Let $G = \mathcal{G}(n,r)$ be a random geometric graph resulting from placing $n$ nodes uniformly at random in the unit square (or the unit disk) and connecting every two nodes if and only if their Euclidean distance is at most $r$. Let $r_\mathrm{con} = \sqrt{\frac{\log n}{\pi n}}(1+o(1))$ be the known critical radius for connectivity when $n \to \infty$. The restricted Delaunay graph RDG$(G)$ is a subgraph of $G$ with the following properties: it is a planar graph and a spanner of $G$, and in particular it contains all the short edges of the Delaunay triangulation of $G$. While in general graphs, the construction of RDG$(G)$ requires $\Theta(n)$ messages, we show that when $r = O(r_\mathrm{con})$ and $G=\mathcal{G}(n,r)$, then with high probability, RDG$(G)$ can be constructed locally in one round of communication with $O(\sqrt{n \log n})$ messages, and with only one-hop neighborhood information. This size of $r$ proves that the existence of long Delaunay edges (an order larger than $r_\mathrm{con}$) in the unit square (disk) does not significantly affect the efficiency with which good routing graphs can be maintained.

Broadcasting is a basic problem of communication in usual networks. Many papers have investigated the construction of minimum broadcast networks, the cheapest possible broadcast network architecture (having the fewest communication lines), in which broadcasting can be accomplished as fast as theoretically possible from any vertex. Since this problem is very difficult, numerous papers have investigated ways to construct sparse networks in which broadcasting can be completed in minimum time from any originator. In this paper we introduce an efficient vertex addition method that allows us to construct sparse networks and improve the best known upper bounds on the broadcast function $B(n)$ for many odd values of $n$. We also give the exact value of $B(127)$.

We propose a self-organizing algorithm that maintains a high degree of connectivity and robustness in unstructured P2P networks. The algorithm is based on the ``jump'' primitive that periodically replaces overlay links at each node. The algorithm produces and maintains P2P networks that closely approximate sparse random graphs. Although the sparse random graphs have low degree, they are well connected and possess low diameter. This makes them good candidates for common P2P networks. Through a series of simulation experiments, we confirm that the jump algorithm leads to a network topology with these desirable properties if we start with an ad-hoc network. P2P networks conditioned through jumps also accomplish search and broadcast operations very effectively as compared to both simulated and real-life systems such as the Gnutella ultrapeer network.

We consider a conditionally Poisson random-graph model in which the mean degrees, ``capacities,'' follow a power-tail distribution with finite mean and infinite variance. Such a graph of size $N$ has a giant component that is supersmall in the sense that the typical distance between vertices is of order $\log\log N$. The shortest paths travel through a core consisting of nodes with high mean degrees. In this paper we derive upper bounds for the distance between two random vertices when an upper part of the core is removed, including the case that the whole core is removed.

We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits.