Network sampling and applications to big data and machine learning
1. Network sampling and applications to big data
and machine learning
Université Paris Nanterre, 200 av. de la République, 92000 Nanterre, FRANCE
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Applications of graph (network) data are increasingly popular in the tech industry. Used in many applications, including statistical analyses and machine
learning, graph data processing is often very costly and pose challenges when scaling. Here we investigate diﬀerent graph sampling algorithms, describe
their eﬃciency, and showcase their application using a basic recommender system problem. We propose Weighted vertex-induced snowball sampling
(WVISS) as it is found to be more eﬃcient than many competing graph sampling algorithms.
2. Graph topology is very diverse
Real-world graphs often possess diﬃcult-to-
model properties (), including the scale-free
property (distribution of degrees have long tails)
and the small-world property (path lengths are
short). Since the 1990s, probabilistic models
have been created to capture these properties,
such as Barabasi and Albert’s (scale-free) or
Watts and Strogatz’s (small-world).
Real-world graphs are often a combination of
both and thus challenging to model. The follow-
ing graph shows distribution of the log-degree
and path lengths for the Twitter graph ().
They both correspond to the scale-free and
4. General WVISS eﬃciency
We measured the precision of WVISS-based esti-
mates on simulations on graphs generated from
probabilistic models and real-world networks.
The following graph shows the design eﬀect of
WVISS based on local clustering (which is the
precision of the estimates obtained compared to
a plan with uniform probabilities of the same
size). A design eﬀect greater than 1 indicates
that it is more cost-eﬀective to run simple sam-
pling than WVISS and vice-versa:
This graph shows that eﬃciency is diﬃcult to
predict and depends very much on the graph
and the sampling strategy chosen.
3. Weighted vertex-induced snowball sampling
Probabilistic sampling involves selecting n units of a population of size N
at random with (inclusion) probability πk for each k ∈ U. Given a variable of
interest taking values yk, an unbiased estimate of its mean can be computed
using Horvitz-Thompson’s formula: ˆ¯y = 1
. The coeﬃcients wk = 1
called sampling weights. Maximum precision is obtained when πk ∝ yk. In
practice, yk is unobserved so inclusion probabilities are computed using some
auxiliary information correlated to yk.
One of the simplest methods for graph sampling is uniform vertex-induced
subgraph sampling, which consists in selecting vertices at random with uniform
probabilities (πk = c), c constant, along with any edge that connects two ver-
tices of the sample. Snowball sampling (unweighted), described in , consists
in selecting vertices using uniform probability and then adding all their neigh-
bors (plus the induced vertices) to the sample. Generally, it uses unweighted
Weighted vertex-induced snowball sampling (WVISS) works in three
phases: ﬁrst, a sample of n0 vertices is drawn randomly with a speciﬁc strategy
based on external information and/or graph topology. Second, all vertices that
are connected to the vertices selected in the ﬁrst phase are added to the sample.
Finally, all edges connecting sampled vertices in the initial graph are added,
which ﬁnalizes the sample graph. The ﬁnal sample size n of the sampled graph
is thus random. The graph on the left illustrates the ﬁrst and the ﬁnal step of the procedure on
an example graph. All estimates for WVISS computations uses sample weights. We show that the
corresponding Horvitz-Thompson weights can be computed in closed form for each vertex k of the
graph, thus providing unbiased estimates for any mean or total of a linear variable:
1 − j∈Bk
(1 − πj)
where Bk is the set of vertices having an edge pointing to vertex k.
5. Application to machine learning (recommendations)
0.00 0.05 0.10 0.15 0.20 0.25
We simulated a graph recommendation problem
using a co-purchases graph of items generated
from a forest-ﬁre model (of order N = 8000 and
fwprobs = 0.15). The goal is to measure the
next purchases of users. Each user who just
purchased object i has a hidden preference for
object j determined by the equation:
(1) preferencej = β1degreej + β2distancei,j
The graph on the left shows the accuracy of
the next 10 purchases prediction (top 10 accu-
racy) for each sampling algorithm for some val-
ues of the sample sizes (expressed in fraction of
N), and β1 = 0.2, β2 = 0.1. WVISS is consis-
tently more accurate than the other sampling
algorithms. Reaching accuracy of 0.5 requires
nearly half as many units with WVISS
than unweighted snowball.
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