Approximate Graph Searching Spatio-Temporal Graphs

Transcription

1 Approximate Graph Searching Spatio-Temporal Graphs Hrishikesh Terdalkar Prof. Arnab Bhattacharya

2 1. Introduction Terminology and Setting

3 What are Graphs V : Vertices E : Edges Av : Vertex Attributes AE : Edge Attributes Labelled G = (V, E)

4 What is.. Querying Matching Graph Searching Given Graph database D D = { G1, G2,.., Gn } Query graph Q Find Occurrences of Q in D GraphGrep Giugno et al. 2002

5 What is.. Graph Indexing Graph Searching Construct Index Paths Structures gindex Yan et al. 2004

6 Subgraph Isomorphism NP-Complete Given graphs G and H. Is there a subgraph G0 G such that G0 H

7 Frequent Subgraph Mining Given Graph database D = { G1, G2,.., Gn } Threshold - δ (0,1) AGM Inokuchi et al Find G = { g subgraph g occurs in at least δ. D graphs in D } gspan Yan et al. 2002

8 2. Principle Techniques Evolution

9 Process Overview Filter Matching Search Build Index Query Processing Verify Prune

10 GraphGrep A Fast and Universal Method for Querying Graphs Giugno et al Properties of Graphs in D Nodes numeric id - id-node string label - label-node Edges undirected unlabelled Terms id-path : list of n id-node (edges between consecutive nodes) label-path : list of n label-node Database Containing 3 graphs D B A C B B A Graph g1 B 3 C 4 (3, 1) is the corresponding id-path A Graph g2 e.g. (C, A) is a label-path in g1 and 1 1 B C 2 3 Graph g3 C B 6 E

11 Basic Steps GraphGrep Giugno et al B C 1. Database Construction foreach - graph Gi - node v find all paths starting at v of length [1, lp ] Path Representation set of label-paths each label-path has set of id-paths Hash Table keys : hash values of label-paths row : number of id-paths associated with a key per graph Fingerprint ( Gi ) Column vector A B Graph g1 Path Representation of graph g1 A = {(1)} AB = {(1,0), (1,2)} AC = {(1,3)} ABCA = {(1,0,3,1),(1,2,3,1)} Key h(ca) g1 g2 g h(abca) h(abcb) 2 2 0

12 Basic Steps GraphGrep Giugno et al B C 2. Database Filtering Parsing Query Graph Glide Representation Build Fingerprint of Query Hashed set of paths label-paths of length < lp A B Query q Example If query has a label-path ABCA prune out g2 and g3 Filtering Prune graphs that clearly do not contain the query Compare fingerprint of query with fingerprint of database Discard a graph if at least one value in its fingerprint is less than corresponding value in fingerprint of query Key h(ca) g1 g2 g h(abca) h(abcb) 2 2 0

13 Basic Steps GraphGrep Giugno et al Exact Matching Example: Match Query q - Graph g1 Look for all matching subgraphs B A Query q C B B A C 1. Select set of paths in g1 with lp = 4 ABCA = { (1,0,3,1), (1, 2, 3, 1) } CB = { (3,0), (3,2) } 2. Combine any list from ABCA with any list from CB (matching positions) ABCACB = { ((1, 0, 3, 1), (3, 0)), ((1, 0, 3, 1), (3, 2)), ((1, 2, 3, 1), (3, 0)), ((1, 2, 3, 1), (3, 2)) } 3. Remove lists from ABCACB if they contain equal id-nodes in non-overlapping positions B Graph g1 Matching: ((1, 0, 3, 1), (3, 2)), ((1, 2, 3, 1), (3, 0))

14 gindex Graph Indexing: A Frequent Structure-based Approach Yan et al Drawbacks of GraphGrep : Example: Sample Query Path is too simple: structural information is lost. Too many paths: the set of paths in a graph database usually is huge. Example: Sample Database Cannot prune (a) and (b) Contain all paths from Query graph

15 Methodology gindex Yan et al Use graph structure instead of path as basic index feature. Frequent Fragments: Major Steps: Index Construction enumerate & select features F for a feature f F Df is set of graphs containing f Query Processing Search enumerate all features in q candidate answer set Cq = graphs having all features of q Verification support(g) = Dg number of graphs g appears in threshold minsup Example : ( minsup = 2 ) Choice of minsup Discriminative Fragments

