arXiv subjects
Publications and source records attributed to Ilkka Norros.
A method for compression of large graphs and non-negative matrices to a block structure is proposed. Szemerédi's regularity lemma is used as heuristic motivation of the significance of stochastic block models. Another ingredient of the method is Rissanen's minimum description length principle (MDL). We propose practical algorithms and provide theoretical results on the accuracy of the method.
A method for compression of large graphs and matrices to a block structure is further developed. Szemerédi's regularity lemma is used as a generic motivation of the significance of stochastic block models. Another ingredient of the method is Rissanen's minimum description length principle (MDL). We continue our previous work on the subject, considering cases of missing data and scaling of algorithms to extremely large size of graphs. In this way it would be possible to find out a large scale structure of a huge graphs of certain type using only a tiny part of graph information and obtaining a compact representation of such graphs useful in computations and visualization.
This paper is focused on a class of spatial birth and death process of the Euclidean space where the birth rate is constant and the death rate of a given point is the shot noise created at its location by the other points of the current configuration for some response function $f$. An equivalent view point is that each pair of points of the configuration establishes a random connection at an exponential time determined by $f$, which results in the death of one of the two points. We concentrate on space-motion invariant processes of this type. Under some natural conditions on $f$, we construct the unique time-stationary regime of this class of point processes by a coupling argument. We then use the birth and death structure to establish a hierarchy of balance integral relations between the factorial moment measures. Finally, we show that the time-stationary point process exhibits a certain kind of repulsion between its points that we call $f$-repulsion.
We propose a new model for peer-to-peer networking which takes the network bottlenecks into account beyond the access. This model can cope with key features of P2P networking like degree or locality constraints together with the fact that distant peers often have a smaller rate than nearby peers. Using a network model based on rate functions, we give a closed form expression of peers download performance in the system's fluid limit, as well as approximations for the other cases. Our results show the existence of realistic settings for which the average download time is a decreasing function of the load, a phenomenon that we call super-scalability.
We propose a new model for peer-to-peer networking which takes the network bottlenecks into account beyond the access. This model allows one to cope with key features of P2P networking like degree or locality constraints or the fact that distant peers often have a smaller rate than nearby peers. We show that the spatial point process describing peers in their steady state then exhibits an interesting repulsion phenomenon. We analyze two asymptotic regimes of the peer-to-peer network: the fluid regime and the hard--core regime. We get closed form expressions for the mean (and in some cases the law) of the peer latency and the download rate obtained by a peer as well as for the spatial density of peers in the steady state of each regime, as well as an accurate approximation that holds for all regimes. The analytical results are based on a mix of mathematical analysis and dimensional analysis and have important design implications. The first of them is the existence of a setting where the equilibrium mean latency is a decreasing function of the load, a phenomenon that we call super-scalability.
P2P systems provide a scalable solution for distributing large files in a network. The file is split into many chunks, and peers contact other peers to collect missing chunks to eventually complete the entire file. The so-called `rare chunk' phenomenon, where a single chunk becomes rare and prevents peers from completing the file, is a threat to the stability of such systems. Practical systems such as BitTorrent overcome this issue by requiring a global search for the rare chunk, which necessitates a centralized mechanism. We demonstrate a new system based on an approximate rare-chunk rule, allowing for completely distributed file sharing while retaining scalability and stability. We assume non-altruistic peers and the seed is required to make only a minimal contribution.
We consider five different peer-to-peer file sharing systems with two chunks, with the aim of finding chunk selection algorithms that have provably stable performance with any input rate and assuming non-altruistic peers who leave the system immediately after downloading the second chunk. We show that many algorithms that first looked promising lead to unstable or oscillating behavior. However, we end up with a system with desirable properties. Most of our rigorous results concern the corresponding deterministic large system limits, but in two simplest cases we provide proofs for the stochastic systems also.
With $M(t):=\sup_{s\in[0,t]}A(s)-s$ denoting the running maximum of a fractional Brownian motion $A(\cdot)$ with negative drift, this paper studies the rate of convergence of $\mathbb {P}(M(t)>x)$ to $\mathbb{P}(M>x)$. We define two metrics that measure the distance between the (complementary) distribution functions $\mathbb{P}(M(t)>\cdot)$ and $\mathbb{P}(M>\cdot)$. Our main result states that both metrics roughly decay as $\exp(-\vartheta t^{2-2H})$, where $\vartheta$ is the decay rate corresponding to the tail distribution of the busy period in an fBm-driven queue, which was computed recently [Stochastic Process. Appl. (2006) 116 1269--1293]. The proofs extensively rely on application of the well-known large deviations theorem for Gaussian processes. We also show that the identified relation between the decay of the convergence metrics and busy-period asymptotics holds in other settings as well, most notably when Gärtner--Ellis-type conditions are fulfilled.
We study the size of the largest clique $ω(G(n,α))$ in a random graph $G(n,α)$ on $n$ vertices which has power-law degree distribution with exponent $α$. We show that for `flat' degree sequences with $α>2$ whp the largest clique in $G(n,α)$ is of a constant size, while for the heavy tail distribution, when $0<α<2$, $ω(G(n,α))$ grows as a power of $n$. Moreover, we show that a natural simple algorithm whp finds in $G(n,α)$ a large clique of size $(1+o(1))ω(G(n,α))$ in polynomial time.
We consider a variant of so called power-law random graph. A sequence of expected degrees corresponds to a power-law degree distribution with finite mean and infinite variance. In previous works the asymptotic picture with number of nodes limiting to infinity has been considered. It was found that an interesting structure appears. It has resemblance with such graphs like the Internet graph. Some simulations have shown that a finite sized variant has similar properties as well. Here we investigate this case in more analytical fashion, and, with help of some simple lower bounds for large valued expectations of relevant random variables, we can shed some light into this issue. A new term, 'communication range random graph' is introduced to emphasize that some further restrictions are needed to have a relevant random graph model for a reasonable sized communication network, like the Internet. In this case a pleasant model is obtained, giving the opportunity to understand such networks on an intuitive level. This would be beneficial in order to understand, say, how a particular routing works in such networks.
Let S(t,t') be the sigma-algebra generated by the differences X(s)-X(s) with s,s' in the interval(t,t'), where (X_t) is the fractional Brownian motion process with Hurst index H between 0 and 1. We prove that for any two distinct t and t' the sigma-algebras S(t-a,t+a) and S(t'-a,t'+a) are asymptotically independent as a tends to 0. We show this in the strong sense that Shannon's mutual information between these two sigma-algebras tends to zero as a tends to 0. Some generalizations and quantitative estimates are provided also.
This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density $λ$ of the sets grows to infinity and the mean volume $ρ$ of the sets tends to zero. Assuming that the volume distribution has a regularly varying tail with infinite variance, we show that the centered and renormalized random field can have three different limits, depending on the relative speed at which $λ$ and $ρ$ are scaled. If $λ$ grows much faster than $ρ$ shrinks, the limit is Gaussian with long-range dependence, while in the opposite case, the limit is independently scattered with infinite second moments. In a special intermediate scaling regime, there exists a nontrivial limiting random field that is not stable.