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Computational Frameworks for Higher-order Network Data Analysis

1. The document discusses computational frameworks for analyzing higher-order network data, where interactions can involve more than two nodes. Real-world systems often involve higher-order interactions that are reduced to pairwise connections. 2. The author presents several datasets involving higher-order interactions and shows that predicting the formation of new higher-order connections is similar to link prediction but considers groups of nodes rather than individual links. Structural properties like edge density and tie strength influence the likelihood of simplicial closure. 3. Models are proposed to score open simplices based on structural features and predict which will transition to closed simplices. Accounting for higher-order structure provides new insights beyond traditional network analysis of pairwise connections.

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1 Computational Frameworks for Higher-order Network Data Analysis Austin R. Benson · Cornell University Texas A&M Institute of Data Science · October 23, 2020 Slides. bit.ly/arb-tamu-20
Graph or network data modeling important complex systems are everywhere. 2 Commerce nodes are products edges link co-purchased products Communications nodes are people/accounts edges show info. exchange Physical proximity nodes are people edges link those that interact in close proximity Drug compounds nodes are substances edge between substances that appear in the same drug
Network data analysis studies the model to gain insight and make predictions about these systems. 3 1. Evolution / changes What new connections will form? (email auto-fill suggestions, rec. systems) 2. Clustering / partitioning / community detection How to find groups of related nodes? (similar products, protein functions) 3. Spreading and traversing How does stuff move over the network? (viruses or misinformation) 4. Ranking Which things are important? (PageRank and its variants)
Real-world systems are composed of“higher-order” interactions that we often reduce to pairwise ones. 4 Commerce nodes are products Several products purchased at once Communications nodes are people/accounts emails often have several recipients,not just one. Physical proximity nodes are people people gather in groups Drug compounds nodes are substances Drugs are composed of several substances
5 What new insights does this give us?
We can ask the same network analysis questions while taking into account the higher-order structure. 6 1. Evolution / changes What new connections will form? (email auto-fill suggestions, rec. systems) 2. Clustering / partitioning / community detection How to find groups of related nodes? (similar products, protein functions) 3. Spreading and traversing How does stuff move over the network? (viruses or misinformation) 4. Ranking Which things are important? (PageRank and its variants)
Higher-order Network Data Analysis 7 w/ R. Abebe, M. Schaub, J. Kleinberg, A. Jadbabaie 1. Temporal evolution of higher-order interactions. Simplicial Closure and Higher-order Link Prediction,PNAS 2018. 2. Clustering in large networks of higher-order interactions. Minimizing Localized Ratio Cuts in Hypergraphs,KDD,2020. 3. Diffusions over higher-order interactions in networks. Random walks on simplicial complexes and the normalized Hodge 1-Laplacian,SIAM Review,2020.
We collected many datasets of timestamped simplices, where each simplex is a subset of nodes. 8 1. Coauthorship in different domains. 2. Emails with multiple recipients. 3. Tags on Q&A forums. 4. Threads on Q&A forums. 5. Contact/proximity measurements. 6. Musical artist collaboration. 7. Substance makeup and classification codes applied to drugs the FDA examines. 8. U.S. Congress committee memberships and bill sponsorship. 9. Combinations of drugs seen in patients in ER visits. https://math.stackexchange.com/q/80181 bit.ly/sc-holp-data
Thinking of higher-order data as a weighted projected graph with filled-in structures is a convenient viewpoint. 9 1 2 3 4 5 6 7 8 9 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 t1 : {1, 2, 3, 4} t2 : {1, 3, 5} t3 : {1, 6} t4 : {2, 6} t5 : {1, 7, 8} t6 : {3, 9} t7 : {5, 8} t8 : {1, 2, 6} Data. Projected graph W. Wij = # of simplices containing nodes i and j.
10 5 83 16 20
11 i j k i j k Warm-up. What’s more common in data? or “Open triangle” each pair has been in a simplex together but all 3 nodes have never been in the same simplex “Closed triangle” there is some simplex that contains all 3 nodes
music-rap-genius NDC-substances NDC-classes DAWN coauth-DBLP coauth-MAG-geology coauth-MAG-history congress-bills congress-committees tags-stack-overflow tags-math-sx tags-ask-ubuntu email-Eu email-Enron threads-stack-overflow threads-math-sx threads-ask-ubuntu contact-high-school contact-primary-school 10 5 10 4 10 3 10 2 10 1 Edge density in projected graph 0.00 0.25 0.50 0.75 1.00 Fractionoftrianglesopen There is lots of variation in the fraction of triangles that are open,but datasets from the same domain are similar. 12See also Patania-Petri-Vaccarino (2017) for similar ideas in collaboration networks.
Dataset domain separation also occurs at the local level. 13 • Randomly sample 100 egonets per dataset and measure log of average degree and fraction of open triangles. • Logistic regression model to predict domain (coauthorship, tags, threads, email, contact). • 75% model accuracy vs. 21% with random guessing.
14 How do new simplices form? Can we predict which simplices will form?
Groups of nodes go through trajectories until finally reaching a“simplicial closure.” 15 t1 : {1, 2, 3, 4} t2 : {1, 3, 5} t3 : {1, 6} t4 : {2, 6} t5 : {1, 7, 8} t6 : {3, 9} t7 : {5, 8} t8 : {1, 2, 6} For this talk, we will focus on simplicial closure on 3 nodes.
Groups of nodes go through trajectories until finally reaching a“simplicial closure event.” 16 Substances in marketed drugs recorded in the National Drug Code directory. We bin weighted edges into “weak” and “strong ties” in the projected graph W. Wij = # of simplices containing nodes i and j. • Weak ties. Wij = 1 (one simplex contains i and j) • Strong ties. Wij > 2 (at least two simplices contain i and j)
Simplicial closure depends on structure in projected graph. 17 • First 80% of the data (in time) ⟶ record configurations of triplets not in closed triangle. • Remainder of data ⟶ find fraction that are now closed triangles. Increased edge density increases closure probability. Increased tie strength increases closure probability. Tension between edge density and tie strength. Left and middle observations are consistent with theory and empirical studies of social networks. [Granovetter 73; Kossinets-Watts 06; Backstrom+ 06; Leskovec+ 08] Closure probability Closure probability Closure probability
Simplicial closure on 4 nodes is similar to on 3 nodes, just“up one dimension.” 18 Increased edge density increases closure probability. Increased simplicial tie strength increases closure probability. Tension b/w edge density simplicial tie strength. Closure probability Closure probability Closure probability
We proposed“higher-order link prediction”as a framework to evaluate models for closure. 19 t1 : {1, 2, 3, 4} t2 : {1, 3, 5} t3 : {1, 6} t4 : {2, 6} t5 : {1, 7, 8} t6 : {3, 9} t7 : {5, 8} t8 : {1, 2, 6} Data. • Observe simplices up to time t. • Predict which groups of > 2 nodes will appear after time t. t We predict structure that graph models would not even consider!
20 Our structural analysis tells us what we should be looking at for prediction. 1. Edge density matters! ⟶ focus our attention on predicting which open triangles become closed triangles (intelligently reduce search space.) 2. Tie strength matters! ⟶ various ways of incorporating this information i j k Wij Wjk Wjk
21 For every open triangle,we assign a score function on first 80% of data based on structural properties. Score s(i, j, k)… 1. is a function of Wij, Wjk, Wjk arithmetic mean, harmonic mean, etc. 2. looks at common neighbors of the three nodes. generalized Jaccard, Adamic-Adar, etc. 3. uses “whole-network” similarity scores on projected graph sum of PageRank or Katz scores amongst edges 4. is learned from data logistic regression model with features i j k Wij Wjk Wjk After computing scores, predict that open triangles with highest scores will be closed triangles in final 20% of data. i j k l m x y r z N(i) = {j, k, l, m, x, y, z} N(j) = {i, k, l, m, r} N(k) = {i, j, l, m}<latexit 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scorep(i, j, k) = (Wp ij + Wp jk + Wp ik)1/p <latexit 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22
23 A few lessons learned from applying these ideas. 1. We can predict pretty well on all datasets using some simple method. → 4x to 107x better than random w/r/t mean average precision depending on the dataset/method (only predicting on open triangles) 2. Thread co-participation and co-tagging on stack exchange are consistently easy to predict with the harmonic mean. 3. Simple averaging Wij, Wjk, and Wik consistently performs well. i j k Wij Wjk Wjk
Generalized means of edges weights are often good predictors of new 3-node simplices appearing. 24 music-rap-genius NDC-substances NDC-classes DAWN coauth-DBLP coauth-MAG-geology coauth-MAG-history congress-bills congress-committees tags-stack-overflow tags-math-sx tags-ask-ubuntu email-Eu email-Enron threads-stack-overflow threads-math-sx threads-ask-ubuntu contact-high-school contact-primary-school harmonic geometric arithmetic p 4 3 2 1 0 1 2 3 4 0 20 40 60 80 Relativeperformance 4 3 2 1 0 1 2 3 4 p 2.5 5.0 7.5 10.0 12.5 Relativeperformance 4 3 2 1 0 1 2 3 4 p 1.0 1.5 2.0 2.5 3.0 3.5 Relativeperformance Good performance from this local information is a deviation from classical link prediction, where methods that use long paths (e.g., PageRank) perform well [Liben-Nowell & Kleinberg 07]. For structures on k nodes, the subsets of size k-1 contain rich information only when k > 2. i j k Wij Wjk Wjk i j k ? scorep(i, j, k) = (Wp ij + Wp jk + Wp ik)1/p <latexit 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If we only need the top-k weighted triangles, we have fast algorithms for finding them. 25 scorep(i, j, k) = (Wp ij + Wp jk + Wp ik)1/p <latexit 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Simple (incorrect) algorithm. 1. Throw out edges with weight < t. 2. Find triangles in remainder. i j k Wij Wjk Wjk Better (correct) algorithm. 1. Dynamically choose threshold. 2. Careful pruning. w/ R. Kumar, P. Liu, M. Charikar
We often only need the top-k weighted triangles,and we have fast algorithms for finding them. 26 <latexit sha1_base64="mGA+XhzsM3MoFa1t7a1hl4NURg0=">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</latexit> dataset # nodes # edges Fast enumeration Fast top-k (k = 1000) (running time in seconds) Spotify co-listens 3.6M 1.93B too long 30 MAG co-authorship 173M 544M 596 16 AMINER co-authorship 93M 324M 255 10 Ethereum transactions 38M 103M 91 33
Higher-order data is pervasive! 27 • Simplicial Closure and Higher-order Link Prediction.Austin R.Benson,Rediet Abebe,Michael T.Schaub,Ali Jadbabaie,and Jon Kleinberg. Proc.Natl.Acad.Sci.U.S.A.,2018. github.com/arbenson/ScHoLP-Tutorial • Retrieving Top Weighted Triangles in Graphs. Raunak Kumar,Paul Liu,Moses Charikar,and Austin R.Benson. Proc.Of WSDM,2020. github.com/raunakkmr/Retrieving-top-weighted-triangles-in-graphs 1. There are commonalities in temporal evolution. Generative models? 2. There is lots of signal in subsets! Unique to higher-order… 3. Please develop neural embeddings to out-perform our baselines. 😁 1 2 3 4 5 6 7 8 9 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1
28 w/ Nate Veldt, J. KleinbergHigher-order Network Data Analysis 1. Temporal evolution of higher-order interactions. Simplicial Closure and Higher-order Link Prediction,PNAS 2018. 2. Clustering in large networks of higher-order interactions. Minimizing Localized Ratio Cuts in Hypergraphs,KDD,2020. 3. Diffusions over higher-order interactions in networks. Random walks on simplicial complexes and the normalized Hodge 1-Laplacian,SIAM Review,2020.
