Persistence of the Jordan center in Random Growing Trees

The Jordan center of a graph is defined as a vertex whose maximum distance to other nodes in the graph is minimal, and it finds applications in facility location and source detection problems. We study properties of the Jordan Center in the case of random growing trees. In particular, we consider a regular tree graph on which an infection starts from a root node and then spreads along the edges of the graph according to various random spread models. For the Independent Cascade (IC) model and the discrete Susceptible Infected (SI) model, both of which are discrete time models, we show that as the infected subgraph grows with time, the Jordan center persists on a single vertex after a finite number of timesteps. Finally, we also study the continuous time version of the SI model and bound the maximum distance between the Jordan center and the root node at any time.Comment: 28 pages, 14 figure

Effect of Number of Users in Multi-level Coded Caching

It has been recently established that joint design of content delivery and storage (coded caching) can significantly improve performance over conventional caching. This has also been extended to the case when content has non-uniform popularity through several models. In this paper we focus on a multi-level popularity model, where content is divided into levels based on popularity. We consider two extreme cases of user distribution across caches for the multi-level popularity model: a single user per cache (single-user setup) versus a large number of users per cache (multi-user setup). When the capacity approximation is universal (independent of number of popularity levels as well as number of users, files and caches), we demonstrate a dichotomy in the order-optimal strategies for these two extreme cases. In the multi-user case, sharing memory among the levels is order-optimal, whereas for the single-user case clustering popularity levels and allocating all the memory to them is the order-optimal scheme. In proving these results, we develop new information-theoretic lower bounds for the problem.Comment: 13 pages; 2 figures. A shorter version is to appear in IEEE ISIT 201

Content Caching and Delivery over Heterogeneous Wireless Networks

Emerging heterogeneous wireless architectures consist of a dense deployment of local-coverage wireless access points (APs) with high data rates, along with sparsely-distributed, large-coverage macro-cell base stations (BS). We design a coded caching-and-delivery scheme for such architectures that equips APs with storage, enabling content pre-fetching prior to knowing user demands. Users requesting content are served by connecting to local APs with cached content, as well as by listening to a BS broadcast transmission. For any given content popularity profile, the goal is to design the caching-and-delivery scheme so as to optimally trade off the transmission cost at the BS against the storage cost at the APs and the user cost of connecting to multiple APs. We design a coded caching scheme for non-uniform content popularity that dynamically allocates user access to APs based on requested content. We demonstrate the approximate optimality of our scheme with respect to information-theoretic bounds. We numerically evaluate it on a YouTube dataset and quantify the trade-off between transmission rate, storage, and access cost. Our numerical results also suggest the intriguing possibility that, to gain most of the benefits of coded caching, it suffices to divide the content into a small number of popularity classes.Comment: A shorter version is to appear in IEEE INFOCOM 201