Degree correlations in scale-free null models

We study the average nearest neighbor degree $a(k)$ of vertices with degree $k$. In many real-world networks with power-law degree distribution $a(k)$ falls off in $k$, a property ascribed to the constraint that any two vertices are connected by at most one edge. We show that $a(k)$ indeed decays in $k$ in three simple random graph null models with power-law degrees: the erased configuration model, the rank-1 inhomogeneous random graph and the hyperbolic random graph. We consider the large-network limit when the number of nodes $n$ tends to infinity. We find for all three null models that $a(k)$ starts to decay beyond $n^{(\tau-2)/(\tau-1)}$ and then settles on a power law $a(k)\sim k^{\tau-3}$, with $\tau$ the degree exponent.Comment: 21 pages, 4 figure

Subgraphs in preferential attachment models

We consider subgraph counts in general preferential attachment models with power-law degree exponent $\tau>2$. For all subgraphs $H$, we find the scaling of the expected number of subgraphs as a power of the number of vertices. We prove our results on the expected number of subgraphs by defining an optimization problem that finds the optimal subgraph structure in terms of the indices of the vertices that together span it and by using the representation of the preferential attachment model as a P\'olya urn model

Predicting the long-term citation impact of recent publications

A fundamental problem in citation analysis is the prediction of the long-term citation impact of recent publications. We propose a model to predict a probability distribution for the future number of citations of a publication. Two predictors are used: The impact factor of the journal in which a publication has appeared and the number of citations a publication has received one year after its appearance. The proposed model is based on quantile regression. We employ the model to predict the future number of citations of a large set of publications in the field of physics. Our analysis shows that both predictors (i.e., impact factor and early citations) contribute to the accurate prediction of long-term citation impact. We also analytically study the behavior of the quantile regression coefficients for high quantiles of the distribution of citations. This is done by linking the quantile regression approach to a quantile estimation technique from extreme value theory. Our work provides insight into the influence of the impact factor and early citations on the long-term citation impact of a publication, and it takes a step toward a methodology that can be used to assess research institutions based on their most recently published work.Comment: 17 pages, 17 figure

Closure coefficients in scale-free complex networks

The formation of triangles in complex networks is an important network property that has received tremendous attention. The formation of triangles is often studied through the clustering coefficient. The closure coefficient or transitivity is another method to measure triadic closure. This statistic measures clustering from the head node of a triangle (instead of from the center node, as in the often studied clustering coefficient). We perform a first exploratory analysis of the behavior of the local closure coefficient in two random graph models that create simple networks with power-law degrees: the hidden-variable model and the hyperbolic random graph. We show that the closure coefficient behaves significantly different in these simple random graph models than in the previously studied multigraph models. We also relate the closure coefficient of high-degree vertices to the clustering coefficient and the average nearest neighbor degree

Switch chain mixing times through triangle counts

Sampling uniform simple graphs with power-law degree distributions with degree exponent $\tau\in(2,3)$ is a non-trivial problem. We propose a method to sample uniform simple graphs that uses a constrained version of the configuration model together with a Markov Chain switching method. We test the convergence of this algorithm numerically in the context of the presence of small subgraphs. We then compare the number of triangles in uniform random graphs with the number of triangles in the erased configuration model. Using simulations and heuristic arguments, we conjecture that the number of triangles in the erased configuration model is larger than the number of triangles in the uniform random graph, provided that the graph is sufficiently large.Comment: 7 pages, 8 figures in the main article. 2 pages, 2 figures in the supplementary materia

Resource sharing in wireless networks with co-location

With more and more demand from devices to use wireless communication networks, there has been an increased interest in resource sharing among operators, to give a better link quality. However, in the analysis of the benefits of resource sharing among these operators, the important factor of co-location is often overlooked. Indeed, often in wireless communication networks, different operators co-locate: they place their base stations at the same locations due to cost efficiency. We therefore use stochastic geometry to investigate the effect of co-location on the benefits of resource sharing. We develop an intricate relation between the co-location factor and the optimal radius to operate the network, which shows that indeed co-location is an important factor to take into account. We also investigate the limiting behavior of the expected gains of sharing, and find that for unequal operators, sharing may not always be beneficial when taking co-location into account

Degree distributions in AB random geometric graphs

In this paper, we provide degree distributions for $AB$ random geometric graphs, in which points of type $A$ connect to the closest $k$ points of type $B$. The motivating example to derive such degree distributions is in 5G wireless networks with multi-connectivity, where users connect to their closest $k$ base stations. It is important to know how many users a particular base station serves, which gives the degree of that base station. To obtain these degree distributions, we investigate the distribution of area sizes of the $k-$th order Voronoi cells of $B$-points. Assuming that the $A$-points are Poisson distributed, we investigate the amount of users connected to a certain $B$-point, which is equal to the degree of this point. In the simple case where the $B$-points are placed in an hexagonal grid, we show that all $k$-th order Voronoi areas are equal and thus all degrees follow a Poisson distribution. However, this observation does not hold for Poisson distributed $B$-points, for which we show that the degree distribution follows a compound Poisson-Erlang distribution in the 1-dimensional case. We then approximate the degree distribution in the 2-dimensional case with a compound Poisson-Gamma degree distribution and show that this one-parameter fit performs well for different values of $k$. Moreover, we show that for increasing $k$, these degree distributions become more concentrated around the mean. This means that $k$-connected $AB$ random graphs balance the loads of $B$-type nodes more evenly as $k$ increases. Finally, we provide a case study on real data of base stations. We show that with little shadowing in the distances between users and base stations, the Poisson distribution does not capture the degree distribution of these data, especially for $k>1$. However, under strong shadowing, our degree approximations perform quite good even for these non-Poissonian location data.Comment: 23 pages, 13 figure

Scale-free graphs with many edges

We develop tail estimates for the number of edges in a Chung-Lu random graph with regularly varying weight distribution. Our results show that the most likely way to have an unusually large number of edges is through the presence of one or more hubs, i.e.\ edges with degree $O(n)$.Comment: 8 page