16 Features gindex Yan et al Do we need to index every frequent fragment? gindex Tree Redundant Fragments Fragment x is redundant wrt feature set F if, Dx f F and f x Df subgraphs of x already capture its essence Discriminative Fragments Fragment x is discriminative wrt F each fragment edge sequence (DFS code) prefix tree codes of discriminative fragments redundant fragments intermediate nodes leaf nodes discriminative fragments Insert / Delete Maintenance Incremental updates Only for involved frequent fragments Single database scan algorithm for index construction if, Dx f F and f x Df contrary to redundant.

17 Graph Matching Variety of Approaches NeMa SAPPER TALE Giugno et al gindex Yan et al Path Index Zhang et al Fan et al Khan et al VELSET + NAGA Neighbor Based Hybrid Index Local Inexact Global Modify Graph Struct SIM-T Kpodjedo et al Problem Variants Graph Edit Distance Freq Pattern Index Exact Match (cubic) Dutta et al Tian et al GraphGrep Statistical Measures Single DB Graph DB Many Graphs

18 Approximate Graph Matching Given Graph database D D = { G1, G2,.., Gn } Query graph Q Find Occurrences of Qi in D Qi similar to Q Similarity various notions

19 Graph Similarity Techniques Overview Edit Distance Feature Extraction Generalization of Graph Isomorphism Aim : Transform one graph to the other by doing a number of operations add, delete substitute nodes or edges reverse edges operations cost Find : Sequence of operations that minimizes cost of matching two graphs Key Idea : Similar graphs probably share few properties degree distribution diameter eigenvalues Extract features and apply a similarity measure Plethora of measures Neighborhood Based Love thy neighbour Key Idea : Two nodes are similar if their neighbors are. In each iteration, nodes exchange similarity scores. Stop at convergence

20 TALE (Neighborhood Index) A Tool for Approximate Large Graph Matching Tian et al Neighborhood Index (NH-Index) Example For the dark node (say v) Indexing Unit: neighborhood of each node Captures local graph structure v.degree = 7 v.nbconnection = 4 Neighborhood : Induced subgraph of a node and its adjacent nodes Properties of the neighborhood : Number of neighbors How the neighbors connect (# of edges between neighbors) Labels of the actual neighbors NH-Index(v) = (label, degree, nbconnection, nbarray[])

21 TALE - Methodology Matching Query Node (Query node Nq, Database node Nd) Exact Matching Nq.label == Nd.label Nq.degree Nd.degree Nq.nbConn Nd.nbConn neighbors of Nq and Nd should match in same way Approximate Matching Nq.label == Nd.label Nq.degree Nd.degree + nbmiss Nq.nbConn Nd.nbConn + nbcmiss Miss(Nd.nbArr[i], Nq.nbArr[i]) nbmiss Node Match Quality : w(nq, Nd) Case of Approximate Matching parameter ρ : % of neighbors of q that can have no corresponding matches in the neighbors of d # of nodes allowed to be missing: nbmiss = ρ * Nq.degree # of connections allowed to miss : nbcmiss = nbmiss * (nbmiss - 1)/2 + (Nq.degree - nbmiss) * nbmiss based on fraction of nodes missing higher w value better node match Neighbor Array Deterministic bit array Size of bit array = Lv

22 TALE - Methodology Index Structure and Matching Index Structure Hybrid Index : 2 Levels B+ Tree (label, degree, nbconn) Fast Search (equality on label, range on degree and nbconn) Second-level index node-id Bitmap index for neighborarray Index Probing Utilize label, degree and nbconn Probe the B+-Tree index Examine obtained list of bitmaps - count nbmiss - conditions for approx. matching Prune Matching Match important nodes Pimp - degree centrality Probe NH-index imp. nodes Node match quality Extend the match Match nearby nodes ( 2-hop)

