽  
Fractal Dimensions ◽  
The Self ◽  
Hadron Structure ◽  
Self Similarity ◽  
Similarity Variable ◽  
Variable Formula ◽  
Entropy Principle ◽  
Fragmentation Processes

Fractal self-similarity of hadron interactions demonstrated by the [Formula: see text]-scaling of inclusive spectra is studied. The scaling regularity reflects fractal structure of the colliding hadrons (or nuclei) and takes into account general features of fragmentation processes expressed by fractal dimensions. The self-similarity variable [Formula: see text] is a function of the momentum fractions [Formula: see text] and [Formula: see text] of the colliding objects carried by the interacting hadron constituents and depends on the momentum fractions [Formula: see text] and [Formula: see text] of the scattered and recoil constituents carried by the inclusive particle and its recoil counterpart, respectively. Based on entropy principle, new properties of the [Formula: see text]-scaling concept are found. They are conservation of fractal cumulativity in hadron interactions and quantization of fractal dimensions characterizing hadron structure and fragmentation processes at a constituent level.

scholarly journals Studies on the Decrease Mechanisms of Typical Complex Networks

2021 ◽  
Author(s):  
Yuhu Qiu ◽  
Tianyang Lyu ◽  
Xizhe Zhang ◽  
Ruozhou Wang
Keyword(s):  
Average Length ◽  
Shortest Paths ◽  
Real Life ◽  
Network Models ◽  
Small World ◽  
Clustering Coefficient ◽  
Network Growth ◽  
Scale Free ◽  
Complex Network Models ◽  
High Degree

Network decrease caused by the removal of nodes is an important evolution process that is paralleled with network growth. However, many complex network models usually lacked a sound decrease mechanism. Thus, they failed to capture how to cope with decreases in real life. The paper proposed decrease mechanisms for three typical types of networks, including the ER networks, the WS small-world networks and the BA scale-free networks. The proposed mechanisms maintained their key features in continuous and independent decrease processes, such as the random connections of ER networks, the long-range connections based on nearest-coupled network of WS networks and the tendency connections and the scale-free feature of BA networks. Experimental results showed that these mechanisms also maintained other topology characteristics including the degree distribution, clustering coefficient, average length of shortest-paths and diameter during decreases. Our studies also showed that it was quite difficult to find an efficient decrease mechanism for BA networks to withstand the continuous attacks at the high-degree nodes, because of the unequal status of nodes.

Modern Diffusion of Products with Complex Network Models

2011 ◽  
pp. 119-133
Author(s):  
Atsushi Tanaka
Keyword(s):  
Complex Networks ◽  
Marketing Strategy ◽  
Complex Network ◽  
Network Models ◽  
Information Propagation ◽  
Multiagent Simulation ◽  
Product Marketing ◽  
Winner Takes All ◽  
Complex Network Models

In this chapter, some important matters of complex networks and their models are reviewed shortly, and then the modern diffusion of products under the information propagation using multiagent simulation is discussed. The remarkable phenomena like “Winner-Takes-All” and “Chasm” can be observed, and one product marketing strategy is also proposed.

scholarly journals Community Detection Based on a Preferential Decision Model

Information ◽  
2020 ◽  
Vol 11 (1) ◽  
pp. 53
Author(s):  
Jinfang Sheng ◽  
Ben Lu ◽  
Bin Wang ◽  
Jie Hu ◽  
Kai Wang ◽  
...  
Keyword(s):  
Community Structure ◽  
Complex Networks ◽  
Community Detection ◽  
Decision Model ◽  
Structural Characteristics ◽  
Real Life ◽  
Stable State ◽  
Ground Truth ◽  
Label Propagation ◽  
Comparison Algorithms

The research on complex networks is a hot topic in many fields, among which community detection is a complex and meaningful process, which plays an important role in researching the characteristics of complex networks. Community structure is a common feature in the network. Given a graph, the process of uncovering its community structure is called community detection. Many community detection algorithms from different perspectives have been proposed. Achieving stable and accurate community division is still a non-trivial task due to the difficulty of setting specific parameters, high randomness and lack of ground-truth information. In this paper, we explore a new decision-making method through real-life communication and propose a preferential decision model based on dynamic relationships applied to dynamic systems. We apply this model to the label propagation algorithm and present a Community Detection based on Preferential Decision Model, called CDPD. This model intuitively aims to reveal the topological structure and the hierarchical structure between networks. By analyzing the structural characteristics of complex networks and mining the tightness between nodes, the priority of neighbor nodes is chosen to perform the required preferential decision, and finally the information in the system reaches a stable state. In the experiments, through the comparison of eight comparison algorithms, we verified the performance of CDPD in real-world networks and synthetic networks. The results show that CDPD not only has better performance than most recent algorithms on most datasets, but it is also more suitable for many community networks with ambiguous structure, especially sparse networks.

scholarly journals Interdependency and Vulnerability of Multipartite Networks under Target Node Attacks

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-16
Author(s):  
Qing Cai ◽  
Mahardhika Pratama ◽  
Sameer Alam
Keyword(s):  
Complex Networks ◽  
Robustness Analysis ◽  
Network Models ◽  
Continuous Phase ◽  
Structure Design ◽  
Network Robustness ◽  
Multilayer Networks ◽  
Component Theory ◽  
Centrality Metrics ◽  
Entire Network

Complex networks in reality may suffer from target attacks which can trigger the breakdown of the entire network. It is therefore pivotal to evaluate the extent to which a network could withstand perturbations. The research on network robustness has proven as a potent instrument towards that purpose. The last two decades have witnessed the enthusiasm on the studies of network robustness. However, existing studies on network robustness mainly focus on multilayer networks while little attention is paid to multipartite networks which are an indispensable part of complex networks. In this study, we investigate the robustness of multipartite networks under intentional node attacks. We develop two network models based on the largest connected component theory to depict the cascading failures on multipartite networks under target attacks. We then investigate the robustness of computer-generated multipartite networks with respect to eight node centrality metrics. We discover that the robustness of multipartite networks could display either discontinuous or continuous phase transitions. Interestingly, we discover that larger number of partite sets of a multipartite network could increase its robustness which is opposite to the phenomenon observed on multilayer networks. Our findings shed new lights on the robust structure design of complex systems. We finally present useful discussions on the applications of existing percolation theories that are well studied for network robustness analysis to multipartite networks. We show that existing percolation theories are not amenable to multipartite networks. Percolation on multipartite networks still deserves in-depth efforts.

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