Random walks on networks is the standard tool for modelling spreading processes in social and biological systems. This first-order Markov approach is used in conventional community detection, ranking and spreading analysis, although it ignores a potentially important feature of the dynamics: where flow moves to may depend on where it comes from. Here we analyse pathways from different systems, and although we only observe marginal consequences for disease spreading, we show that ignoring the effects of second-order Markov dynamics has important consequences for community detection, ranking and information spreading. For example, capturing dynamics with a second-order Markov model allows us to reveal actual travel patterns in air traffic and to uncover multidisciplinary journals in scientific communication. These findings were achieved only by using more available data and making no additional assumptions, and therefore suggest that accounting for higher-order memory in network flows can help us better understand how real systems are organized and function.

We develop an analytical model for the accretion and gravitational drag on a point mass that moves hypersonically in the midplane of a gaseous disk with a Gaussian vertical density stratification. Such a model is of interest for studying the interaction between a planet and a protoplanetary disk, as well as the dynamical decay of massive black holes in galactic nuclei. The model considers that the flow is ballistic, and gives fully analytical expressions for both the accretion rate onto the point mass, and the gravitational drag it suffers. The expressions are further simplified by taking the limits of a thick, and of a thin disk. The results for the thick disk reduce correctly to those for a uniform density environment (Cantó et al. 2011). We find that for a thin disk (small vertical scaleheight compared to the gravitational radius) the accretion rate is proportional to the mass of the moving object and to the surface density of the disk, while the drag force is independent of the velocity of the object. The gravitational deceleration of the hypersonic perturber in a thin disk was found to be independent of its parameters (i.e. mass or velocity) and depends only on the surface mass density of the disk. The predictions of the model are compared to the results of three-dimensional hydrodynamical simulations, with a reasonable agreement.

To better understand the organization of overlapping modules in large networks with respect to flow, we introduce the map equation for overlapping modules. In this information-theoretic framework, we use the correspondence between compression and regularity detection. The generalized map equation measures how well we can compress a description of flow in the network when we partition it into modules with possible overlaps. When we minimize the generalized map equation over overlapping network partitions, we detect modules that capture flow and determine which nodes at the boundaries between modules should be classified in multiple modules and to what degree. With a novel greedy-search algorithm, we find that some networks, for example, the neural network of the nematode Caenorhabditis elegans, are best described by modules dominated by hard boundaries, but that others, for example, the sparse European-roads network, have an organization of highly overlapping modules.

As the number of scientific journals has multiplied, journal rankings have become increasingly important for scientific decisions. From submissions and subscriptions to grants and hirings, researchers, policy makers, and funding agencies make important decisions with influence from journal rankings such as the ISI journal impact factor. Typically, the rankings are derived from the citation network between a selection of journals and unavoidably depend on this selection. However, little is known about how robust rankings are to the selection of included journals. Here we compare the robustness of three journal rankings based on network flows induced on citation networks. They model pathways of researchers navigating scholarly literature, stepping between journals and remembering their previous steps to different degree: zero-step memory as impact factor, one-step memory as Eigenfactor, and two-step memory, corresponding to zero-, first-, and second-order Markov models of citation flow between journals. We conclude that higher-order Markov models perform better and are more robust to the selection of journals. Whereas our analysis indicates that higher-order models perform better, the performance gain for the secondorder Markov model comes at the cost of requiring more citation data over a longer time period.Science builds on previous science in a recursive quest for new knowledge (1-3). Researchers put great effort into finding the best work by other researchers and into achieving maximum visibility of their own work. Therefore, they both search for good work and seek to publish in prominent journals. Inevitably, where researchers publish becomes a proxy for how good their work is, which in turn influences decisions regarding hiring, promotion, and tenure, as well as university rankings and academic funding (4, 5). As a consequence, researchers depend on the perceived importance of the journals they publish in. While actually reading the work published in a journal is the only way to qualitatively evaluate the scientific content, different metrics are nevertheless used to quantitatively assess the importance of scientific journals (6-13). In different ways, the metrics extract information from the network of citations between articles published in the journals.In this paper, we analyze three flow-based journal rankings (12-14) that at different order of approximations seek to capture the pathways of researchers navigating scholarly literature. Specifically, the metrics measure the journal visit frequency of random walk processes that correspond to zero-, first-, and second-order Markov models. That is, given a citation network between journals and a random walker following the citations, movements in a zero-order model are independent of the cur- * Electronic address: ludvig.bohlin@physics.umu.se; Corresponding author † Electronic address: a.viamontes.esquivel@physics.umu.se ‡ Electronic address: andrea.lancichinetti@physics.umu.se § Electronic address: martin.rosvall@physics.umu.se rently visited journal, moveme...

To better understand the inner workings of information spreading, network researchers often use simple models to capture the spreading dynamics. But most models only highlight the effect of local interactions on the global spreading of a single information wave, and ignore the effects of interactions between multiple waves. Here we take into account the effect of multiple interacting waves by using an agent-based model in which the interaction between information waves is based on their novelty. We analyzed the global effects of such interactions and found that information that actually reaches nodes reaches them faster. This effect is caused by selection between information waves: slow waves die out and only fast waves survive. As a result, and in contrast to models with non-interacting information dynamics, the access to information decays with the distance from the source. Moreover, when we analyzed the model on various synthetic and real spatial road networks, we found that the decay rate also depends on the path redundancy and the effective dimension of the system. In general, the decay of the information wave frequency as a function of distance from the source follows a power law distribution with an exponent between -0.2 for a two-dimensional system with high path redundancy and -0.5 for a tree-like system with no path redundancy. We found that the real spatial networks provide an infrastructure for information spreading that lies in between these two extremes. Finally, to better understand the mechanics behind the scaling results, we provide analytical calculations of the scaling for a one-dimensional system.