23 SIGMA A Set-cover-based Inexact Graph Matching Algorithm Mongiovi et al Generic Steps Multiset-Multicover : NP-Hard, but.. Pre-processing Filtering Matching Greedy Approximation Algorithm with tight bounded error (by Hn Harmonic Number) Multiset = (A, m) where, multiplicity m : A ℕ (A, m) (B, n) = (A B, min(m, n)) (A, m) (B, n) = (A B, (m + n)) Multiset Multicover Feature set F edges of query Collections of such sets whose removal will allow exact matching Covering the missing features of the graph with overlapping multisets Example: Exact Matching Features = Edges (graphs with size 1)

24 SIGMA Example - Multiset-Multicover Mongiovi et al G contains a copy of Q with two deletions Features of Q FQ = { F1, F2, F3, F4 } Fi = Features containing edge i Minimum Cover of FQ - FG of is cardinality 2 at least two deletions needed {F1, F2} = a possible cover G is a candidate to match Q with edges 1 and 2 deleted Greedy Multiset-Multicover (Y, S) Ans φ while ( Y φ ) do X argmaxa S A Y Y Y-X Ans Ans { X } return Ans Example: Matching with 2-deletions query Q and a graph G Features = connected subgraphs with exactly 2 edges

25 SAPPER Subgraph Indexing and Approximate Matching in Large Graphs Zhang et al Edge Edit Distance # of edge modifications needed to transform one graph to another Bloom Filter ( B ) used to qualify an occurrence of q θ threshold (max EED allowed) AI(q, θ) = Set of approx. matches v in G, Ni (v, G) = { u G path of i edges between u and v } Space-efficient probabilistic data structure L-bit vector set of m independent hash functions {f1,, fm} Used to determine whether an element x is member of a set X fi : X [1, L] ℤ Hybrid Neighborhood Unit Index (HNU) ( label(v), degree(v), labels(n1(v, G)), labels(n2(v, G)) ) Properties No false negatives Small rate ( 1%) of false positives

26 SAPPER - Methodology Index Construction - Query Processing - Matching Zhang et al Bloom Filter - Parameters Error rate depends on L, X and m Optimal m = 0.7 * L / X for 1% L / X = 9.6 X = labels in N2(v, G) X d2 where d = avg degree in G WLOG L = 9.6 d2, m = 7 Query Processing and Matching Index Construction In HNU(v), labels N2(v, G) collected L-bit bloom filter is built during index construction time Time Complexity: Per vertex : O( d + m * d2 + L ) = O( md2 ) Total : O(md2 VG ) Vertex Matching Candidate matches vq q to vertices in G based on HNU Constructing Spanning Trees of q Randomly generate a set of spanning trees of q Generating a matching order of graphs in AI(q, θ) Find matches of spanning trees based on vertices match Use ST matches to match the approximate graphs DFS based order on matching these graphs Final graph matching

27 3. Graph Indexing and Searching Advances

28 NeMa Fast Graph Search with Label Similarity Khan et al Neighborhood based matching Neighborhood Vectorization Convert h-hop neighborhood into multi-dimensional vector RG(u) = { v, wu(v) } (where wu = αd(u, v)) Cost Metric Measure the quality of the matches Example: h = 2, α = 0.5 costs of matching individual query nodes ( cost of matching node labels and neighborhoods within h hops. (h = 2)) C( φ ) = [ ΔL+ ΔL ] Handles structural and label noise RG(a) = { b, 0.5, c, 0.5, d, 0.25 } Minimum Cost Subgraph Matching a b Given, query graph Q and a data graph G c d e Identify, top-k matches of query graph with minimum costs in target graph NP-Hard Hard to approximate Heuristic method efficient

29 NeMa - Methodology Example - Iterative Inference Algorithm Khan et al Query Graph Q Target Graph G a a v2 b v4 c b u2 a c b u5 a c b d d u7 u9 c d u10 e c e Ui (v, u) = Inference Cost in iteration i (query node v, candidate node u) Suppose, we ve already determined candidate matches (v) h = 1, i = 0 (v2) = {u2, u5, u9} (v4) = {u7, u10} U0 (v2, u5) = U0 (v2, u9) = 0 U0 (v4, u10) < U0 (v4, u7) U1(v2, u5) < U1 (v2, u9)