Graph minimum s-t cuts are fundamental. 29 minimizeS⇢V cut(S) subject to s 2 S, t /2 S.<latexit 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1 3 2 4 5 6 7 8 s t • Maximum flow / min s-t cut [Ford, Fulkerson, Dantzig 1950s] • Densest subgraph [Goldberg 84; Shang+ 18] • Graph-based semi-supervised learning algorithms [Blum-Chawla 01] • Local graph clustering [Andersen-Lang 08; Oreccchia-Zhu 14; Veldt+ 16] poly-time algorithms!
Real-world systems are composed of“higher-order” interactions that we can model with hypergraphs. 30 H = (V, E), edge e 2 E is a subset of V (e ⇢ V)<latexit 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1 2 3 4 5 V = {1, 2, 3, 4, 5} E = {{1, 2, 3}, {2, 4, 5}}<latexit 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What is a hypergraph minimum s-t cut? 31 s t Should we treat the 2/2 split differently from the 1/3 split? Historically, no. [Lawler 73, Ihler+ 93] More recently, yes. [Li-Milenkovic 17, Veldt-Benson-Kleinberg 20] edge in a graph size-3 hyperedges “Only one way to split a triangle” [Benson+ 16; Li-Milenkovic 17; Yin+ 17] Must be split 1/1.
We model hypergraph cuts with splitting functions. 32 s t Given a cut defined by S, we incur penalty of at each hyperedge e. Hypergraph minimum s-t cut problem. Cardinality-based splitting functions. S<latexit 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cutH(S) = f (2) + f (1)<latexit 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<latexit 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we(e S) <latexit 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minimizeS⇢V P e2E we(e S) ⌘ cutH(S) subject to s 2 S, t /2 S. <latexit sha1_base64="vCSQ5hxLftoc4zdzUNdXcsthqGM=">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</latexit> Non-negativity we(A) 0. Non-split ignoring we(e) = we(;) = 0. C-B we(A) = f (min(|A|, |Ae|)).
Cardinality-based splitting functions appear throughout the literature. 33 [Lawler 73; Ihler+ 93; Yin+ 17] [Hu-Moerder 85; Heuer+ 18] [Agarwal+ 06; Zhou+ 06; Benson+ 16] [Yaros- Imielinski 13] [Li-Milenkovic 18] <latexit sha1_base64="gX/87S67KKdqKR6T9Qjl8vcDiHo=">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</latexit> All-or-nothing we(A) = ( 0 if A 2 {e, ;} 1 otherwise Linear penalty we(A) = min{|A|, |eA|} Quadratic penalty we(A) = |A| · |eA| Discount cut we(A) = min{|A|↵ , |eA|↵ } L-M submodular we(A) = 1 2 + 1 2 · min n 1, |A| b↵|e|c , |eA| b↵|e|c o
We solve hypergraph cut problems with graph reductions. 34 1/21/2 1/2 1 1 1 1 ∞ ∞ ∞ ∞ ∞∞ Gadgets (expansions) model a hyperedge with a small graph. clique expansion star expansion Lawler gadget [1973]hyperedge In a graph reduction, we first replace all hyperedges with graph gadgets... s t s t s t s t … then solve the (min s-t cut) problem exactly on the graph, and finally convert the solution to a hypergraph solution. Quadratic penalty f(i) = i ( |e| – i ) Linear penalty f(i) = i All-or-nothing f(0) = 0,o/w f(i) = 1
b We made a new gadget for C-B splitting functions. 35 This gadget models min(|A|, |eA|, b). Theorem [Veldt-Benson-Kleinberg 20a]. Nonnegative linear combinations of the C-B gadget can model any submodular cardinality-based splitting function. See also Graph Cuts for Minimizing Robust Higher Order Potentials,Kohli et al.,2008. <latexit sha1_base64="beQz4cdyY+p8N+9L01TDcNAiwcQ=">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</latexit> C-B we(A) = f (min(|A|, |eA|)). (F is submodular on X if F(A B) + F(A [ B)  F(A) + F(B) for any A, B ✓ X.)<latexit 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36 Theorem [Veldt-Benson-Kleinberg 20a]. The hypergraph min s-t cut problem with a cardinality-based splitting function is graph-reducible (via gadgets) if and only if the splitting function is submodular. Cardinality-based splitting functions. s t S<latexit 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cutH(S) = f (2) + f (1)<latexit 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Submodularity is key to efficient algorithms. What happens when the splitting function isn’t submodular? Can we use some other algorithm? <latexit sha1_base64="vCSQ5hxLftoc4zdzUNdXcsthqGM=">AAAIS3icfVXdbts2FLa7Ncm8v2a73A27wEMSyI6dIksyIEC6FsUKtFs2J20B08go6UgiTFIqScX2CD3FbrdH2gPsOXY37GKHtrPYTlYBtijyfN/55wkLwY3tdP6s33vv/ftr6xsfND786ONPPn2w+dkrk5c6gosoF7l+EzIDgiu4sNwKeFNoYDIU8DocPvHnr69AG56rczspYCBZqnjCI2Zx63KzvkZDSLlyTPBU7VYNamFs3fe5ailIUeiK20lFvqKZKVgErtPeP4hkRejV6BK2H+8QmsJb0mkTShegBk23BAlzzVWKcLKKnxPADjmZLynIwk4MWL+1SPik9e27DDghyTaVXG1TgY5a8phQ7RcBudkIWTQ0gpmMwPx0Z6fdoKDia78vH2x12p3pQ24vuvPFVm3+nF1urgU0zqNSgrIRUpt+t1PYgWPa8kgABrI0gBYPWQp9XComwQzcNGMVaeJOTJJc409ZMt1tLEKQR7PJEouzLCwF0+Pl3TDPh3hiqiV83xeEUaUMQUMc6FJAjMaJFDNiM7kPK+KlTY4GjquitKCimYFJKYjNiS8bEnMNkRUTsmyl5cNfAsUjSDSLAiaNZDYLCu69CiQbQgRCzOz1ooKHmumJdy4fmcCnJdV5qWITFMxa0Mogymo+DkzGCjBBwm0QMRH579hjCpFbyfTQ/B9rW4JleDiNqQDrzsvEwk8QVw4j8fCo8zAUqHdRwmaQagBVuenLy4wybmFFJhQlVM7/L0g0miSztjDf7O1htbaNRW4YRxlTKbSjXO69LcH4XjN73a8PjveP9wxIjiUYYnPJ1giz0fJOtLhqhdi4oKdyjw63Zq8G9WFk2Ng+Pg2aijxkguIn9bBTUKbUcBrnAkvjFNs6ymM4oRoEG19jczR+ubz6592B80nyyV7K6Nl5jykfXA0KRuiAZNglNGGSi0kMCSuFrRw1yfV6uSBM4iugajQXlRnMIMQnnfZxEGGnWow2E9gMqMCOTeIplp1Ebqrs2FOdzsDO7PaxCw8G1apTTwHbT0NvIsNcPEOX3IzFVO6Hly8qp7wKySsnK8fRXNoDe5cwbsSrkHAOmevwgF4ZYjpt6VN6t4JVDb1nL31IrhWcd5fC58Jx5Yy4UeKFZ2j3HCV9DJgoMlbdmPrz85Wox6kAHmWtWezvOsFEG7x4lm8O6WkWsyx7PJWoic6qytM5GkpHZ/vVrbKQL3DUxHch5gfVsopdOg6Z7mPx0SzMx45e+f9mg2b+hiIZ8DSzeO8eHhSWNMl5BoRFtmSCIKxBh3hD+AEA4ya5fprkKY5JpiIgIdgR9q+XJaiMmGkYGzNVzQYOHU/Q6rS7IJvX6F6Wa4wOzimSK4JFRQQklhgeg0cs+LXVrf4jwdHw6J0keurJlAWD4OdLd3Wa3F682m93D9qdH/e3To/mk2aj9kXty9p2rVs7rJ3Wvqud1S5qUV3Wf63/Vv99/Y/1v9b/Xv9nJnqvPsd8Xlt6Nu7/C+Rx6YI=</latexit> Non-negativity we(A) 0. Non-split ignoring we(e) = we(;) = 0. C-B we(A) = f (min(|A|, |Ae|)).
37 Unlike graph min s-t cut, hypergraph min s-t cut can be NP-hard. w1 = 1 0 1 2 w2 ?? Reducible/Submodular NP-hard Unknown Hard Reducible w3 3 2.5 2 1.5 1 0.5 1 1.5 2 2.5 w2 0.5 w2 w3 w4 4 3 2 1 0 1 1.5 2 2.5 1 2 3 max hyperedge size 4 or 5 max hyperedge size 6 or 7 max hyperedge size 8 or 9 Theorem [Veldt-Benson-Kleinberg 20a]. For C-B splitting functions, Open Question: For 4-uniform hypergraphs, is there an efficient algorithm to find the minimum s-t cut with no 2-2 splits (w1 = 1, w2 = ∞). s t cutH(S) = f (2) + f (1) = w2 + 1<latexit 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38 How can we use this framework to enable new data science algorithms?
G = (V,E) is a graph. R ⊆ V (Reference or seed set). Finds a “good” cluster S “near” R. 39 Background.Local clustering has been studied extensively in graphs,but not much in hypergraphs.
Rewards high overlap with R. Penalizes nodes outside R. R(S) = cut(S) vol(S R) "vol(S ¯R) Max Flow.Quot.Imp. (Lang,Rao,2004) Flow-Improve (Andersen,Lang 2008) Local-Improve (Orecchia,Allen-Zhou 2014) SimpleLocal (Veldt,Gleich,Mahoney 2016) FlowSeed (Veldt,Klymko,Gleich 2019) Great survey paper! (Fountoulakis et al.2020) 40 Background.Flow-based methods minimize a localized variant of conductance. Rewards contained clusters vol(T) = sum of degrees in T. minimize node sets S FAST ALGORITHMS FOR EXACT MINIMIZATION!
s t 2 4 4 7 3 4 7 3 2 1 3 2 6 4 5 7 8 9 10Set R 4 1 3 2 6 4 5 7 8 9 10 s t 2 4 4 7 3 4 7 3 2 1 3 2 6 4 5 7 8 9 10Set R 4 [Andersen-Lang 08,Orecchia-Zhou 14,Veldt+ 16] Construct G’ R(S) < ↵ () min s-t cut of G0 < ↵vol(R) Compute min s-t cut of G’. 41 Connect R to a source node s ; edges weighted with respect to 𝛼. Connect VR to a sink node t ; edges weighted with respect to β = 𝛼ε. Is R(S) < ↵ for any S? Background.Flow methods repeatedly solve min-cut problems on an auxiliary graph.
