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The ability of complex networks to synchronize themselves is limited by available coupling resources. Zhang and Strogatz show that allowing temporal variation in the network structure can lead to synchronization even when stable synchrony is impossible in any static network under the fixed budget.

We study the electric field around a continuous one-dimensional loop of static charge, under the assumption that the charge is distributed uniformly along the loop. For rectangular or stadium-shaped loops in the plane, we find that the electric field can undergo a symmetry-breaking pitchfork bifurcation as the loop is elongated; the field can have either one or three zeros, depending on the loop's aspect ratio. For knotted charge distributions in three-dimensional space, we compute the electric field numerically and compare our results to previously published theoretical bounds on the number of equilibrium points around charged knots. Our computations reveal that the previous bounds are far from sharp. The numerics also suggest conjectures for the actual minimum number of equilibrium points for all charged knots with five or fewer crossings. In addition, we provide the first images of the equipotential surfaces around charged knots and visualize their topological transitions as the level of the potential is varied.

Collective self-organized behavior can be observed in a variety of systems such as colloids and microswimmers. Here O’Keeffe et al. propose a model of oscillators which move in space and tend to synchronize with neighboring oscillators and outline five types of collective self-organized states.

Contagion, broadly construed, refers to anything that can spread infectiously from peer to peer. Examples include communicable diseases, rumors, misinformation, ideas, innovations, bank failures, and electrical blackouts. Sometimes, as in the 1918 Spanish flu epidemic, a contagion mutates at some point as it spreads through a network. Here, using a simple susceptible-infected model of contagion, we explore the downstream impact of a single mutation event. Assuming that this mutation occurs at a random node in the contact network, we calculate the distribution of the number of “descendants,” d, downstream from the initial “patient zero” mutant. We find that the tail of the distribution decays as d^{−2} for complete graphs, random graphs, small-world networks, networks with block-like structure, and other infinite-dimensional networks. This prediction agrees with the observed statistics of memes propagating and mutating on Facebook and is expected to hold for other effectively infinite-dimensional networks, such as the global human contact network. In a wider context, our approach suggests a possible starting point for a mesoscopic theory of contagion. Such a theory would focus on the paths traced by a spreading contagion, thereby furnishing an intermediate level of description between that of individual nodes and the total infected population. We anticipate that contagion pathways will hold valuable lessons, given their role as the conduits through which single mutations, innovations, or failures can sweep through a network as a whole.

Previous network analyses of several languages revealed a unique set of structural characteristics. One of these characteristics—the presence of many smaller components (referred to as islands)—was further examined with a comparative analysis of the island constituents. The results showed that Spanish words in the islands tended to be phonologically and semantically similar to each other, but English words in the islands tended only to be phonologically similar to each other. The results of this analysis yielded hypotheses about language processing that can be tested with psycholinguistic experiments, and offer insight into cross-language differences in processing that have been previously observed.

The incubation period for typhoid, polio, measles, leukemia and many other diseases follows a right-skewed, approximately lognormal distribution. Although this pattern was discovered more than sixty years ago, it remains an open question to explain its ubiquity. Here, we propose an explanation based on evolutionary dynamics on graphs. For simple models of a mutant or pathogen invading a network-structured population of healthy cells, we show that skewed distributions of incubation periods emerge for a wide range of assumptions about invader fitness, competition dynamics, and network structure. The skewness stems from stochastic mechanisms associated with two classic problems in probability theory: the coupon collector and the random walk. Unlike previous explanations that rely crucially on heterogeneity, our results hold even for homogeneous populations. Thus, we predict that two equally healthy individuals subjected to equal doses of equally pathogenic agents may, by chance alone, show remarkably different time courses of disease.

Do regional boundaries defined by governments respect the more natural ways that people interact across space? This paper proposes a novel, fine-grained approach to regional delineation, based on analyzing networks of billions of individual human transactions. Given a geographical area and some measure of the strength of links between its inhabitants, we show how to partition the area into smaller, non-overlapping regions while minimizing the disruption to each person's links. We tested our method on the largest non-Internet human network, inferred from a large telecommunications database in Great Britain. Our partitioning algorithm yields geographically cohesive regions that correspond remarkably well with administrative regions, while unveiling unexpected spatial structures that had previously only been hypothesized in the literature. We also quantify the effects of partitioning, showing for instance that the effects of a possible secession of Wales from Great Britain would be twice as disruptive for the human network than that of Scotland.