We show that the space in which scientific, technological and economic activities interplay with each other can be mathematically shaped using techniques from statistical physics of networks. We build a holistic view of the innovation system as the tri-layered network of interactions among these many activities (scientific publication, patenting, and industrial production in different sectors), also taking into account the possible time delays. Within this construction we can identify which capabilities and prerequisites are needed to be competitive in a given activity, and even measure how much time is needed to transform, for instance, the technological know-how into economic wealth and scientific innovation, being able to make predictions with a very long time horizon. We find empirical evidence that, at the aggregate scale, technology is the best predictor for industrial and scientific production over the upcoming decades.

We study the relationship between the performance of firms and their technological portfolios using tools borrowed from complexity science. In particular, we ask whether the accumulation of knowledge and capabilities associated with a coherent set of technologies leads firms to experience advantages in terms of productive efficiency. To this end, we analyze both the balance sheets and the patenting activity of about 70 thousand firms that have filed at least one patent over the period 2004-2013. We define a measure of corporate coherent diversification, based on the bipartite network linking companies with the technological fields in which they patent, and relate it to firm performance in terms of labor productivity. Our measure favors technological portfolios that can be decomposed into large blocks of closely related fields over portfolios with the same breadth of scope, but a more scattered diversification structure. We find that the coherent diversification of firms is quantitatively related with their economic performance and captures relevant information about their productive structure. In particular, we prove on a statistical basis that a naive definition of technological diversification can explain labor productivity only as a proxy of size and coherent diversification. This approach can be used to investigate possible synergies within firms and to recommend viable partners for mergers and acquisitions.

The flourishing of fake news is supported by recommendation algorithms of online social networks, which, based on previous user activity, provide content adapted to their preferences and so create filter bubbles. We introduce an analytically tractable voter model with personalized information, in which an external field tends to align the agent's opinion with the one she held more frequently in the past. Our model shows a surprisingly rich dynamics despite its simplicity. An analytical mean-field approach, confirmed by numerical simulations, allows us to build a phase diagram and to predict if and how consensus is reached. Remarkably, polarization can be avoided only for weak interaction with personalized information and if the number of agents is below a threshold. We compute analytically this critical size, which depends on the interaction probability in a strongly nonlinear way.

The rank-size plots of a large number of different physical and socio-economic systems are usually said to follow Zipf's law, but a unique framework for the comprehension of this ubiquitous scaling law is still lacking. Here we show that a dynamical approach is crucial: during their evolution, some systems are attracted towards Zipf's law, while others present Zipf's law only temporarily and, therefore, spuriously. A truly Zipfian dynamics is characterized by a dynamical constraint, or coherence, among the parameters of the generating PDF, and the number of elements in the system. A clear-cut example of such coherence is natural language. Our framework allows us to derive some quantitative results that go well beyond the usual Zipf's law: (i) earthquakes can evolve only incoherently and thus show Zipf's law spuriously; this allows an assessment of the largest possible magnitude of an earthquake occurring in a geographical region. (ii) We prove that Zipfian dynamics are not additive, explaining analytically why US cities evolve coherently, while world cities do not. (iii) Our concept of coherence can be used for model selection, for example, the Yule-Simon process can describe the dynamics of world countries' GDP. (iv) World cities present spurious Zipf's law and we use this property for estimating the maximal population of an urban agglomeration.