A new family of graphs, entangled networks, with optimal properties in many respects, is introduced. By definition, their topology is such that it optimizes synchronizability for many dynamical processes. These networks are shown to have an extremely homogeneous structure: degree, node distance, betweenness, and loop distributions are all very narrow. Also, they are characterized by a very interwoven (entangled) structure with short average distances, large loops, and no well-defined community structure. This family of nets exhibits an excellent performance with respect to other flow properties such as robustness against errors and attacks, minimal first-passage time of random walks, efficient communication, etc. These remarkable features convert entangled networks in a useful concept, optimal or almost optimal in many senses, and with plenty of potential applications in computer science or neuroscience.

An efficient and relatively fast algorithm for the detection of communities in complex networks is introduced. The method exploits spectral properties of the graph Laplacian matrix combined with hierarchical-clustering techniques, and includes a procedure to maximize the "modularity" of the output. Its performance is compared with that of other existing methods, as applied to different well-known instances of complex networks with a community-structure, both computer-generated and from the real-world. Our results are in all the tested cases, at least as good as the best ones obtained with any other methods, and faster in most of the cases than methods providing similarquality results. This converts the algorithm in a valuable computational tool for detecting and analyzing communities and modular structures in complex networks.

Why are large, complex ecosystems stable? Both theory and simulations of current models predict the onset of instability with growing size and complexity, so for decades it has been conjectured that ecosystems must have some unidentified structural property exempting them from this outcome. We show that trophic coherence-a hitherto ignored feature of food webs that current structural models fail to reproduce-is a better statistical predictor of linear stability than size or complexity. Furthermore, we prove that a maximally coherent network with constant interaction strengths will always be linearly stable. We also propose a simple model that, by correctly capturing the trophic coherence of food webs, accurately reproduces their stability and other basic structural features. Most remarkably, our model shows that stability can increase with size and complexity. This suggests a key to May's paradox, and a range of opportunities and concerns for biodiversity conservation.food webs | May's paradox | diversity-stability debate | dynamical stability | complex networks I n the early seventies, Robert May addressed the question of whether a generic system of coupled dynamical elements randomly connected to each other would be stable. He found that the larger and more interconnected the system, the more difficult it would be to stabilize (1, 2). His deduction followed from the behavior of the leading eigenvalue of the interaction matrix, which, in a randomly wired system, grows with the square root of the mean number of links per element. This result clashed with the received wisdom in ecology-that large, complex ecosystems were particularly stable-and initiated the "diversity-stability debate" (3-6). Indeed, Charles Elton had expressed the prevailing view in 1958: "the balance of relatively simple communities of plants and animals is more easily upset than that of richer ones; that is, more subject to destructive oscillations in populations, especially of animals, and more vulnerable to invasions" (7). Even if this description were not accurate, the mere existence of rainforests and coral reefs seems incongruous with a general mathematical principle that "complexity begets instability," and has become known as May's paradox.One solution might be that the linear stability analysis used by May and many subsequent studies does not capture essential characteristics of ecosystem dynamics, and much work has gone into exploring how more accurate dynamical descriptions might enhance stability (5,8,9). However, as ever-better ecological data are gathered, it is becoming apparent that the leading eigenvalues of matrices related to food webs (networks in which the species are nodes and the links represent predation) do not exhibit the expected dependence on size or link density (10). Food webs must, therefore, have some unknown structural feature that accounts for this deviation from randomness-irrespectively of other stabilizing factors.We show here that a network feature we call trophic coherence accounts for much of the varianc...

We report on some recent developments in the search for optimal network topologies. First we review some basic concepts on spectral graph theory, including adjacency and Laplacian matrices, and paying special attention to the topological implications of having large spectral gaps. We also introduce related concepts as "expanders", Ramanujan, and Cage graphs. Afterwards, we discuss two different dynamical features of networks: synchronizability and flow of random walkers and so that they are optimized if the corresponding Laplacian matrix have a large spectral gap. From this, we show, by developing a numerical optimization algorithm that maximum synchronizability and fast random walk spreading are obtained for a particular type of extremely homogeneous regular networks, with long loops and poor modular structure, that we call entangled networks. These turn out to be related to Ramanujan and Cage graphs. We argue also that these graphs are very good finite-size approximations to Bethe lattices, and provide almost or almost optimal solutions to many other problems as, for instance, searchability in the presence of congestion or performance of neural networks. Finally, we study how these results are modified when studying dynamical processes controlled by a normalized (weighted and directed) dynamics; much more heterogeneous graphs are optimal in this case. Finally, a critical discussion of the limitations and possible extensions of this work is presented.

We demonstrate the confinement of acoustic phonons in ultrathin silicon layers and study its effect on electron mobility. We develop a model for confined acoustic phonons in an ideal single-layer structure and in a more realistic three-layer structure. Phonon quantization is recovered, and the dispersion relations for distinct phonon modes are computed. This allows us to obtain the confined phonon scattering rates and, using Monte Carlo simulations, to compute the electron mobility in ultrathin silicon on insulator inversion layers. Thus, comparing the results with those obtained using the bulk phonon model, we are able to conclude that it is very important to include confined acoustic phonon models in the electron transport simulations of ultrathin devices, if we want to reproduce the actual behavior of electron transport in silicon layers of nanometric thickness.