![Graphulo- 18
As an iterative algorithm
• Strategy: start with the whole graph and iteratively remove
edges that don’t find the k-truss criteria
• Adjacency Matrix (A) = ETE – diag(ETE)
• Algorithm:
– R ← E*A
– x ← find(( R = 2 )*𝟏 < k − 2) % x is edges preventing a k-truss
– While x is not empty, do:
• E 𝑥 ← E(x, ∶) % get the edges to remove
• E ← E(xc, ∶) % keep only the complementary edges
• R ← E(xc, ∶)*A % remove the rows associated with non-truss edges
• R ← R−E * [E 𝑥E 𝑥
𝑇− ( diag(E 𝑥E 𝑥
𝑇) ) ] %update R
• x ← find(( R==2 )*𝟏< k−2 ) %update x
• GraphBLAS kernels required: SpGEMM, SpMV](https://image.slidesharecdn.com/141205-graphuloingraphblas-141212080804-conversion-gate01/85/141205-graphulo-ingraphblas-18-320.jpg)
141205 graphulo ingraphblas
- 1. Graphulo- 1 Graph Analytics in GraphBLAS Jeremy Kepner, Vijay Gadepally, Ben Miller 2014 December This material is based upon work supported by the National Science Foundation under Grant No. DMS-1312831. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. G R A P H U L O
- 2. Graphulo- 2 Outline • Introduction • Degree Filtered Breadth First Search • K-Truss • Jaccard Coefficient • Non-Negative Matrix Factorization • Summary
- 3. Graphulo- 3 Graphulo Goals • Primary Goal – Open source Apache Accumulo Java library that enables many graph algorithms in Accumulo • Additional Goals – Enable a wide range of graph algorithms with a small number of functions on a range of graph schemas – Efficient and predictable performance; minimize maximum run time – Instructive and useful example programs; well written spec – Small and tight code base – Minimal external dependencies – Fully documented at graphulo.mit.edu – Accepted to Accumulo Contrib – Drive Accumulo features (e.g., temporary tables, split API, user defined functions, …) – Focus on localized analytics within a neighborhood, as opposed to whole table analytics
- 4. Graphulo- 4 Plan • Phase 1: Graph Mathematics Specification – Define library mathematics – Define example applications and data sets • Phase 2: Graph Mathematics Prototype – Implement example applications in Accumulo prototyping environment – Verify that example applications can be effectively implemented • Phase 3: Java Implementation – Implement in Java and test at scale
- 5. Graphulo- 5 GraphBLAS • The GraphBLAS is an effort to define standard building blocks for graph algorithms in the language of linear algebra – More information about the group: http://istc-bigdata.org/GraphBlas/ • Background material in book by J. Kepner and J. Gilbert: Graph Algorithms in the Language of Linear Algebra. SIAM, 2011 • Draft GraphBLAS functions: – SpGEMM, SpM{Sp}V, SpEWiseX, Reduce, SpRef, SpAsgn, Scale, Apply • Goal: show that these functions can perform the types of analytics that are often applied to data represented in graphs GraphBLAS is a natural starting point Graphulo Mathematics
- 6. Graphulo- 6 Examples of Graph Problems Algorithm Class Description Algorithm Examples Exploration & Traversal Algorithms to traverse or search vertices Depth First Search, Breadth First Search Centrality & Vertex Nomination Finding important vertices or components within a graph Betweenness Centrality, K-Truss sub graph detection Similarity Finding parts of a graph which are similar in terms of vertices or edges Graph Isomorphism, Jaccard Index, Neighbor matching Community Detection Look for communities (areas of high connectedness or similarity) within a graph Topic Modeling, Non-negative matrix factorization, Principle Component Analysis Prediction Predicting new or missing edges Link Prediction Shortest Path Finding the shorted distance between two vertices Floyd Warshall, Bellman Ford, A* algorithm, Johnson’s algorithm
