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Scalable K-Core Decomposition for Static Graphs
Using a Dynamic Graph Data Structure
Alok Tripathy, Fred Hohman, Duen Horng Chau, and Oded Green
Georgia Institute of Technology
Abstract—The k-core of a graph is a metric used in a
wide range of applications, including social network analytics,
visualization, and graph coloring. We present two new parallel
and scalable algorithms for finding the maximal k-core in a
graph. Unlike past approaches, our new algorithms do not rebuild
the graph in every iteration – rather, they use a dynamic graph
data structure and avoid one of the largest performance penalties
of k-core – pruning vertices and edges. We also show how to
extend our algorithms to support k-core edge decomposition for
different size k-cores found in the graph. While our new algo-
rithms are architecture independent, our implementations target
NVIDIA GPUs. When comparing our algorithms against several
highly optimized algorithms, including the sequential igraph
implementation and the multi-thread ParK implementation, our
new algorithms are significantly faster. For finding the maximal
k-core in the graph, our new algorithm can be up-to 58×faster
the igraph and up-to 4×faster than ParK executed on a 36 core
(72 thread) system. For the k-core decomposition algorithm, we
saw even greater and more consistent speedups for our algorithm
where it was up-to 130×faster than igraph and up-to 8×faster
than ParK. Our algorithms were executed on an NVIDIA P100
GPU.
I. INTRODUCTION
Network graphs are now a ubiquitous data type and model
many natural and synthetic phenomena in our modern world.
However, analyzing graph data to gain insight into a network
remains challenging. In a recent online survey conducted to
gather information about how graphs are used in practice,
researchers discovered that graph analysts rated scalability
and visualization as the most pressing issues to address [1].
Modern day graphs can easily grow to billions of vertices
and edges; therefore, as graphs grow in size and become
more complex, the need for scalable sense-making algorithms
becomes critical for gaining insight into modern day large
graphs.
Modern day graph algorithms, for example edge decomposi-
tion algorithms based on fixed points of degree peeling, show
strong potential in helping people explore unfamiliar graph
data [2]. This decomposition, based on the well-studied k-
core decomposition, has been shown to be useful for graph
exploration, navigation, and visualization [3]. The heart of
this edge decomposition algorithm requires computing the
maximal k-core for a graph. From graph theory, the k-core of a
graph is a maximal subgraph in which all vertices have degree
at least k.k-core is not only vital to edge decomposition algo-
rithms, but also powers a diverse set of graph exploration tools
and systems with applications in large-scale visualization [4],
[5], graph clustering [6], hierarchical structure analysis [5],
and graph mining [7]. It has been shown that k-core can
be computed in linear time by iteratively removing minimum
degree vertices from a graph using a separate list of vertices
per degree [8]. This process of removing minimum degree
vertices is commonly called pruning, and it is the primary
computation by which k-core and edge decompositions rely
on.
In this paper, we present two fast and scalable algorithms
for finding the maximal k-core of a graph, and extend these to
two edge decomposition algorithms for breaking down a graph
into smaller subgraphs based on the k-core sizes. Our new
algorithms do not require rebuilding the graph after pruning in
each iteration of edge composition. Rather, we use a dynamic
graph data structure to avoid one of the largest performance
penalties of k-core decomposition.
While our new algorithms are architecture independent,
our implementations target NVIDIA GPUs. Furthermore, we
run extensive experiments on a wide range of graphs, with
different topological properties and scales, to evaluate our
algorithms. We compare against the current state-of-the-art
results found in literature, including the highly optimized
sequential igraph implementation and a multi-thread ParK
implementation [9].
Contributions
In summary, the contributions of this paper are as follows:
•Scalable, maximal k-core algorithms. We introduce two
fast and scalable algorithms for finding the maximal k-core of
a graph. Both use a dynamic graph data structure to avoid the
penalty of rebuilding the graph after each pruning phase of the
algorithm. The first has parallel bottlenecks, but would likely
perform well on a sequential processor. The latter performs
much better in parallel and on a GPU. When compared
with a sequential igraph implementation and a multi-threaded
ParK[9] implementation with 72 threads, our second algorithm
can be up to 58×faster than igraph and up to 4×faster than
ParK (though it is sometimes slower than ParK).
•Scalable k-core decomposition. We introduce two dif-
ferent k-core decomposition algorithms for breaking down
the graph into smaller subgraphs for different k-core sizes.
These algorithms also use a dynamic graph data structure.
