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Geometric scale-free random graphs are popular models for networks that exhibit as heavy-tailed degree distributions, small-worldness and high clustering. In these models, vertices have weights that cause the heavy-tailed degrees and are embedded in a metric space so that close-by groups of vertices tend to cluster. The interplay between the vertex weights and positions heavily affects the local structure of the random graph, in particular the occurrence of subgraph patterns, but the dependencies in these structures and weights make them difficult to analyze. In this paper we investigate subgraph counts using a \textit{divide et impera} strategy: first counting the number of subgraphs in specific classes of vertices; then computing which class yields maximum contribution. Interestingly, the scaling behavior of induced and general subgraphs in such geometric heavy-tailed random graphs is closely related to the solution of a mixed-integer linear program which also shows that subgraphs appear predominantly on vertices with some prescribed degrees and inter-distances. Finally, we derive precise asymptotics for trees and Hamiltonian subgraphs.

Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $\mathbb{R}^d$. A graph $\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points $(v,u)$ with $v\in\mathcal{V}$ and $u\in\mathcal{U}$ independently with probability $g(v-u)$, where $g$ is a non-increasing radial function, and then connecting two points $v_1,v_2\in\mathcal{V}$ if and only if they have a joint neighbor $u\in\mathcal{U}$. This gives rise to a random intersection graph on $\mathbb{R}^d$. Local properties of the graph, including the degree distribution, are investigated and quantified in terms of the intensities of the underlying Poisson processes and the function $g$. Furthermore, the percolation properties of the graph are characterized and shown to differ depending on whether $g$ has bounded or unbounded support.

We consider the problem of detecting whether a power-law inhomogeneous random graph contains a geometric community, and we frame this as an hypothesis testing problem. More precisely, we assume that we are given a sample from an unknown distribution on the space of graphs on n vertices. Under the null hypothesis, the sample originates from the inhomogeneous random graph with a heavy-tailed degree sequence. Under the alternative hypothesis, $k = o(n)$ vertices are given spatial locations and connect between each other following the geometric inhomogeneous random graph connection rule. The remaining $n-k$ vertices follow the inhomogeneous random graph connection rule. We propose a simple and efficient test, which is based on counting normalized triangles, to differentiate between the two hypotheses. We prove that our test correctly detects the presence of the community with high probability as $n \to \infty$, and identifies large-degree vertices of the community with high probability., Comment: 30 pages, 3 figures

In the past decade, geometric network models have received vast attention in the literature. These models formalize the natural idea that similar vertices are likely to connect. Because of that, these models are able to adequately capture many common structural properties of real-world networks, such as self-invariance and high clustering. Indeed, many real-world networks can be accurately modeled by positioning vertices of a network graph in hyperbolic spaces. Nevertheless, if one observes only the network connections, the presence of geometry is not always evident. Currently, triangle counts and clustering coefficients are the standard statistics to signal the presence of geometry. In this paper we show that triangle counts or clustering coefficients are insufficient because they fail to detect geometry induced by hyperbolic spaces. We therefore introduce a novel triangle-based statistic, which weighs triangles based on their strength of evidence for geometry. We show analytically, as well as on synthetic and real-world data, that this is a powerful statistic to detect hyperbolic geometry in networks., Comment: 11 pages, 2 figures, 1 table

Many real-world networks were found to be highly clustered, and contain a large amount of small cliques. We here investigate the number of cliques of any size k contained in a geometric inhomogeneous random graph: a scale-free network model containing geometry. The interplay between scale-freeness and geometry ensures that connections are likely to form between either high-degree vertices, or between close by vertices. At the same time it is rare for a vertex to have a high degree, and most vertices are not close to one another. This trade-off makes cliques more likely to appear between specific vertices. In this paper, we formalize this trade-off and prove that there exists a typical type of clique in terms of the degrees and the positions of the vertices that span the clique. Moreover, we show that the asymptotic number of cliques as well as the typical clique type undergoes a phase transition, in which only k and the degree-exponent tau are involved. Interestingly, this phase transition shows that for small values of tau, the underlying geometry of the model is irrelevant: the number of cliques scales the same as in a non-geometric network, Comment: 23 pages, 3 figures

Recently, different communities in computer science, telecommunication and control systems have devoted a huge effort towards the design of energy efficient solutions for data transmission and network management. This paper collocates along this research line and presents a novel energy efficient strategy conceived for Ethernet networks. The proposed strategy combines the statistical properties of the network traffic with the opportunities offered by the IEEE 802.3az amendment to the Ethernet standard, called Energy Efficient Ethernet (EEE). This strategy exploits the possibility of predicting the incoming traffic from the analysis of the current data flow, which typically presents a self-similar behavior. Based on the prediction, Ethernet links can then be put in a low power consumption state for the intervals of time in which traffic is expected to be of low intensity. Theoretical bounds are derived that detail how the performance figures depend on the parameters of the designed strategy and scale with respect to the traffic load. Furthermore, simulations results, based on both real and synthetic traffic traces, are presented to prove the effectiveness of the strategy, which leads to considerable energy savings at the cost of only a limited bounded delay in data delivery., Comment: 11 pages, 7 figures, journal paper, submitted to IEEE Transactions on Communications