- 7. Graphulo- 7 Accumulo Graph Schema Variants • Adjacency Matrix (directed/undirected/weighted graphs) – row = start vertex; column = vertex; value = edge weight • Incidence Matrix (multi-hyper-graphs) – row = edge; column = vertices associated with edge; value = weight • D4M Schema – Standard: main table, transpose table, column degree table, row degree table, raw data table – Multi-Family: use 1 table with multiple column families – Many-Table: use different tables for different classes of data • Single-Table – use concatenated v1|v2 as a row key, and isolated v1 or v2 row key implies a degree Graphulo should work with as many of Accumulo graph schemas as is possible
- 8. Graphulo- 8 Algorithms of Interest • Degree Filtered Breadth First Search – Very common graph analytic • K-Truss – Finds the clique-iness of a graph • Jaccard Coefficient – Finds areas of similarity in a graph • Topic Modeling through Non-negative matrix factorization – Provides a quick topic model of a graph
- 9. Graphulo- 9 Outline • Introduction • Degree Filtered Breadth First Search • K-Truss • Jaccard Coefficient • Non-Negative Matrix Factorization • Summary
- 10. Graphulo- 10 Degree Filtered Breadth First Search • Used for searching in a graph starting from a root node – Very often, popular nodes can significantly slow down the search process and may not lead to results of interest • A degree filtered breadth first search, first filters out high degree nodes and then performs a BFS on the remaining graph • A graph G=(V,E) can be represented by an adjacency matrix A where A(i,j)=1 if there is an edge between vi and vj • Alternately, one can represent a graph G using an incidence matrix representation E where E(i,j) = 1 if there is an edge from vi -> vj and -1 if the edge is from ej -> ei • The Degree Filtered BFS can be computed using either representation
- 11. Graphulo- 11 Adjacency Matrix based Degree Filtered BFS • Uses the adjacency matrix representation of a graph G to perform the BFS. • Algorithm Inputs: v0: Starting vertex set k: number of hops to go T: Accumulo table of graph adjacency matrix Tin = sum(Tadj,1).'; % Accumulo table in-degree Tout = sum(Tadj,2); % Accumulo table of out-degree dmin: minimum allowable degree dmax: maximum allowable degree • Algorithm Output: Ak: adjacency matrix of sub-graph
- 12. Graphulo- 12 Adjacency Matrix based Degree Filtered BFS • The algorithm begins by retaining vertices whose degree are between dmin and dmax • Algorithm: vk = v0; % Initialize seed set for i=1:k uk = Row(dmin ≥ str2num(Tout(vk,:)) ≤ dmax); % Check dmin and dmax Ak = T(uk,:); % Get graph of uk vk = Col(Ak); % Neighbors of uk end
- 13. Graphulo- 13 Incidence Matrix based Degree Filtered BFS • Uses the incidence matrix representation of a graph G to perform the BFS. • Algorithm Inputs v0: starting vertex set k: number of hops to go T: Accumulo table of graph incidence matrix Tcol = sum(Tadj,1).'; % Accumulo table column degree Trow = sum(Tadj,2); % Accumulo table row degree dmin: minimum allowable degree dmax: maximum allowable degree • Algorithm Output Ek: adjacency matrix of sub-graph ( sum of edge weights) ( node degree)
- 14. Graphulo- 14 Incidence Matrix based Degree Filtered BFS • The algorithm begins by retaining vertices whose degree are between dmin and dmax • Algorithm: vk = v0; % Initialize seed set for i=1:k uk = Row(dmin ≥ str2num(Tcol(vk,:)) ≤ dmax); % Check dmin and dmax Ek = T(Row(T(:,uk)),:); % Get graph of uk vk = Col(Ek); % Get neighbors of uk end
- 15. Graphulo- 15 Outline • Introduction • Degree Filtered Breadth First Search • K-Truss • Jaccard Coefficient • Non-Negative Matrix Factorization • Summary
- 16. Graphulo- 16 K-Truss • A graph is a k-truss if each edge is part of at least k-2 triangles • A generalization of a clique (a k-clique is a k-truss), ensuring a minimum level of connectivity within the graph • Traditional technique to find a k-truss subgraph: – Compute the support for every edge – Remove any edges with support less than k-2 and update the list of edges – When all edges have support of at least k-2, we have a k-truss Example 3-truss