Our first algorithm uses a large number of small edge udpates,
whereas our second algorithm uses a small number of large
edge updates. As a GPU supports thousands of lightweight
threads, our second algorithm performs better in our GPU-
based experiments. Specifically, it is up to 130×faster than
igraph and up to 8×faster than ParK.
II. BACKGROU ND
A. k-core Applications and Computation
k-core was first introduced for studying social net-
works [10], but has since seen great attention in a diverse
set of other domains. Applications for k-core for graph data
include large-scale visualization [4], [5], graph clustering [6],
hierarchical structure analysis [5], and graph mining [7].
Other applications that use k-core for understanding particular
domains whose data is represented as network graphs include
bioinformatics [11], [12], [13], identifying and understanding
Internet structure [14], studying the spreading of economic
crises [15], identifying influential spreaders in a complex
network [16], and recently, revealing hierarchical cortical
organization of the human brain [17].
From graph theory, the k-core of a graph is a maximal
subgraph in which all vertices a have degree of at least k. In
recent work using k-core, igraph, a network analysis library, is
used to perform edge decompsition [3]. However, the igraph
implementation of k-core is still sequential [18]. There is,
though, a recent multi-threaded implementation of k-core,
called ParK, which improves upon the state-of-the-art [9] by
significantly reducing the working set size and minimizing
the number of random accesses to the data structure. In our
work, we compare our algorithms against both the igraph
implementation and ParK and show that we outperform both
these algorithms. There is a GPU implementation of k-core
vertex decomposition [19] we could not compare as it is not
open-source.
B. Edge Decompositions
Edge decompositions based on fixed points of degree peel-
ing divide large graphs into an ordered set of subgraphs that is
dependent only upon the topology of the graph [2]. In general,
the edge decomposition is computed by finding the maximal
k-core of a graph, removing the recently found k-core from
the original graph, and repeating until the original graph is
empty. Whereas computing a maximal k-core of a graph relies
on graph pruning, computing an edge decomposition relies on
multiple maximal k-core computations. Each maximal k-core
computed is a fixed point of degree peeling of the graph, i.e.,
if one were to re-run the edge decomposition on a particular
maximal k-core, the decomposition would simply return the
original graph, therefore each maximal k-core found in an
edge decomposition is fixed. A result of this is that the edge
decomposition is deterministic — a useful property for sense-
making graph algorithms. In the existing literature for applying
edge decompositions to large scale graph visualization and
exploration, each maximal k-core is called a graph layer [4],
[3]. Graph layers help users identify potentially important
substructures (e.g., quasi-cliques, multi-partite-cores), by au-
tomatically separating such patterns from the majority of the
graph. Our algorithms and implementations presented in this
TABL E I: List of symbols and notations used by our algorithm
Symbols and Notations
Symbol Description
GInput graph.
V(G)Set of vertices in the input graph.
E(G)Set of edges in the input graph.
b
GSeparate graph, used for storage in HKS/HDS.
QQueue of vertices removed per peel.
VbBatch of vertices to delete.
EbBatch of edges to delete.
K K-core of G
Notations and Fields
color[v]True if vwill be pruned this iteration in HKS/HDS, false otherwise.
flag [v]True if vdoes not exist in Gin HKO/HDO, false otherwise.
peel Max degree value used to determine whether to prune a vertex.
paper will allow future researchers to decompose large graphs
faster to better understand networks and gain insights into their
complex internal structure.
C. Dynamic Graph Data Structures
Dynamic graph data structures, as opposed to static graph
data structures, deal with graphs that change. Dynamic can
imply that these changes are temporal. Yet, in fact changes
can also be structural, meaning that the structure of the graph
changes. This is the case with k-core where vertices and
edges are pruned from the graph repeatedly. This process
can also be found for other problems such k-truss and [20].
Dynamic graph data structures, described below, can help
avoid recreating the graph in every step of the computation as
was done in [2], [4], [3]. Using a dynamic graph data structure
avoids these overheads.
The Hornet [21] data structure is a dynamic graph data
structure designed for dealing with fast and parallel updates
to the graph. Specifically, Hornet was designed to process
numerous insertion and deletion of a large number of vertices
and edges. These types of operations are especially important
for practical purposes as an inefficient dynamic graph data
structure will greatly reduce the overall performance of the
graph algorithm. Hornet supports over 150 million updates per
second on current GPU systems. While our implementation
uses the Hornet data structure, we note that our algorithm can
be implemented with other dynamic graph data structures so
long as they support vertex and edge insertions and deletions.
The reader is referred to [21] for more details on the Hornet
data structure and a wider literature survey of dynamic graph
data structures.
