Absence of percolation in graphs based on stationary point processes with degrees bounded by two

We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollob'as that the bidirectional $k$-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k=2

Absence of percolation in graphs based on stationary point processes with degrees bounded by two

We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollob\'as that the bidirectional $k$-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for $k=2$.Comment: 16 pages, 3 figure

G+G^{+} Method in Action: New Classes of Nonnegative Matrices with Results

The $G^{+}$ method is a new method, a powerful one, for the study of (homogeneous and nonhomogeneous) products of nonnegative matrices -- for problems on the products of nonnegative matrices. To study such products, new classes of matrices are introduced: that of the sum-positive matrices, that of the $\left[ \Delta \right]$-positive matrices on partitions (of the column index sets), that of the $g_{k}^{+}$-matrices... On the other hand, the $g_{k}^{+}$-matrices lead to necessary and sufficient conditions for the $k$-connected graphs. Using the $G^{+}$ method, we prove old and new results (Wielandt Theorem and a generalization of it, etc.) on the products of nonnegative matrices -- mainly, sum-positive, $\left[ \Delta \right]$-positive on partitions, irreducible, primitive, reducible, fully indecomposable, scrambling, or Sarymsakov matrices, in some cases the matrices being, moreover, $g_{k}^{+}$-matrices (not only irreducible)

On the Strength of Connectivity of Inhomogeneous Random K-out Graphs

Random graphs are an important tool for modelling and analyzing the underlying properties of complex real-world networks. In this paper, we study a class of random graphs known as the inhomogeneous random K-out graphs which were recently introduced to analyze heterogeneous sensor networks secured by the pairwise scheme. In this model, first, each of the $n$ nodes is classified as type-1 (respectively, type-2) with probability $0<\mu<1$ (respectively, $1-\mu)$ independently from each other. Next, each type-1 (respectively, type-2) node draws 1 arc towards a node (respectively, $K_n$ arcs towards $K_n$ distinct nodes) selected uniformly at random, and then the orientation of the arcs is ignored. From the literature on homogeneous K-out graphs wherein all nodes select $K_n$ neighbors (i.e., $\mu=0$), it is known that when $K_n \geq2$, the graph is $K_n$-connected asymptotically almost surely (a.a.s.) as $n$ gets large. In the inhomogeneous case (i.e., $\mu>0$), it was recently established that achieving even 1-connectivity a.a.s. requires $K_n=\omega(1)$. Here, we provide a comprehensive set of results to complement these existing results. First, we establish a sharp zero-one law for $k$-connectivity, showing that for the network to be $k$-connected a.a.s., we need to set $K_n = \frac{1}{1-\mu}(\log n +(k-2)\log\log n + \omega(1))$ for all $k=2, 3, \ldots$. Despite such large scaling of $K_n$ being required for $k$-connectivity, we show that the trivial condition of $K_n \geq 2$ for all $n$ is sufficient to ensure that inhomogeneous K-out graph has a connected component of size $n-O(1)$ whp