30 SIM-T Using local similarity measures for efficient approximate matching Kpodjedo et al Local (neighborhood) Searches Take a potential solution Check its immediate neighbors (solutions very similar except for very few minor details) Hope to find an improved solution Tabu Search Metaheuristic search method Enhances the performance of local search by relaxing its basic rule At each step worsening moves can be accepted if no improving move is available e.g. the search is stuck at a strict local minimum prohibitions ( tabu) to discourage the search from coming back to previously-visited solutions SIM-T : Two-Phase Algorithm Greedy procedure Greedy-Sim Followed by tabu search High accuracy when initialized with a potentially good solution Greedy-Sim Step-by-step matching between V1 and V2 (vertex sets of G1 and G2) Iteratively insert a new node match into configuration Choice of node match to be inserted follows greedy criterion Greedy score: gr( x1, x2 ) = δ0 ( x1, x2 ) + B S ( x1, x2 ) where, - δ0 ( x1, x2 ) is # of new perfect matches B ( B 1 ) is a real number used to weigh the similarity of x1 and x2 S(x1, x2) - similarity value of x1 and x2

31 Statistical Measures Preliminaries Null Hypothesis : (H0) General statement that there is no relationship between two phenomena under consideration. Statistical Inference : The process of deducing properties of an underlying distribution by analysis of data. Gamma Distribution : k > 0 (shape), θ > 0 (scale) Two-parameter family of continuous probability distributions. Chi-square Distribution : k degrees of freedom Distribution of a sum of the squares of k independent standard normal random variables. ( a special case of the gamma distribution )

32 Statistical Significance Measures and Tests p-value probability that the phenomenon happened by chance probability of occurrence of events at least as extreme as this one. z-score z-score (aka, a standard score) indicates how many standard deviations a data point is above from the mean eg, in a coin toss experiment, p-value of observing 8 heads in a series of 10 tosses = probability of getting 8, 9 or 10 heads 2 (chi-square) log-likelihood (G2) Category of statistical tests. Follows chi-square distribution. Category of statistical tests. Often comparable to 2 tests. Numerous variations. Log of the ratio of two likelihoods as the test statistic.

33 Motivation: Graph Setting p-value Vertices represented by label set pertaining to their neighbors v2 G { A, B, EA } q2 Q { A, A, B } q5 Q { A, B } Using structural similarity (3 nghbrs) v2 can be matched with q2 or q3 Using labels v2 can be matched with q2 or q5 Best match of v2 is with q2 q2 provides a higher degree of match Computing the p-value is practically inefficient (exponential # of possibilities) Graph Matching Vertices with higher degrees of match Higher statistical significance Better candidates for matching

34 Pearson s chi-square 2 goodness-of-fit test Difference between the expected and observed occurrences of outcome values Good approximation of p-value Σ i (Oi-Ei)2 Ei

35 Properties Pearson s Chi-Square Statistic Follows chi-square distribution Degrees of freedom : # { possible outcomes } - 1 Higher the 2 Lower p-value Higher statistical significance More similar subgraph subgraph with larger 2

36 Example Pearson s Chi-Square Statistic Publish fliers in three different colours H0 : No relation in which fliers are taken Two degrees of freedom (R - 1) * (C - 1) = (3-1) * (2-1) = 2 Fliers Pink Light Blue Neon Pink Taken Not Taken = (32-24)2 / 24 + (38-36)2/ 36 + (20-30)2/30 + (8-16)2/16 + (22-24)2/24 + (30-20)2/20 = 82/ / / / / /20 = 64/24 + 4/ / /16 + 4/ /20 = Chi-square distribution table probability p < Highly Significant

37 VELSET + NAGA Neighbor-Aware Search using the Chi-Square Statistics Dutta et al Subgraph Similarity based on Statistical Significance ( 2 ) VELSET: VErtex Label Similarity on Edge Triplets Accuracy NAGA: Neighbor-Aware Greedy Algorithm Efficiency Input Large database graph G, Query graph q Number of matches k Output top-k subgraphs most similar to q Example Input graph G, query Q

