Hannu Reittu

On a conditionally Poissonian graph process

Random (pseudo)graphs G N with the following structure are studied: first, independent and identically distributed capacities Λ i are drawn for vertices i = 1, …, N; then, each pair of vertices (i, j) is connected, independently of the other pairs, with E(i, j) edges, where E(i, j) has distribution Poisson(Λ i Λ j / ∑ k=1 N Λ k ). The main result of the paper is that when P(Λ1 > x) ≥ x −τ+1, where τ ∈ (2, 3), then, asymptotically almost surely, G N has a giant component, and the distance between two randomly selected vertices of the giant component is less than (2 + o(N))(log log N)/(-log (τ − 2)). It is also shown that the cases τ > 3, τ ∈ (2, 3), and τ ∈ (1, 2) present three qualitatively different connectivity architectures.

On the stability of two-chunk file-sharing systems

We consider five different peer-to-peer file-sharing systems with two chunks, assuming non-altruistic peers who leave the system immediately after downloading the second chunk. Our aim is to find chunk selection algorithms that have provably stable performance with any input rate. We show that many algorithms that first looked promising lead to unstable or oscillating behaviour. However, we end up with a system with desirable properties. Most of our rigorous results concern the corresponding deterministic large system limits, but in the two simplest cases we provide proofs for the stochastic systems also.

Network models with a 'soft hierarchy': a random graph construction with loglog scalability

Power law random graphs with infinite variance degree distribution are shown to possess a fascinating architecture with a softly hierarchical core network. The core provides loglog-scalable connectivity properties for the whole giant component of the graph. The core is also shown to be robust against attacks to the very heart of the architecture, the top layers of the hierarchy. The structures similarities and dissimilarities with the Internets autonomous system graph are discussed. The main significance of a soft hierarchy is seen, however, in the context of structural aspects of large future networks.

On uncoordinated file distribution with non-altruistic downloaders

We consider a BitTorrent-like file sharing system, where the peers interested in downloading a large file join an overlay network. The seed node possessing the file stays in the system, whereas all other peers are non-altruistic in the sense that they leave the system as soon as they have downloaded the whole file. We consider a flash crowd scenario, where the peers join the overlay simultaneously. We show that the chunk selection algorithm is critical, propose an analytic approach to the process, and find that the encounters can be restricted to neighbours in a Chord overlay without losing much in performance.

Relying on randomness PlanetLab experiments with distributed file-sharing protocols

In this paper we present and evaluate a fully distributed file-sharing system architecture. Initially, a seeder node splits a large file into moderate-sized chunks and offers it for download. The seeder and all peers interested in downloading the file join a Chord-based overlay and contact each other randomly for chunk transfers. We report and discuss PlanetLab tests on this system as well as on our modified Chord implementation which it is based on. We give an algorithm for finding uniformly random peers in the overlay and another for estimating the distribution of chunk copies. Both algorithms turn out to have a significant effect on the performance, the first one in particular.

Architectural features of the power-law random graph model of Internet: notes on soft hierarchy, vulnerability and multicasting

The so-called power-law random graph (PLRG) has recently turned out to be an interesting model for very large, Internet-like networks. The resulting network possesses a kind of “soft hierarchy”, where large nodes play the role of a core network providing very good connectivity properties. In this paper, we study with simulations several aspects of the PLRG model. We check to what extent the shortest paths between randomly selected nodes really go through the core, propose a new way of adding link capacity diversity to the core by what we call natural routing, study the vulnerability of the network against node losses, and discuss the implications of the model to the efficiency of multicasting.