- 17. Graphulo- 17 K Truss in Terms of Matrices • If E is the unoriented incidence matrix (rows are edges and columns are vertices) of graph G, and A is the associated adjacency matrix • If G is a k-truss, the following must be satisfied: – AND((E*A == 2) * 1 > k – 2) – where AND is the logical and operation • Why? – E*A: each row of the result is the sum of rows in A associated with the two vertices of an edge in G – E*A == 2: Result is 1 where vertex pair of edge have a common neighbor – (E*A ==2) * 1 : Result is the sum of number of common neighbors for vertices of each edge – (E*A ==2) * 1 > k – 2: Result is 1 if more common neighbors than k-2
- 18. Graphulo- 18 As an iterative algorithm • Strategy: start with the whole graph and iteratively remove edges that don’t find the k-truss criteria • Adjacency Matrix (A) = ETE – diag(ETE) • Algorithm: – R ← E*A – x ← find(( R = 2 )*𝟏 < k − 2) % x is edges preventing a k-truss – While x is not empty, do: • E 𝑥 ← E(x, ∶) % get the edges to remove • E ← E(xc, ∶) % keep only the complementary edges • R ← E(xc, ∶)*A % remove the rows associated with non-truss edges • R ← R−E * [E 𝑥E 𝑥 𝑇− ( diag(E 𝑥E 𝑥 𝑇) ) ] %update R • x ← find(( R==2 )*𝟏< k−2 ) %update x • GraphBLAS kernels required: SpGEMM, SpMV
- 19. Graphulo- 19 For example: find a 3-truss of G For 3 truss, k=3 1 2 3 4 e1 e2 e3 e5 e4 5 e6 3 truss SubGraph given by
- 20. Graphulo- 20 Outline • Introduction • Degree Filtered Breadth First Search • K-Truss • Jaccard Coefficient • Non-Negative Matrix Factorization • Summary
- 21. Graphulo- 21 Jaccard Index • The Jaccard coefficient measures the neighborhood overlap of two vertices in an unweighted, undirected graph • Expressed as (for vertices vi and vj), where N is the neighbors: • Given the connection vectors (a column or row in the adjacency matrix A) for vertices vi and vj (denoted as ai and aj) the numerator and denominator can be expressed as ai Taj where the we replace multiplication with the AND operator in the numerator and the OR operator in the denominator • This gives us: – Where ./ represents the element by element division
- 22. Graphulo- 22 Algorithm to Find Jaccard Index • Using the standard operations, A2 AND is the same as A2 • Also, the inclusion-exclusion principle gives us a way to compute A2 OR when we have the degrees of the vertex neighbors di and dj: A2 OR = Σdi + Σdj - A2 AND • So, an algorithm to compute the Jaccard in linear algebraic terms would be: – Initialize J to A2: J = triu(A2) %Take upper triangular portion – Remove diagonal of J: J = J-diag(J) – For each non zero entry in J given by index i and j that correspond to vertices vi and vj: Jij = Jij/(di + dj – Jij)
- 23. Graphulo- 23 Example Jaccard Calculation 1 2 3 4 5
- 24. Graphulo- 24 Efficiently Computing triu(A2) • Since only the upper triangular part of A2 is needed, we can exploit the symmetry of the matrix A, and its lack of nonzero values on the diagonal, to only compute what is necessary • Let A=(L+U), where L and U are strictly lower and upper triangular, respectively – Note that L = UT, since A is symmetric • Then A2 = (UT)2+UTU+UUT+U2 = (UT)2+(UUT)T+UUT+U2 – Note that (UT)2 is lower triangular and U2 is upper triangular • Then triu(A2) can be efficiently computed as follows: – U ← triu(A) – X ← U*UT – X ← triu(X) + tril(X)T + U*U • Now triu(X) is the same as triu(A2)
