A Novel Algorithmic Framework for the Subgraph Isomorphism Problem

A Novel Algorithmic Framework for the Subgraph Isomorphism Problem Introduction Dense Subgraph Discovery Background/Motivation • Graph isomorphism is a problem of testing whether 2 graphs are identical • Central problem in algorithmics for 50+ years • Applied in social networks, image processing, and computational biology nodes in subgraph S: nodes in subgraph S: " User User 4 4 User IGIG User 5 Data and Results User User 22 Synthetic: • Random Power Law Graph of 60 nodes as networks • Planted Isomorphism (PI) of varying size • Evaluation metrics : !! value & Relative Frobenius Difference • X-axis: One standard deviation of noise = x-value G1 - Power Law, Size: 60, W∈[0,100] | SI - Random, W[20.0,80.0] | ε = 3.0 8/5/2019 8/5/2019 G1 - Power|Law, Size:2 60, W∈[0,100] | SI - Random, ε = 3.0 • Network: 2 ∈ [0, 100] PI: ∈ [20, 80] W[20.0,80.0] | & = | 3.0 User IGIG User User FBFB User User IGIG User 8/5/2019 User IGIG User 130 280 130 280 G1 - Power Law, Size: Algorithm: 60, W∈[0,100] |DS SI - Random, W[20.0,80.0] | ε = 3.0 The colored nodes and green edges represent an isomorphic mapping. !! value for TDS !! value for DS !! F1 value value 0.6 8/5/2019 0.7 0.5 0.3 8/4/2019 G1 | SI - Random, W[20.0,80.0] | ε = 3.0 0.2- Power Law, Size: 60, W∈[0,100] G1 - Power Law, Size: 60, W∈[0,100] | SI - Random, W[20.0,80.0] | ε = 3.0 G1 - Power Law, Size: 60, W∈[0,100] | SI - Random, W[20.0,80.0] | ε = 3.0 0.3 0 1 2 3 4 5 0 6 1 2 3 4 5 6 G1 - Power Law, Size:Range 60, W∈[0,100] | SI within - Random, Weight of Noise σ W[20.0,80.0] | ε = 3.0 0.5 Weight of Noise σ W[20.0,80.0] G1 - Power Law, Size: Range 60, W∈[0,100] | SI -within Random, | ε = 3.0 Range of Noise within 1σ G1 - Power Law, Size: 60, W∈[0,100] | SIWeight - Random, W[20.0,80.0] | ε = 3.0 G1 - Power Law, Size: 60, W∈[0,100] | SIWeight - Random, W[20.0,80.0] | ε = 3.0 Range of Noise within 1σ Subgraph Isomorphism Algorithm 1 0.8 F1 value '8 =0.7 .8 , 08 0.6 1 2 3 4 Weight Range of Noise within 1σ 0.4 0.2 0 24 nodes1 0.7 1 2 3 4 5 Weight Range of Noise within σ 3 4 5 Favorite-Reply 5 0.6 0.4 0.2 1 2 3 4 5 Result: The exact nodes and edges in the green subgraph represent the mapping of nodes and edges between "! and "" that yields an approximate subgraph isomorphism. 6 WeightWeight Range of Noise within σ Range of Noise within 1σ file:///Users/vsachin/OneDrive - Bellarmine College Preparatory/Jay/Boston RISE Research/Poster/Plots/TDS/F1.html 1/1 0.4 Relative Frobenius Difference (DS) 1.05235 (0.1429) Relative Frobenius Difference (TDS) 0.968025 (0.1787) 0.3 6 1.14843 (0.10910) 1.066875 (0.0903) 0 1 2 3 4 Weight Range of Noise within 1σ 5 6 1.103275 (0.043) Conclusion Data Analysis • Synthetic: TDS had better !! and Frobenius norm values - better near-cliques file:///Users/vsachin/OneDrive - Bellarmine College Preparatory/Jay/Boston RISE Research/Poster/Plots/DS/F1.html 1/1 Preparatory/Jay/Boston RISE Research/Poster/Plots/DS/F1.html file:///Users/vsachin/OneDrive - Bellarmine College lead to better isomorphisms • Twitter: TDS also achieves lower relative Frobenius norm values than DS – high triangle count in ego networks '6 = (.6 , 06 ) 24 nodes 6 Weight Range of Noise within 1σ 1.21998 (0.0436) Weight Range of Noise within 1σ 4 0.8 1/1 Twitter Data Sets (First 7 days) 2 3 0 file:///Users/vsachin/OneDrive - Bellarmine College Preparatory/Jay/Boston RISE Research/Poster/Plots/DS/F1.html Favorite-Quote 2 0 0.5 Weight Range of Noise within 1σ 18 nodes 1 6 Favorite-Mention Step 3: Apply a variety of Dense Subgraph Discovery algorithms to find near-cliques in the product graph. 