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

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

The mixing of polymers and nanoparticles is opening pathways for engineering flexible composites that exhibit advantageous electrical, optical, or mechanical properties. Recent advances reveal routes to exploit both enthalpic and entropic interactions so as to direct the spatial distribution of nanoparticles and thereby control the macroscopic performance of the material. For example, by tailoring the particle coating and size, researchers have created self-healing materials for improved sustainability and self-corralling rods for photovoltaic applications. A challenge for future studies is to create hierarchically structured composites in which each sublayer contributes a distinct function to yield a mechanically integrated, multifunctional material.

http://science.sciencemag.org/content/sci/314/5802/1107.full.pdf

Nanoparticle Polymer Composites: Where Two Small Worlds Meet

Dealloying is a common corrosion process during which an alloy is 'parted' by the selective dissolution of the most electrochemically active of its elements. This process results in the formation of a nanoporous sponge composed almost entirely of the more noble alloy constituents. Although considerable attention has been devoted to the morphological aspects of the dealloying process, its underlying physical mechanism has remained unclear. Here we propose a continuum model that is fully consistent with experiments and theoretical simulations of alloy dissolution, and demonstrate that nanoporosity in metals is due to an intrinsic dynamical pattern formation process. That is, pores form because the more noble atoms are chemically driven to aggregate into two-dimensional clusters by a phase separation process (spinodal decomposition) at the solid-electrolyte interface, and the surface area continuously increases owing to etching. Together, these processes evolve porosity with a characteristic length scale predicted by our continuum model. We expect that chemically tailored nanoporous gold made by dealloying Ag-Au should be suitable for sensor applications, particularly in a biomaterials context.

/pdf/evolution-of-nanoporosity-in-dealloying-1vysl3gdw0.pdf

Evolution of nanoporosity in dealloying

Enzymes synthesized by hyperthermophiles (bacteria and archaea with optimal growth temperatures of >80°C), also called hyperthermophilic enzymes, are typically thermostable (i.e., resistant to irreversible inactivation at high temperatures) and are optimally active at high temperatures. These enzymes share the same catalytic mechanisms with their mesophilic counterparts. When cloned and expressed in mesophilic hosts, hyperthermophilic enzymes usually retain their thermal properties, indicating that these properties are genetically encoded. Sequence alignments, amino acid content comparisons, crystal structure comparisons, and mutagenesis experiments indicate that hyperthermophilic enzymes are, indeed, very similar to their mesophilic homologues. No single mechanism is responsible for the remarkable stability of hyperthermophilic enzymes. Increased thermostability must be found, instead, in a small number of highly specific alterations that often do not obey any obvious traffic rules. After briefly discussing the diversity of hyperthermophilic organisms, this review concentrates on the remarkable thermostability of their enzymes. The biochemical and molecular properties of hyperthermophilic enzymes are described. Mechanisms responsible for protein inactivation are reviewed. The molecular mechanisms involved in protein thermostabilization are discussed, including ion pairs, hydrogen bonds, hydrophobic interactions, disulfide bridges, packing, decrease of the entropy of unfolding, and intersubunit interactions. Finally, current uses and potential applications of thermophilic and hyperthermophilic enzymes as research reagents and as catalysts for industrial processes are described.

Hyperthermophilic Enzymes: Sources, Uses, and Molecular Mechanisms for Thermostability

Traditionally the dispersion of particles in polymeric materials has proven difficult and frequently results in phase separation and agglomeration. We show that thermodynamically stable dispersion of nanoparticles into a polymeric liquid is enhanced for systems where the radius of gyration of the linear polymer is greater than the radius of the nanoparticle. Dispersed nanoparticles swell the linear polymer chains, resulting in a polymer radius of gyration that grows with the nanoparticle volume fraction. It is proposed that this entropically unfavorable process is offset by an enthalpy gain due to an increase in molecular contacts at dispersed nanoparticle surfaces as compared with the surfaces of phase-separated nanoparticles. Even when the dispersed state is thermodynamically stable, it may be inaccessible unless the correct processing strategy is adopted, which is particularly important for the case of fullerene dispersion into linear polymers.

https://science.sciencemag.org/content/sci/311/5768/1740.full.pdf

General strategies for nanoparticle dispersion.