We generalized local flow-based techniques to the hypergraph setting 42 • We introduce localized hypergraph conductance • We can minimize it exactly with our hypergraph min s-t cuts framework • Strongly-local runtime! (Only depends on size of seed set) • Normalized cut improvement guarantees The analysis provides even new guarantees for the graph case!
43 We define hypergraph s-t cut problems similar to the ones used in the graph case. t 𝜀𝛼dj r j s 𝛼dr R minimize S⇢V HLCR,"(S) = cutH(S) volH(S R) "volH(S ¯R) cutH(S) = cut from C-B splitting function volH(X) = X i2X X e2E 1 = sum of hypergraph deg <latexit 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Hypergraph cut function Encourage overlap with reference set. Discourage overlap outside reference set ⟶ di = # hyperedges node r is in volH(S) = X i2S di <latexit sha1_base64="E541uqeWvYyixS/NzXbACyuddlM=">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</latexit> Theorem [Veldt-Benson-Kleinberg 20b]. We can repeatedly solve min hypergraph s-t cut problems with different 𝛼 to exactly minimize the hypergraph localized conductance (HLC) exactly.
We carefully apply graph reduction techniques to growing subsets of the hypergraph. 44 s R t s t 𝛼 Theorem [Veldt-Benson-Kleinberg 20b]. Strong locality. Can make this algorithm run in time proportional to the size of seed set (does not look at the full hypergraph).
<latexit sha1_base64="ff4QhRAHZfL2qp7+qW21R4vDGq8=">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</latexit> If a target set T ⇢ V satisfies vol(T R) vol(T) vol(¯T R) vol(¯T) + Then (S)  1 (T) where S is the set returned by our algorithm. 45 We prove new normalized cut guarantees that are new even for the graph case. (S) = cut(S) vol(S) + cut(S) vol(S)<latexit sha1_base64="+2zPfSQ7UOPKTHfrSfqqRxABg7w=">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</latexit> Normalized cut is another ratio-cut objective related to conductance. If T overlaps enough with seed set R... ...then our output has normalized cut almost as good as T. Theorem [Veldt-Benson-Kleinberg 20b]. Normalized cut improvement.
schem e ocam l tcl m dx com m on- lisp verilog lotus- notes xslt- 1.0 plone typo3 abap sitecore m arklogic wolfram - m athem atica alfresco axapta vhdl sparql prolog netsuite racket spring- integration xslt- 2.0 m ule wso2 system - verilog wso2esb google- sheets- form ula stata xpages netlogo openerp data.table google- bigquery docusignapi aem codenam eone dax cypher julia sapui5 ibm - m obilefirst office- js jq apache- nifi 0.2 0.4 0.6 0.8 F1Scores HyperLocal TN/BN FlowSeed 46 • 15M StackOverflow questions (nodes), answered by 1.1M users (hyperedges). • mean hyperedge size 23.7, max hyperedge size ~ 60k. • Tags provide ground truth cluster labels. • Delta-linear splitting function wi = min(i, 5000). HyperLocal Clique-Expansion + FlowSeed Neighborhood Baselines
47 Cluster |T| time (s) HyperLocal Baseline1 Baseline2 Amazon Fashion 31 3.5 0.83 0.77 0.6 All Beauty 85 30.8 0.69 0.60 0.28 Appliances 48 9.8 0.82 0.73 0.56 Gift Cards 148 6.5 0.86 0.75 0.71 Magazine Subscriptions 157 14.5 0.87 0.72 0.56 Luxury Beauty 1581 261 0.33 0.31 0.17 Software 802 341 0.74 0.52 0.24 Industrial & Scientific 5334 503 0.55 0.49 0.15 Prime Pantry 4970 406 0.96 0.73 0.36 <latexit 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• 2.3M Amazon products (nodes), reviewed by 4.3M users (hyperedges). • mean hyperedge size > 17, max hyperedge size ~9.3k. • Product categories provide ground truth cluster labels. • All-or-nothing penalty (wi = 1). F1 recovery scores given a handful of nodes from the ground truth cluster T.
48 Gadget reductions sometimes create dense graphs, which can make computations expensive. Theorem [Veldt-Benson-Kleinberg 20c].Any submodular C-B splitting function can be 𝜀-approx with log r / 𝜀 splitting functions (instead of r, r = hyperedge size). And one specific case… • r = 60k clique expansion only need O(r / √𝜀) instead of O(r2) 0 1 2 3 0 2 4 6 8 10 e0 1 e00 1 e0 2 e00 2 e0 3 e00 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 4 3 0 1 2 3 0 2 4 6 8 10 e0 e00 5 5 5 5 5 5 5 5 5 5 5 5 9
We can now model and use hypergraph min s-t cuts. 49 • Hypergraph Cuts with General Splitting Functions.Nate Veldt,Austin R.Benson,and Jon Kleinberg. arXiv:2001.02817,2020. • Localized Flow-Based Clustering in Hypergraphs.Nate Veldt,Austin R.Benson,and Jon Kleinberg. Proc.Of KDD,2020. github.com/nveldt/HypergraphFlowClustering • Augmented Sparsifiers for Generalized Hypergraph Cuts.Nate Veldt,Austin R.Benson,and Jon Kleinberg. arXiv:2007.08075,2020. 1. A model for hypergraph cuts. C-B splitting functions that depend on # of nodes on small side of the cut 2. Algorithm for min s-t cuts with submodular C-B splitting functions. Graph-reducible if and only if C-B splitting function is submodular 3. Applications to local hypergraph clustering. Strong locality lets us scale to large hypergraphs with large hyperedges s t s t s t w2 = 0.5 (NP-hard) w2 = 1.5 (poly-time via graph reduction) w2 = 2.5 (?)
50 w/ M.Schaub,A.Jadbabaie, G.Lippner,and P.HornHigher-order Network Data Analysis 1. Temporal evolution of higher-order interactions. Simplicial Closure and Higher-order Link Prediction,PNAS 2018. 2. Clustering in large networks of higher-order interactions. Minimizing Localized Ratio Cuts in Hypergraphs,KDD,2020. 3. Diffusions over higher-order interactions in networks. Random walks on simplicial complexes and the normalized Hodge 1-Laplacian,SIAM Review,2020.
Background.Graph Laplacians,diffusions,and spectral graph theory underly many graph data methods. 51 Low-dimensional embeddings [Belkin-Niyogi 02; Coifman-Lafon 06] Personalized PageRank [Andersen-Chung-Lang 08; Gleich 15] <latexit sha1_base64="QQhZuRTVdivo8Dt7YLhgdqyYsKI=">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</latexit> N ⇥ x1 x2 ⇤ = ⇥ 1x1 2x2 ⇤ Norm. Lap. N = D-1/2LD-1/2. Random walk Lap. LD-1. D = diagonal degree matrix, A = adjacency matrix, L = D – A is graph Laplacian.. <latexit sha1_base64="POsukIjAahItuceyxYGW6/9e3Cc=">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</latexit> ( I + LD 1 )x = v
52 What is a “higher-order” Laplacian?
Background. By interpreting our data as a simplicial complex,we can get higher-order Laplacians. 53 • (Abstract) simplicial complex X: if A ∈ X and B ⊆ A, then B ∈ X. • Graph G = (V, E) as a simplicial complex: X = V ⋃ E. • Can induce a complex from higher-order interactions. <latexit sha1_base64="LJ5+N3kgPFW8cDVOT8x5Tlih5vo=">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</latexit> L0 = 0T 0 + B1BT 1 = D A graph Laplacian operates on nodes L1 = BT 1 B1 + B2BT 2 Hodge Laplacian operates on oriented edges B1 maps edges to nodes, B2 maps triangles to edges. See Hodge Laplacians on Graphs by Lek-Heng Lim.
We spent a lot of time getting the normalization and connections to random walks right. 54 Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian. Michael T.Schaub,Austin R.Benson,Paul Horn,Gabor Lippner,and Ali Jadbabaie. SIAM Review,2020.
Flow embeddings are the higher-order analog of diffusion maps. 55 <latexit sha1_base64="t3a3BIllEKBJoLIHn+m/2Gk55Ew=">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</latexit> N1 ⇥ x1 x2 ⇤ = ⇥ 1x1 2x2 ⇤ for normalized Hodge 1-Laplacian N1.
Flow embeddings are the higher-order analog of diffusion maps. 56 <latexit sha1_base64="t3a3BIllEKBJoLIHn+m/2Gk55Ew=">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</latexit> N1 ⇥ x1 x2 ⇤ = ⇥ 1x1 2x2 ⇤ for normalized Hodge 1-Laplacian N1. A B
Flow embeddings are the higher-order analog of diffusion maps. 57 <latexit sha1_base64="t3a3BIllEKBJoLIHn+m/2Gk55Ew=">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</latexit> N1 ⇥ x1 x2 ⇤ = ⇥ 1x1 2x2 ⇤ for normalized Hodge 1-Laplacian N1. A B C
The holes correspond to the idea of homology in algebraic topology. 58 A B First eigenvectors of the graph Laplacian capture (near) connected components, or zeroth-order homology. First eigenvectors of the Hodge Laplacian capture (near) topological holes, or first-order homology A good reference is Hodge Laplacians on Graphs by Lek-Heng Lim.
We also have simplicial Personalized PageRank. 59 <latexit sha1_base64="G87jKzztl/ZL8VW+UOYg28uFc2o=">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</latexit> ( I + L1)x = v for (asymmetric) normalized Hodge 1-Laplacian L1.
Abstract simplicial complexes are another way to model and analyze higher-order network data. 60 • Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian.Michael T.Schaub,Austin R. Benson,Paul Horn,Gabor Lippner,and Ali Jadbabaie.SIAM Review,2020. • Graph-based Semi-Supervised & Active Learning for Edge Flows. Junteng Jia,Michael T.Schaub,Santiago Segarra,and Austin R.Benson.Proc.of KDD,2019. github.com/000Justin000/ssl_edge 1. Algebraic topology provides the computational framework. 2. The hard part is getting a normalization scheme that connects the Hodge Laplacian to diffusions and “respects the topology.” 3. We can apply these ideas to graph algorithms based on random walks. BA Edge View initial position initial position Lifted space upper adjacent walk lower adjacent walk forward backward C
61 THANKS! Austin R. Benson Slides. bit.ly/arb-TAMU-20 http://cs.cornell.edu/~arb @austinbenson arb@cs.cornell.edu Computational frameworks for higher-order data analysis. A B C Supported by ARO MURI, ARO Award W911NF19- 1-0057, NSF Award DMS-1830274, and JP Morgan Chase & Co. Lots of data available at https://www.cs.cornell.edu/~arb/data/

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Computational Frameworks for Higher-order Network Data Analysis

  • 1.
