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In retrospect average path length between any two nodes in the network — measured as the smallest number of edges needed to connect the nodes — scales as the logarithm of the total number Twenty years of of nodes. In other words, randomness is suffi- cient to explain the small-world phenomenon popularized as ‘six degrees of separation’3,4: the network science idea that everyone in the world is connected to everyone else through a chain of, at most, six The idea that everyone in the world is connected to everyone else by just six degrees mutual acquaintances. of separation was explained by the ‘small-world’ network model 20 years ago. However, random construction fell short What seemed to be a niche finding turned out to have huge consequences. of capturing the local cliquishness of nodes observed in real-world networks. Cliquishness is measured quantitatively by the clustering ALESSANDRO VESPIGNANI from contagion processes to information coefficient of a node, which is defined as the diffusion. It soon became apparent that the ratio of the number of links between a node’s n 1998, Watts and Strogatz1 introduced paper had ushered in a new era of research that neighbours and the maximum number of such the ‘small-world’ model of networks, would lead to the establishment of network links. In real-world networks, node clustering which describes the clustering and short science as a multidisciplinary field. is clearly exemplified by the axiom ‘the friends Iseparations of nodes found in many real- Before Watts and Strogatz published their of my friends are my friends’: the probability life networks. I still vividly remember the paper, the archetypical network-generation of three people being friends with each other discussion I had with fellow statistical physi- algorithms were based on construction in a social network, for example, is generally cists at the time: the model was seen as sort processes such as those described by the much higher than would be predicted by a of interesting, but seemed to be merely an Erdös–Rényi model2. These processes are model network constructed using the simple, exotic departure from the regular, lattice-like characterized by a lack of knowledge of the stochastic process. network structures we were used to. But the principles that guide the creation of connec- To overcome the dichotomy between more the paper was assimilated by scientists tions (edges) between nodes in networks, randomness and cliquishness, Watts and from different fields, the more it became clear and make the simple assumption that pairs Strogatz proposed a model whose starting that it had deep implications for our under- of nodes can be connected at random with a point is a regular network that has a large standing of dynamic behaviour and phase given connection probability. Such a process clustering coefficient. Stochasticity is then transitions in real-world phenomena ranging generates random networks, in which the introduced by allowing links to be rewired at random between nodes, with a fixed probabil- ity of rewiring (p) for all links. By tuning p, the model effectively interpolates between a reg- a b ular lattice (p → 0) and a completely random network (p → 1). At very small p values, the resulting network is a regular lattice and therefore has a high clustering coefficient. However, even at small p, short cuts appear between distant nodes in the lattice, dramatically reducing the average shortest path length (Fig. 1). Watts and Stro- gatz showed that, depending on the number of nodes5, it is possible to find networks that have 1 Figure 1 | The small-world network model. In 1998, Watts and Strogatz described a model that a large clustering coefficient and short aver- helps to explain the structures of networks in the real world. a, They started with a regular network, age distances between nodes for a broad range depicted here as nodes connected in a triangular lattice in which each node is connected to six other of p values, thus reconciling the small-world nodes. b, They then allowed links between nodes to be rewired at random, with a fixed probability of rewiring for all links. As the probability increases, an increasing number of short cuts (red lines) phenomenon with network cliquishness. connect distant nodes in the network. This generates the small-world effect: all nodes in the network Watts and Strogatz’s model was initially can be connected by passing along a small number of links between nodes, but neighbouring nodes regarded simply as the explanation for six are connected to one another, forming clustered cliques. (Adapted from Samay/Vespignani.) degrees of separation. But possibly its most important impact was to pave the way for