Series Preface. Preface. Rigidity Theory. Generic and Abstract Rigidity B. Servatius, H. Servatius. Rigidity of Molecular Structures: Generic and Geometric Analysis W. Whiteley. Tensegrity Structures: Why Are They Stable? R. Connelly. The Role of Tensegrity in Distance Geometry T.F. Havel. Applications to Networks. Comparison of Connectivity and Rigidity Percolation C.F. Moukarzel, P.M. Duxbury. Rigidity Percolation on Trees P.L. Leath, C. Zeng. Rigidity as an Emergent Property of Random Networks: A Statistical Mechanical View P.M. Goldbart. Granular Matter Instability: A Structural Rigidity Point of View C.F. Moukarzel. Rigidity and Memory in a Simple Glass P. Chandra, L.B. Ioffe. Constraint Theory, Stiffness Percolation and the Rigidity Transition in Network Glasses J.C. Phillips. Topologically Disordered Networks of Rigid Polytopes: Applications to Noncrystalline Solids and Constrained Viscous Sintering P.K. Gupta. Rigidity Constraints in Amorphization of Singly- and Multiply-Polytopic Structures L.W. Hobbs, et al. Floppy Modes in Crystalline and Amorphous Silicates M.T. Dove, et al. Generic Rigidity of Networks Glasses M.F. Thorpe, et al. Rigidity Transition in Chalcogenide Glasses P. Boolchand, et al. Rigidity, Fragility, Bond Models and the `Energy Landscape' for Covalent Glassformers C.A. Angell. Entropic Rigidity B. Joos, et al. Applications to Proteins. Molecular Dynamics and Normal Mode Analysis of Biomolecular Rigidity D.A. Case. Efficient Stochastic Global Optimization for Protein Structure Prediction Y. Zhou, R. Abagyan. Flexible and Rigid Regions in Proteins D.J. Jacobs, et al. Flexibly Screening for Molecules Interacting with Proteins V. Schnecke, L.A. Kuhn. Studying Macromolecular Motions in a Database Framework: From Structure to Sequence M. Gerstein, et al. List of Participants. Index.

Rigidity theory and applications

An analysis of a prototypical percolation model (the fuse network) for breakdown in quenched random systems is given. The breakdown voltage and the topology of the eventual breakdown path are studied analytically and numerically. New scaling concepts, based on the most critical defect in the network, combined with standard percolation scaling ideas, lead to a complete picture of the strength of the network. The mean breakdown strength and the distribution of breakdown strengths are derived in the different concentration regimes. The breakdown path is described by new order parameters on approach to ${p}_{c}$. One, the number of bonds broken in the breakdown process, is studied in detail. Many models and physical systems should show an analogous behavior and simplified models for two of these problems, brittle fracture and dielectric breakdown in solids, are discussed.

Breakdown properties of quenched random systems: The random-fuse network.

It is demonstrated that nanocomposites exhibiting reduced viscosity and multifunctional performance enhancements may be fabricated using simple processing procedures. These behaviors are elucidated by analysis of the effects of dispersed organic (fullerene) nanoparticles and inorganic (magnetite) nanoparticles on the behavior of polystyrene, demonstrating that simple spherical nanoparticles can induce a range of unexpected behavior due to nanoscale effects. In general, multifunctional performance improvements including enhanced mechanical, electrical, magnetic and thermal degradation properties as well as reduced viscosity are promoted when simple design guidelines are followed. These guidelines are tabulated.

Multifunctional nanocomposites with reduced viscosity

Two percolation models for electrical breakdown in quenched random media, a fuse-wire network and a dielectric network, are introduced and studied. A combination of Lifshitz and scaling arguments leads to a size dependence given by $\frac{{V}_{b}}{L}\ensuremath{\sim}\frac{a(p)}{[1+b(p){(\mathrm{ln}L)}^{\ensuremath{\beta}}]}$, where $\frac{\ensuremath{\beta}=1}{(d\ensuremath{-}1)}$ for the fuse network and $\ensuremath{\beta}=1$ for the dielectric network. Simulations support this hypothesis in the 2D fuse network. We argue that any finite fraction of quenched defects qualitatively reduces the breakdown strength of a wide variety of electrical and mechanical systems in both two and three dimensions.

Phillip M. Duxbury

Papers

General strategies for nanoparticle dispersion.

Rigidity theory and applications

Breakdown properties of quenched random systems: The random-fuse network.

Multifunctional nanocomposites with reduced viscosity

Size effects of electrical breakdown in quenched random media.