    1 Computational Frameworks for Higher-orderNetwork Data Analysis Austin R. Benson · Cornell University Texas A&M Institute of Data Science · October 23, 2020 Slides. bit.ly/arb-tamu-20
  • 2.
    Graph or networkdata modeling important complex systems are everywhere. 2 Commerce nodes are products edges link co-purchased products Communications nodes are people/accounts edges show info. exchange Physical proximity nodes are people edges link those that interact in close proximity Drug compounds nodes are substances edge between substances that appear in the same drug
  • 3.
    Network data analysisstudies the model to gain insight and make predictions about these systems. 3 1. Evolution / changes What new connections will form? (email auto-fill suggestions, rec. systems) 2. Clustering / partitioning / community detection How to find groups of related nodes? (similar products, protein functions) 3. Spreading and traversing How does stuff move over the network? (viruses or misinformation) 4. Ranking Which things are important? (PageRank and its variants)
  • 4.
    Real-world systems arecomposed of“higher-order” interactions that we often reduce to pairwise ones. 4 Commerce nodes are products Several products purchased at once Communications nodes are people/accounts emails often have several recipients,not just one. Physical proximity nodes are people people gather in groups Drug compounds nodes are substances Drugs are composed of several substances
  • 5.
    5 What new insightsdoes this give us?
  • 6.
    We can askthe same network analysis questions while taking into account the higher-order structure. 6 1. Evolution / changes What new connections will form? (email auto-fill suggestions, rec. systems) 2. Clustering / partitioning / community detection How to find groups of related nodes? (similar products, protein functions) 3. Spreading and traversing How does stuff move over the network? (viruses or misinformation) 4. Ranking Which things are important? (PageRank and its variants)
  • 7.
    Higher-order Network DataAnalysis 7 w/ R. Abebe, M. Schaub, J. Kleinberg, A. Jadbabaie 1. Temporal evolution of higher-order interactions. Simplicial Closure and Higher-order Link Prediction,PNAS 2018. 2. Clustering in large networks of higher-order interactions. Minimizing Localized Ratio Cuts in Hypergraphs,KDD,2020. 3. Diffusions over higher-order interactions in networks. Random walks on simplicial complexes and the normalized Hodge 1-Laplacian,SIAM Review,2020.
  • 8.
    We collected manydatasets of timestamped simplices, where each simplex is a subset of nodes. 8 1. Coauthorship in different domains. 2. Emails with multiple recipients. 3. Tags on Q&A forums. 4. Threads on Q&A forums. 5. Contact/proximity measurements. 6. Musical artist collaboration. 7. Substance makeup and classification codes applied to drugs the FDA examines. 8. U.S. Congress committee memberships and bill sponsorship. 9. Combinations of drugs seen in patients in ER visits. https://math.stackexchange.com/q/80181 bit.ly/sc-holp-data
  • 9.
    Thinking of higher-orderdata as a weighted projected graph with filled-in structures is a convenient viewpoint. 9 1 2 3 4 5 6 7 8 9 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 t1 : {1, 2, 3, 4} t2 : {1, 3, 5} t3 : {1, 6} t4 : {2, 6} t5 : {1, 7, 8} t6 : {3, 9} t7 : {5, 8} t8 : {1, 2, 6} Data. Projected graph W. Wij = # of simplices containing nodes i and j.
  • 10.
  • 11.
    11 i j k i j k Warm-up.What’s more common in data? or “Open triangle” each pair has been in a simplex together but all 3 nodes have never been in the same simplex “Closed triangle” there is some simplex that contains all 3 nodes
  • 12.
    music-rap-genius NDC-substances NDC-classes DAWN coauth-DBLP coauth-MAG-geology coauth-MAG-history congress-bills congress-committees tags-stack-overflow tags-math-sx tags-ask-ubuntu email-Eu email-Enron threads-stack-overflow threads-math-sx threads-ask-ubuntu contact-high-school contact-primary-school 10 5 10 4 103 10 2 10 1 Edge density in projected graph 0.00 0.25 0.50 0.75 1.00 Fractionoftrianglesopen There is lots of variation in the fraction of triangles that are open,but datasets from the same domain are similar. 12See also Patania-Petri-Vaccarino (2017) for similar ideas in collaboration networks.
  • 13.
    Dataset domain separationalso occurs at the local level. 13 • Randomly sample 100 egonets per dataset and measure log of average degree and fraction of open triangles. • Logistic regression model to predict domain (coauthorship, tags, threads, email, contact). • 75% model accuracy vs. 21% with random guessing.
  • 14.
    14 How do newsimplices form? Can we predict which simplices will form?
  • 15.
    Groups of nodesgo through trajectories until finally reaching a“simplicial closure.” 15 t1 : {1, 2, 3, 4} t2 : {1, 3, 5} t3 : {1, 6} t4 : {2, 6} t5 : {1, 7, 8} t6 : {3, 9} t7 : {5, 8} t8 : {1, 2, 6} For this talk, we will focus on simplicial closure on 3 nodes.
  • 16.
    Groups of nodesgo through trajectories until finally reaching a“simplicial closure event.” 16 Substances in marketed drugs recorded in the National Drug Code directory. We bin weighted edges into “weak” and “strong ties” in the projected graph W. Wij = # of simplices containing nodes i and j. • Weak ties. Wij = 1 (one simplex contains i and j) • Strong ties. Wij > 2 (at least two simplices contain i and j)
  • 17.
    Simplicial closure dependson structure in projected graph. 17 • First 80% of the data (in time) ⟶ record configurations of triplets not in closed triangle. • Remainder of data ⟶ find fraction that are now closed triangles. Increased edge density increases closure probability. Increased tie strength increases closure probability. Tension between edge density and tie strength. Left and middle observations are consistent with theory and empirical studies of social networks. [Granovetter 73; Kossinets-Watts 06; Backstrom+ 06; Leskovec+ 08] Closure probability Closure probability Closure probability
  • 18.
    Simplicial closure on4 nodes is similar to on 3 nodes, just“up one dimension.” 18 Increased edge density increases closure probability. Increased simplicial tie strength increases closure probability. Tension b/w edge density simplicial tie strength. Closure probability Closure probability Closure probability
  • 19.
    We proposed“higher-order linkprediction”as a framework to evaluate models for closure. 19 t1 : {1, 2, 3, 4} t2 : {1, 3, 5} t3 : {1, 6} t4 : {2, 6} t5 : {1, 7, 8} t6 : {3, 9} t7 : {5, 8} t8 : {1, 2, 6} Data. • Observe simplices up to time t. • Predict which groups of > 2 nodes will appear after time t. t We predict structure that graph models would not even consider!
  • 20.
    20 Our structural analysistells us what we should be looking at for prediction. 1. Edge density matters! ⟶ focus our attention on predicting which open triangles become closed triangles (intelligently reduce search space.) 2. Tie strength matters! ⟶ various ways of incorporating this information i j k Wij Wjk Wjk
  • 21.
    21 For every opentriangle,we assign a score function on first 80% of data based on structural properties. Score s(i, j, k)… 1. is a function of Wij, Wjk, Wjk arithmetic mean, harmonic mean, etc. 2. looks at common neighbors of the three nodes. generalized Jaccard, Adamic-Adar, etc. 3. uses “whole-network” similarity scores on projected graph sum of PageRank or Katz scores amongst edges 4. is learned from data logistic regression model with features i j k Wij Wjk Wjk After computing scores, predict that open triangles with highest scores will be closed triangles in final 20% of data. i j k l m x y r z N(i) = {j, k, l, m, x, y, z} N(j) = {i, k, l, m, r} N(k) = {i, j, l, m}<latexit 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scorep(i, j, k) = (Wp ij + Wp jk + Wp ik)1/p <latexit 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  • 22.
  • 23.
    23 A few lessonslearned from applying these ideas. 1. We can predict pretty well on all datasets using some simple method. → 4x to 107x better than random w/r/t mean average precision depending on the dataset/method (only predicting on open triangles) 2. Thread co-participation and co-tagging on stack exchange are consistently easy to predict with the harmonic mean. 3. Simple averaging Wij, Wjk, and Wik consistently performs well. i j k Wij Wjk Wjk
  • 24.
    Generalized means ofedges weights are often good predictors of new 3-node simplices appearing. 24 music-rap-genius NDC-substances NDC-classes DAWN coauth-DBLP coauth-MAG-geology coauth-MAG-history congress-bills congress-committees tags-stack-overflow tags-math-sx tags-ask-ubuntu email-Eu email-Enron threads-stack-overflow threads-math-sx threads-ask-ubuntu contact-high-school contact-primary-school harmonic geometric arithmetic p 4 3 2 1 0 1 2 3 4 0 20 40 60 80 Relativeperformance 4 3 2 1 0 1 2 3 4 p 2.5 5.0 7.5 10.0 12.5 Relativeperformance 4 3 2 1 0 1 2 3 4 p 1.0 1.5 2.0 2.5 3.0 3.5 Relativeperformance Good performance from this local information is a deviation from classical link prediction, where methods that use long paths (e.g., PageRank) perform well [Liben-Nowell & Kleinberg 07]. For structures on k nodes, the subsets of size k-1 contain rich information only when k > 2. i j k Wij Wjk Wjk i j k ? scorep(i, j, k) = (Wp ij + Wp jk + Wp ik)1/p <latexit 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  • 25.
    If we onlyneed the top-k weighted triangles, we have fast algorithms for finding them. 25 scorep(i, j, k) = (Wp ij + Wp jk + Wp ik)1/p <latexit 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Simple (incorrect) algorithm. 1. Throw out edges with weight < t. 2. Find triangles in remainder. i j k Wij Wjk Wjk Better (correct) algorithm. 1. Dynamically choose threshold. 2. Careful pruning. w/ R. Kumar, P. Liu, M. Charikar
  • 26.
