HDRF: Stream-Based Partitioning for Power-Law Graphs

HDRF: Stream-Based Partitioning
for Power-Law Graphs.
travel grant from
Fabio Petroni, L. Querzoni, K. Daudjee, S. Kamali, G. Iacoboni

Graphs
I large amount of real data can be represented as a graph.
Vertices
Edges
G(V,E)
I the problem of optimally partitioning a graph while
maintaining load balance is important in several contexts.
CIKM 2015. October 19-23, 2015. Melbourne, Australia. 2 of 22

Distributed Graph Computing Frameworks
I support e cient computation on really large graphs.
B graph analytics; data mining and machine learning tasks.
I input data partitioning can have a signiﬁcant impact on the
performance of the graph computation.
B a↵ects network usage, memory occupation, synchronization.

Balanced Graph Partitioning
vertex-cutedge-cut
I partition G into smaller components of (ideally) equal size
edge-cut
vertex disjoint partitions
vertex-cut
edge disjoint partitions
I a vertex can be cut in multiple ways and span several
partitions while a cut edge connects only two partitions.

Vertex-Cut
computation steps are
associated with edges
v-cut perform better on
power law graphs
create less storage and
network overhead
(Gonzalez et al., 2012)
I modern distributed graph computing
frameworks use vertex-cut.

Power-Law Graphs
I characteristic of real graphs: power-law degree distribution.
B most vertices have few connections while a few have many.
10
0
101
10
2
10
3
10
4
10
5
10
6
10
0
10
1
10
2
10
3
10
4
10
5
numberofvertices
degree
crawl
Twitter
connection
2010
log-log scale
I the probability that a vertex has degree d is P(d) / d ↵
.
I ↵ controls the “skewness” of the degree distribution.

Balanced Vertex-Cut Graph Partitioning
I v 2 V vertex; e 2 E edge; p 2 P partition.
I A(v) set of partitions where vertex v is replicated.
I 1 tolerance to load imbalance.
I the size |p| of partition p is its edge cardinality.
minimize replicas
reduce (1) bandwidth, (2) memory
usage and (3) synchronization
balance the load
eﬃcient usage of available
computing resources
min
1
|V |
X
v2V
|A(v)| s.t. max
p2P
|p| <
|E|
|P|
I the objective function is the replication factor (RF).
B average number of replicas per vertex.

Streaming Setting
I input data is a list of edges, consumed in streaming fashion,
requiring only a single pass.
partitioning
algorithm
stream of edges
graph
3 handle graphs that don’t ﬁt in the main memory.
3 impose minimum overhead in time.
3 scalable, easy parallel implementations.
7 assignment decision taken cannot be later changed.

Algorithms Taxonomy
Grid PDS HDRFDBH Greedy
hashing
stream-based
vertex-cut graph
partitioning
Hashing
constrained
partitioning
(Gonzalez et al., 2012)(Jain et al., 2013) (Petroni et al., 2015)(Xie et al., 2014)
history
aware
(Jain et al., 2013)(Gonzalez et al., 2012)
history
agnostic

HDRF: High Degree are Replicated First
I favor the replication of high-degree vertices.
I the number of high-degree vertices in power-law graphs is
very low.
I overall reduction of the replication factor.

HDRF: High Degree are Replicated First
I in the context of robustness to network failure.
I if few high-degree vertices are removed from a power-law
graph then it is turned into a set of isolated clusters.
I focus on the locality of low-degree vertices.

The HDRF Algorithm
vertex without replicas
vertex with replicas
case 1
vertices not assigned to partitions
incoming edge
e

The HDRF Algorithm
case 1
place e in the least loaded partition
incoming edge
e

The HDRF Algorithm
case 2
only one vertex has been assigned
case 1
incoming edge
e
e

The HDRF Algorithm
case 2
place e in the partition
case 1
incoming edge
e
e

The HDRF Algorithm
case 2
case 1
incoming edge
e
e
case 3
vertices assigned, common partition
e

The HDRF Algorithm
case 2
case 1
incoming edge
e
e
case 3
place e in the intersection
e

Create Replicas
case 4
empty intersection
e

Create Replicas
case 4
least loaded partition in the union
case 4
empty intersection
e
e
standard Greedy solution

Create Replicas
case 4
least loaded partition in the union
case 4
empty intersection
e
case 4
replicate vertex with highest degree
e
δ(v1) > δ(v2)
e
standard Greedy solution
HDRF

Equivalent Formulation: Maximize The Score
I computing a score C(e, p) for all partitions p 2 P.
I assigns e to the partition that maximizes C(e, p).
C(e, p) = CREP(e, p) + CBAL(p)
I the balance term breaks ties in replication term .
I this may not be enough to ensure load balance.
control the importance of the balance term.
balanced partitions even when classical greedy fails.