- 25. Graphulo- 25 triu, tril, diag as element-wise products • A Hadamard (entrywise) matrix product can be used to implement functions that extract the upper- and lower- triangular parts of a matrix in the GraphBLAS framework • To implement triu, tril, and diag on a matrix A, we perform A 1 • Where = f(i,j) is a user defined multiply function that operates on indices of the non-zero element of A – For triu(A) = A 1, the upper triangle, f(i,j) = {A(i,j): i ≤ j , 0 otherwise} – For tril(A) = A 1, the lower triangle, f(i,j) = {A(i,j): i ≥ j, 0 otherwise} – For diag(A) = A 1, the diagonal, f(i,j) = {A(i,j): i==j, 0 otherwise} • triu, tril, and diag all represent GraphBLAS utility functions than can be built with user defined multiplication capabilities found in the GraphBLAS
- 26. Graphulo- 26 Outline • Introduction • Degree Filtered Breadth First Search • K-Truss • Jaccard Coefficient • Non-Negative Matrix Factorization • Summary
- 27. Graphulo- 27 Topic Modeling • Common tool for individuals working with big data – Quick summarization – Understanding of common themes in dataset – Used extensively in recommender systems and similar systems • Common techniques: Latent dirichlet allocation, Latent semantic analysis, Non-negative matrix factorization (NMF) • Non-negative matrix factorization is a (relatively) recent algorithm for matrix factorization that has the property that the results will be positive • NMF applied on a matrix Amxn: – where W, H are the resultant matrices and k is the number of desired topics • Columns of W can be considered as basis for matrix A and rows of H being the associated weights needed to reconstruct A (or vice versa)
- 28. Graphulo- 28 NMF through Iteration • One way to compute the NMF is through an iterative technique known as alternating least squares given below: • A challenge implementing the above is in determining the matrix inverse (essentially the solution of a least squares problem for alternating W and H)
- 29. Graphulo- 29 Matrix Inversion through Iteration • A (not too common) way to solve a least squares problem is to use the relation that • In matrix notation, • Thus, to compute the least squares solution, we can use an algorithm as below:
- 30. Graphulo- 30 Combining NMF and matrix inversion • The previous two slides can be combined to provide an algorithm that uses only GraphBLAS kernels to determine the factorization of a matrix A (which can be a matrix representation of a graph)
- 31. Graphulo- 31 Mapping to GraphBLAS • In order to implement the NMF using the formulation, the functions necessary are: – SpRef/SpAsgn – SpGEMM – SpEWiseX – Scale – Reduce – Addition/Subtraction (can be realized over (min,+) semiring with scale operator) • Challenges: – Major challenge is making sure pieces are sparse. The matrix inversion process may lead to dense matrices. Looking at other ways to solve the least squares problem through QR factorization (however same challenge applies) – Complexity of the proposed algorithm is quite high
- 32. Graphulo- 32 Summary • The GraphBLAS effort aims to standardize the kernels used to express graph algorithms in terms of linear algebraic operations • One of the important aspects in standardizing these kernels is in the ability to perform common graph algorithms • This presentation hightlights the applicability of the current GraphBLAS kernels applied to four popular analytics: – Degree Filtered Breadth First Search – K-Truss – Jaccard Index – Non-negative matrix factorization
Editor's Notes
- #2: Title Page
- #20: A = \begin{bmatrix} 1 & 1 & 0 & 0 & 0\\ 0 & 1 & 1 & 0 & 0\\ 1 & 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 1 \end{bmatrix} A = \begin{bmatrix} 0 & 1 & 1 & 1 & 0\\ 1 & 0 & 1 & 0 & 1\\ 1 & 1 & 0 & 1 & 0\\ 1 & 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 \end{bmatrix} x=6\\ x^{c} = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \end{bmatrix}^{T} E(x^{c},:) = \begin{bmatrix} 1 & 1 & 0 & 0 & 0\\ 0 & 1 & 1 & 0 & 0\\ 1 & 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 0 & 0 \end{bmatrix}
- #22: J = \frac{| N(v_{i}) \cap N(v_{j}) |}{| N(v_{i}) \cup N(v_{j}) |}