    We often onlyneed the top-k weighted triangles,and we have fast algorithms for finding them. 26 <latexit sha1_base64="mGA+XhzsM3MoFa1t7a1hl4NURg0=">AAAJZXicfVVbc9tEFHYNlMZQaIHhhQe2JO6kHduR7DipmelMWtpCZxp6cXqZiTJlJR1JO96L2F01Dho982t4hd/CL+BvcFa2W9sNaBJ7vTrf+c79hDlnxnre3xeaH3z40cWPL220Pvn08mefX7n6xQujCh3B80hxpV+F1ABnEp5bZjm8yjVQEXJ4GU5+cO9fvgFtmJJH9iyHE0FTyRIWUYtXr682SRBCymRpaVhwqqtSkyjCv6oVWJXrgkMrphYZLLlOgk0iVQxmdoQ4rY8PqLEEZCFA11oXV4jvbk22yPbWhNwmvud5WzdIELSCSLDYad7m+ka52x1WLTJ/rhNy7jEQBbcM3S2ELPtVGVXlti6kZDIllgkgTBIDkZKxuVHVHHOK1jhXliVnJFJdF06Q5p3WQW/vsD74vdHgrqOxMLWlVYpwJdPKiXhO2+GdH50CWthMaZOxfMk0f38wUzLc3T2sv0d77nrPAe8cPvz5/rPzsNfJaI4b9Ge4/nDocDXhfZuBhkIQq6k0NHJhNQurb82N9gY1buS7y4GDtceWRhPyGDOecHWKUbGYkzkaGXpzyr5fIwcO6fVqaBAqa5WoYxaAjN8WxOsrm17Pqx/y/sGfHzYb8+fJ66sXO0GsIiwHaSNOjTn2vdyelFRjBjlgYRUGcrSTpnCMR0kFmJOyruWKtPEmJonS+C8tqW9byxDUo+nZipaFrdPV21CpCb4x1Qr+2LWKwWoNMcJxxzkco3E8VZrZTPRhTbywya2Tksm8wOKJZgYmBcfqJq6hSMw0RJafkVUrLZv81pEsggQT0KHCCGqzTs6cVx1BJxAB5zN7nShnoab6zDmnTk0nRC2pVgXWcyenFrMoDaKsZtOOyWgOppMw24koj9zv2GFyrqygemL+S2tPgKX4so4pB1seFYmFZxBj00N87ZZ3LeTIuyyBZZhqAFmV9ZeTOc2YhTWZkBdQle5zSaLVJpm1ufl+ZwfbqmdcacI0yqhMoRcpsfNrAaYuzR1/bzjqj3YMCIalG+IUEd1TzEbXOdFlshviSANdyw32N2df2OMYRoojz8WnFaRchZQH+DNwsANs9ULDQaw4lsYBDrwIR9ftQAOn0wVWuT5bzfeRf1K6JLlkr2T0ydEYexHvNEg4RQcExTYJEioYP4shoTiiqjIwyeK8WhAmcRVQtdrLZAYzCPFtrzfq4FBE0lRTjs2ABHZqEqdi1UnUHUg7daoOZuDS3DzGLhyeVOtO3QNsPw3jMxEq/gBdKmdaTFU+PnxUldJRCFaVoioZmhuMwZ4njBfxOiScQ+YcDjAuQkynLVxKzydYZxg/OHQhWRAc+SvhK8NpVRr+jsQJz9DlQ5R0MaA8z2j1ztRfHq5FPU45sCjrzmJ/3hu3EnDwrE4O4dQsZ1mMWSqQKZhVlVNXBqEog9l99V5ZiEe4hOPzEPMX1SrFzWAaUn2MxRdkoZqWwRv32W4FmZtQJAOWZhbn7v4wt6RNjjIgONgLygnCWsEEJ4TX6w9h2n67m9rkHm48KiMgIdhT7F8nS5CMmDqMrRlVG9dvraDr9XwQ7QV6jAsLo+N2LC51LCrCIbHEsBgcYsmvTb96qwRXw+B/lejak1oLBsHtF399m7x/eNHv+cOe93R38+DufNNcanzT+K6x3fAb+42Dxk+NJ43njaj5e/OP5p/Nvy79s3F546uNr2eizQtzzJeNlWfj238B8AUo0g==</latexit> dataset # nodes # edges Fast enumeration Fast top-k (k = 1000) (running time in seconds) Spotify co-listens 3.6M 1.93B too long 30 MAG co-authorship 173M 544M 596 16 AMINER co-authorship 93M 324M 255 10 Ethereum transactions 38M 103M 91 33
  • 27.
    Higher-order data ispervasive! 27 • Simplicial Closure and Higher-order Link Prediction.Austin R.Benson,Rediet Abebe,Michael T.Schaub,Ali Jadbabaie,and Jon Kleinberg. Proc.Natl.Acad.Sci.U.S.A.,2018. github.com/arbenson/ScHoLP-Tutorial • Retrieving Top Weighted Triangles in Graphs. Raunak Kumar,Paul Liu,Moses Charikar,and Austin R.Benson. Proc.Of WSDM,2020. github.com/raunakkmr/Retrieving-top-weighted-triangles-in-graphs 1. There are commonalities in temporal evolution. Generative models? 2. There is lots of signal in subsets! Unique to higher-order… 3. Please develop neural embeddings to out-perform our baselines. 😁 1 2 3 4 5 6 7 8 9 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1
  • 28.
    28 w/ Nate Veldt,J. KleinbergHigher-order Network Data Analysis 1. Temporal evolution of higher-order interactions. Simplicial Closure and Higher-order Link Prediction,PNAS 2018. 2. Clustering in large networks of higher-order interactions. Minimizing Localized Ratio Cuts in Hypergraphs,KDD,2020. 3. Diffusions over higher-order interactions in networks. Random walks on simplicial complexes and the normalized Hodge 1-Laplacian,SIAM Review,2020.
  • 29.
    Graph minimum s-tcuts are fundamental. 29 minimizeS⇢V cut(S) subject to s 2 S, t /2 S.<latexit 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1 3 2 4 5 6 7 8 s t • Maximum flow / min s-t cut [Ford, Fulkerson, Dantzig 1950s] • Densest subgraph [Goldberg 84; Shang+ 18] • Graph-based semi-supervised learning algorithms [Blum-Chawla 01] • Local graph clustering [Andersen-Lang 08; Oreccchia-Zhu 14; Veldt+ 16] poly-time algorithms!
  • 30.
    Real-world systems arecomposed of“higher-order” interactions that we can model with hypergraphs. 30 H = (V, E), edge e 2 E is a subset of V (e ⇢ V)<latexit 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1 2 3 4 5 V = {1, 2, 3, 4, 5} E = {{1, 2, 3}, {2, 4, 5}}<latexit 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  • 31.
    What is ahypergraph minimum s-t cut? 31 s t Should we treat the 2/2 split differently from the 1/3 split? Historically, no. [Lawler 73, Ihler+ 93] More recently, yes. [Li-Milenkovic 17, Veldt-Benson-Kleinberg 20] edge in a graph size-3 hyperedges “Only one way to split a triangle” [Benson+ 16; Li-Milenkovic 17; Yin+ 17] Must be split 1/1.
  • 32.
    We model hypergraphcuts with splitting functions. 32 s t Given a cut defined by S, we incur penalty of at each hyperedge e. Hypergraph minimum s-t cut problem. Cardinality-based splitting functions. S<latexit 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cutH(S) = f (2) + f (1)<latexit 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sha1_base64="JdV0NHpso/GwwYvqd/CeIvys+E4=">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</latexit> <latexit 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we(e S) <latexit 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minimizeS⇢V P e2E we(e S) ⌘ cutH(S) subject to s 2 S, t /2 S. <latexit sha1_base64="vCSQ5hxLftoc4zdzUNdXcsthqGM=">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</latexit> Non-negativity we(A) 0. Non-split ignoring we(e) = we(;) = 0. C-B we(A) = f (min(|A|, |Ae|)).
  • 33.
    Cardinality-based splitting functionsappear throughout the literature. 33 [Lawler 73; Ihler+ 93; Yin+ 17] [Hu-Moerder 85; Heuer+ 18] [Agarwal+ 06; Zhou+ 06; Benson+ 16] [Yaros- Imielinski 13] [Li-Milenkovic 18] <latexit sha1_base64="gX/87S67KKdqKR6T9Qjl8vcDiHo=">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</latexit> All-or-nothing we(A) = ( 0 if A 2 {e, ;} 1 otherwise Linear penalty we(A) = min{|A|, |eA|} Quadratic penalty we(A) = |A| · |eA| Discount cut we(A) = min{|A|↵ , |eA|↵ } L-M submodular we(A) = 1 2 + 1 2 · min n 1, |A| b↵|e|c , |eA| b↵|e|c o
  • 34.
    We solve hypergraphcut problems with graph reductions. 34 1/21/2 1/2 1 1 1 1 ∞ ∞ ∞ ∞ ∞∞ Gadgets (expansions) model a hyperedge with a small graph. clique expansion star expansion Lawler gadget [1973]hyperedge In a graph reduction, we first replace all hyperedges with graph gadgets... s t s t s t s t … then solve the (min s-t cut) problem exactly on the graph, and finally convert the solution to a hypergraph solution. Quadratic penalty f(i) = i ( |e| – i ) Linear penalty f(i) = i All-or-nothing f(0) = 0,o/w f(i) = 1
  • 35.
