Self-similarity, small-world, scale-free scaling, disassortativity, and robustness in hierarchical lattices

In this paper, firstly, we study analytically the topological features of a family of hierarchical lattices (HLs) from the view point of complex networks. We derive some basic properties of HLs controlled by a parameter $q$. Our results show that scale-free networks are not always small-world, and support the conjecture that self-similar scale-free networks are not assortative. Secondly, we define a deterministic family of graphs called small-world hierarchical lattices (SWHLs). Our construction preserves the structure of hierarchical lattices, while the small-world phenomenon arises. Finally, the dynamical processes of intentional attacks and collective synchronization are studied and the comparisons between HLs and Barab{\'asi}-Albert (BA) networks as well as SWHLs are shown. We show that degree distribution of scale-free networks does not suffice to characterize their synchronizability, and that networks with smaller average path length are not always easier to synchronize.Comment: 26 pages, 8 figure

Complex network analysis and nonlinear dynamics

This chapter aims at reviewing complex network and nonlinear dynamical models and methods that were either developed for or applied to socioeconomic issues, and pertinent to the theme of New Economic Geography. After an introduction to the foundations of the field of complex networks, the present summary introduces some applications of complex networks to economics, finance, epidemic spreading of innovations, and regional trade and developments. The chapter also reviews results involving applications of complex networks to other relevant socioeconomic issue

Characterization of complex networks: A survey of measurements

Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements. Important related issues covered in this work comprise the representation of the evolution of complex networks in terms of trajectories in several measurement spaces, the analysis of the correlations between some of the most traditional measurements, perturbation analysis, as well as the use of multivariate statistics for feature selection and network classification. Depending on the network and the analysis task one has in mind, a specific set of features may be chosen. It is hoped that the present survey will help the proper application and interpretation of measurements.Comment: A working manuscript with 78 pages, 32 figures. Suggestions of measurements for inclusion are welcomed by the author

Maturation trajectories of cortical resting-state networks depend on the mediating frequency band

The functional significance of resting state networks and their abnormal manifestations in psychiatric disorders are firmly established, as is the importance of the cortical rhythms in mediating these networks. Resting state networks are known to undergo substantial reorganization from childhood to adulthood, but whether distinct cortical rhythms, which are generated by separable neural mechanisms and are often manifested abnormally in psychiatric conditions, mediate maturation differentially, remains unknown. Using magnetoencephalography (MEG) to map frequency band specific maturation of resting state networks from age 7 to 29 in 162 participants (31 independent), we found significant changes with age in networks mediated by the beta (13–30 Hz) and gamma (31–80 Hz) bands. More specifically, gamma band mediated networks followed an expected asymptotic trajectory, but beta band mediated networks followed a linear trajectory. Network integration increased with age in gamma band mediated networks, while local segregation increased with age in beta band mediated networks. Spatially, the hubs that changed in importance with age in the beta band mediated networks had relatively little overlap with those that showed the greatest changes in the gamma band mediated networks. These findings are relevant for our understanding of the neural mechanisms of cortical maturation, in both typical and atypical development.This work was supported by grants from the Nancy Lurie Marks Family Foundation (TK, SK, MGK), Autism Speaks (TK), The Simons Foundation (SFARI 239395, TK), The National Institute of Child Health and Development (R01HD073254, TK), National Institute for Biomedical Imaging and Bioengineering (P41EB015896, 5R01EB009048, MSH), and the Cognitive Rhythms Collaborative: A Discovery Network (NFS 1042134, MSH). (Nancy Lurie Marks Family Foundation; Autism Speaks; SFARI 239395 - Simons Foundation; R01HD073254 - National Institute of Child Health and Development; P41EB015896 - National Institute for Biomedical Imaging and Bioengineering; 5R01EB009048 - National Institute for Biomedical Imaging and Bioengineering; NFS 1042134 - Cognitive Rhythms Collaborative: A Discovery Network