The domination number of on-line social networks and random geometric graphs

We consider the domination number for on-line social networks, both in a stochastic network model, and for real-world, networked data. Asymptotic sublinear bounds are rigorously derived for the domination number of graphs generated by the memoryless geometric protean random graph model. We establish sublinear bounds for the domination number of graphs in the Facebook 100 data set, and these bounds are well-correlated with those predicted by the stochastic model. In addition, we derive the asymptotic value of the domination number in classical random geometric graphs

Rank-based attachment leads to power law graphs

We investigate the degree distribution resulting from graph generation models based on rank-based attachment. In rank-based attachment, all vertices are ranked according to a ranking scheme. The link probability of a given vertex is proportional to its rank raised to the power -a, for some a in (0,1). Through a rigorous analysis, we show that rank-based attachment models lead to graphs with a power law degree distribution with exponent 1+1/a whenever vertices are ranked according to their degree, their age, or a randomly chosen fitness value. We also investigate the case where the ranking is based on the initial rank of each vertex; the rank of existing vertices only changes to accommodate the new vertex. Here, we obtain a sharp threshold for power law behaviour. Only if initial ranks are biased towards lower ranks, or chosen uniformly at random, we obtain a power law degree distribution with exponent 1+1/a. This indicates that the power law degree distribution often observed in nature can be explained by a rank-based attachment scheme, based on a ranking scheme that can be derived from a number of different factors; the exponent of the power law can be seen as a measure of the strength of the attachment

Geometric evolution of complex networks with degree correlations

We present a general class of geometric network growth mechanisms by homogeneous attachment in which the links created at a given time t are distributed homogeneously between a new node and the existing nodes selected uniformly. This is achieved by creating links between nodes uniformly distributed in a homogeneous metric space according to a Fermi-Dirac connection probability with inverse temperature β and general time-dependent chemical potential μ(t). The chemical potential limits the spatial extent of newly created links. Using a hidden variable framework, we obtain an analytical expression for the degree sequence and show that μ(t) can be fixed to yield any given degree distributions, including a scale-free degree distribution. Additionally, we find that depending on the order in which nodes appear in the network—its history—the degree-degree correlations can be tuned to be assortative or disassortative. The effect of the geometry on the structure is investigated through the average clustering coefficient ⟨c⟩. In the thermodynamic limit, we identify a phase transition between a random regime where ⟨c⟩→ 0 when β 0 when β>βc

De la structure croissante des réseaux complexes : approche de la géométrie des réseaux

L’internet, le cerveau humain et bien d’autres sont des systèmes complexes ayant un grand nombre d’éléments qui interagissent fortement entre eux selon leur structure. La science des réseaux complexes, qui associe ces éléments et interactions respectivement à des noeuds et liens d’un graphe, permet aujourd’hui de mieux les comprendre grâce aux types d’analyses quantitatives qu’elle rend possible. D’une part, elle permet de définir une variété de propriétés structurelles menant vers une classification de ces systèmes. D’autre part, la compréhension de l’émergence de ces propriétés grâce à certains modèles stochastiques de réseaux devient réalité. Dans les dernières années, un effort important a été déployé pour identifier des mécanismes d’évolution universels pouvant expliquer la structure des réseaux complexes réels. Ce mémoire est consacré à l’élaboration d’un de ces mécanismes de croissance universels basé sur la théorie de la géométrie des réseaux complexes qui stipule que les réseaux sont des objets abstraits plongés dans des espaces métriques de similarité où la distance entre les noeuds affecte l’existence des liens. Au moyen de méthodes d’analyse avancées, la caractérisation complète de ce mécanisme a été établie et permet le contrôle de plusieurs propriétés structurelles des réseaux ainsi générés. Ce mécanisme général pourrait expliquer, du moins de manière effective, la structure d’un nombre important de systèmes complexes dont la formation est, encore aujourd’hui, mal comprise.The internet and the human brain among others are complex systems composed of a large number of elements strongly interacting according to their specific structure. Nowadays, network science, which construes these elements and interactions respectively as nodes and links of a graph, allows a better understanding of these systems thanks to the quantitative analysis it oers. On the one hand, network science provides the definition of a variety of structural properties permitting their classification. On the other hand, it renders possible the investigation of the emergence of these properties via stochastic network models. In recent years, considerable efforts have been deployed to identify universal evolution mechanisms responsible for the structure of real complex networks. This memoir is dedicated to one of these universal growth mechanisms based on the network geometry theory which prescribes that real networks are abstract objects embedded in similarity metric spaces where the distance between nodes aect the existence of the links. Thanks to advanced analysis methods, the complete characterization of the mechanism has been achieved and allows the control of structural properties over a wide range. This general mechanism could explain, at least effectively, the structure of a number of complex systems for which the evolution is still poorly understood