72 research outputs found
Subgraphs in preferential attachment models
We consider subgraph counts in general preferential attachment models with
power-law degree exponent . For all subgraphs , 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
From trees to graphs: collapsing continuous-time branching processes
Continuous-time branching processes (CTBPs) are powerful tools in random
graph theory, but are not appropriate to describe real-world networks, since
they produce trees rather than (multi)graphs. In this paper we analyze
collapsed branching processes (CBPs), obtained by a collapsing procedure on
CTBPs, in order to define multigraphs where vertices have fixed out-degree
. A key example consists of preferential attachment models (PAMs), as
well as generalized PAMs where vertices are chosen according to their degree
and age. We identify the degree distribution of CBPs, showing that it is
closely related to the limiting distribution of the CTBP before collapsing. In
particular, this is the first time that CTBPs are used to investigate the
degree distribution of PAMs beyond the tree setting.Comment: 18 pages, 3 figure
The dynamics of power laws: Fitness and aging in preferential attachment trees
Continuous-time branching processes describe the evolution of a population
whose individuals generate a random number of children according to a birth
process. Such branching processes can be used to understand preferential
attachment models in which the birth rates are linear functions. We are
motivated by citation networks, where power-law citation counts are observed as
well as aging in the citation patterns. To model this, we introduce fitness and
age-dependence in these birth processes. The multiplicative fitness moderates
the rate at which children are born, while the aging is integrable, so that
individuals receives a finite number of children in their lifetime. We show the
existence of a limiting degree distribution for such processes. In the
preferential attachment case, where fitness and aging are absent, this limiting
degree distribution is known to have power-law tails. We show that the limiting
degree distribution has exponential tails for bounded fitnesses in the presence
of integrable aging, while the power-law tail is restored when integrable aging
is combined with fitness with unbounded support with at most exponential tails.
In the absence of integrable aging, such processes are explosive.Comment: 41 pages, 10 figure
Universality of the local limit of preferential attachment models
We study preferential attachment models where vertices enter the network with
i.i.d. random numbers of edges that we call the out-degree. We identify the
local limit of such models, substantially extending the work of Berger et
al.(2014). The degree distribution of this limiting random graph, which we call
the random P\'{o}lya point tree, has a surprising size-biasing phenomenon. Many
of the existing preferential attachment models can be viewed as special cases
of our preferential attachment model with i.i.d. out-degrees. Additionally, our
models incorporate negative values of the preferential attachment fitness
parameter, which allows us to consider preferential attachment models with
infinite-variance degrees. Our proof of local convergence consists of two main
steps: a P\'olya urn description of our graphs, and an explicit identification
of the neighbourhoods in them. We provide a novel and explicit proof to
establish a coupling between the preferential attachment model and the
P\'{o}lya urn graph. Our result proves a density convergence result, for fixed
ages of vertices in the local limit
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