    b We made anew gadget for C-B splitting functions. 35 This gadget models min(|A|, |eA|, b). Theorem [Veldt-Benson-Kleinberg 20a]. Nonnegative linear combinations of the C-B gadget can model any submodular cardinality-based splitting function. See also Graph Cuts for Minimizing Robust Higher Order Potentials,Kohli et al.,2008. <latexit sha1_base64="beQz4cdyY+p8N+9L01TDcNAiwcQ=">AAAHvnicfVVbb9s2FFa6rem8S9PtcS/sAg9JIDt2iizJgADuWhQr0GLZnLQFzCClpCOJMElpJBXLJfRD97afskNf2tjJRkASRZ7vO3cyKgU3ttf7e+PeZ59/cX/zwZetr77+5tuHW4++e2OKSsdwERei0O8iZkBwBReWWwHvSg1MRgLeRuNnfv/tNWjDC3VupyVcSpYpnvKYWVy62vpAI8i4ckzwTO01LWqhtu5Z59eG/ERzU7IYXK97cBjLhtDryRXsPN0lpyTdoZKrHSqQ25KnhGo/CcliAQiNWDw2gpn84+7ubrdFQSVLXVdb271ubzbI7Ul/MdkOFuPs6tH9kCZFXElQNkZqM+r3SnvpmLY8FoDGVwbQ4jHLYIRTxSSYSzcLUkPauJKQtND4KEtmq62bEOTRbLrC4iyLKsF0vboaFcUYd0yzgh/5HBhVyQg0JKGuBCRonMgKzW0uD2BNvLLp8aXjqqwsqHhuYFoJYgviM0USriG2YkpWrbR8/CFUPIZUszhk0khm87Dk3qtQsjHEIMTcXi8qeKSZnnrniokJfVoyXVQqMWHJrAWtDKKs5nVoclaCCVNuw5iJ2P8nHlOKwkqmx+a/WLsSLMPNWUwFWHdepRb+hKRxGInHx73HkUC9NyVsDpkGUI2bfbzMJOcW1mQiUUHj/PuGRKtNcmtL88v+PlZr11jkhjrOmcqgGxdy/68KjC9vs9//+fDk4GTfgORYghEWvexMMBsd70SHq06EvQJ6JvfkaHv+aVEfRoa95OPTopkoIiYo/lIPG4AylYZBUggsjQF2UlwkcEo1CFYvsQUav1peo/P+pfNJ8sleyejZ+ZApH1wNCibogGTYJTRlkotpAimrhG0cNelyvloQJvUV0LTaN5UZzCAkp73uSRhjp1qMNhPYDKjA1ib1FKtOIjdVtvZUgznYmb0RduHhZbPu1HPA9tMwnMqoEC/QJTdnMY37/fWrximvQvLGycZxNJcOwd4ljAvJOiRaQBY6PGBYRZhOW/mU3q1gXcPwxWsfkqWC8/5K+FxUN86IT0q88BztXqKkjwETZc6aT6a+f7kW9SQTwOO8M4/9XTuYaIMHz+rJIT3NzSzLIc8kaqLzqvJ0jkbS0fl6c6ss5Cs83ZO7EIuNZlXFHq0jpkdYfDSPitrRa/9ut2juTyiSA89yi+fu0WFpSZuc50BYbCsmCMJadIwnhL8AoG6T5WiT53gzMRUDicBOsH+9LEFlxMzC2JqrarcImRF0et0+yPYSPcwLjdHhKiOFIlhUREBqieEJeMQNv7b7zUcSvBqe/C+JnnkyY8Eg+Pulv36b3J68Oej2D7u9Pw62B8eLm+ZB8EPwY7AT9IOjYBD8FpwFF0Ec/LNxf+PhxtbmYDPdlJvFXPTexgLzfbAyNut/AfzCt34=</latexit> C-B we(A) = f (min(|A|, |eA|)). (F is submodular on X if F(A B) + F(A [ B)  F(A) + F(B) for any A, B ✓ X.)<latexit sha1_base64="jx6llVBabtrhi3TShW9c6Ptv2Kc=">AAAHkXicfVXfb9s2EFa7re60H02Xx72wiw0knezYKbIkAwq4WRGsWItls9MGiIyMkk4WYZJSSaqxK+g/3D+wf2Ov28OOkrNYbjY9SNTxvvt4x4/HIONMm37/jzt3P/r4k3ut+5+6n33+xZcPNh5+9VqnuQrhLEx5qs4DqoEzCWeGGQ7nmQIqAg5vgtkPdv7NO1CapXJsFhlMBJ1KFrOQGjRdbsTb7ZM2YZroPBBplHOqSCpJ+xyNMWmfbD8jfkgzcrxDviX1X179+RzeWkNtP95pkzhVhMoFaT/zyDHxMaAGg07n7d7O5cZWv9evHvLhYLAcbDnL5/Ty4b1NP0rDXIA0IadaXwz6mZkUVBkWcihdP9eQ0XBGp3CBQ0kF6ElRFaQkHbRE1YLiVBpSWd1VCMZRdNGIUhga2PTnTWuQpjOc0aXrNjlNfDgpmMxyAzKsKeOcE5MSW2cSMQWh4QvS5DVs9t6TLIRY0dCjQgtqEi9jdp2emb3vThXNEk/QGYTA+Y2pXpWFcxYoqhY2hfRKewFGnqo0l5H2MmoMKKkRbxSbezqhGWgvZsYLKQ/tf2QxGU+NoGqm/ytqT4ChOFlVjoMpxnls4FeIykJB9Oiw/yjgyLvqYRKYKgBZFtXH+lwlzMCaT8BzKAv7XvFwOyQxJtPf7+4amPe0wdgwDxMqp9ALU7H7NgdtBat3B9/tH+0d7WoQDHUdoIxF94qZpGuT6DLZDVD9oCq/Jwdb9cf1bUEpng5bH9ef8jSgHCVsfAsbgtS5gmGUchTAEM9GmEbw1FfA6fwam+LimyK6GA8mhd04K4DGLp+OR1Ta4iqQcIUJCCqjwo+pYHwRQUxzbsrC1/H1uCkSHVtVlG5nlUzjDkL0tN878kLBkBRlwVHySGDmOrYhmklibF+auQ01rMGFfnyBZ21/Uq4n9RzwkCkYLUSQ8hNMqaij6LL4+dXLspCWQrCyEGXBcLn+CMxtzmiI1iHBErLksIAR9gZsVrnd0tsJ1hlGJ69sSa4JxoNG+YpgXhaa35BY5xpdvEBPWwPKs4SWN0v97cVa1aMpBxYm3br2t83gRmtsL83+IGyY1V0WIzYVyOTXqrLhCj8QhV/byw9kIV5iv45uQywnyibFY38eUHWB4vOTIJ0X/jv77rh+onIOJAE2TQx214P9zJAOGSdAaGhyygnCXH+GHaLf29uHeYdcPx3yHO8aKkMgAZgrPL/WF9t5RHRVRrem6riEVAG6/d4AROcaPUpShdVhcmpvDxQV4RAbolkEFrGS19ag/DcIXgBP/jeIqjKpopS2CniNDNYvjQ8Hr/d6A1zeL3tbw8PlhXLf+dr5xtl2Bs6BM3R+dE6dMyd0fnf+dP5y/m5tto5aw9Zx7Xr3zhKz6TSe1k//AIESoIo=</latexit><latexit 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  • 36.
    36 Theorem [Veldt-Benson-Kleinberg 20a].The hypergraph min s-t cut problem with a cardinality-based splitting function is graph-reducible (via gadgets) if and only if the splitting function is submodular. Cardinality-based splitting functions. s t S<latexit 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cutH(S) = f (2) + f (1)<latexit 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Submodularity is key to efficient algorithms. What happens when the splitting function isn’t submodular? Can we use some other algorithm? <latexit sha1_base64="vCSQ5hxLftoc4zdzUNdXcsthqGM=">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</latexit> Non-negativity we(A) 0. Non-split ignoring we(e) = we(;) = 0. C-B we(A) = f (min(|A|, |Ae|)).
  • 37.
    37 Unlike graph mins-t cut, hypergraph min s-t cut can be NP-hard. w1 = 1 0 1 2 w2 ?? Reducible/Submodular NP-hard Unknown Hard Reducible w3 3 2.5 2 1.5 1 0.5 1 1.5 2 2.5 w2 0.5 w2 w3 w4 4 3 2 1 0 1 1.5 2 2.5 1 2 3 max hyperedge size 4 or 5 max hyperedge size 6 or 7 max hyperedge size 8 or 9 Theorem [Veldt-Benson-Kleinberg 20a]. For C-B splitting functions, Open Question: For 4-uniform hypergraphs, is there an efficient algorithm to find the minimum s-t cut with no 2-2 splits (w1 = 1, w2 = ∞). s t cutH(S) = f (2) + f (1) = w2 + 1<latexit 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  • 38.
    38 How can weuse this framework to enable new data science algorithms?
  • 39.
    G = (V,E)is a graph. R ⊆ V (Reference or seed set). Finds a “good” cluster S “near” R. 39 Background.Local clustering has been studied extensively in graphs,but not much in hypergraphs.
  • 40.
    Rewards high overlap withR. Penalizes nodes outside R. R(S) = cut(S) vol(S R) "vol(S ¯R) Max Flow.Quot.Imp. (Lang,Rao,2004) Flow-Improve (Andersen,Lang 2008) Local-Improve (Orecchia,Allen-Zhou 2014) SimpleLocal (Veldt,Gleich,Mahoney 2016) FlowSeed (Veldt,Klymko,Gleich 2019) Great survey paper! (Fountoulakis et al.2020) 40 Background.Flow-based methods minimize a localized variant of conductance. Rewards contained clusters vol(T) = sum of degrees in T. minimize node sets S FAST ALGORITHMS FOR EXACT MINIMIZATION!
  • 41.
    s t 2 4 4 7 3 4 7 3 2 1 3 2 6 4 5 7 8 9 10Set R 4 1 3 2 6 4 5 7 8 9 10 s t 2 4 4 7 3 4 7 3 2 1 3 2 6 4 5 7 8 9 10Set R 4 [Andersen-Lang 08,Orecchia-Zhou 14,Veldt+16] Construct G’ R(S) < ↵ () min s-t cut of G0 < ↵vol(R) Compute min s-t cut of G’. 41 Connect R to a source node s ; edges weighted with respect to 𝛼. Connect VR to a sink node t ; edges weighted with respect to β = 𝛼ε. Is R(S) < ↵ for any S? Background.Flow methods repeatedly solve min-cut problems on an auxiliary graph.
  • 42.
    We generalized localflow-based techniques to the hypergraph setting 42 • We introduce localized hypergraph conductance • We can minimize it exactly with our hypergraph min s-t cuts framework • Strongly-local runtime! (Only depends on size of seed set) • Normalized cut improvement guarantees The analysis provides even new guarantees for the graph case!
  • 43.
    43 We define hypergraphs-t cut problems similar to the ones used in the graph case. t 𝜀𝛼dj r j s 𝛼dr R minimize S⇢V HLCR,"(S) = cutH(S) volH(S R) "volH(S ¯R) cutH(S) = cut from C-B splitting function volH(X) = X i2X X e2E 1 = sum of hypergraph deg <latexit 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sha1_base64="1b3WAyFb1xHR0eMss1XSCe103wc=">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</latexit> Hypergraph cut function Encourage overlap with reference set. Discourage overlap outside reference set ⟶ di = # hyperedges node r is in volH(S) = X i2S di <latexit sha1_base64="E541uqeWvYyixS/NzXbACyuddlM=">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</latexit> Theorem [Veldt-Benson-Kleinberg 20b]. We can repeatedly solve min hypergraph s-t cut problems with different 𝛼 to exactly minimize the hypergraph localized conductance (HLC) exactly.
  • 44.
    We carefully applygraph reduction techniques to growing subsets of the hypergraph. 44 s R t s t 𝛼 Theorem [Veldt-Benson-Kleinberg 20b]. Strong locality. Can make this algorithm run in time proportional to the size of seed set (does not look at the full hypergraph).
  • 45.
