Capacity and scale-free dynamics of evolving wireless networks

Many large-scale random graphs (e.g., the Internet) exhibit complex topology, nonhomogeneous spatial node distribution, and preferential attachment of new nodes. Current topology models for ad-hoc networks mostly consider a uniform spatial distribution of nodes and do not capture the dynamics of evolving, real-world graphs, in which nodes "gravitate" toward popular locations and self-organize into non-uniform clusters. In this thesis, we first investigate two constraints on scalability of ad-hoc networks network reliability and node capacity. Unlike other studies, we analyze network resilience to node and link failure with an emphasis on the growth (i.e., evolution) dynamics of the entire system. Along the way, we also study important graph-theoretic properties of ad-hoc networks (including the clustering coefficient and the expected path length) and strengthen our generic understanding of these systems. Finally, recognizing that under existing uniform models future ad-hoc networks cannot scale beyond trivial sizes, we argue that ad-hoc networks should be modeled from an evolution standpoint, which takes into account the well-known "clustering" phenomena observed in all real-world graphs. This model is likely to describe how future ad-hoc networks will self-organize since it is well documented that information content distribution among end-users (as well as among spatial locations) is non-uniform (often heavy-tailed). Results show that node capacity in the proposed evolution model scales to larger network sizes than in traditional approaches, which suggest that non-uniformly clustered, self-organizing, very large-scale ad-hoc networks may become feasible in the future

Preferential attachment in growing spatial networks

We obtain the degree distribution for a class of growing network models on flat and curved spaces. These models evolve by preferential attachment weighted by a function of the distance between nodes. The degree distribution of these models is similar to the one of the fitness model of Bianconi and Barabasi, with a fitness distribution dependent on the metric and the density of nodes. We show that curvature singularities in these spaces can give rise to asymptotic Bose-Einstein condensation, but transient condensation can be observed also in smooth hyperbolic spaces with strong curvature. We provide numerical results for spaces of constant curvature (sphere, flat and hyperbolic space) and we discuss the conditions for the breakdown of this approach and the critical points of the transition to distance-dominated attachment. Finally we discuss the distribution of link lengths.Comment: 9 pages, 12 figures, revtex, final versio

Spatial preferential attachment networks: Power laws and clustering coefficients

We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent $\tau>2$. The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value $\tau=3$. Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1006 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Crossover from Scale-Free to Spatial Networks

In many networks such as transportation or communication networks, distance is certainly a relevant parameter. In addition, real-world examples suggest that when long-range links are existing, they usually connect to hubs-the well connected nodes. We analyze a simple model which combine both these ingredients--preferential attachment and distance selection characterized by a typical finite `interaction range'. We study the crossover from the scale-free to the `spatial' network as the interaction range decreases and we propose scaling forms for different quantities describing the network. In particular, when the distance effect is important (i) the connectivity distribution has a cut-off depending on the node density, (ii) the clustering coefficient is very high, and (iii) we observe a positive maximum in the degree correlation (assortativity) which numerical value is in agreement with empirical measurements. Finally, we show that if the number of nodes is fixed, the optimal network which minimizes both the total length and the diameter lies in between the scale-free and spatial networks. This phenomenon could play an important role in the formation of networks and could be an explanation for the high clustering and the positive assortativity which are non trivial features observed in many real-world examples.Comment: 4 pages, 6 figures, final versio