A new mechanism leading to scale-free networks is proposed in this Letter. It is shown that, in many cases of interest, the connectivity power-law behavior is neither related to dynamical properties nor to preferential attachment. Assigning a quenched fitness value x i to every vertex, and drawing links among vertices with a probability depending on the fitnesses of the two involved sites, gives rise to what we call a good-get-richer mechanism, in which sites with larger fitness are more likely to become hubs (i.e., to be highly connected).

A celebrated and controversial hypothesis suggests that some biological systems -parts, aspects, or groups of them-may extract important functional benefits from operating at the edge of instability, halfway between order and disorder, i.e. in the vicinity of the critical point of a phase transition. Criticality has been argued to provide biological systems with an optimal balance between robustness against perturbations and flexibility to adapt to changing conditions, as well as to confer on them optimal computational capabilities, huge dynamical repertoires, unparalleled sensitivity to stimuli, etc. Criticality, with its concomitant scale invariance, can be conjectured to emerge in living systems as the result of adaptive and evolutionary processes that, for reasons to be fully elucidated, select for it as a template upon which further layers of complexity can rest. This hypothesis is very suggestive as it proposes that criticality could constitute a general and common organizing strategy in biology stemming from the physics of phase transitions. However, despite its thrilling implications, this is still in its embryonic state as a well-founded theory and, as such, it has elicited some healthy skepticism. From the experimental side, the advent of high-throughput technologies has created new prospects in the exploration of biological systems, and empirical evidence in favor of criticality has proliferated, with examples ranging from endogenous brain activity and gene-expression patterns, to flocks of birds and insect-colony foraging, to name but a few. Some pieces of evidence are quite remarkable, while in some other cases empirical data are limited, incomplete, or not fully convincing. More stringent experimental set-ups and theoretical analyses are certainly needed to fully clarify the picture. In any case, the time seems ripe for bridging the gap between this theoretical conjecture and its empirical validation. Given the profound implications of shedding light on this issue, we believe that it is both pertinent and timely to review the state of the art and to discuss future strategies and perspectives. Appendix A: Generic Scale invariance 23 Appendix B: Probabilistic models and statistical criticality 24 Appendix C: Adaptation and evolution towards criticality 24 Appendix D: Other putatively critical living systems 25 References 25 arXiv:1712.04499v2 [cond-mat.stat-mech]

We present a pedagogical introduction to self-organized criticality (SOC), unraveling its connections with nonequilibrium phase transitions. There are several paths from a conventional critical point to SOC. They begin with an absorbing-state phase transition (directed percolation is a familiar example), and impose supervision or driving on the system two commonly used methods are extremal dynamics, and driving at a rate approaching zero. We illustrate this in sandpiles, where SOC is a consequence of slow driving in a system exhibiting an absorbing-state phase transition with a conserved density. Other paths to SOC, in driven interfaces, the Bak-Sneppen model, and selforganized directed percolation, are also examined. We review the status of experimental realizations of SOC in light of these observations. I IntroductionThe label \self-organized" is applied indiscriminately in the current literature to ordering or pattern formation amongst many i n teracting units. Implicit is the notion that the phenomenon of interest, be it scale invariance, cooperation, or supra-molecular organization (e.g., micelles), appears spontaneously. That, of course, is just how the magnetization appears in the Ising model but we don't speak of \self-organized magnetization." After nearly a century of study, w e've come to expect the spins to organize the zero-eld magnetization below T c is no longer a surprise. More generally, s p o n taneous organization of interacting units is precisely what we seek, to explain the emergence of order in nature. We can expect many more surprises in the quest to discover what kinds of order a given set of interactions lead to. All will be self-organized, there being no outside agent on hand to impose order! \Self-organized criticality" (SOC) carries greater speci city, because criticality usually does not happen spontaneously: various parameters have t o b e t u n e d to reach the critical point. Scale-invariance in natural systems, far from equilibrium, isn't explained merely by showing that the interacting units can exhibit scale invariance at a point in parameter space one has to show how the system is maintained (or maintains itself) a t t h e critical point. (Alternatively one can try to show t h a t there is generic scale invariance, that is, that criticality appears over a region of parameter space with nonzero measure 1, 2 ].) \SOC" has been used to describe spontaneous scale invariance in general this would seem to embrace random walks, as well as fractal growth 3], diffusive annihilation (A + A ! 0 and related processes), and nonequilibrium surface dynamics 4]. Here we r estrict the term to systems that are attracted to a critical (scale-invariant) stationary state the chief examples are sandpile models 5]. Another class of realizations, exempli ed by the Bak-Sneppen model 6], involve extremal dynamics (the unit with the extreme value of a certain variable is the next to change). We will see that in many examples of SOC, there is a choice between global supervision (an odd state of a airs fo...

Hallmarks of criticality, such as power-laws and scale invariance, have been empirically found in cortical-network dynamics and it has been conjectured that operating at criticality entails functional advantages, such as optimal computational capabilities, memory and large dynamical ranges. As critical behaviour requires a high degree of fine tuning to emerge, some type of self-tuning mechanism needs to be invoked. Here we show that, taking into account the complex hierarchical-modular architecture of cortical networks, the singular critical point is replaced by an extended critical-like region that corresponds-in the jargon of statistical mechanics-to a Griffiths phase. Using computational and analytical approaches, we find Griffiths phases in synthetic hierarchical networks and also in empirical brain networks such as the human connectome and that of Caenorhabditis elegans. Stretched critical regions, stemming from structural disorder, yield enhanced functionality in a generic way, facilitating the task of self-organizing, adaptive and evolutionary mechanisms selecting for criticality.

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.