Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com

Axiomatic Construction of Hierarchical Clustering in Asymmetric Networks

This paper considers networks where relationships between nodes are represented by directed dissimilarities. The goal is to study methods for the determination of hierarchical clusters, i.e., a family of nested partitions indexed by a connectivity parameter, induced by the given dissimilarity structures. Our construction of hierarchical clustering methods is based on defining admissible methods to be those methods that abide by the axioms of value - nodes in a network with two nodes are clustered together at the maximum of the two dissimilarities between them - and transformation - when dissimilarities are reduced, the network may become more clustered but not less. Several admissible methods are constructed and two particular methods, termed reciprocal and nonreciprocal clustering, are shown to provide upper and lower bounds in the space of admissible methods. Alternative clustering methodologies and axioms are further considered. Allowing the outcome of hierarchical clustering to be asymmetric, so that it matches the asymmetry of the original data, leads to the inception of quasi-clustering methods. The existence of a unique quasi-clustering method is shown. Allowing clustering in a two-node network to proceed at the minimum of the two dissimilarities generates an alternative axiomatic construction. There is a unique clustering method in this case too. The paper also develops algorithms for the computation of hierarchical clusters using matrix powers on a min-max dioid algebra and studies the stability of the methods proposed. We proved that most of the methods introduced in this paper are such that similar networks yield similar hierarchical clustering results. Algorithms are exemplified through their application to networks describing internal migration within states of the United States (U.S.) and the interrelation between sectors of the U.S. economy.Comment: This is a largely extended version of the previous conference submission under the same title. The current version contains the material in the previous version (published in ICASSP 2013) as well as material presented at the Asilomar Conference on Signal, Systems, and Computers 2013, GlobalSIP 2013, and ICML 2014. Also, unpublished material is included in the current versio

Complex networks analysis in socioeconomic models

This chapter aims at reviewing complex networks 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 adds insights on the statistical mechanical approach, and on the most relevant computational aspects for the treatment of these systems. As the most frequently used model for interacting agent-based systems, a brief description of the statistical mechanics of the classical Ising model on regular lattices, together with recent extensions of the same model on small-world Watts-Strogatz and scale-free Albert-Barabasi complex networks is included. Other sections of the chapter are devoted to applications of complex networks to economics, finance, spreading of innovations, and regional trade and developments. The chapter also reviews results involving applications of complex networks to other relevant socioeconomic issues, including results for opinion and citation networks. Finally, some avenues for future research are introduced before summarizing the main conclusions of the chapter.Comment: 39 pages, 185 references, (not final version of) a chapter prepared for Complexity and Geographical Economics - Topics and Tools, P. Commendatore, S.S. Kayam and I. Kubin Eds. (Springer, to be published