Equivalent Formulation: Maximize The Score
I computing a score C(e, p) for all partitions p 2 P.
I assigns e to the partition that maximizes C(e, p).
C(e, p) = CREP(e, p) + · CBAL(p)
the balance term breaks ties in replication term.
this may not be enough to ensure load balance.
I controls the importance of the balance term .
I balanced partitions even when classical greedy fails.

Ordered Stream Of Edges
I without

I without
single partition
all the graph in a

I with

Experiments - Settings
I standalone partitioner.
B VGP, a software package for one-pass vertex-cut balanced
graph partitioning.
B measure the performance: replication and balancing.
I GraphLab.
B HDRF has been integrated in GraphLab PowerGraph 2.2.
B measure the impact on the execution time of graph
computation in a distributed graph computing frameworks.
I stream of edges in random order.

Experiments - Datasets
I real-word
graphs.
I synthetic
graphs.
1M vertices
60M to 3M edges
Dataset |V | |E|
MovieLens 10M 80.6K 10M
Netﬂix 497.9K 100.4M
Tencent Weibo 1.4M 140M
twitter-2010 41.7M 1.47B
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103
104
105
numberofvertices
degree
α = 1.8
α = 2.2
α = 2.6
α = 3.0
α = 3.4
α = 4.0

Results - Synthetic Graphs Replication Factor
I 128 partitions
4
8
1.8 2.0 2.4 2.8 3.2 3.6 4.0
replicationfactor
alpha
HDRF
PDS |P|=133
grid |P|=121
greedy
DBH
skewed homogeneous

I 128 partitions
4
8
1.8 2.0 2.4 2.8 3.2 3.6 4.0
replicationfactor
alpha
HDRF
PDS |P|=133
grid |P|=121
greedy
DBH
skewed homogeneous
less dense
less edges
easyer to partition

I 128 partitions
4
8
1.8 2.0 2.4 2.8 3.2 3.6 4.0
replicationfactor
alpha
HDRF
PDS |P|=133
grid |P|=121
greedy
DBH
skewed homogeneous
less dense
less edges
easier to partition
more dense
more edges
diﬃcult to partition

I 128 partitions
4
8
1.8 2.0 2.4 2.8 3.2 3.6 4.0
replicationfactor
alpha
HDRF
PDS |P|=133
grid |P|=121
greedy
DBH
skewed homogeneous
less dense
less edges
easier to partition
more dense
more edges
diﬃcult to partition
exploit
power-law

Results - Real-Word Graphs Replication Factor
I 133 partitions
1
2
4
8
16
Tencent Weibo Netflix MovieLens 10M twitter-2010
replicationfactor
PDS
7.9
11.3 11.5
8.0
greedy
2.8
8.1
10.7
5.9
DBH
1.5
7.1
12.9
6.8
HDRF
1.3
4.9
9.1
4.8
1
2
4
8
16
replicationfactor
PDS
greedy
DBH
HDRF

Results - Real-Word Graphs Replication Factor
I 133 partitions
1
2
4
8
16
replicationfactor
PDS
7.9
11.3 11.5
8.0
greedy
2.8
8.1
10.7
5.9
DBH
1.5
7.1
12.9
6.8
HDRF
1.3
4.9
9.1
4.8
1
2
4
8
16
replicationfactor
PDS
greedy
DBH
HDRF
HDRF achieves a replication factor
about 40% smaller than DBH,
more than 50% smaller than Greedy,
almost 3× smaller than PDS,
more than 4× smaller than Grid and
almost 14× smaller than Hashing.

Results - Load Relative Standard Deviation
I MovieLens 10M
0
1
2
3
4
5
6
7
8 16 32 64 128 256
loadrelativestandarddeviation(%)
partitions
PDS
grid
greedy
DBH
HDRF
hashing

Results - Graph Algorithm Runtime Speedup
I SGD algorithm for collaborative ﬁltering on Tencent Weibo.
1
1.5
2
2.5
3
32 64 128
speedup(×)
partitions
greedy
PDS
1
1.5
2
2.5
3
32 64 128
speedup(×)
partitions
greedy
PDS |P|=31,57,133
I the speedup is proportional to both:
B the advantage in replication factor.
B the actual network usage of the algorithm.

Conclusion
I HDRF is a one-pass vertex-cut graph partitioning algorithm.
I based on a greedy vertex-cut approach that leverages
information on vertex degrees.
I we provide a theoretical analysis of HDRF with an
average-case upper bound for the vertex replication factor.
I experimental study shows that:
B HDRF provides the smallest replication factor with close to
optimal load balance.
B HDRF signiﬁcantly reduces the time needed to perform
computation on graphs.
I the stand-alone software package for one-pass v-cut balanced
partitioning at https://github.com/fabiopetroni/VGP.

Thank you!
Questions?
Fabio Petroni
Sapienza University of Rome, Italy
Current position:
PhD Student in Engineering in Computer Science
Research Interests:
data mining, machine learning, big data
petroni@dis.uniroma1.it

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