Specifically, batches are a sequence of multiple updates
made to the graph at a time. In other words, the order in
which the updates are processed is not important. In the case
of k-core we will show that all the vertices and edges pruned
in every iteration can actually be placed into a single batch
and deleted concurrently.
III. k-CO RE NUMBER ALGORITHMS
In this section we present our new algorithms for finding
the largest k-core in a graph. While there has been exten-
sive research in designing algorithms for finding the k-core
numbers, our algorithms are unique as these are the first, to
the best of our knowledge, to take advantage of a dynamic
graph data structure. Whereas many past algorithms needed to
rebuild the graph for each iteration of the k-core search, our
algorithm utilizes a data structure designed for dynamic graphs
FIG . 1: Finding deg =peel = 1 vertices
FIG . 2: Incrementing peel = 1 when no deg = 1 vertices
remain
that is highly optimized for edge insertions and deletions. This
allows us to avoid the overhead of rebuilding the graph in each
iteration of the algorithm. As we will show in this section, in
each iteration of k-core, edges that no longer meet the k-core
requirements are pruned from the graph using these dynamic
graph operations.
In the next section (Sec.IV) we show that these operations
are in fact even more important as the k-core algorithm
will be executed many times. Here we will show two dif-
ferent approaches for finding the k-core number of a static
graph using dynamic graph operations. Each approach offers
a different set of advantages and disadvantages, primarily
regarding parallelization. Our first algorithm opts for a large
number of small edge batches, and small edge batches need
fewer cores for parallelization. There is also a fair amount of
synchronization in our first algorithm.
Our second algorithm makes use of a small number of
large edge batches. These large batches are more easily par-
allelizable. Furthermore, our second algorithm does not need
as much synchronization, making it better-suited for parallel
acceleration.
A. First Algorithm
At a high level, our first algorithm works as follows. It
starts off by finding the k-core number of Gby incrementally
removing vertices and their incident edges that do meet the
core requirement. Specifically, the algorithm starts with k= 1.
This value is incremented as described below. When a vertex
is marked for deletion from G, its respective incident edges
will also be removed. We begin by repeatedly removing all
vertices with degree less than 1and their edges until there are
no longer degree less than 1vertices in Gas part of a single
batch. Once there are no longer vertices with degree 1in G,
we begin removing vertices with degree at most 2. This is
illustrated in Figures 1 and 2.
The removal process is continued until there are no more
vertices and edges in the graph. While it might seem that the
graph is empty and as such we have lost the largest k-core, we
in fact have the set of vertices and edges removed in the last
Algorithm 1 K-Core Slow - first algorithm for finding the
maximal k-core.
1: peel ←1
2: Q← {}
3: b
G←({},{})
4: while |V(G)|>0do
5: color[v]←0∀v∈V(G)
6: Vb← {}
7: parallel for v∈V(G)do
8: if deg[v]≤peel then
9: color[v]←1
10: Vb.enqueue(v)
11: end parallel for
12:
13: if |Vb|>0then
14: Eb← {}
15: parallel for (u, v) : u∈Vb, v ∈adj(u)do
16: if color[u]∧color[v]then
17: Eb.enqueue((u, v))
18: end parallel for
19: G.delete edges(Eb)
20: G.delete vertices(Vb)
21: b
G.insert vertices(Vb)
22: b
G.insert edges(Eb)
23: Q←Q∪Vb
24: else
25: peel ←peel + 1
26: Q← {}
27: return (induced subgraph(
b
G, Q), peel)
Algorithm 2 K-Core Optimized - second algorithm for finding
the maximal k-core.
1: peel ←1
2: Q← {}
3: num activ e =|V(G)|
4: color[v]←0∀v∈V(G)
5: deg[v]←G.deg(v)∀v∈V(G)
6: while num activ e > 0do
7: Vb← {}
8: parallel for v∈V(G)∧!flag [v]do
9: if deg[v]≤peel then
10: flag [v]←1
11: Vb.enqueue(v)
12: end parallel for
13: Q←Q∪Vb
14: num activ e ←num active − |Vb|
15:
16: if |Vb|>0then
17: parallel for (u, v) : u∈Vb, v ∈adj(u)do
18: deg[u]←deg[u]−1
19: deg[v]←deg[v]−1
20: end parallel for
21: else
22: peel ←peel + 1
23: Q← {}
24: return (induced subgraph(G, Q), peel)
(and previous) iteration. That is, if we reach an empty graph
with peel =k, then all vertices removed when searching for
vertices with degree at most kform the largest k-core of G. To
find these vertices, whenever we remove vertices with degree
peel, we insert these vertices into a queue Qas shown on
line 23 in Algorithm 1. This queue is emptied when we begin
searching for vertices with degree peel + 1 on line 25. Thus,
once there are no vertices in Gremaining, the vertices in the
queue form the largest k-core of G, and the k-core number of
Gis peel.
