90 research outputs found

### 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

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