38 VELSET + NAGA Framework 1. Creation of Different Lists (IL, LNL) Inverted lists mapping vertex labels to vertices Label-neighbor-list (LNL) information of neighbors 3a. Statistical Significance 2 for each VPs is computed 3b. Statistical Significance NAGA VELSET u G : Consider LNLu as set u G : triplet of vertices (x, u, y) LNLu and LNLu : divided in sliding window of size 2. v3 (A,B,C) (C,B,D), (D,B,A) Degree of match - s0, s1, s2 q5 (A,C,B) Triplets matched for pairs VP Degree of match - s0, s1, s2 2. Vertex Pair Construction (VP) v Q fnd g G with NAGA - exact same labels VELSET - similar labels Jaccard Similarity Expected Values Let p = ( 1-1/L)d where, d = degree(u) P(s0) = p2 P(s1) = 2. (1 - p). (p) P(s2) = (1 -p)2 4. Generating Approximate Match Pick Highest 2 pair (u, v) Consider neighbors-pairs of u, v repeat

39 Performance Comparison NeMa, SIM-T, VELSET, NAGA YAGO Dataset F1 Score Running Time (seconds) Query Scenario NeMa SIM-T VELSET NAGA NeMa SIM-T VELSET NAGA Exact Match Noisy Edges Noisy Labels Combined Average

40 Performance Comparison NeMa, SIM-T, VELSET, NAGA IMDB Dataset F1 Score Running Time (seconds) Query Scenario NeMa SIM-T VELSET NAGA NeMa SIM-T VELSET NAGA Exact Match Noisy Edges Noisy Labels Combined Average

41 Performance Comparison NeMa, SIM-T, VELSET, NAGA Overall Performance YAGO Methods IMDB F1 (%) Running Time (s) Indexing Time (s) F1 (%) Running Time (s) Indexing Time (s) VELSET , NAGA NeMa ~10, ~10,000 SIM-T

42 igq - New Perspective Indexing Query Graphs to Speedup Graph Query Processing Wang et al DB Graph Index CS(g) = { g1, g2, g3, g4 } CS(g) - Answer(G) = { g3, g4 } igq Query Index (SUB) g G Answer(G) = { g1, g2 } Properties Utilize knowledge from previously executed queries Can be incorporated with existing algorithms Subgraph Isomorphism Test Answer Answer(g) = Answer { g1, g2 } Process (Subgraph query g) igq index - start off empty query processing parallelized candidate set CS(g) as usual check relation to prev queries subgraph of prev query graphs supergraph

43 igq - New Perspective Indexing Query Graphs to Speedup Graph Query Processing Wang et al DB Graph Index CS(g) = { g1, g2, g3, g4 } CS(g) Answer(G) = { g1 } igq Query Index (SUP) Answer(g) G g Answer(G) = { g1, g5 } Properties Subgraph Isomorphism Test Utilize knowledge from previously executed queries Can be incorporated with existing algorithms Process (Subgraph query g) igq index - start off empty query processing parallelized candidate set CS(g) as usual check relation to prev queries subgraph of prev query graphs supergraph

44 4. Spatio-Temporal Graphs Future

45 What are.. Spatio-Temporal Graphs G = (V, E) Edges and Vertices may appear, disappear with time label vector with location and time dimensions Lv = ( _, _, _, l, ts, te ) Le = ( _, _, _, t s, t e ) Attributes Location Time

46 CHEBY Indexing Spatio-Temporal Trajectories with Chebyshev Polynomials Cai et al { Pm (t) = cos(m cos 1 (t)) } Spatio-Temporal Trajectory: Discrete (t1, v1 ),..., (tn, vn) Theorem 1: Orthogonality Chebyshev Polynomials P0(t),, Pm(t) Chebyshev Polynomials: P0 (t) = 1 P1 (t) = t P2 (t) = 2t2-1 P3 (t) = 4t3-3t P4 (t) = 8t4-8t2 + 1 Chebyshev approximation almost identical to the minimax polynomial (optimal) easy to compute Approximating Any Function Theorem 1 CPs can be used as a base for approximating any function. Given a function f(t), f(t) = c0p0 + + cmpm