Note that while the queue has the vertices in the largest k-
core of G, we do not have the edges of the k-core. The edges
of the k-core are the edges in the induced subgraph formed
by the vertices of the k-core. However, Gis now empty, so
it is impossible to determine what those edges are. To solve
this, we maintain an additional graph
b
G.
b
Gbegins empty, and
whenever a vertex and its edges are deleted from G, we insert
them into
b
G, as depicted on lines 21 and 22. This way, once
each vertex and edge is removed from G,
b
Gwill be equal to
what Gwas at the beginning of the algorithm. We can find
the induced subgraph in
b
Gto find the largest k-core.
There are certain implementation details that are critical for
this algorithm to be parallelized. For instance, when we iterate
over all vertices to look for those of degree peel =k, we do
not remove vertices immediately. Rather, we simply color any
vertex with degree at most kfor now. Then, in a separate
loop, we iterate over all edges in G. Any edge with at least
one colored endpoint is added to a batch to be deleted.
B. Second Algorithm
This algorithm performs much better on multithreaded sys-
tems due to its improved parallelism and reduced synchroniza-
tion.
Similar to the previous algorithm, we start off this algorithm
in the same way as above, initializing peel = 1 and finding
all vertices with degree peel = 1.
Previously we found all vertices with degree peel = 1, and
delete them with their incident edges from G. Now, however,
we do not delete the vertices and edges. Instead, when we find
a vertex u, we flag uand decrement the degree of uand all
of u’s neighbors. Vertices that are flagged are not considered
to be present in G, although they are not explicitly deleted.
This is depicted in Algorithm 2.
The remainder of this algorithm is similar to the first. We
terminate once all vertices in Ghave been flagged, analogous
to terminating when all vertices in Ghave been deleted. Once
the algorithm terminates, the vertices flagged with the most
recent peel form the largest k-core in G. To keep track of
those vertices, we maintain a queue of removed vertices that
empties each time peel is incremented. The k-core in Gwill
then be the subgraph of Ginduced by the vertices in the queue,
and the k-core number is peel. Note that we do not need
b
G
as we do not remove any vertices or edges from G.
C. Complexity Analysis
1) Work Complexity: We note that, for both algorithms,
every edge is accessed exactly only once. The only time an
edge is accessed is prior to its removal. Each edge is in fact
accessed only after the source vertex of the edge has been
marked for removal; this can happen no more than once. Thus,
the work complexity for both is O(cV +E), where cis the
number of iterations in the algorithm. In most cases cis small.
Note that in the worst case, we have O(V)iterations, since
each iteration enqueues at least one vertex. The worst-case
work complexity for both algorithms is, thus, O(V2+E).
2) Storage Complexity: The first algorithm has a few O(V)
arrays and O(E)arrays.
b
Gwill also contribute space. In theory
it should not as every edge inserted into
b
Ghas been deleted
from G, but in practice extra space is used to prevent excessive
reallocating. This is an extra O(E)space in worst-case. Thus,
the overall storage complexity adds up to O(V+E)in worst-
case. Our second algorithm only uses arrays of length O(V),
so it has O(V)storage complexity.
IV. k-COR E DECOMPOSITION ALGORITHMS
Every edge in the graph can belong to several k-cores (of
different sizes). The k-core edge decomposition of a graph
Algorithm 3 K-Core Decomposition Slow - first algorithm
for finding the k-core decomposition algorithm.
1: b
G←({},{})
2: while |V(G)|>0do
3: K, k num ←K coreNum1(G,
b
G)
4: parallel for e∈E(K)do
5: peels[e]←k num
6: end parallel for
7: b
G.delete edges(E(K))
8: b
G.delete vertices(V(K))
9: swap(G,
b
G)
10: return peels[]
Algorithm 4 K-Core Decomposition Optimized - first algo-
rithm for finding the k-core decomposition algorithm.
1: while |V(G)|>0do
2: K, k num ←K coreNum2(G)
3: parallel for e∈E(K)do
4: peels[e]←k num
5: end parallel for
6: G.delete edges(E(K))
7: G.delete vertices(V(K))
8: return peels[]
will find the largest k-core that each edge belongs to. The
decomposition for a specific value of kis also known as the
peel.
