scholarly journals Growth and Optimality in Network Evolution

2011 ◽  
Vol 17 (4) ◽  
pp. 281-291 ◽  
Author(s):  
Markus Brede
Keyword(s):  
Complex Networks ◽  
Time Scales ◽  
Power Law ◽  
Path Length ◽  
Network Evolution ◽  
Hierarchical Organization ◽  
Optimization Process ◽  
Assembly Process ◽  
Rich Variety ◽  
Mixing Patterns

We investigate networks whose evolution is governed by the interaction of a random assembly process and an optimization process. In the first process, new nodes are added one at a time and form connections to randomly selected old nodes. In between node additions, the network is rewired to minimize its path length. For time scales at which neither the assembly nor the optimization processes are dominant, we find a rich variety of complex networks with power law tails in the degree distributions. These networks also exhibit nontrivial clustering, a hierarchical organization, and interesting degree-mixing patterns.

scholarly journals p-adic numbers encode complex networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Hao Hua ◽  
Ludger Hovestadt
Keyword(s):  
Complex Networks ◽  
Random Graph ◽  
Genetic Interaction ◽  
Hierarchical Structures ◽  
Number System ◽  
Hierarchical Organization ◽  
Natural Representation ◽  
Multimodal Distributions ◽  
Number Sequences ◽  
Er Model

AbstractThe Erdős-Rényi (ER) random graph G(n, p) analytically characterizes the behaviors in complex networks. However, attempts to fit real-world observations need more sophisticated structures (e.g., multilayer networks), rules (e.g., Achlioptas processes), and projections onto geometric, social, or geographic spaces. The p-adic number system offers a natural representation of hierarchical organization of complex networks. The p-adic random graph interprets n as the cardinality of a set of p-adic numbers. Constructing a vast space of hierarchical structures is equivalent for combining number sequences. Although the giant component is vital in dynamic evolution of networks, the structure of multiple big components is also essential. Fitting the sizes of the few largest components to empirical data was rarely demonstrated. The p-adic ultrametric enables the ER model to simulate multiple big components from the observations of genetic interaction networks, social networks, and epidemics. Community structures lead to multimodal distributions of the big component sizes in networks, which have important implications in intervention of spreading processes.

scholarly journals Power-law distribution of degree–degree distance: A better representation of the scale-free property of complex networks

2020 ◽  
Vol 117 (26) ◽  
pp. 14812-14818 ◽  
Author(s):  
Bin Zhou ◽  
Xiangyi Meng ◽  
H. Eugene Stanley
Keyword(s):  
Complex Networks ◽  
Complex Network ◽  
Power Law ◽  
Real World ◽  
Preferential Attachment ◽  
Statistical Tests ◽  
Finite Size ◽  
Distance Distribution ◽  
Scale Free ◽  
Degree Distance

Whether real-world complex networks are scale free or not has long been controversial. Recently, in Broido and Clauset [A. D. Broido, A. Clauset,Nat. Commun.10, 1017 (2019)], it was claimed that the degree distributions of real-world networks are rarely power law under statistical tests. Here, we attempt to address this issue by defining a fundamental property possessed by each link, the degree–degree distance, the distribution of which also shows signs of being power law by our empirical study. Surprisingly, although full-range statistical tests show that degree distributions are not often power law in real-world networks, we find that in more than half of the cases the degree–degree distance distributions can still be described by power laws. To explain these findings, we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted, two-way selection process. The model does not always produce solid power-law distributions but predicts that the degree–degree distance distribution exhibits stronger power-law behavior than the degree distribution of a finite-size network, especially when the network is dense. We test the strength of our model and its predictive power by examining how real-world networks evolve into an overly dense stage and how the corresponding distributions change. We propose that being scale free is a property of a complex network that should be determined by its underlying mechanism (e.g., preferential attachment) rather than by apparent distribution statistics of finite size. We thus conclude that the degree–degree distance distribution better represents the scale-free property of a complex network.

scholarly journals Degree difference: a simple measure to characterize structural heterogeneity in complex networks

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Amirhossein Farzam ◽  
Areejit Samal ◽  
Jürgen Jost
Keyword(s):  
Complex Networks ◽  
Structural Properties ◽  
Degree Sequence ◽  
Structural Heterogeneity ◽  
Local Network ◽  
Basic Unit ◽  
Local Geometry ◽  
Scale Free ◽  
Mixing Patterns ◽  
Topological Robustness

AbstractDespite the growing interest in characterizing the local geometry leading to the global topology of networks, our understanding of the local structure of complex networks, especially real-world networks, is still incomplete. Here, we analyze a simple, elegant yet underexplored measure, ‘degree difference’ (DD) between vertices of an edge, to understand the local network geometry. We describe the connection between DD and global assortativity of the network from both formal and conceptual perspective, and show that DD can reveal structural properties that are not obtained from other such measures in network science. Typically, edges with different DD play different structural roles and the DD distribution is an important network signature. Notably, DD is the basic unit of assortativity. We provide an explanation as to why DD can characterize structural heterogeneity in mixing patterns unlike global assortativity and local node assortativity. By analyzing synthetic and real networks, we show that DD distribution can be used to distinguish between different types of networks including those networks that cannot be easily distinguished using degree sequence and global assortativity. Moreover, we show DD to be an indicator for topological robustness of scale-free networks. Overall, DD is a local measure that is simple to define, easy to evaluate, and that reveals structural properties of networks not readily seen from other measures.

