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 Interplay between $$k$$-core and community structure in complex networks

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Irene Malvestio ◽  
Alessio Cardillo ◽  
Naoki Masuda
Keyword(s):  
Community Structure ◽  
Complex Networks ◽  
Random Graph ◽  
Shell Structure ◽  
Crucial Role ◽  
Configuration Model ◽  
Graph Models ◽  
Random Graph Models ◽  
Empirical Networks

Abstract The organisation of a network in a maximal set of nodes having at least k neighbours within the set, known as $$k$$ k -core decomposition, has been used for studying various phenomena. It has been shown that nodes in the innermost $$k$$ k -shells play a crucial role in contagion processes, emergence of consensus, and resilience of the system. It is known that the $$k$$ k -core decomposition of many empirical networks cannot be explained by the degree of each node alone, or equivalently, random graph models that preserve the degree of each node (i.e., configuration model). Here we study the $$k$$ k -core decomposition of some empirical networks as well as that of some randomised counterparts, and examine the extent to which the $$k$$ k -shell structure of the networks can be accounted for by the community structure. We find that preserving the community structure in the randomisation process is crucial for generating networks whose $$k$$ k -core decomposition is close to the empirical one. We also highlight the existence, in some networks, of a concentration of the nodes in the innermost $$k$$ k -shells into a small number of communities.

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.

A New Method for Extracting the Hierarchical Organization of Networks

2017 ◽  
Vol 16 (05) ◽  
pp. 1359-1385 ◽  
Author(s):  
Weihua Zhan ◽  
Jihong Guan ◽  
Zhongzhi Zhang
Keyword(s):  
Complex Networks ◽  
Networked Systems ◽  
Hierarchical Organization ◽  
New Method ◽  
Network Data ◽  
Large Complex

Extracting the hierarchical organization of networks is currently a pressing task for understanding complex networked systems. The hierarchy of a network is essentially defined by the heterogeneity of link densities of communities at different scales. Here, we define a top-level partition (TLP) as a bipartition of the network (or a sub-network) such that no top-level community (TLC) runs across the two parts. It has been found that a TLP generally has a higher modularity than a non-top-level (TLP) partition when their TLCs have similar sizes and when the link densities of neighboring levels are well separated from each other. A spectral TLP procedure is proposed here to search for TLPs of a network (or sub-network). To extract the hierarchical organization of large complex networks, an algorithm called TLPA has been developed based on the TLP. Experiments have shown that the method developed in this research extract hierarchy accurately from network data.

scholarly journals Quantifying the compressibility of complex networks

2021 ◽  
Vol 118 (32) ◽  
pp. e2023473118
Author(s):  
Christopher W. Lynn ◽  
Danielle S. Bassett
Keyword(s):  
Complex Networks ◽  
Resource Constraints ◽  
Rate Distortion ◽  
Hierarchical Organization ◽  
Human Cognition ◽  
Minimal Amount ◽  
Amount Of Information ◽  
Rate Distortion Theory ◽  
Network Properties ◽  
Biological Entities

Many complex networks depend upon biological entities for their preservation. Such entities, from human cognition to evolution, must first encode and then replicate those networks under marked resource constraints. Networks that survive are those that are amenable to constrained encoding—or, in other words, are compressible. But how compressible is a network? And what features make one network more compressible than another? Here, we answer these questions by modeling networks as information sources before compressing them using rate-distortion theory. Each network yields a unique rate-distortion curve, which specifies the minimal amount of information that remains at a given scale of description. A natural definition then emerges for the compressibility of a network: the amount of information that can be removed via compression, averaged across all scales. Analyzing an array of real and model networks, we demonstrate that compressibility increases with two common network properties: transitivity (or clustering) and degree heterogeneity. These results indicate that hierarchical organization—which is characterized by modular structure and heterogeneous degrees—facilitates compression in complex networks. Generally, our framework sheds light on the interplay between a network’s structure and its capacity to be compressed, enabling investigations into the role of compression in shaping real-world networks.

Hierarchical Organization in Smooth Dynamical Systems

2005 ◽  
Vol 11 (4) ◽  
pp. 493-512 ◽  
Author(s):  
Martin N. Jacobi
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Hierarchical Structures ◽  
Hierarchical Organization ◽  
Necessary And Sufficient ◽  
Deterministic Dynamical System ◽  
Functional Components ◽  
Different Levels

This article is concerned with defining and characterizing hierarchical structures in smooth dynamical systems. We define transitions between levels in a dynamical hierarchy by smooth projective maps from a phase space on a lower level, with high dimensionality, to a phase space on a higher level, with lower dimensionality. It is required that each level describe a self-contained deterministic dynamical system. We show that a necessary and sufficient condition for a projective map to be a transition between levels in the hierarchy is that the kernel of the differential of the map is tangent to an invariant manifold with respect to the flow. The implications of this condition are discussed in detail. We demonstrate two different causal dependences between degrees of freedom, and how these relations are revealed when the dynamical system is transformed into global Jordan form. Finally these results are used to define functional components on different levels, interaction networks, and dynamical hierarchies.

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