Computing the k-core decomposition is, however, more
computationally demanding that finding the maximal k-core,
as it also includes just finding the maximal k-core in addition
to the proceeding k-cores. This is especially true for large
graphs with millions of vertices and billions of edges. For such
graphs the number of iterations to run k-core is equal to the
number of graph layers produced by the edge decomposition,
which could be in the hundreds [2], [4] The current state of the
art algorithms generally do not process large graphs quickly
due to several key constraints: 1) they are not taking full
advantage of massively multi-threaded systems and 2) they
primarily use inefficient data structures for representing the
graph. If the data structure is storage efficient, such as CSR,
it is typically immutable and will require rebuilding the graph
in every iteration. Other mutable data structures such as edge
lists are possible but then these lose locality and have limited
parallel scalability.
A. K-core Decomposition
The following outlines the approach we take for decom-
posing the graph into various k-cores. First of all, find the
maximal k-core of the graphs (in terms of k) using either
Algorithm 1 or Algorithm 2. Then, for all edges in the this
k-core, set their peel value to k. Then remove the k-core from
the graph while storing these edges in a map peels[] with the
peel values. These three steps are done in an iterative manner
until the graph is empty. These steps are also illustrated in our
k-core decomposition algorithms: Algorithm 3 and Algorithm
4.
Our first k-core decomposition algorithm uses the first k-
core number algorithm, and our second k-core decomposition
algorithm uses the second k-core number algorithm. Differ-
ences in our k-core decomposition algorithm are because of
the corresponding maximal k-core number algorithms. Our
first algorithm uses small vertex and edge batches for our
dynamic graph data structure. When accelerated on a multi-
threaded system, this will show poor scalability, although this
performs well on a small number of cores (or threads). Our
second algorithm, on the other hand, has fewer but larger
batches of vertices and edges, and more SIMD parallelization,
making it well-suited for GPU acceleration.
Note that our dynamic graph data structure is a critical com-
ponent for these algorithms. Both algorithms make heavy use
of dynamic graph operations. Namely, they use a considerable
amount of edge and vertex insertion and deletion. Performing
these algorithms on a data structure meant for static graphs
would prove to be a bottleneck for the algorithms, since a
lot of time will be spent on copying, memory allocation, and
sequentially accessing each edge or vertex in the batch. Thus,
a dynamic graph data structure not only is the appropriate
structure, but also one that is imperative for performance for
both algorithms. As the data structure we will use for exper-
imentation is best-suited for the GPU, our second algorithm
will show to benefit the most from the data structure since it
has more SIMD parallelization.
B. First Algorithm
Our first k-core decomposition algorithm is an adaptation of
our first k-core number algorithm. We refer to this algorithm
as K-Core Decomposition Slow. We repeatedly call the first
k-core number algorithm, modify G, call the k-core algorithm
on the modified version of G, and repeat. In this algorithm we
use two graphs, the first graph is the original input graph and
the second graph will be all the vertices and edges removed
from the first graph during the pruning process for finding
the maximal k-core. Thus, the second graph inserts the edges
deleted from the first graph–both of these are dynamic graph
operations.
The algorithm is made up of the following steps. 1) Finding
the largest k-core - first, we find the largest the largest k-core
in Gsimply by calling our first k-core number algorithm. 2)
Setting peel values - in the next phase, we iterate over the
edges found by the maximal k-core algorithm and mark them
with the value of that k-core (lines 4−6of 3)–this array
stores the output of the algorithm.3) Removing k-core from
G-after marking all the edges found in the current maximal
k-core, we remove them from the second graph
b
G(lines 7−8
of 3). Recall, that in the process of finding the maximal k-
core, we have removed all the edges from Gand such it is
empty. This leads us to our final phase. 4) Reiterating on the
remainder of the graph - thus, to continue finding k-cores,
we need to continue iterating over the remaining edges and
we will swap the vertices and edges in Gand
b
G1.
C. Second Algorithm
Our second k-core decomposition algorithm uses our second
maximal k-core algorithm, K-Core Optimized. We refer to
this decomposition as the optimized algorithm, or K-Core
Decomposition Optimized as it removes many unnecessary
1In practice, this inexpensive and is equivalent to pointer swapping.
operations/ Much like the first algorithm, we use a maximal
k-core number algorithm, set peel values, remove the k-core,
and reiterate until the graph is empty. Yet, the individual step
are slightly different as we only require a single graph (rather
than the two graphs needed by K-Core Decomposition Slow).