The N-star network evolution model

2019 ◽  
Vol 56 (2) ◽  
pp. 416-440 ◽  
Author(s):  
István Fazekas ◽  
Csaba Noszály ◽  
Attila Perecsényi
Keyword(s):  
Power Law ◽  
Preferential Attachment ◽  
Network Evolution ◽  
Evolution Model ◽  
Asymptotic Power ◽  
Star Network ◽  
Power Law Distributions

AbstractA new network evolution model is introduced in this paper. The model is based on cooperations of N units. The units are the nodes of the network and the cooperations are indicated by directed links. At each evolution step N units cooperate, which formally means that they form a directed N-star subgraph. At each step either a new unit joins the network and it cooperates with N − 1 old units, or N old units cooperate. During the evolution both preferential attachment and uniform choice are applied. Asymptotic power law distributions are obtained both for in-degrees and for out-degrees.

INFORMATION ASYMMETRY FLOWING IN COMPLEX NETWORKS

2012 ◽  
Vol 26 (31) ◽  
pp. 1250183
Author(s):  
CHEN-XI SHAO ◽  
HUI-LING DOU ◽  
BING-HONG WANG
Keyword(s):  
Symmetry Breaking ◽  
Complex Networks ◽  
Information Asymmetry ◽  
Complex Network ◽  
Information Flow ◽  
Weak Link ◽  
Network Evolution ◽  
Long Tail ◽  
Network Nodes ◽  
Concept Of Information

The concept of information asymmetry in complex networks is introduced on the basis of information asymmetry in economics and symmetry breaking. Information flowing between two nodes on a link is bidirectional, whose size is closely related to traffic dynamics on the network. Based on asymmetric information theory, we proposed information flow between network nodes is asymmetrical. We designed two methods to calculate the amount of information flow based on two mechanisms of complex network. Unequal flow of two opposite directions on the same link proved information asymmetry exists in the complex network. A complex network evolution model based on symmetry breaking is established, which is a truthful example for complex network mimicking nature. The evolution mechanism of symmetry breaking can best explain the phenomenon of the weak link and long tail theory in complex network.

THE ROLE OF UNBOUNDED TIME-SCALES IN GENERATING LONG-RANGE MEMORY IN ADDITIVE MARKOVIAN PROCESSES

2013 ◽  
Vol 12 (02) ◽  
pp. 1340002 ◽  
Author(s):  
SALVATORE MICCICHÈ ◽  
FABRIZIO LILLO ◽  
ROSARIO N. MANTEGNA
Keyword(s):  
Time Scales ◽  
Power Law ◽  
Planck Equation ◽  
Markovian Process ◽  
Quantum Potential ◽  
Autocorrelated Process ◽  
Markovian Processes ◽  
Exactly Solvable ◽  
The Continuum

Any additive stationary and continuous Markovian process described by a Fokker–Planck equation can also be described in terms of a Schrödinger equation with an appropriate quantum potential. By using such analogy, it has been proved that a power-law correlated stationary Markovian process can stem from a quantum potential that (i) shows an x-2 decay for large x values and (ii) whose eigenvalue spectrum admits a null eigenvalue and a continuum part of positive eigenvalues attached to it. In this paper we show that such two features are both necessary. Specifically, we show that a potential with tails decaying like x-μ with μ < 2 gives rise to a stationary Markovian process which is not power-law autocorrelated, despite the fact that the process has an unbounded set of time scales. Moreover, we present an exactly solvable example where the potential decays as x-2 but there is a gap between the continuum spectrum of eigenvalues and the null eigenvalue. We show that the process is not power law autocorrelated, but by decreasing the gap one can arbitrarily well approximate it. A crucial role in obtaining a power-law autocorrelated process is played by the weights [Formula: see text] giving the contribution of each time-scale contribute to the autocorrelation function. In fact, we will see that such weights must behave like a power-law for small energy values λ. This is only possible if the potential VS(x) shows a x-2 decay to zero for large x values.

Effect of landslides on the structural characteristics of land-cover based on complex networks

2017 ◽  
Vol 31 (22) ◽  
pp. 1750156 ◽  
Author(s):  
Jing He ◽  
Chuan Tang ◽  
Gang Liu ◽  
Weile Li
Keyword(s):  
Land Cover ◽  
Complex Networks ◽  
Power Law ◽  
Satellite Images ◽  
Structural Characteristics ◽  
Power Law Distribution ◽  
Graph Topology ◽  
Linear Coefficient ◽  
Area Distribution ◽  
Before And After

Landslides have been widely studied by geologists. However, previous studies mainly focused on the formation of landslides and never considered the effect of landslides on the structural characteristics of land-cover. Here we define the modeling of the graph topology for the land-cover, using the satellite images of the earth’s surface before and after the earthquake. We find that the land-cover network satisfies the power-law distribution, whether the land-cover contains landslides or not. However, landslides may change some parameters or measures of the structural characteristics of land-cover. The results show that the linear coefficient, modularity and area distribution are all changed after the occurence of landslides, which means the structural characteristics of the land-cover are changed.

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