    <latexit sha1_base64="ff4QhRAHZfL2qp7+qW21R4vDGq8=">AAAIZXicfVXdbhtFFLYLtGahtAHEDRdMiS0lwXbsVCEJUiVLVBWRWhFqJ63kscLs7tndkWdmtzOzsc1qr3kabuFZeAJegzO2k9pOYKX1jmfOd86c7/z5meDGdjp/V+998OFH9x/UPvY++fThZ48eb31+YdJcB3AepCLVb31mQHAF55ZbAW8zDUz6At744x/d+Zsr0IanamBnGYwkixWPeMAsbl1uVQlVKVchKEtOI8KIZToGSwy+9QGhJvfd8qJODCJMxMF41IeYq4IJHqu90qORZkFBLUytHxVXqSh3EBiwjLzeLTf2d0tCY3hH7sBQn+liUN4NXB4i/DuC5i3zKKjw5gqDBBSp0yzhO/1dQsWNhS7qceJo1h0OdkmdTBLQQOr9OuGG2ATmvmqwuVYQEn9GkFvCRJxqbhPZvny83Wl35g+5veguF9uV5XN2uXW/ScM0yCVyGghmzLDbyeyoYNryQAASlhvIWDBmMQxxqZgEMyrmsSxJA3dCEqUaX4zJfNdbhaAezWZrWgrL/FwwPV3f9dN0jCemXMMPXaoYlUsfaQibOhcQ4uWu/T2ADfHcRsejgqsst6CCxQWjXBCbEpdQJOQaAitmZP2Wlo9/ayoegItDk0kjmU2aGXdeNSUbQwBCLO7rRAX3NdMz51w6MU0ftcQ6zVVomhmzFrQyiLKaT5smYRmYZsRtM2AicP9Dh8lEaiXTY/NfWtvSZYJH55wKsMUgjyy8hrAskIknx50nvkC7qxKYHrEGUGUx/ziZScItbMj4IoeycL8rEl6DJNZm5of9fUzltrGoG6ZBwlQM7SCV++9yMK4KzX73+8OTg5N9A5JjsfpYabI1wWi0nBMtrlo+ljToudzTo+3Fx6OORoYl7/jxaCxSnwlMfksdrAfK5Bp6YSowNXpY8EEawjOqQbDpNTbFy6+n13DQHRUuSC7YaxE9G/SZcuRqUDBBByTDEqQRk1zMQohYLiyWm4mu1+sJYSKXAaXXWDVmMIIQPuu0T5qB5Gg01kxgMaABOzWRU7HuJOqmyk6dqt4CXJi9IVbh4ajcdOo5YPlp6M+kn4oX6FKx0GLK4udXL8tCOROSl4UsC47XpX2wdwnjRrgJ8ZeQpQ0H6GOrxA6cu5DebWDTQv/FK0fJtYFBd42+wp+WhRHvjTjhBbo4RUnHARNZwsr3V/31dIP1MBbAg6S14P6uEwy0wcaz3jmkU7MaZdnnsURLdJFVTh12VlnQxX55Ky3kSxxC4V2I5UG5bmKPTrHHDzH5aOKn04Jeud+GRxPXoUgCPE4s9t2jw8ySBsGOT1hgcyYIwjw6xg7RaR8cwrRBrp8GeY4DlKkACM6ACdavkyVojJg5jd7CVMMjZK6g1Wl3QTau0f0k1cgOVzFJ1XxSCIhwNPIQHGLFr+1ueaMER8PT/1Wi557MtSAJJc6X7uY0ub24OGh3D9udXw62e8fLSVOrfF35trJT6VaOKr3KT5WzynklqP5e/aP6Z/WvB//UHta+rH21EL1XXWK+qKw9tW/+Ba2e9dU=</latexit> If atarget set T ⇢ V satisfies vol(T R) vol(T) vol(¯T R) vol(¯T) + Then (S)  1 (T) where S is the set returned by our algorithm. 45 We prove new normalized cut guarantees that are new even for the graph case. (S) = cut(S) vol(S) + cut(S) vol(S)<latexit sha1_base64="+2zPfSQ7UOPKTHfrSfqqRxABg7w=">AAAFDXicjVNLbxMxEHbbACW8Wjhy2VJVaiFESQFRDpUqQBWVqCgkfUjdqHi9s4kV70O2t01q+TfwR5A4cUNcuXND8GMYJynNpj1gKdnxzDffPDwTZIIrXav9mpqeKV25em32evnGzVu378zN391TaS4Z7LJUpPIgoAoET2BXcy3gIJNA40DAftB95ez7xyAVT5Om7mfQimk74RFnVKPqaG7Tzzp8ubHirXt+JCkzPss13q3xj1PhBO/RpRY/RVoX1TTsij2aW6xVa4PjXRTqI2GRjM7O0fzMFz9MWR5DopmgSh3Wa5luGSo1ZwJs2c8VZJR1aRsOUUxoDKplBgVbbwk1oRelEn+J9gbacQ+kkbRvi6pYqX4coHNMdUdN2pzyUpumQS6o7BXZgjTtokVNJKqjtZbhSZZrSNgwzygXnk4913wv5BKYFn2vGKF7+rgtadYZkmnePRU8kFT2XR3piaqoDs1AVRgVrBJxjbhByQK0aeaRhg8QWiMhXFirLQQCaccRugNtCZBYM/g4zEmHa5jABCIHa9z/OGKsuGa9ZVy7XWWF/HeaDZq4TkhI4ISlcUyT0PgRjbnohxDRXGicGhWdyQVvFQ1aX14aj6WwVAjXa9UXFRZzjIn9Efj+yK97KkKGmHaB4rhrDfj0yOwnuueYNoa+Rj08xLl71vqHTbFIhL4GnDcJDZyFVGxiPWZIoqx5t/3WmsQFiLk1sTWDVjdAXwZGRTjpEoxcRjGcQyMPFO5l7tbtvwI0NrddP874m/VC60zQs0aJ8xgOPPQ2W4h0HaAi61B7nunHLVsu++NPg5uMWA09HUTGXWzRjgt+bncXJMAVr08u9EVhb7Vaf1Jdff90cePlaNlnyX3ygCyTOnlONsgbskN2CSOfyU/ym/wpfSp9LX0rfR9Cp6dGPvdI4ZR+/AUEuMqZ</latexit> Normalized cut is another ratio-cut objective related to conductance. If T overlaps enough with seed set R... ...then our output has normalized cut almost as good as T. Theorem [Veldt-Benson-Kleinberg 20b]. Normalized cut improvement.
  • 46.
    schem e ocam l tcl m dx com m on- lisp verilog lotus- notes xslt-1.0 plone typo3 abap sitecore m arklogic wolfram - m athem atica alfresco axapta vhdl sparql prolog netsuite racket spring- integration xslt- 2.0 m ule wso2 system - verilog wso2esb google- sheets- form ula stata xpages netlogo openerp data.table google- bigquery docusignapi aem codenam eone dax cypher julia sapui5 ibm - m obilefirst office- js jq apache- nifi 0.2 0.4 0.6 0.8 F1Scores HyperLocal TN/BN FlowSeed 46 • 15M StackOverflow questions (nodes), answered by 1.1M users (hyperedges). • mean hyperedge size 23.7, max hyperedge size ~ 60k. • Tags provide ground truth cluster labels. • Delta-linear splitting function wi = min(i, 5000). HyperLocal Clique-Expansion + FlowSeed Neighborhood Baselines
  • 47.
    47 Cluster |T| time(s) HyperLocal Baseline1 Baseline2 Amazon Fashion 31 3.5 0.83 0.77 0.6 All Beauty 85 30.8 0.69 0.60 0.28 Appliances 48 9.8 0.82 0.73 0.56 Gift Cards 148 6.5 0.86 0.75 0.71 Magazine Subscriptions 157 14.5 0.87 0.72 0.56 Luxury Beauty 1581 261 0.33 0.31 0.17 Software 802 341 0.74 0.52 0.24 Industrial & Scientific 5334 503 0.55 0.49 0.15 Prime Pantry 4970 406 0.96 0.73 0.36 <latexit 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• 2.3M Amazon products (nodes), reviewed by 4.3M users (hyperedges). • mean hyperedge size > 17, max hyperedge size ~9.3k. • Product categories provide ground truth cluster labels. • All-or-nothing penalty (wi = 1). F1 recovery scores given a handful of nodes from the ground truth cluster T.
  • 48.
    48 Gadget reductions sometimescreate dense graphs, which can make computations expensive. Theorem [Veldt-Benson-Kleinberg 20c].Any submodular C-B splitting function can be 𝜀-approx with log r / 𝜀 splitting functions (instead of r, r = hyperedge size). And one specific case… • r = 60k clique expansion only need O(r / √𝜀) instead of O(r2) 0 1 2 3 0 2 4 6 8 10 e0 1 e00 1 e0 2 e00 2 e0 3 e00 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 4 3 0 1 2 3 0 2 4 6 8 10 e0 e00 5 5 5 5 5 5 5 5 5 5 5 5 9
  • 49.
    We can nowmodel and use hypergraph min s-t cuts. 49 • Hypergraph Cuts with General Splitting Functions.Nate Veldt,Austin R.Benson,and Jon Kleinberg. arXiv:2001.02817,2020. • Localized Flow-Based Clustering in Hypergraphs.Nate Veldt,Austin R.Benson,and Jon Kleinberg. Proc.Of KDD,2020. github.com/nveldt/HypergraphFlowClustering • Augmented Sparsifiers for Generalized Hypergraph Cuts.Nate Veldt,Austin R.Benson,and Jon Kleinberg. arXiv:2007.08075,2020. 1. A model for hypergraph cuts. C-B splitting functions that depend on # of nodes on small side of the cut 2. Algorithm for min s-t cuts with submodular C-B splitting functions. Graph-reducible if and only if C-B splitting function is submodular 3. Applications to local hypergraph clustering. Strong locality lets us scale to large hypergraphs with large hyperedges s t s t s t w2 = 0.5 (NP-hard) w2 = 1.5 (poly-time via graph reduction) w2 = 2.5 (?)
  • 50.
    50 w/ M.Schaub,A.Jadbabaie, G.Lippner,and P.HornHigher-orderNetwork Data Analysis 1. Temporal evolution of higher-order interactions. Simplicial Closure and Higher-order Link Prediction,PNAS 2018. 2. Clustering in large networks of higher-order interactions. Minimizing Localized Ratio Cuts in Hypergraphs,KDD,2020. 3. Diffusions over higher-order interactions in networks. Random walks on simplicial complexes and the normalized Hodge 1-Laplacian,SIAM Review,2020.
  • 51.
    Background.Graph Laplacians,diffusions,and spectral graphtheory underly many graph data methods. 51 Low-dimensional embeddings [Belkin-Niyogi 02; Coifman-Lafon 06] Personalized PageRank [Andersen-Chung-Lang 08; Gleich 15] <latexit sha1_base64="QQhZuRTVdivo8Dt7YLhgdqyYsKI=">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</latexit> N ⇥ x1 x2 ⇤ = ⇥ 1x1 2x2 ⇤ Norm. Lap. N = D-1/2LD-1/2. Random walk Lap. LD-1. D = diagonal degree matrix, A = adjacency matrix, L = D – A is graph Laplacian.. <latexit sha1_base64="POsukIjAahItuceyxYGW6/9e3Cc=">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</latexit> ( I + LD 1 )x = v
  • 52.
    52 What is a“higher-order” Laplacian?
  • 53.
