Extracting hierarchical organization of complex networks by dynamics towards synchronization

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.

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Detecting Hierarchical Organization in Complex Networks by Nearest Neighbor Correlation

Nature Inspired Cooperative Strategies for Optimization (NICSO 2007) - Studies in Computational Intelligence ◽

10.1007/978-3-540-78987-1_44 ◽

2008 ◽

pp. 487-494 ◽

Cited By ~ 1

Author(s):

Chao Long Wang ◽

Ka Wai Au ◽

Ching King Chan ◽

Hon Wai Lau ◽

K. Y. Szeto

Keyword(s):

Complex Networks ◽

Nearest Neighbor ◽

Hierarchical Organization

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Growth and Optimality in Network Evolution

Artificial Life ◽

10.1162/artl_a_00039 ◽

2011 ◽

Vol 17 (4) ◽

pp. 281-291 ◽

Cited By ~ 5

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.

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A New Method for Extracting the Hierarchical Organization of Networks

International Journal of Information Technology & Decision Making ◽

10.1142/s021962201450028x ◽

2017 ◽

Vol 16 (05) ◽

pp. 1359-1385 ◽

Cited By ~ 1

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.

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Quantifying the compressibility of complex networks

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.2023473118 ◽

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.

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Evolutionary significance of metabolic network properties

Journal of The Royal Society Interface ◽

10.1098/rsif.2011.0652 ◽

2011 ◽

Vol 9 (71) ◽

pp. 1168-1176 ◽

Cited By ~ 15

Author(s):

Georg Basler ◽

Sergio Grimbs ◽

Oliver Ebenhöh ◽

Joachim Selbig ◽

Zoran Nikoloski

Keyword(s):

Complex Networks ◽

Structural Properties ◽

Protein Interactions ◽

Biological Networks ◽

Metabolic Networks ◽

Functional Organization ◽

Hierarchical Organization ◽

Clustering Coefficient ◽

Evolutionary Significance ◽

System Architectures

Complex networks have been successfully employed to represent different levels of biological systems, ranging from gene regulation to protein–protein interactions and metabolism. Network-based research has mainly focused on identifying unifying structural properties, such as small average path length, large clustering coefficient, heavy-tail degree distribution and hierarchical organization, viewed as requirements for efficient and robust system architectures. However, for biological networks, it is unclear to what extent these properties reflect the evolutionary history of the represented systems. Here, we show that the salient structural properties of six metabolic networks from all kingdoms of life may be inherently related to the evolution and functional organization of metabolism by employing network randomization under mass balance constraints. Contrary to the results from the common Markov-chain switching algorithm, our findings suggest the evolutionary importance of the small-world hypothesis as a fundamental design principle of complex networks. The approach may help us to determine the biologically meaningful properties that result from evolutionary pressure imposed on metabolism, such as the global impact of local reaction knockouts. Moreover, the approach can be applied to test to what extent novel structural properties can be used to draw biologically meaningful hypothesis or predictions from structure alone.

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