1) Finding the largest k-core - We find the largest the
largest k-core in Gsimply by calling our HKO k-core number
algorithm (line 2of 4). 2) Setting peel values - Our second
k-core number algorithm also returns both the k-core number
and the corresponding k-core. Thus, we can simply iterate
over all edges of the k-core in parallel and assign the k-core
number as the peel for each edge. Once again, this array
stores the output of the algorithm.3) Removing k-core from
G-Removing the k-core from Gis simpler in our second
k-core decomposition algorithm as no edges were removed in
the maximal k-core functions. This leads to fewer and larger
edge removals from our dynamic data structure. 4) Reiterating
on the remainder of the graph - After removing the k-core
from G, we can simply reiterate on G. This step is also simpler
for our second k-core decomposition algorithm.
D. Comparison
1) Complexity Analysis: First, note that both our new
decomposition algorithms give the same output and have
the same number of iterations. The number of iterations is
dependent on the graph topology and is independent of the
algorithm.
The complexity of a single iteration of each k-core de-
composition algorithm is dominated by the finding the k-
core number algorithm, as there is not much work done
outside of calls to k-core number functions. Thus, both k-
core decomposition algorithms have a work complexity of
O(V2+E)for one iteration.
2) Algorithm Scalability: Both algorithms have a good
amount of parallelism available. Specifically, the second k-
core decomposition algorithm tends to scale better as it uses
a smaller number of large batches used for the dynamic
graph data structure. In contrast, the first algorithm uses a
larger number of small batches. For massively multi-threaded
systems, using smaller batches can underutilize the system due
to workload imbalance, synchronization, and communications
overheads.
E. Sequential Vs. Parallel Algorithms
In this section we presented two algorithms for k-core
decomposition. Throughout the discussion, we did not discuss
the architecture on which these algorithms are executed as
they are in fact architecture agnostic. Specifically, our im-
plementation of these algorithms targets NVIDIA’s GPU, a
massively multithreaded system that can support thousands
of light weight threads. Effective utilization of this system
requires good load-balancing and ensuring that this is enough
work to be dispatched to these threads. In practice, we found
that our first algorithm lacked the scalability needed to utilize
thousands of threads in each phase of the algorithm. This
led to the development of the second algorithm which in
TABL E II: Systems used in experiments.
Architecture Micro-architecture Processor Frequency Cores Threads LL-Cache / Total Bandwidth DRAM Size & Type
CPU x86-64 Broadwell Xeon E5-2695 2.1 GHz 36 72 45 MB LLC 1007GB DDR4
GPU Tesla Pascal P100 PCIe 1.13 GHz 56 3584 732GB/s Bandwidth 16GB HBM2
TABL E III: List of networks used in experiments.
Name |V| |E|k-core number
dblp-author 5,425,964 8,649,002 10
patentcite 3,774,769 16,518,947 64
soc-LiveJournal1 4,847,571 42,851,237 372
soc-pokec-relationships 1,632,804 22,301,964 29
trackers 27,665,731 140,613,747 438
wikipedia-link-de 3,225,566 65,759,634 829
practice significantly outperforms the first. For a sequential
implementation it could be that the first algorithm will perform
as well as the second. However, we are unable to compare
these as we do not have an optimal data structure for the
CPU.
V. EXPERIMENTAL SET UP
In the following section we highlight our experimental setup
for checking the performance of our new algorithms. This
includes the systems, networks, and various benchmarks used.
Our new algorithms were benchmarked on an NVIDIA
P100 GPU connected to an Intel Xeon E5-2695 with 36 cores
72 threads (details in Table II). The P100 is a Pascal based
GPU with 56 SMs and 64 SPs per SM, for a total of 3584
SPs (lightweight threads). The P100 has a total of 16GB of
HBM2 memory. The Intel Xeon E5-2695 is a Broadwell based
processor running at 2.1 GHz with 45MB L3 cache. The server
consists of two such processors with a total of 1TB of memory.
While our new algorithms are architecture independent, the
final implementation targets the GPU as the Hornet [21] data
structure we used targets the GPU.
A. Dataset
For experiments we use a wide range of graphs taken from
the Konect graph database [22], though some of these graphs
are originally from SNAP [23]. Details of these graphs can
be found in Table III. Note, we also include the size of the
maximal k-core for each of these graphs. The benchmarks,
detailed below, require the input in a different format (edge
list and binary format). We ensure that benchmarks receive
the correct inputs and also do not include the time needed to
read the file or pre-process it.
