The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.

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Community detection in graphs

http://cds.cern.ch/record/413720/files/CM-P00060268.pdf

A planar diagram theory for strong interactions

This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Spin glasses: Experimental facts, theoretical concepts, and open questions

Statistical physics has proven to be a fruitful framework to describe phenomena outside the realm of traditional physics. Recent years have witnessed an attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. A wide list of topics are reviewed ranging from opinion and cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, and social spreading. The connections between these problems and other, more traditional, topics of statistical physics are highlighted. Comparison of model results with empirical data from social systems are also emphasized.

Statistical physics of social dynamics

This book contains a detailed and self-contained presentation of the replica theory of infinite range spin glasses. The authors also explain recent theoretical developments, paying particular attention to new applications in the study of optimization theory and neural networks. About two-thirds of the book are a collection of the most interesting and pedagogical articles on the subject.

Spin Glass Theory and Beyond

Ultrametricity is a simple topological concept, but its appearance in the language of physicists is recent. This review provides all the elementary background (from mathematics, taxonomy, and statistical physics) and surveys the main fields of development (spin glasses, optimization theory). Static and dynamic aspects are covered. From present knowledge, one can already draw some tentative conclusions on the common causes for the occurrence of ultrametric structures in nature. Some perspectives on unresolved problems in physics and biology are also presented.

Ultrametricity for physicists

Spin Glass Theory And Beyond: An Introduction To The Replica Method And Its Applications

A probability distribution has been proposed recently by one of us as an order parameter for spin-glasses We show that this probability depends on the particular realization of the couplings even in the thermodynamic limit, and we study its distribution We also show that the space of states has an ultrametric topology

Nature of the Spin-Glass Phase

A probability distribution has been proposed recently by one of us as an order parameter for spin glasses. We show that this probability depends on the particular realization of the couplings even in the thermodynamic limit, and we study its distribution. We also show that the space of states has an ultrametric topology Etude d'une distribution de probabilite, proposee recemment, comme parametre d'onde pour les verres de spin: la probabilite depend de la realisation particuliere des couplages, meme a la limite thermodynamique. Etude de la distribution de cette probabilite. Mise en evidence de la topologie ultrametrique de l'espace des etats

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Miguel Angel Virasoro

Papers

Spin Glass Theory and Beyond

Ultrametricity for physicists

Spin Glass Theory And Beyond: An Introduction To The Replica Method And Its Applications

Nature of the Spin-Glass Phase

Replica symmetry breaking and the nature of the spin glass phase