The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. 
Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. 
The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. 
The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.

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Statistical mechanics of complex networks

Block copolymers are macromolecules composed of sequences, or blocks, of chemically distinct repeat units. The development of this field originated with the discovery of termination-free anionic polymerization, which made possible the sequential addition of monomers to various carbanion-ter­ minated ("living") linear polymer chains. Polymerization of just two dis­ tinct monomer types (e.g. styrene and isoprene) leads to a class of materials referred to as AB block copolymers. Within this class, a variety of molec­ ular architectures is possible. For example, the simplest combination, obtained by the two-step anionic polymerization of A and B monomers, is an (A-B) dioblock copolymer. A three-step reaction provides for the preparation of (ABA) or (BAB) triblock copolymer. Alternatively, "living" diblock copolymers can be reacted with an n-functional coupling agent to produce (A-B)n star-block copolymers, where n = 2 constitutes a triblock copolymer. Several representative (A-B)n block copolymer architectures

Block Copolymer Thermodynamics: Theory and Experiment

Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

We study the statistical properties of a variety of diverse real-world networks. We present evidence of the occurrence of three classes of small-world networks: (a) scale-free networks, characterized by a vertex connectivity distribution that decays as a power law; (b) broad-scale networks, characterized by a connectivity distribution that has a power law regime followed by a sharp cutoff; and (c) single-scale networks, characterized by a connectivity distribution with a fast decaying tail. Moreover, we note for the classes of broad-scale and single-scale networks that there are constraints limiting the addition of new links. Our results suggest that the nature of such constraints may be the controlling factor for the emergence of different classes of networks.

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Classes of small-world networks

The field of phase transitions and critical phenomena continues to be active in research, producing a steady stream of interesting and fruitful results. It has moved into a central place in condensed matter studies. Statistical physics, and more specifically, the theory of transitions between states of matter, more or less defines what we know about 'everyday' matter and its transformations. The major aim of this serial is to provide review articles that can serve as standard references for research workers in the field, and for graduate students and others wishing to obtain reliable information on important recent developments.

Phase Transitions and Critical Phenomena

The critical point

1. Preface 2. Historical Survey 3. Fluids: Classical Theory 4. Light Scattering and Correlations: Classical Theory 5. The Onsager Revolution 6. Reconciliation 7. Renormalization Group 8. Appendix: Related Topics 9. References 10. Index Contents Preface Foreword: About the Author and the Subject 1 Historical Survey 1.1 The Development of Statistical Mechanics 1.2 Gibbs Ensembles 1.3 Non-interacting and Interacting Systems 1.4 The Classical Period 1.5 The Onsager Revolution 1.6 Reconciliation 1.7 Renormalization Group: Respectability General Notation References 2 Fluids: Classical Theory 2.1 Thermodynamic Background 2.2 Stability of a Phase: First-order Transitions 2.3 Theory of van der Waals 2.4 Critical Behaviour of a van der Waals Fluid (T > Tc) 2.5 Critical Behaviour of a van der Waals Fluid (T

The Critical Point: A Historical Introduction To The Modern Theory Of Critical Phenomena

The present paper discusses the problem of making the most effective use of the coefficients of series expansions for the Ising model and excluded volume problem in estimating critical behavior. It is shown that after initial irregularities the coefficients appear to settle down to a smooth asymptotic behavior. Alternative methods of analysis are considered for the provision of a steady series of approximations to the critical point. Numerical conclusions are drawn for particular lattices for which additional terms have recently become available.

Use of Series Expansions for the Ising Model Susceptibility and Excluded Volume Problem

The present paper is concerned with the detailed calculation of the partition functions for a two-dimensional quadratic lattice from the matrix derived previously. A brief discussion is given of the general methods of calculation available. The method here employed is the expansion of the eigenvector and eigenvalue as power series about zero temperature. This is readily applicable to finite matrices of quite high order, and, when a suitable notation has been introduced, to the infinite case. It is found more convenient to deal with the general problem of an unsymmetric net with different interactions in the two directions; for a ferromagnetic in the absence of a magnetic field a symmetry relation observed empirically enables terms to be derived successively, and it is assumed that this solution is equivalent to Onsager’s (1944). The method is applied in the presence of a magnetic field, and several terms of a generalized series are deduced. Several terms are hence obtained of a series for the spontaneous magnetization. An inspection of the generalized series leads one to conjecture that the specific heat curve becomes continuous in the presence of a magnetic field. A rearrangement of the terms of the generalized series enables several terms of the high-temperature expansion to be deduced. Finally, the results are applied to the theory of binary solid solutions; the solubility curve for the two substances is formally closely related to the spontaneous magnetization. The separation into two phases is established, and the corresponding specific heat singularities are analyzed.

Cyril Domb

Papers

Phase Transitions and Critical Phenomena

The critical point

The Critical Point: A Historical Introduction To The Modern Theory Of Critical Phenomena

Use of Series Expansions for the Ising Model Susceptibility and Excluded Volume Problem

Order-disorder statistics. II. A two-dimensional model