First Passage Percolation on Inhomogeneous Random Graphs

We investigate first passage percolation on inhomogeneous random graphs. The random graph model G(n,kappa) we study is the model introduced by Bollob\'as, Janson and Riordan, where each vertex has a type from a type space S and edge probabilities are independent, but depending on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distribution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal weight path, properly normalized follows a central limit theorem. We handle the cases where lambda(n)->lambda is finite or infinite, under the assumption that the average number of neighbors lambda(n) of a vertex is independent of the type. The paper is a generalization the paper by Bhamidi, van der Hofstad and Hooghiemstra, where FPP is explored on the Erdos-Renyi graphs

Degrees and distances in random and evolving Apollonian networks

This paper studies Random and Evolving Apollonian networks (RANs and EANs), in d dimension for any d>=2, i.e. dynamically evolving random d dimensional simplices looked as graphs inside an initial d-dimensional simplex. We determine the limiting degree distribution in RANs and show that it follows a power law tail with exponent tau=(2d-1)/(d-1). We further show that the degree distribution in EANs converges to the same degree distribution if the simplex-occupation parameter in the n-th step of the dynamics is q_n->0 and sum_{n=0}^infty q_n =infty. This result gives a rigorous proof for the conjecture of Zhang et al. that EANs tend to show similar behavior as RANs once the occupation parameter q->0. We also determine the asymptotic behavior of shortest paths in RANs and EANs for arbitrary d dimensions. For RANs we show that the shortest path between two uniformly chosen vertices (typical distance), the flooding time of a uniformly picked vertex and the diameter of the graph after n steps all scale as constant times log n. We determine the constants for all three cases and prove a central limit theorem for the typical distances. We prove a similar CLT for typical distances in EANs

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

In this paper we study weighted distances in scale-free spatial network models: hyperbolic random graphs (HRG), geometric inhomogeneous random graphs (GIRG) and scale-free percolation (SFP). In HRGs, $n=\Theta(\mathrm{e}^{R/2})$ vertices are sampled independently from the hyperbolic disk with radius $R$ and two vertices are connected either when they are within hyperbolic distance $R$, or independently with a probability depending on the hyperbolic distance. In GIRGs and SFP, each vertex is given an independent weight and location from an underlying measured metric space and $\mathbb{Z}^d$, respectively, and two vertices are connected independently with a probability that is a function of their distance and weights. We assign i.i.d. weights to the edges of the random graphs and study the weighted distance between two uniformly chosen vertices. In SFP, we study the weighted distance from the origin of vertex-sequences with norm tending to infinity. In particular, we study the case when the parameters are so that the degree distribution in the graph follows a power law with exponent $\tau\in(2,3)$ (infinite variance), and the edge-weight distribution is such that it produces an explosive age-dependent branching process with power-law offspring distribution. We show that in all three models, typical distances within the giant/infinite component converge in distribution, solving an open question in [Explosion and distances in scale-free percolation (2017)]. The main tools of our proof are to couple the models to infinite versions, to follow the shortest paths to infinity and to connect these paths using weight-dependent percolation on the graphs: delete edges attached to vertices with higher weight with higher probability. We realise this using the edge-weights: only short edges connected to high weight vertices will stay, yielding arbitrarily short upper bounds for the connections.Comment: 49 pages, 4 figure

Degree distribution of shortest path trees and bias of network sampling algorithms

In this article, we explicitly derive the limiting degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the asymptotics of the degree distribution for large degrees of this tree and compare it to the degree distribution of the original graph. We perform this analysis for the complete graph with edge weights that are powers of exponential random variables (weak disorder in the stochastic mean-field model of distance), as well as on the configuration model with edge-weights drawn according to any continuous distribution. In the latter, the focus is on settings where the degrees obey a power law, and we show that the shortest path tree again obeys a power law with the same degree power-law exponent. We also consider random $r$-regular graphs for large $r$, and show that the degree distribution of the shortest path tree is closely related to the shortest path tree for the stochastic mean-field model of distance. We use our results to shed light on an empirically observed bias in network sampling methods. This is part of a general program initiated in previous works by Bhamidi, van der Hofstad and Hooghiemstra [Ann. Appl. Probab. 20 (2010) 1907-1965], [Combin. Probab. Comput. 20 (2011) 683-707], [Adv. in Appl. Probab. 42 (2010) 706-738] of analyzing the effect of attaching random edge lengths on the geometry of random network models.Comment: Published at http://dx.doi.org/10.1214/14-AAP1036 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Uniform mixing time for Random Walk on Lamplighter Graphs

