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

An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of $^{4}\mathrm{He}$.

Theory of Dynamic Critical Phenomena

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

The Renormalization group and the epsilon expansion

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

We prove that to all orders in perturbation expansion, the critical exponents of a phase transition in a $d$-dimensional ($4ldl6$) system with short-range exchange and a random quenched field are the same as those of a ($d\ensuremath{-}2$)-dimensional pure system. Heuristic arguments are given to discuss both this result and the random-field Ising model for $2ldl6$.

Lowering of Dimensionality in Phase Transitions with Random Fields

We report in detail a study of the critical dynamics of a model ferromagnet in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions both above and below ${T}_{c}$, for small $\ensuremath{\epsilon}$. The precession of spins in the local magnetic field plays a major role in the dynamics. Its effect can be ignored at 6 or more dimensions, as many authors have previously observed. At a dimension slightly less than 6, the effect is nontrivial but tractable by perturbation theory. We carry out a renormalization-group analysis and determine the dynamic exponent $z=4\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}=1+\frac{d}{2}$, accurate to $O(\ensuremath{\epsilon})$. We also solve the equation of motion to obtain the spin response function. The frequency and damping of spin waves below ${T}_{c}$ are determined. Forms of equations of motion and the extent of universality in critical dynamics are discussed.

Critical dynamics of ferromagnets in 6 − ε dimensions: General discussion and detailed calculation

The dynamic critical behavior of a continuum analog of the kinetic Ising model is studied using a generalization of Wilson's expansion methods. Results are found which disagree with the mode-mode coupling approach and the conventional (Van Hove) theory.

Calculation of Dynamic Critical Properties Using Wilson's Expansion Methods

It is argued on the basis of a new interface model that the lower critical dimension of random-field Ising systems is two, in agreement with simple domain estimates.

Shang-keng Ma

Papers

Lowering of Dimensionality in Phase Transitions with Random Fields

Critical dynamics of ferromagnets in 6 − ε dimensions: General discussion and detailed calculation

Calculation of Dynamic Critical Properties Using Wilson's Expansion Methods

Roughening and Lower Critical Dimension in the Random-Field Ising Model

Temperature Dependence of the Spin Susceptibility of a Nearly Ferromagnetic Fermi Liquid