B. Benchmarks
We compare the performance of our algorithms with two
highly optimized implementations. Specifically, we compare
our algorithms against the sequential algorithm found in [24]
which is also implemented in the igraph library [18]–we refer
to this algorithm simply as igraph in our comparison. We
also compare against the multi-threaded ParK algorithm [9]
which extends the igraph algorithm the multi-threaded system.
ParK was able to use all 36 cores and 72 threads of the CPU
used in our experiments. In many cases, ParK shows good
scalability in comparison to its sequential counterpart, yet our
new algorithms are significantly faster.
TABL E IV: k-core number times in seconds.
Name HKO HKS ParK igraph
dblp-author 0.028 0.731 0.105 1.633
58×2.2×15.×1×
patentcite 0.147 2.953 0.253 3.825
26×1.3×15×1×
soc-LiveJournal1 0.838 OOM 0.549 6.191
7.4×OOM 11.3×1×
soc-pokec-relationships 0.174 4.331 0.155 2.586
15×0.6×16.6×1×
trackers 13.160 OOM 3.052 20.693
1.6×OOM 6.8×1×
wikipedia-link-de 1.987 OOM 0.764 3.954
2×OOM 5.1 1×
The igraph library has an implementation of the BZ algo-
rithm as part of its coreness function. However, recall that
BZ computes the k-core vertex decomposition of G. That is,
it returns an array with the largest k-core each vertex in Gis
contained in, not each edge. The k-core number implementa-
tion with the igraph library simply calls this coreness function
and returns the largest value in the array. The k-core edge
decomposition implementation with igraph uses the above as
the k-core number implementation, removes the corresponding
k-core from G, and repeats iteratively.
C. New Algorithms
Recall, for both finding the maximal k-core algorithm and
for the k-core decomposition we presented two algorithms. In
both cases, the second algorithm was developed due to some
performance flaws found in the first algorithm.We show results
for both the slower and faster instances and compare their
performance behavior in addition to the external benchmarks
we used. We note that the slower of our implementations
might in fact be suitable for a sequential or multi-threaded
environment when the number of executing threads is not to
high.
For finding the maximal k-core we denote our two algo-
rithms, Algorithm 1 and Algorithm 2 as HKS and HKO,
respectively. HKS refers to Hornet K-Core Slow and HKO
refers Hornet K-Core Optimized. For the k-core decomposi-
tion we denote our two algorithms, Algorithm 3 and Algorithm
4 as HDS and HDO, respectively. HDS refers to Hornet
Decomposition Slow and HKO refers Hornet Decomposition
Optimized.
D. Hornet
Our implementation uses the highly-optimized Hornet data
structure [21]. Hornet is the fastest dynamic graph data struc-
ture available for shared-memory systems and can handle over
150 million edges per second for large updates on the GPU
used in our experiments.
VI. PERFORMANCE ANA LYSIS
a) Execution Time Analysis: Table IV and Table V depict
the execution for finding the maximal k-core of the graph
and for k-core decomposition, respectively. In both tables we
compare our new algorithms against the multi-threaded ParK
(a) wikipedia −de (b) trackers (c) soc −LiveJ ournal1
FIG . 3: Time per peel for k-core decomposition. Notice that the peels are placed in reversed order as the decomposition starts
with the largest peels first. Lower is better.
FIG . 4: Batch Size vs. Time (ms.) on soc −pokec −
relationships. This plot highlights that smaller batch
updates are slow and are the performance bottlenecks
for HDS as they under-utilize the GPU. Its through this
analysis that we designed HKO and HDO to overcome
the small batch problem.
(a) HDO (b) HDS
FIG . 5: GPU Utilization vs. Time (sec.) for patentcite nor-
malized across execution.
algorithm and the sequential igraph algorithm. Note, all these
timings ignore the time it takes to read the input from disk
and the time it takes to initialize the graphs. Instead we focus
on the time it takes to do the k-core algorithms. In our tables,
OOM refers to the execution running out of memory. This only
happened with the slower of our two algorithms, so we did not
investigate. In [21], it was shown the creating a Hornet data
structure is not much more expensive than creating a CSR data
structure. Also, the time it takes to build the graphs is fairly
small in comparison to the time spent finding the maximal
k-core or decomposing the graph. We discuss in more detail
in below (and show experimental results in Fig. 5). Lastly,
Figure 3 depicts the execution time taken per iteration for both
HDO and ParK for the k-core decomposition. Lower times are
better. For the higher peels (beginning of the decomposition)
our algorithm can be more than 10×faster than ParK. In the
TABL E V: k-core decomposition times in seconds.
