1. Efficient Triangle Motif Counting in Large Scale Complex Networks with GPUs
Hakan Kardeş
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2. Introduction
Many systems are being modeled as complex networks to understand local and global characteristics of these systems.
Studying network models of these systems provides a new direction towards understanding biological, chemical, technological or social systems in a better way.
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3. Complex Networks Everywhere
Aspirin
Yeast protein interaction network
An Internet Web
Co-author network
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4. Why Graph Mining and Searching?
In many cases, systems under investigation are very large and the corresponding graphs have large number of nodes/edges requiring graph mining techniques to derive information from the graph.
Several graph mining techniques have been developed to extract useful information from graph representation and analyze various features of complex networks.
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5. Why is Triangle Counting important?
Clustering coefficient
Transitivity ratio
Social Network Analysis fact: “Friends of friends are friends” A C B
[WF94)]
Hidden Thematic Structure of the Web (Eckmann et al. PNAS [EM02])
Motif Detection, (e.g., [YPSB05] )
Web Spam Detection (Becchetti et.al. KDD ’08 [BBCG08])
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6. Related Work
Hakan Kardes, and M. H. Gunes. Structural Graph Indexing for Mining Complex Networks. IEEE ICDCS 2010 Workshop on Simplifying Complex Networks for Practitioners, Genoa, ITALY, June
Our paper in which we count all star, triangle, complete bipartite and clique structures.
Matthieu Latapy Main-memory triangle computations for very large (sparse (power-law)) graphs. Theor. Comput. Sci. 407, 1-3 (November 2008),
Survey paper, focused on space complexity
Charalampos Tsourakakis, Petros Drineas, Eirinaios Michelakis, Ioannis Koutis, Christos Faloutsos, "Spectral Counting of Triangles in Power-Law Networks via Element-Wise Sparsification," Social Network Analysis and Mining, International Conference on Advances in, pp , 2009 International Conference on Advances in Social Network Analysis and Mining, 2009
relies on the spectral properties of power-law networks, focused on power-law networks
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7. Related Work
Luca Becchetti, Paolo Boldi, Carlos Castillo, and Aristides Gionis Efficient algorithms for large-scale local triangle counting. ACM Transactions on Knowl. Discov. Data 4, 3, Article 13 (October 2010), 28 pages.
They count the number of triangles for a given node.
Charalampos E. Tsourakakis, U Kang, Gary L. Miller, and Christos Faloutsos. Doulion: Counting triangles in massive graphs with a coin.In Knowledge Discovery and Data Mining (KDD '09)
Belkacem Serrour, Alex Arenas, Sergio Gomez Detecting communities of triangles in complex networks using spectral optimization.
Bill Andreopoulos, Christof Winter, Dirk Labudde and Michael Schroeder. Triangle network motifs predict complexes by complementing high-error interactomes with structural information. BMC Bioinformatics 2009 
The graph indexing studies can
be mainly categorized into two categories, namely, path-based
and structure-based approaches.
Path-based graph indexing approaches use path expressions
as indexing features such as GraphGrep [19] and Daylight
[11]. GraphGrep enumerates all paths in the graph up
to the length maxL. Then, it looks for each gi whether it
contains all paths up to MaxL for a graph query qi. A
significant feature of path-based approaches is that paths can
be manipulated easier than general graphs. However, as Yan
et. al. indicated, path is a simple structure loosing structural
information of a graph, and hence false positive ratio of pathbased
methods would be very high [22]. In addition, the
number of paths in a graph database increases exponentially
making path-based methods impractical for very large graphs.
Alternatively, structure-based graph indexing approaches
identifies subgraphs to be indexed as in gIndex [22]. gIndex
first searches for the frequent subgraphs in the graph, then
indexes these frequent structures. An issue in this case is
that frequent subgraph discovery increases complexity and
exponential number of frequent fragments may exist under
low frequency support. Therefore, in their study, they limit
the number of nodes and index frequent structures up to 10
nodes.
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8. Methodology

9. Star
We first index the star structure where a node has multiple neighbors as shown in below figures.
All star structures within a graph G = (V,E) are represented as s(vi , nsi) where vi ∈ V and nsi is the set of all neighbors of vi.
We index maximal star structures for each node. v1
ns1
ns2
ns2 v1
ns1
ns3 vi
ns1
ns2
nsn .