47 CHEBY Properties of Chebyshev Polynomials Cai et al Approximation of Time-Series Calculation of coefficients Theorem 2 : Gauss-Chebyshev Formula General Assumptions about Time Series i. ii. iii. iv. Same-length Each time series of same length (N) Power-of-2 N = 2k for some integer k Same-set Each series occurs at same set of time points {t1,, tn } (need not be uniform width) Same-set-uniformly-spaced uniform width (ti - ti-1 ) = (ti+1 - ti ) Indexing 1-dim s-t trajectory i.e., time serie (TS) Given collection of TS of length N Represent it by Chebyshev approximation of degree m << N n = m + 1 coeffs in multi-dim index

48 Example - CHEBY Time Series of the Opening Stock Price Cai et al Fortune500 company called ALCOA Time Period (6480 days) Feb 28, 1978 to Oct 24, 2003

49 TG-CSA Compressed Suffix-Array Strategy for Temporal-Graph Indexing Brisaboa et. al Suffix Array sorted array of all suffixes of a string Compressed Suffix Array (CSA) compressed data structure for pattern matching general class of data structure that improve on the suffix array enable quick search for an arbitrary string with relatively small index Temporal-Graph CSA TGCSA representation consists of a bitmap B structures D and Ψ from the icsa (integer based CSA)

50 Example - TG-CSA Structures Involved in the creation of TGCSA Brisaboa et. al Example: Suppose, we have a Temporal Graph with v = 5 vertices numbered 1,, 5 τ = 8 time instants numbered 1,, 8 Five Contacts (u, v, ts, te) (1, 3, 1, 8), (1, 4, 5, 8), (2, 1, 1, 5), (4, 3, 7, 8), and (4, 5, 5, 7)

51 Problem Approximate Exact Spatio Temporal

52 Thank You! Questions? Hrishikesh Terdalkar

53 References I Yan, Xifeng, and Jiawei Han. "gspan: Graph-based substructure pattern mining." Data Mining, ICDM Proceedings IEEE International Conference on. IEEE, Giugno, Rosalba, and Dennis Shasha. "Graphgrep: A fast and universal method for querying graphs." Pattern Recognition, Proceedings. 16th International Conference on. Vol. 2. IEEE, Yan, Xifeng, Philip S. Yu, and Jiawei Han. "Graph indexing: a frequent structure-based approach." Proceedings of the 2004 ACM SIGMOD international conference on Management of data. ACM, Fan, Wenfei, et al. "Graph pattern matching: from intractable to polynomial time." Proceedings of the VLDB Endowment (2010): Tian, Yuanyuan, and Jignesh M. Patel. "Tale: A tool for approximate large graph matching." Data Engineering, ICDE IEEE 24th International Conference on. IEEE, Mongiovi, Misael, et al. "Sigma: a set-cover-based inexact graph matching algorithm." Journal of bioinformatics and computational biology 8.02 (2010): Zhang, Shijie, Jiong Yang, and Wei Jin. "Sapper: Subgraph indexing and approximate matching in large graphs." Proceedings of the VLDB Endowment (2010):

54 References II Khan, Arijit, et al. "Nema: Fast graph search with label similarity." Proceedings of the VLDB Endowment. Vol. 6. No. 3. VLDB Endowment, Kpodjedo, Segla, Philippe Galinier, and Giulio Antoniol. "Using local similarity measures to efficiently address approximate graph matching." Discrete Applied Mathematics 164 (2014): Wang, Jing, Nikos Ntarmos, and Peter Triantafillou. "Indexing Query Graphs to Speed Up Graph Query Processing." (2016). Dutta, Sourav, Pratik Nayek, and Arnab Bhattacharya. "Neighbor-Aware Search for Approximate Labeled Graph Matching using the Chi-Square Statistics." Proceedings of the 26th International Conference on World Wide Web. International World Wide Web Conferences Steering Committee, Cai, Yuhan, and Raymond Ng. "Indexing spatio-temporal trajectories with Chebyshev polynomials." Proceedings of the 2004 ACM SIGMOD international conference on Management of data. ACM, Brisaboa, Nieves R., et al. "A compressed suffix-array strategy for temporal-graph indexing." International Symposium on String Processing and Information Retrieval. Springer International Publishing, 2014.

55 Appendix I

56 Gamma and Chi-square Distributions Gamma Distribution Chi-square Distribution

57

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