How to calculate the fractal dimension of a complex network: the box covering algorithm
TLDR
It is argued that the algorithms presented provide a solution close to optimal and that another algorithm that can significantly improve this result in an efficient way does not exist.Abstract:
Covering a network with the minimum possible number of boxes can reveal interesting features for the network structure, especially in terms of self-similar or fractal characteristics. Considerable attention has been recently devoted to this problem, with the finding that many real networks are self-similar fractals. Here we present, compare and study in detail a number of algorithms that we have used in previous papers towards this goal. We show that this problem can be mapped to the well-known graph colouring problem and then we simply can apply well-established algorithms. This seems to be the most efficient method, but we also present two other algorithms based on burning which provide a number of other benefits. We argue that the algorithms presented provide a solution close to optimal and that another algorithm that can significantly improve this result in an efficient way does not exist. We offer to anyone that finds such a method to cover his/her expenses for a one-week trip to our lab in New York (details in http://jamlab.org).read more
Citations
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Journal ArticleDOI
Critical phenomena in complex networks
TL;DR: A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolations, phenomena near epidemic thresholds, condensation transitions,critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned.
Book
Complex Networks: Structure, Robustness and Function
Reuven Cohen,Shlomo Havlin +1 more
TL;DR: This chapter discusses random network models, which are based on the Erdos-Renyi models, and their application in the context of complex networks, where distances in scale-free networks are small and distances in complex networks are large.
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A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks
TL;DR: It is shown that a modified percolation theory can define a set of hierarchically organized modules made of strong links in functional brain networks, which are far from being small-world but which suggest a natural solution to the paradox of efficient information flow in the highly modular structure of the brain.
Journal ArticleDOI
From first-passage times of random walks in confinement to geometry-controlled kinetics
TL;DR: In this article, the authors present a general theory which allows one to accurately evaluate the mean first-passage time (FPT) for regular random walks in bounded domains, and its extensions to related firstpassage observables such as splitting probabilities and occupation times.
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Small-world to fractal transition in complex networks: a renormalization group approach.
TL;DR: It is shown that renormalization group (RG) theory applied to complex networks is useful to classify network topologies into universality classes in the space of configurations and allows us to extract information on the distribution of shortcuts in real-world networks.
References
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Book
Computers and Intractability: A Guide to the Theory of NP-Completeness
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Journal ArticleDOI
Statistical mechanics of complex networks
TL;DR: In this paper, a simple model based on the power-law degree distribution of real networks was proposed, which was able to reproduce the power law degree distribution in real networks and to capture the evolution of networks, not just their static topology.
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The Structure and Function of Complex Networks
TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.