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10. Star
Nodes: a, b, c, d, e, f.
Edges: (a,b), (a,d), (a,f), (b,e), (c,f), (d,f)
Star Structures:
Algorithm:
First build a star structure s(v,ø) for each node v ∈ V, without any neighbors.
Then, for each edge e(a, b) ∈ E, append neighbor sets of nodes a and b to the other one.
Finally, remove star structures s(v,ns) that have less than two neighbors. a b f a f b c e d a a b d e f d c f
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11. Triangle
Algorithm:
Find second hop neighbors of ‘a’ by iterating over the ns set
Then, take the intersection of second hop neighbors of ‘a’ and ns set.
Grow the triangle set for each isi ԑ is. a
ns1
ns2
nsn .
ls1
ls2
lsn .
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12. CUDA
For the parallel algorithm, I will use CUDA.
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13. CUDA
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14. Experiments

15. Possible Datasets for Experiments
Router-level Internet topology (around 2.3 M nodes and 4M edges)
the routing data on the Internet network (around 124K nodes and 207K edges)
a mobile phone graph. (around 2.7M nodes and 6M edges)
Will be requested from the authors of “Structure of neighborhoods in a large social network”
Biological Data
Wikipedia graph (around 1.6M nodes and 18.5M edges)
I haven’t decided how to do it yet.
I will generate sample graphs with different number of triangles
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16. Results Triangle Counting CPU vs. GPU: Execution Time no. of nodes
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17. no. of edges(while no. of nodes is constant
Results
Triangle Counting CPU vs. GPU:
Execution Time
no. of edges(while no. of nodes is constant
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18. Results Triangle Counting with different triangle sizes:
Execution Time
No. of triangles
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19. Results Triangle Counting with different block sizes: Execution Time
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20. Future Work

21. Structural Graph Indexing(SGI)
We propose an alternative structural indexing approach to search and process queries efficiently even in very large graphs.
As indexing features, we use commonly observed graph structures: star, complete bipartite, triangle and clique.
These structures are ubiquitous in biological, chemical, technological, social, and many other complex networks.
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22. Structural Models . . 3-Star (K1,3) n-Star (K1,n)dn d2 v1 . d3 v1 v3 d4 d2 d1 v2 d3
3-Star (K1,3) n-Star (K1,n)
2*3-Complete Bipartite(K2,3) *4-Complete Bipartite(K3,4) v1 vm dn d1 . v1 v3 v2 v4 vn .
Redraw in powerpoint and animate
m*n-Complete Bipartite(Km,n)
Triangle(K3) Clique (K4) n-Clique(Kn)
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23. Structural Graph Indexing
An important feature of these structures is that each one is comprised from the previous one where clique contains complete bipartite structures and complete bipartite contains star structures.
So, we can index these structures within the original graph in a consecutive manner. We first identify star structures, and then the complete-bipartite and clique structures from the preceding ones.
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24. Structural Graph Indexing
An important difference of our approach from the previous studies is that we does not limit the size of subgraph considered in indexing. We index all maximal graphs that match the structure formulation. For instance, a maximal clique is a clique that cannot be extended by adding one more vertex from the graph. However, the substructure size in indexing may be limited when needed, since maximal clique search is known to be NP-complete.
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25. Complete Bipartite
The second structure we index is complete bipartite, shown in below figures.
A complete bipartite graph G = (V1 ∪ V2,E) is a bipartite graph such that V1 and V2 are two distinct sets and for any two vertices vi ∈ V1 and vj ∈ V2, then there is an edge between them (i.e., ∃ e∗ (vi,vj ) ∈ E). v1 vm dn d1 . v1 v3 d4 d2 d1 v2 d3 v1 v2 d3 d2 d1
The complete bipartite graph with partitions of size |V 1| = m and |V 2| = n is denoted as K(m, n). Note that, star structure is a special case of a bipartite graph (not necessarily complete) where m = 1. Moreover, finding complete bipartite subgraph K(m,n) with maximal number of edges m.n is an NP-complete problem [15].
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26. Complete Bipartite
Complete bipartite structure is ubiquitous in many complex networks.
protein-protein interaction networks (Thomas et. al.)
the Internet (Fay et. al.)
We index all complete bipartite structures in the graph G using indexed star structures.
