The acquaintance time of (percolated) random geometric graphs

In this paper, we study the acquaintance time \AC(G) defined for a connected graph $G$. We focus on \G(n,r,p), a random subgraph of a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from $[0,1]^2$, and two vertices are adjacent with probability $p$ if the Euclidean distance between them is at most $r$. We present asymptotic results for the acquaintance time of \G(n,r,p) for a wide range of $p=p(n)$ and $r=r(n)$. In particular, we show that with high probability \AC(G) = \Theta(r^{-2}) for G \in \G(n,r,1), the "ordinary" random geometric graph, provided that $\pi n r^2 - \ln n \to \infty$ (that is, above the connectivity threshold). For the percolated random geometric graph G \in \G(n,r,p), we show that with high probability \AC(G) = \Theta(r^{-2} p^{-1} \ln n), provided that p n r^2 \geq n^{1/2+\eps} and p < 1-\eps for some \eps>0

Acquaintance time of random graphs near connectivity threshold

Benjamini, Shinkar, and Tsur stated the following conjecture on the acquaintance time: asymptotically almost surely $AC(G) \le p^{-1} \log^{O(1)} n$ for a random graph $G \in G(n,p)$, provided that $G$ is connected. Recently, Kinnersley, Mitsche, and the second author made a major step towards this conjecture by showing that asymptotically almost surely $AC(G) = O(\log n / p)$, provided that $G$ has a Hamiltonian cycle. In this paper, we finish the task by showing that the conjecture holds in the strongest possible sense, that is, it holds right at the time the random graph process creates a connected graph. Moreover, we generalize and investigate the problem for random hypergraphs

Spatial networks with wireless applications

Many networks have nodes located in physical space, with links more common between closely spaced pairs of nodes. For example, the nodes could be wireless devices and links communication channels in a wireless mesh network. We describe recent work involving such networks, considering effects due to the geometry (convex,non-convex, and fractal), node distribution, distance-dependent link probability, mobility, directivity and interference.Comment: Review article- an amended version with a new title from the origina

Collision-free network exploration

International audienc

Temporal connectivity in finite networks with non-uniform measures

Soft Random Geometric Graphs (SRGGs) have been widely applied to various models including those of wireless sensor, communication, social and neural networks. SRGGs are constructed by randomly placing nodes in some space and making pairwise links probabilistically using a connection function that is system specific and usually decays with distance. In this paper we focus on the application of SRGGs to wireless communication networks where information is relayed in a multi hop fashion, although the analysis is more general and can be applied elsewhere by using different distributions of nodes and/or connection functions. We adopt a general non-uniform density which can model the stationary distribution of different mobility models, with the interesting case being when the density goes to zero along the boundaries. The global connectivity properties of these non-uniform networks are likely to be determined by highly isolated nodes, where isolation can be caused by the spatial distribution or the local geometry (boundaries). We extend the analysis to temporal-spatial networks where we fix the underlying non-uniform distribution of points and the dynamics are caused by the temporal variations in the link set, and explore the probability a node near the corner is isolated at time $T$. This work allows for insight into how non-uniformity (caused by mobility) and boundaries impact the connectivity features of temporal-spatial networks. We provide a simple method for approximating these probabilities for a range of different connection functions and verify them against simulations. Boundary nodes are numerically shown to dominate the connectivity properties of these finite networks with non-uniform measure.Comment: 13 Pages - 4 figure

Topology of Social and Managerial Networks

With the explosion of innovative technologies in recent years, organizational and man- agerial networks have reached high levels of intricacy. These are one of the many complex systems consisting of a large number of highly interconnected heterogeneous agents. The dominant paradigm in the representation of intricate relations between agents and their evolution is a network (graph). The study of network properties, and their implications on dynamical processes, up to now mostly focused on locally defined quantities of nodes and edges. These methods grounded in statistical mechanics gave deep insight and explanations on real world phenomena; however there is a strong need for a more versatile approach which would rely on new topological methods either separately or in combination with the classical techniques. In this thesis we approach this problem introducing new topological methods for network analysis relying on persistent homology. The results gained by the new methods apply both to weighted and unweighted networks; showing that classi- cal connectivity measures on managerial and societal networks can be very imprecise and extending them to weighted networks with the aim of uncovering regions of weak connectivity. In the first two chapters of the thesis we introduce the main instruments that will be used in the subsequent chapters, namely basic techniques from network theory and persistent homology from the field of computational algebraic topology. The third chapter of the thesis approaches social and organizational networks studying their con- nectivity in relation to the concept of social capital. Many sociological theories such as the theory of structural holes and of weak ties relate social capital, in terms of profitable managerial strategies and the chance of rewarding opportunities, to the topology of the underlying social structure. We review the known connectivity measures for social networks, stressing the fact that they are all local measures, calculated on a node’s Ego network, i.e considering a nodes direct contacts. By analyzing real cases it, nevertheless, turns out that the above measures can be very imprecise for strategical individuals in social networks, revealing fake brokerage opportunities. We, therefore, propose a new set of measures, complementary to the existing ones and focused on detecting the position of links, rather than their density, therefore extending the standard approach to a mesoscopic one. Widening the view from considering direct neighbors to considering also non-direct ones, using the “neighbor filtration”, we give a measure of height and weight for structural holes, obtaining a more accurate description of a node’s strategical position within its contacts. We also provide a refined version of the network efficiency measure, which collects in a compact form the height of all structural holes. The methods are implemented and have been tested on real world organizational and managerial networks. In pursuing the objective of improving the existing methods we faced some technical difficulties which obliged us to develop new mathematical tools. The fourth chapter of the thesis deals with the general problem of detecting structural holes in weighted networks. We introduce thereby the weight clique rank filtration, to detect particular non-local structures, akin to weighted structural holes within the link-weight network fabric, which are invisible to existing methods. Their properties divide weighted networks in two broad classes: one is characterized by small hierarchi- cally nested holes, while the second displays larger and longer living inhomogeneities. These classes cannot be reduced to known local or quasi local network properties, because of the intrinsic non-locality of homology, and thus yield a new classification built on high order coordination patterns. Our results show that topology can provide novel insights relevant for many-body interactions in social and spatial networks. In the fifth chapter of the thesis, we develop new insights in the mathematical setting underlying multipersistent homology. More specifically we calculate combinatorial resolutions and efficient Gro ̈bner bases for multipersistence homology modules. In this new frontier of persistent homology, filtrations are parametrized by multiple elements. Using multipersistent homology temporal networks can be studied and the weight filtration and neighbor filtration can be combined

Aspects of random graphs

The present report aims at giving a survey of my work since the end of my PhD thesis "Spectral Methods for Reconstruction Problems". Since then I focussed on the analysis of properties of different models of random graphs as well as their connection to real-world networks. This report's goal is to capture these problems in a common framework. The very last chapter of this thesis about results in bootstrap percolation is different in the sense that the given graph is deterministic and only the decision of being active for each vertex is probabilistic; since the proof techniques resemble very much results on random graphs, we decided to include them as well. We start with an overview of the five random graph models, and with the description of bootstrap percolation corresponding to the last chapter. Some properties of these models are then analyzed in the different parts of this thesis