19,622 research outputs found
On sampling nodes in a network
Random walk is an important tool in many graph mining applications including estimating graph parameters, sampling portions of the graph, and extracting dense communities. In this paper we consider the problem of sampling nodes from a large graph according to a prescribed distribution by using random walk as the basic primitive. Our goal is to obtain algorithms that make a small number of queries to the graph but output a node that is sampled according to the prescribed distribution. Focusing on the uniform distribution case, we study the query complexity of three algorithms and show a near-tight bound expressed in terms of the parameters of the graph such as average degree and the mixing time. Both theoretically and empirically, we show that some algorithms are preferable in practice than the others. We also extend our study to the problem of sampling nodes according to some polynomial function of their degrees; this has implications for designing efficient algorithms for applications such as triangle counting
Estimating graph parameters with random walks
An algorithm observes the trajectories of random walks over an unknown graph
, starting from the same vertex , as well as the degrees along the
trajectories. For all finite connected graphs, one can estimate the number of
edges up to a bounded factor in
steps, where
is the relaxation time of the lazy random walk on and
is the minimum degree in . Alternatively, can be estimated in
, where is
the number of vertices and is the uniform mixing time on
. The number of vertices can then be estimated up to a bounded factor in
an additional steps. Our
algorithms are based on counting the number of intersections of random walk
paths , i.e. the number of pairs such that . This
improves on previous estimates which only consider collisions (i.e., times
with ). We also show that the complexity of our algorithms is optimal,
even when restricting to graphs with a prescribed relaxation time. Finally, we
show that, given either or the mixing time of , we can compute the
"other parameter" with a self-stopping algorithm
Degree Ranking Using Local Information
Most real world dynamic networks are evolved very fast with time. It is not
feasible to collect the entire network at any given time to study its
characteristics. This creates the need to propose local algorithms to study
various properties of the network. In the present work, we estimate degree rank
of a node without having the entire network. The proposed methods are based on
the power law degree distribution characteristic or sampling techniques. The
proposed methods are simulated on synthetic networks, as well as on real world
social networks. The efficiency of the proposed methods is evaluated using
absolute and weighted error functions. Results show that the degree rank of a
node can be estimated with high accuracy using only samples of the
network size. The accuracy of the estimation decreases from high ranked to low
ranked nodes. We further extend the proposed methods for random networks and
validate their efficiency on synthetic random networks, that are generated
using Erd\H{o}s-R\'{e}nyi model. Results show that the proposed methods can be
efficiently used for random networks as well
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