For each star structure s(a,ns) where a ∈ V and ns is the neighbor set of the node a, we identify the maximal complete bipartite involving the node ‘a’.
Complete bipartite structure is ubiquitous in many complex networks. For example, [4] examines the structure of protein-protein interaction networks and showed that the graph of all protein-protein interactions is made up of complete bipartite structures containing two disjoint sets of nodes in which each node in one set is connected to every node in the other set. Additionally, bipartite graphs are very relevant in the Internet, as such subgraphs correspond to nodes that have identical sets of neighbors. Studying those nodes would be of particular interest to understand the structure of the Internet as these are duplicated structures in the graph [8].
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27. Complete Bipartite Algorithm:
Find second hop neighbors of ‘a’ by iterating over the ns set and unifying them under Lcan set that indicates candidates for the left side of the complete bipartite while the ns set is the candidate set for the right hand side.
Then, find a K2,n and then grow it to Km,n. In finding K2,n , iterate over each candidate node in the Lcan and determine the neighbor intersection with a. If the intersection set is larger than two, then these nodes belong to the right hand side.
Grow the K2,n by finding all nodes in the left hand side (i.e., Lcan) that has the right hand side nodes (i.e., Rnew) as a neighbor. a
ns1
ns2
nsn .
ls1
ls2
lsn .
Right can. set
Left
can. set
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28. Clique Finally, we index clique structures shown in below figures.
A clique in graph G = (V,E) is a subset of the vertex set (i.e., C ⊆ V ) such that there are edges between all node pairs (i.e., ∀(ci, cj) ∈ C, ∃e(ci,cj) ∈ E, when i ≠ j).
We index all maximal clique structures in the graph using complete bipartite structures. v1 vn v2 v3 v4 . v1 v2 v3 v1 v3 v2 v4
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29. Clique This structure has been observed and utilized in many fields.
computational biology
protein structure prediction (Samudrala et. al.)
electronic circuits (Cong et. al.)
chemicals in a chemical database (Rhodes et. al.)
For example, in computational biology many problems can be solved by finding maximal or all cliques within the graph. Similarly, [18] models protein structure prediction as a problem of finding cliques in a graph whose vertices represent positions of subunits of the protein; [6] finds a hierarchical partition of an electronic circuit into smaller subunits using cliques; and [16] uses cliques to describe chemicals in a chemical database that have a high degree of similarity with a target structure.
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30. Clique Algorithm: Set1 v1 Set2 d1 d2 v2 d3 v3 d4
First get the set of nodes from each complete bipartite k(m,n) and look for cliques that are formed by those nodes.
The clique search algorithm works recursively on each node from the k(m,n) as the pivot node in the L1 set and considers other nodes as candidate nodes in the L2 set.
The function, moves each node from the L2 set to the L1 set if it is connected to all nodes in the L1 and then recursively tries to grow the structure with remaining nodes as candidates.
When there are no more candidates to consider in L2 set then a clique has been identified.
Set1 v1 v3 d4 d2 d1 v2 d3 v1
Set2 d1 d2 d3 d4
Note that, any clique larger than three nodes in the graph G will be indexed as multiple bipartite structures. Hence, we do not need to consider all nodes in the graph when indexing maximal clique structures.
Note that, this algorithm is not optimal and better solutions for finding all cliques are proposed in [5], [17]. v2 v3
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31. Where to Submit
Advances in Social Network Analysis and Mining (ASONAM 2011):
Full paper submission deadline is March 1, 2011.
Full paper manuscripts must be with a maximum length of 8 pages (using the IEEE two- column template).
Kaohsiung, Taiwan 7/25-7/27
Workshop on Large-scale Data Mining: Theory and Applications (LDMTA 2011)
Workshop on Mining and Learning with Graphs (MLG 2011)
Workshop on Social Network Mining and Analysis (SNAKDD 2011)
Full paper submission deadline is May 4-10, 2011.
Full paper manuscripts must be with a maximum length of 10 pages (using the ACM two- column template).
San Diego, CA 8/21-8/24
Simplifying Network Science for Practitioners: (SIMPLEX 2011)
Full paper submission deadline is Jan 31, 2011 – Feb
Full paper manuscripts must be with a maximum length of 6-10 pages (using the IEEE two- column template).
Minneapolis, Minnesota, USA 6/20-6/24
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32. Questions
SIMPLEX’10

33. Thank you