Significance: The architecture of the human brain underlies human behavior and is extremely complex with multiple scales interacting with one another. However, research efforts are typically focused on a single spatial scale. We explored the spatial multiscale organization of the human brain by using two high-quality datasets with connectomes at five anatomical resolutions for 84 healthy subjects. We found that the zoomed-out layers remain self-similar and that a geometric network model, where distances are not Euclidean, predicts the observations by application of a renormalization protocol. Our results prove that the same principles organize brain connectivity at different scales and lead to efficient decentralized communication. Implications extend to debates, like criticality in the brain, and applications, including tools for brain simulation.

Significance: Branching processes underpin the complex evolution of many real systems. However, network models typically describe network growth in terms of a sequential addition of nodes. Here, we measured the evolution of real networks—journal citations and international trade—over a 100-y period and found that they grow in a self-similar way that preserves their structural features over time. This observation can be explained by a geometric branching growth model that generates a multiscale unfolding of the network by using a combination of branching growth and a hidden metric space approach. Our model enables multiple practical applications, including the detection of optimal network size for maximal response to an external influence and a finite-size scaling analysis of critical behavior.

The dynamics of epidemic spreading is often reduced to the single control parameter R₀ (reproduction-rate), whose value, above or below unity, determines the state of the contagion. If, however, the pathogen evolves as it spreads, R₀ may change over time, potentially leading to a mutation-driven spread, in which an initially sub-pandemic pathogen undergoes a breakthrough mutation. To predict the boundaries of this pandemic phase, we introduce here a modeling framework to couple the inter-host network spreading patterns with the intra-host evolutionary dynamics. We find that even in the extreme case when these two process are driven by mutually independent selection forces, mutations can still fundamentally alter the pandemic phase-diagram. The pandemic transitions, we show, are now shaped, not just by R₀, but also by the balance between the epidemic and the evolutionary timescales. If mutations are too slow, the pathogen prevalence decays prior to the appearance of a critical mutation. On the other hand, if mutations are too rapid, the pathogen evolution becomes volatile and, once again, it fails to spread. Between these two extremes, however, we identify a broad range of conditions in which an initially sub-pandemic pathogen can breakthrough to gain widespread prevalence.

Information spreading has been studied for decades, but its underlying mechanism is still under debate, especially for those ones spreading extremely fast through the Internet. By focusing on the information spreading data of six typical events on Sina Weibo, we surprisingly find that the spreading of modern information shows some new features, i.e., either extremely fast or slow, depending on the individual events. To understand its mechanism, we present a susceptible-accepted-recovered model with both information sensitivity and social reinforcement. Numerical simulations show that the model can reproduce the main spreading patterns of the six typical events. By this model, we further reveal that the spreading can be speeded up by increasing either the strength of information sensitivity or social reinforcement. Depending on the transmission probability and information sensitivity, the final accepted size can change from continuous to discontinuous transition when the strength of the social reinforcement is large. Moreover, an edge-based compartmental theory is presented to explain the numerical results. These findings may be of significance on the control of information spreading in modern society.

A great deal of significant progress has been seen in the study of information spreading on populations of networked individuals. A common point in many of the past studies is that there is only one transition in the phase diagram of the final accepted size versus the transmission probability. However, whether other factors alter this phenomenology is still under debate, especially for the case of information spreading through many channels and platforms. In the present study, we adopt a two-layered network to represent the interactions of multiple channels and propose a Susceptible-Accepted-Recovered information spreading model. Interestingly, our model shows a novel double transition including a continuous transition and a following discontinuous transition in the phase diagram, which originates from two outbreaks between the two layers of the network. Furthermore, we reveal that the key factors are a weak coupling condition between the two layers, a large adoption threshold, and the difference of the degree distributions between the two layers. Moreover, we also test the model in the coupled empirical social networks and find similar results as in the synthetic networks. Then, an edge-based compartmental theory is developed which fully explains all numerical results. Our findings may be of significance for understanding the secondary outbreaks of information in real life.

The geometric renormalization technique for complex networks has successfully revealed the multiscale self-similarity of real network topologies and can be applied to generate replicas at different length scales. Here, we extend the geometric renormalization framework to weighted networks, where the intensities of the interactions play a crucial role in their structural organization and function. Our findings demonstrate that the weighted organization of real networks exhibits multiscale self-similarity under a renormalization protocol that selects the connections with the maximum weight across increasingly longer length scales. We present a theory that elucidates this symmetry, and that sustains the selection of the maximum weight as a meaningful procedure. Based on our results, scaled-down replicas of weighted networks can be straightforwardly derived, facilitating the investigation of various size-dependent phenomena in downstream applications.