Name HDO HDS ParK igraph
dblp-author 0.635 6.184 1.595 82.066
129.2×13.3×51.5×1×
patentcite 5.200 91.481 13.294 331.538
63.8×3.6×25×1×
soc-LiveJournal1 60.755 OOM 487.112 1572.985
25.9×OOM 3.3×1×
soc-pokec-relationships 2.756 50.049 6.488 235.790
85.6×4.7×36.3×1×
trackers 1006.954 OOM 1148.638 4725.317
4.692×OOM 4.113×1×
wikipedia-link-de 266.923 OOM 1397.323 3003.166
11.3×OOM 2.149 1×
later iterations, our algorithm is still faster though the speedup
is not as high as there is less work and we are unable to utilize
the GPU to its full capabilities.
We start off by analyzing the slower of our algorithms, HDS
and HKS. HDS is faster than the igraph by about 3.5× −13×,
but is 2× −4×worse than ParK. HKS is occasionally faster
than igraph, and if so only be 1.1× −2×, and is worth than
ParK by 6× −32×. The key bottleneck in HKS is the large
number of updates made to Hornet, the dynamic graph data
structure. This also becomes a bottleneck in HDS, as HDS
calls HKS numerous times. Small batches underutilize the
GPU and add considerable overheads such as kernel launch
overhead. This leads to the slowdowns in comparison to ParK.
To overcome these slowdowns algorithms HDO and HKO
were designed to avoid these performance penalties.
HDO performs quite well compared to the CPU algorithms,
2× −8×faster than ParK (on 36 cores / 72 threads) and
11× −130×faster than the sequential and optimized igraph al-
gorithm. HKO also performs well, ranging between 2× −58×
faster than igraph, and is sometimes faster than ParK by
2× −4×. Unlike HDS, HDO is able to fully utilize the GPU.
We discuss the GPU utilization below.
Note that while HKO occasionally beats ParK, HDO is
consistently faster than both ParK and igraph by a fair amount.
The reason HDO generally outperforms ParK, despite having
a higher work complexity, is the massive amount of paral-
lelization available and the dynamic graph data structure. The
benefits of the dynamic data structure is only available to
HDO which actually does the edge deletions–HKO does not
remove any edges. Without this dynamic graph data structure,
deletions end up being prohibitive and require rebuilding the
graph which is what ParK is doing. This leads to HDO our
performing better than ParK for k-core decomposition.
b) HDS and HKS analysis: To verify that the bottleneck
for HDS and HKS is numerous small batches, we observe
the time taken by the batches within HKS as a function of
batch size in Figure 4. Note that this is not linear. Most of
the batches fall under 200000 edges, and that many of these
batches take roughly 10ms, no less time than some batches of
size 600000−800000. This implies that the execution of these
batches is dominated by overheads such as kernel instantiation.
c) GPU Utilization: We also plot GPU utilization for
the patentcite network for both HDS and HDO, Fig. 5. To
measure the times, we run the nvidia-smi tool concurrently to
our algorithm and sample the utilization. As this adds some
overhead to the execution time, we normalize 2the execution
time between 0 and 1. For both HDS and HDO, these plots
also include the time it takes to load the graph from disk,
create the graph in memory, and write the results to disk. For
this reason, its possible to the GPU at 0% utillization at the
beginning and end of the execution. Note, due to normalization
its not possible to see that the HDO’s execution time is roughly
12×lower than the that of HDS.
For the HDO algorithm, the GPU utilization is consistently
above 80% and is many cases is above 90%. In the later phases
of HDO, the utilization goes down as the graph has been
greatly reduced and few vertices and edges remain. This is
not the case for the slower HDS algorithm which suffers from
lower utilization due to a larger number of small updates made
to the graph. The sample points for HDS fluctuate in the range
of 20% −75% utilization.
VII. CONCLUSIONS
In this paper we presented several new algorithms for find-
ing both the maximal k-core of a graph as well as decomposing
the graph into smaller subgraphs based on various k-core
values. Finding the maximal k-core and decomposing the
graphs into smaller subgraphs are ubiqitious and used across a
wide range of problems: structural analysis, visualization, and
graph clustering. For finding the maximal k-core we showed
that our new algorithm can be up-to 4×faster than ParK and
58×faster than igraph. For the k-core decomposition our new
algorithm was upto 8×faster than ParK and 130×faster than
igraph. As graphs continue to grow in size faster and more
scalable solutions become necessary. Our new algorithms meet
these requirements and can be executed on a massively multi-
threaded systems. We show a detailed performance analysis on
an NVIDIA GPU and show that our algorithm can be executed
across thousands of threads.
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