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NEWS & VIEWS RESEARCH

studies of the effect of network structure on a their paper by saying: “We hope that our work 4. Guare, J. Six Degrees of Separation (Vintage, 1990). 5. Barthélémy, M. & Amaral, L. A. N. Phys. Rev. Lett. 82, wide range of dynamic phenomena. Another will stimulate further studies of small-world 3180–3183 (1999). paper was also pivotal: in 1999, Barabási and networks.” Perhaps no statement has ever been 6. Barabási, A.-L. & Albert, R. Science 286, 509–512 Albert proposed the ‘preferential-attachment’ more prophetic. ■ (1999). 6 7. Pastor-Satorras, R., Castellano, C., Van Mieghem, P. & network model , which highlighted that the Vespignani, A. Rev. Mod. Phys. 87, 925–979 (2015). probability distribution describing the number Alessandro Vespignani is in the Network 8. Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y. & of connections that form between nodes in Science Institute and the Laboratory for the Zhou, C. Phys. Rep. 469, 93–153 (2008). 9. Albert, R., Jeong, H. & Barabási, A.-L. Nature 401, real-world networks is often characterized Modeling of Biological and Sociotechnical 130–131 (1999). by ‘heavy-tailed’ distributions, instead of the Systems, Northeastern University, Boston, 10. Sporns, O., Chialvo, D. R., Kaiser, M. & Hilgetag, C. C. Poisson distribution predicted by random Massachusetts 02115, USA. Trends Cogn. Sci. 8, 418–425 (2004). 11. Newman, M. E. J. SIAM Rev. 45, 167–256 (2003). networks. The broad spectrum of emergent e-mail: [email protected] 12. Porter, M. A., Onnela, J. P. & Mucha, P. J. Not. Am. behaviour and phase transitions encapsulated Math. Soc. 56, 1082–1097 (2009); go.nature. 1. Watts, D. J. & Strogatz, S. H. Nature 393, 440–442 in networks that have clustered connectedness com/2jg9dgq (1998). 13. Fortunato, S. Phys. Rep. 486, 75–174 (2010). (as in Watts and Strogatz’s model) and hetero- 2. Erdös, P. & Rényi, A. Publ. Math. 6, 290–297 (1959). geneous connectedness (as in the preferential- 3. Milgram, S. Psychol. Today 1, 61–67 (1967). This article was published online on 19 June 2018. attachment model) attracted the attention of scientists from many fields. A string of discoveries followed, highlighting STRUCTURAL BIOLOGY how the complex structure of such networks underpins real-world systems, with implica- tions for network robustness, the spreading of epidemics, information flow and the syn- A complex story of chronization of collective behaviour across networks7,8. For example, the small-world con- nectivity pattern proved to be the key to under- receptor signalling standing the structure of the World Wide Web9 and how anatomical and functional areas of the G-protein-coupled receptors activate different G-protein types to trigger brain communicate with each other10. Other divergent signalling pathways. Four structures of receptor–G-protein complexes structural properties of networks came under shed light on this selectivity. See Articles p.547, p.553 & p.559 & Letter p.620 the microscope soon after11–13, such as modu- larity and the concept of structural motifs, all of which helped scientists to characterize and MICHAEL J. CAPPER & DANIEL WACKER particular. But much less is known about how understand the architecture of living and arti- GPCRs selectively activate inhibitory Gα pro- ficial systems, from subcellular networks to bout one-third of all drugs, including teins, which include Gi1, Gi2, Gi3 and Go, and are ecosystems and the Internet. opioid painkillers, antihistamines and collectively known as Gi/o. The current generation of network research many antipsychotics, target members The four papers in this issue report structures cross-fertilizes areas that benefit from unprec- Aof a family of proteins called G-protein- of Gi/o-bound GPCRs obtained using cryo­ edented computing power, big data sets and coupled receptors (GPCRs)1. This reflects the electron microscopy: Koehl et al.2 (page 547) new computational modelling techniques, and fact that GPCRs are important in almost all report the structure of the µ-opioid receptor 3 thus provides a bridge between the dynamics aspects of human physiology, and suggests bound to Gi1; Draper-Joyce et al. (page 559) of individual nodes and the emergent proper- that many more of them will be promising describe the adenosine A1 receptor in complex 4 ties of macroscopic networks. But the imme- drug targets for numerous diseases. GPCRs with Gi2; García-Nafría et al. (page 620) report diacy and the simplicity of the small-world and span the cell membrane and convert myriad the 5HT1B receptor bound to Go; and Kang preferential-attachment models still under- extra­cellular signals, including neurotrans- et al.5 (page 553) reveal the structure of the light pin our understanding of network topology. mitter molecules, hormones, and even light, receptor rhodopsin in complex with Gi1. The Indeed, the relevance of these models to differ- into a cellular response by activating cellular G-protein activation cycle involves the bind- ent areas of science laid the foundation of the G proteins and other transducer proteins. ing and release of nucleotides to and from the multi­disciplinary field now known as network Four papers2–5 in this issue help to unravel the G proteins, and all of the reported structures science. mystery of how GPCRs selectively activate a capture the receptors bound to the nucleotide- Integrating knowledge and methodologies particular group of G proteins known as Gi/o, free state of their respective G proteins. from fields as disparate as the social sciences, and provide clues that might aid the design of In some respects, the four structures are physics, biology, computer science and applied improved GPCR-targeting drugs. similar to those of the previously published 7,8 mathematics was not easy. It took several years Although more than 800 GPCRs are encoded GPCR–Gs complexes , probably because to find common ground, agree on definitions in the human genome, they couple to only a Gs- and Gi/o-containing complexes have the and reconcile and appreciate the different small number of intracellular signal trans- same overall conformation at the stage of the approaches that each field had adopted to ducers, including 16 Gα proteins6. The latter G-protein activation cycle captured by the study networks. This is still a work in progress, proteins assemble with Gβ and Gγ proteins to structures. Nevertheless, the Gi/o-containing presenting all the difficulties and traps inher- form heterotrimeric G proteins. The G-protein structures reveal striking differences at the ent in interdisciplinary work. However, in the complex disassembles on activation by GPCRs, receptor–G-protein interface when com- past 20 years a vibrant network-science com- whereupon the various subunits activate differ- pared with the Gs-containing structures. For munity has emerged, with its own prestigious ent signalling pathways. For instance, stimu- example, there are no interactions between journals, research institutes and conferences latory Gα proteins (known as Gs) increase the receptors and the Gβ subunits in the attended by thousands of scientists. cellular levels of cyclic AMP molecules, which Gi/o-containing structures. By the 20th anniversary of the paper, more regulate various cellular processes. Structures The four structures uncover several key 7,8 than 18,000 papers have cited the model, which of Gs-bound GPCRs have been reported that interactions at the GPCR–Gi/o interface medi- is now considered to be one of the benchmark have begun to elucidate the general activa- ated by the α5 helix — an α-helix structure network topologies. Watts and Strogatz closed tion mechanism of Gα proteins, and of Gs in in the carboxy terminus of Gα subunits. It is

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