Suppose that \CG is a finite, connected graph and $X$ is a lazy random walk on \CG. The lamplighter chain $X^\diamond$ associated with $X$ is the random walk on the wreath product \CG^\diamond = \Z_2 \wr \CG, the graph whose vertices consist of pairs $(f,x)$ where $f$ is a labeling of the vertices of \CG by elements of $\Z_2$ and $x$ is a vertex in \CG. There is an edge between $(f,x)$ and $(g,y)$ in \CG^\diamond if and only if $x$ is adjacent to $y$ in \CG and $f(z) = g(z)$ for all $z \neq x,y$. In each step, $X^\diamond$ moves from a configuration $(f,x)$ by updating $x$ to $y$ using the transition rule of $X$ and then sampling both $f(x)$ and $f(y)$ according to the uniform distribution on $\Z_2$; $f(z)$ for $z \neq x,y$ remains unchanged. We give matching upper and lower bounds on the uniform mixing time of $X^\diamond$ provided \CG satisfies mild hypotheses. In particular, when \CG is the hypercube $\Z_2^d$, we show that the uniform mixing time of $X^\diamond$ is $\Theta(d 2^d)$. More generally, we show that when \CG is a torus $\Z_n^d$ for $d \geq 3$, the uniform mixing time of $X^\diamond$ is $\Theta(d n^d)$ uniformly in $n$ and $d$. A critical ingredient for our proof is a concentration estimate for the local time of random walk in a subset of vertices.Comment: 2 figures, 27 pages. We added a new section containing a detailed proof of that our conditions hold for the torii $Z_n^d$, uniformly in $n$ and $d

Four universal growth regimes in degree-dependent first passage percolation on spatial random graphs I

One-dependent first passage percolation is a spreading process on a graph where the transmission time through each edge depends on the direct surroundings of the edge. In particular, the classical iid transmission time $L_{xy}$ is multiplied by $(W_xW_y)^\mu$, a polynomial of the expected degrees $W_x, W_y$ of the endpoints of the edge $xy$, which we call the penalty function. Beyond the Markov case, we also allow any distribution for $L_{xy}$ with regularly varying distribution near $0$. We then run this process on three spatial scale-free random graph models: finite and infinite Geometric Inhomogeneous Random Graphs, and Scale-Free Percolation. In these spatial models, the connection probability between two vertices depends on their spatial distance and on their expected degrees. We show that as the penalty-function, i.e., $\mu$ increases, the transmission time between two far away vertices sweeps through four universal phases: explosive (with tight transmission times), polylogarithmic, polynomial but strictly sublinear, and linear in the Euclidean distance. The strictly polynomial growth phase here is a new phenomenon that so far was extremely rare in spatial graph models. The four growth phases are highly robust in the model parameters and are not restricted to phase boundaries. Further, the transition points between the phases depend non-trivially on the main model parameters: the tail of the degree distribution, a long-range parameter governing the presence of long edges, and the behaviour of the distribution $L$ near $0$. In this paper we develop new methods to prove the upper bounds in all sub-explosive phases. Our companion paper complements these results by providing matching lower bounds in the polynomial and linear regimes.Comment: 78 page

First Passage Percolation on Inhomogeneous Random Graphs

We investigate first passage percolation on inhomogeneous ran- dom graphs. The random graph model G(n, κ) we study is the model introduced by Bollob´as, Janson and Riordan in, where each vertex has a type from a type space S and edge probabilities are indepen- dent, but depending on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distri- bution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal weight path, properly normalized follows a central limit theorem. We handle the cases where ˜λn → ˜λ is finite or infinite, under the assumption that the average number of neighbors ˜λn of a vertex is independent of the type. The paper is a generalization of written by Bhamidi, van der Hofstad and Hooghiemstra, where FPP is explored on the Erd˝os-R´enyi graphs

Generating hierarchial scale free graphs from fractals

Motivated by the hierarchial network model of E. Ravasz, A.-L. Barabasi, and T. Vicsek, we introduce deterministic scale-free networks derived from a graph directed self-similar fractal $\Lambda$. With rigorous mathematical results we verify that our model captures some of the most important features of many real networks: the scale free and the high clustering properties. We also prove that the diameter is the logarithm of the size of the system. Using our (deterministic) fractal $\Lambda$ we generate random graph sequence sharing similar properties.Comment: 4 figure