Connectivity of inhomogeneous random graphs

We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for G(n, p), when p = c log n/n. We draw n independent points X_i from a general distribution on a separable metric space, and let their indices form the vertex set of a graph. An edge (i,j) is added with probability min(1, \K(X_i,X_j) log n/n), where \K \ge 0 is a fixed kernel. We show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined.Comment: 13 pages. To appear in Random Structures and Algorithm

On connectivity and robustness of random graphs with inhomogeneity

The study of threshold functions has a long history in random graph theory. It is known that the thresholds for minimum degree k, k-connectivity, as well as k-robustness coincide for a binomial random graph. In this paper we consider an inhomogeneous random graph model, which is obtained by including each possible edge independently with an individual probability. Based on an intuitive concept of neighborhood density, we show two sufficient conditions guaranteeing k-connectivity and k-robustness, respectively, which are asymptotically equivalent. Our framework sheds some light on extending uniform threshold values in homogeneous random graphs to threshold landscapes in inhomogeneous random graphs

Next nearest neighbour Ising models on random graphs

This paper develops results for the next nearest neighbour Ising model on random graphs. Besides being an essential ingredient in classic models for frustrated systems, second neighbour interactions interactions arise naturally in several applications such as the colour diversity problem and graphical games. We demonstrate ensembles of random graphs, including regular connectivity graphs, that have a periodic variation of free energy, with either the ratio of nearest to next nearest couplings, or the mean number of nearest neighbours. When the coupling ratio is integer paramagnetic phases can be found at zero temperature. This is shown to be related to the locked or unlocked nature of the interactions. For anti-ferromagnetic couplings, spin glass phases are demonstrated at low temperature. The interaction structure is formulated as a factor graph, the solution on a tree is developed. The replica symmetric and energetic one-step replica symmetry breaking solution is developed using the cavity method. We calculate within these frameworks the phase diagram and demonstrate the existence of dynamical transitions at zero temperature for cases of anti-ferromagnetic coupling on regular and inhomogeneous random graphs.Comment: 55 pages, 15 figures, version 2 with minor revisions, to be published J. Stat. Mec

On the Giant Component of Geometric Inhomogeneous Random Graphs

In this paper we study the threshold model of geometric inhomogeneous random graphs (GIRGs); a generative random graph model that is closely related to hyperbolic random graphs (HRGs). These models have been observed to capture complex real-world networks well with respect to the structural and algorithmic properties. Following comprehensive studies regarding their connectivity, i.e., which parts of the graphs are connected, we have a good understanding under which circumstances a giant component (containing a constant fraction of the graph) emerges. While previous results are rather technical and challenging to work with, the goal of this paper is to provide more accessible proofs. At the same time we significantly improve the previously known probabilistic guarantees, showing that GIRGs contain a giant component with probability 1 - exp(-?(n^{(3-?)/2})) for graph size n and a degree distribution with power-law exponent ? ? (2, 3). Based on that we additionally derive insights about the connectivity of certain induced subgraphs of GIRGs

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

Inhomogeneous percolation models for spreading phenomena in random graphs

Percolation theory has been largely used in the study of structural properties of complex networks such as the robustness, with remarkable results. Nevertheless, a purely topological description is not sufficient for a correct characterization of networks behaviour in relation with physical flows and spreading phenomena taking place on them. The functionality of real networks also depends on the ability of the nodes and the edges in bearing and handling loads of flows, energy, information and other physical quantities. We propose to study these properties introducing a process of inhomogeneous percolation, in which both the nodes and the edges spread out the flows with a given probability. Generating functions approach is exploited in order to get a generalization of the Molloy-Reed Criterion for inhomogeneous joint site bond percolation in correlated random graphs. A series of simple assumptions allows the analysis of more realistic situations, for which a number of new results are presented. In particular, for the site percolation with inhomogeneous edge transmission, we obtain the explicit expressions of the percolation threshold for many interesting cases, that are analyzed by means of simple examples and numerical simulations. Some possible applications are debated.Comment: 28 pages, 11 figure