18 nodes 0.6 0 1 12 nodes 0.8 0.4 0 24 nodes 1224nodes nodes 0.8 0.6 PI Size 18 nodes 1.2 0.9 0 Step 2: Using the two input graphs, generate an unweighted product graph whose vertices are represented 0.5 by the cartesian product of $! and $" and whose edges are based on the similarity of edge weights in "! and "" . PI Size 18 nodes 0.3 6 5 12 nodes 12 nodes 1 1.2 Relative Frobenius Difference Relative Frobenius value Step 1: Generate or identify two networks with the 0.9 same set of nodes. 0.4 1.4 1 0 Relative Frobenius Difference for TDS Relative Frobenius Difference for DS 0.4 '6 = (.6 , 06 ) 0.6 0.4 0.6 G1 - Power Law, Size: 60, W∈[0,100] | SI - Random, W[20.0,80.0] 8/5/2019 | ε = 3.0 24 nodes 0.7 0.8 0.4 18 nodes 0.8 0.9 0.5 if and only Edges in %! and %" are said to be similar8/5/2019 0.6 if their edge weights are within some error threshold & of each other. 0.3 0.9 18 nodes 0.7 0.7 0.3 12 24 nodes nodes 24 nodes 0.8 12 nodes PI Size 18 nodes 1 12 nodes 24 nodes 1 0.9 0.8 0.5 12 nodes PI Size 18 nodes 1 0.9 F1 value G1 - Power Law, Size: S:60,Algorithm: W∈[0,100] | SI - TDS Random, W[20.0,80.0] | ε = 3.0 G1 - Power Law, Size: 60, W∈[0,100] | SI - Random, W[20.0,80.0] | ε = 3.0 G1 - Power Law, Size: 60, W∈[0,100] | SI - Random, W[20.0,80.0] | ε = 3.0 Subgraph Isomorphism • Need to find the maximum subset of vertices in1 the networks that creates a 1:1 isomorphic mapping • Known to be an NP-Hard problem Formal Problem Twitter: • Data set of user interactions on Twitter from February 2018 • Largest 4 ego-networks of size ∈ [0,100] • Average & standard deviation of results evaluated • Undirected graphs with edge weights equal to # of times each G1 - Power Law, Size: 60, W∈[0,100] | SI - Random, W[20.0,80.0] | ε = 3.0 interaction occurs 8/4/2019 G1 - Power Law, Size: S:60, W∈[0,100] | SI - Random, W[20.0,80.0] | ε = 3.0 !! F1 value value User FBFB User User IGIG User F1 value User FBFB User F1 value User FBFB User |4 3 | |3| • Employs a greedy-peeling approach, removing vertex of lowest triangle edge degree at each iteration • ( )+ runtime ! − approximagon guarantee • '*+,-$./$0 User FBFB User '7 = .7 , 07 |2 3 | |3| Relative Frobenius Relative Frobenius Difference value '#$%&'(() TDS Example Tsourakakis’s[2] Triangle Densest Subgraph Problem (TDS) • Maximize the ratio of triangles to • Employs a greedy-peeling approach, removing vertex of lowest edge degree at each iteration • ( ) + + runtime User User 11 ! - approximation guarantee • Example: Depicted below are two graphs representing User how the same set of users are connected on both User 3 3 Facebook and Instagram 130, 155 130, 155 DS Example Charikar’s[1] Densest Subgraph Problem (DS) • Maximize the ratio of edges to 1/1 file:///Users/vsachin/OneDrive - Bellarmine College Preparatory/Jay/Boston RISE Research/Poster/Plots/TDS/Frobenius.html Shortcomings • Specific mapping between "! and "" isomorphisms often incorrect • Sparse implanted isomorphisms and isomorphisms within dense background networks hard to extract file:///Users/vsachin/OneDrive - Bellarmine College Preparatory/Jay/Boston RISE Research/Poster/Plots/DS/Frobenius.html Next steps • Test algorithms with subgraph isomorphisms and networks of different graph types • Generate a weighted product graph based on better edge weight similarity • Employ other DSD algorithms efficient at extracting near-cliques 1/1 1/1