In this chapter, the mathematically simplest random graph, the Bernoulli random graph, is introduced. Each of the possible edges is present, independently, with the same probability, so that the model is one of a network entirely without structure. To start with, the structure of the graph in the neighbourhood of a point is investigated and is shown to be very similar to that of a branching process with Poisson-distributed offspring numbers. Explicit bounds on the accuracy of the approximation are derived, using the Poisson approximation techniques derived in Chapter 7. The classical threshold theorem for the existence of a giant component is then established; the precision of the neighbourhood approximation simplifies the proof. The counts of small subgraphs are then investigated, and a subgraph threshold theorem is proved. Finally, the distribution of the length (in graph distance) of the shortest path between two vertices is investigated. These grow logarithmically with the number of points, if the expected degree of a vertex is kept constant. Once again, the approximation of the neighbourhood structure is a key element in the proofs, and the statement of the main theorem involves the Laplace transform of the distribution of the limit random variable associated with the approximate branching process.

Many networks are not completely known; the only access to them is by taking samples. This chapter presents methods for deducing information about the whole network from samples. First, some classical sampling methods are briefly considered; random sampling, with and without replacement, stratified sampling and the Horvitz–Thompson estimator. Then sampling methods based on the network structure are introduced, including two-level sampling, induced subgraph sampling, star and snowball sampling and traversal sampling. The differences between the structure of sample networks and those of the parent network are illustrated for some simple models. Finally, the problem of assessing whether a particular network sample is `interesting’ is discussed; interesting, in that it differs from what might be expected of a typical network sample.

When the King’s Men took over the residency of the Blackfriars indoor theatre in 1609, they embarked upon a period of conscious reinvention of company identity, combining their existing resources and reputation with a new playing space, audience, financial approach, and performative sensibility. The company made music a key part of this reinvention, drawing on its new assets to increase the presence and variety of original music in its plays. By 1642 they still held a reputation for musical innovation, performing newly written music for both new and older plays in the repertory; their musical settings were still newly circulated in manuscript and print into the 1650s. This chapter explores these musical reinventions, examining dramatic impacts of altered lyrics, melody, harmony, accompaniment, and performance style.

This category includes a hemispheric culture industry and audience even while it maintains identification with US identities. Central is Latin music and dance across multiple genres and styles blending African, South American, Mexican, and Caribbean traditions. Midcentury, several performers achieved celebrity with music that sparked mainstream dance crazes like the mambo and cha-cha. Latin music has also influenced American jazz and rock ’n’ roll, exemplified by Latin jazz recordings and Ritchie Valens’s rock stardom. Latin chord progressions and rhythms continued to influence rock and pop, especially in salsa and 1970s boogaloo. Since then, more artists have achieved mainstream musical popularity while Latino and Latina actors and directors have likewise become nationally celebrated. Other forms have flourished, too: activist Chicano theatre, mural culture, lowrider car culture, and fashion expressions of pachucos and pachucas dating back to the 1930s are all counterculture means of signifying ethnic pride and resistance.