    Background. By interpretingour data as a simplicial complex,we can get higher-order Laplacians. 53 • (Abstract) simplicial complex X: if A ∈ X and B ⊆ A, then B ∈ X. • Graph G = (V, E) as a simplicial complex: X = V ⋃ E. • Can induce a complex from higher-order interactions. <latexit sha1_base64="LJ5+N3kgPFW8cDVOT8x5Tlih5vo=">AAAIAnicfVVbbxtFFHYK1GW5pfDIy5RghIrtrFNCEqRIKVRVi1IRsNNWyqRhdvfs7shz2c7MNg6jfePX8IZ4RfwP/g1nbIfaTspovTueOef75lwnqQS3Lo7/Wbvx1tvv3Gzfejd67/0PPvxo/fbHT62uTQrHqRbaPE+YBcEVHDvuBDyvDDCZCHiWjL8P+89egbFcq5G7qOBUskLxnKfM4dLZ+t+HZzH5Yp/EL0Yx+Yp8dzbA58WI7JMHpEfuE/qyZhl1MHG+MKwqySGrBEs5U0RXYJgDS7QiSmdgG0qjw7NBgJuC4GsKuYUPQtKqZMpp6fcJlQGdyvvNIsEjnRXwBgIKsiq9NhyUg6whgKK2ic7WN+J+PB3k6mQwn2y05uPo7PbNLs10WkvESQWz9mQQV+7UM+N4KqCJaG2hYumYFXCCU8Uk2FM/dXRDOriSkVwb/ClHpqvRogriGHaxhOIdS2rBzGR5NdF6jDtowxJliKNVtUzAQNY1tYAMDycKtNyVcgtWxGuX7556rqragUpnB8xrQZwmIdok4wZSJy7I8ikdH//aVTyF3LC0y6SVzJXdigerupKNIQUhZucNooInhpmLYJw+t90EUQqja5XZbsWcA6MsajnDJ11bsgpsN+eumzKRhv9Z0KmEdpKZsX0Tal+CY7g59akA50d17uBnjLVHT9zZje8kmBbjRQlXQmEAVOOnnyBzXnIHKzKJqKHx4b0gEXVI6Vxlv93cxNzrW4fYMEkxQwvop1puvqzBhhKxm4Nvtve29jYtSI6VlGDhyN45RqMXjOhx1Uuw3sBM5e7tbMw+EQ1uZFiPwT8RLYROmKD4lwa1A1C2NnCQaYGpcYDVmGIF7VMDgk0udTUefjm9TkaDUx+CFIK9FNGj0ZCp4FwDCs7RAMlU5mnOJBcXGeSsFq7x1OaX8+WEsHnIgCbqLJJZjCBk+3F/r5tKjqTYAQQWAxK4ic0DxLKRiE2VmwSog5myt3dPsAq3T5tVox4Alp+B4YVMtHiIJvkZim38j08OG68CheSNl43neFw6BHedMC5kqyrJXGXOERSGdYLhdHUI6fUEqwzDh0+CSy4JRoMl9/lk0ngrXpME4Zm2f4ySwQdMYMdrXh/1l8crXs8KATwtezPfX7eDgbbYeJY7hwwwi1GWQ15IZKKzrApwnibS09l6cyUt5CHeENl1GvONZpniLp0kzJxg8tEy0RNPX4V3J6Jl6FCkBF6UDvvuznblSIeMSiAsdTUTBNUiOsYOEfe3tmHSIZejQx7g7cZUCiQBd471G2QJkhE7dWM0o+pEhEwBenF/ALJzqT0stUHvcFWE6wGTigjIHbE8g6CxYNfGoPkPBK+Ge/8LYqaWTFHQCQ3eL4PV2+Tq5OlWf7Ddj3/6euNgd37T3Gp92vqs9WVr0NppHbQetY5ax6107fO1H9aGa6P2b+3f23+0/5yJ3lib63zSWhrtv/4FksDOKA==</latexit> L0 = 0T 0 + B1BT 1 = D A graph Laplacian operates on nodes L1 = BT 1 B1 + B2BT 2 Hodge Laplacian operates on oriented edges B1 maps edges to nodes, B2 maps triangles to edges. See Hodge Laplacians on Graphs by Lek-Heng Lim.
  • 54.
    We spent alot of time getting the normalization and connections to random walks right. 54 Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian. Michael T.Schaub,Austin R.Benson,Paul Horn,Gabor Lippner,and Ali Jadbabaie. SIAM Review,2020.
  • 55.
    Flow embeddings arethe higher-order analog of diffusion maps. 55 <latexit sha1_base64="t3a3BIllEKBJoLIHn+m/2Gk55Ew=">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</latexit> N1 ⇥ x1 x2 ⇤ = ⇥ 1x1 2x2 ⇤ for normalized Hodge 1-Laplacian N1.
  • 56.
    Flow embeddings arethe higher-order analog of diffusion maps. 56 <latexit sha1_base64="t3a3BIllEKBJoLIHn+m/2Gk55Ew=">AAAH0nicfVVbb9xEFHYKNGW5NIU3eJmSLEKVd7PeEtIgRVqJqmqlFAK7vUhxtIztY3u0czEz48aJ5QfEK+L/8T/4AZxZb9p4E5gHey7nO5fvnDMTFZwZOxr9vXHrvfc/uL1558PeRx9/8undrXufvTSq1DG8iBVX+nVEDXAm4YVllsPrQgMVEYdX0eIHd/7qDWjDlJzZ8wJOBc0kS1lMLW7Nt/7a+XEehBFkTNaRoFazqqnmAfmaVPNxCDJ5u0sOyZocCTkaSug8aBGXyzFCSQe7Q1KliVRaUM4uICFPVZIBCQZHtOA0ZlQS58fOcL61PRqOloNcnwSryba3Gsfze7f9MFFxKUDamFNjToJRYU9rqi2LOTS9sDRQ0HhBMzjBqaQCzGm9JK4hfdxJlr6lSlqy3O1dhaAeTc87WmpLo5JTXXV3I6UWeGKaDv7E5cXIUkSgIfF1ySFB53imNLO5GMOaeGnTR6c1k0VpQcatg2nJiVXEZY8kTENs+TnpemnZ4sKXLIZU09inwiDvuV8wF5Uv6AJi4Lz114lyFmmqz11w6sz4EWrJtCplYvyCWgtaGr/NnG9yWoDxU2b9mPLYrROHKbiyguqF+S+tQwGW4uGSUw62npWphV8gaWpk4v6j0f0IM7+4KmFzyDSAbOrlz8mc5czCmkzES2hq970i0euT3NrCfL+7a6EaGou6oYpzKjMYxkrs/laCcSVvdoPv9g7GB7sGBMPOiLARxOAMszFwQQyYHETYP6CXcg/3t9tfL3Q0Uuwvx08vzLiKKA9xGTrYBKQpNUwSxbE0JthdsUrgMNTAaXWJVeh8t7xOZsFp7ZLkkt3J6PFsSqUjV4OEMwxAUOynMKWC8fMEUlpy29ShSS/n3YIwqauApte/asxgBiE5HA0P/FgwNJppyrEZ0ICtTOpUdINE3aG0lVM1acG1eXCCXbh32qwH9Riw/TRMz0Wk+BMMqW61mKb+6flRU0tnQrCmFk3N0N1wCvYmYdxI1iHRCrKy4QDTMsJ02tKl9GYD6xamT547Si4NzIIOfXVUNbXh74w44RZdP0NJxwHlRU6bd67++myN9STjwOJ80HJ/0wkm2uDF0705hFNzNctiyjKBlsK2qpy6OoxEHbb7zbWyEEfLm/cGxOqg6Zp4EFYR1SdYfGEeqaoO37hvvxfm7oYiObAst3jv7u8VlvTJLAdCY1tSThDWCxd4Q4yG4z2o+uRy9MljfK2ojIFEYM+wf50sQWPELGnstab6PUKWCgajYQCif4me5kojO0xmREmCRUU4pJYYloBDXIlrO2jeKsGn4eH/KtHLSJZakIQG35dg/TW5Pnk5HgZ7w9HP4+3Jt6uX5o73pfeV940XePvexHvqHXsvvNj7Z+OLje2Nnc3Z5sXm75t/tKK3NlaYz73O2PzzXy26wH8=</latexit> N1 ⇥ x1 x2 ⇤ = ⇥ 1x1 2x2 ⇤ for normalized Hodge 1-Laplacian N1. A B
  • 57.
    Flow embeddings arethe higher-order analog of diffusion maps. 57 <latexit sha1_base64="t3a3BIllEKBJoLIHn+m/2Gk55Ew=">AAAH0nicfVVbb9xEFHYKNGW5NIU3eJmSLEKVd7PeEtIgRVqJqmqlFAK7vUhxtIztY3u0czEz48aJ5QfEK+L/8T/4AZxZb9p4E5gHey7nO5fvnDMTFZwZOxr9vXHrvfc/uL1558PeRx9/8undrXufvTSq1DG8iBVX+nVEDXAm4YVllsPrQgMVEYdX0eIHd/7qDWjDlJzZ8wJOBc0kS1lMLW7Nt/7a+XEehBFkTNaRoFazqqnmAfmaVPNxCDJ5u0sOyZocCTkaSug8aBGXyzFCSQe7Q1KliVRaUM4uICFPVZIBCQZHtOA0ZlQS58fOcL61PRqOloNcnwSryba3Gsfze7f9MFFxKUDamFNjToJRYU9rqi2LOTS9sDRQ0HhBMzjBqaQCzGm9JK4hfdxJlr6lSlqy3O1dhaAeTc87WmpLo5JTXXV3I6UWeGKaDv7E5cXIUkSgIfF1ySFB53imNLO5GMOaeGnTR6c1k0VpQcatg2nJiVXEZY8kTENs+TnpemnZ4sKXLIZU09inwiDvuV8wF5Uv6AJi4Lz114lyFmmqz11w6sz4EWrJtCplYvyCWgtaGr/NnG9yWoDxU2b9mPLYrROHKbiyguqF+S+tQwGW4uGSUw62npWphV8gaWpk4v6j0f0IM7+4KmFzyDSAbOrlz8mc5czCmkzES2hq970i0euT3NrCfL+7a6EaGou6oYpzKjMYxkrs/laCcSVvdoPv9g7GB7sGBMPOiLARxOAMszFwQQyYHETYP6CXcg/3t9tfL3Q0Uuwvx08vzLiKKA9xGTrYBKQpNUwSxbE0JthdsUrgMNTAaXWJVeh8t7xOZsFp7ZLkkt3J6PFsSqUjV4OEMwxAUOynMKWC8fMEUlpy29ShSS/n3YIwqauApte/asxgBiE5HA0P/FgwNJppyrEZ0ICtTOpUdINE3aG0lVM1acG1eXCCXbh32qwH9Riw/TRMz0Wk+BMMqW61mKb+6flRU0tnQrCmFk3N0N1wCvYmYdxI1iHRCrKy4QDTMsJ02tKl9GYD6xamT547Si4NzIIOfXVUNbXh74w44RZdP0NJxwHlRU6bd67++myN9STjwOJ80HJ/0wkm2uDF0705hFNzNctiyjKBlsK2qpy6OoxEHbb7zbWyEEfLm/cGxOqg6Zp4EFYR1SdYfGEeqaoO37hvvxfm7oYiObAst3jv7u8VlvTJLAdCY1tSThDWCxd4Q4yG4z2o+uRy9MljfK2ojIFEYM+wf50sQWPELGnstab6PUKWCgajYQCif4me5kojO0xmREmCRUU4pJYYloBDXIlrO2jeKsGn4eH/KtHLSJZakIQG35dg/TW5Pnk5HgZ7w9HP4+3Jt6uX5o73pfeV940XePvexHvqHXsvvNj7Z+OLje2Nnc3Z5sXm75t/tKK3NlaYz73O2PzzXy26wH8=</latexit> N1 ⇥ x1 x2 ⇤ = ⇥ 1x1 2x2 ⇤ for normalized Hodge 1-Laplacian N1. A B C
  • 58.
    The holes correspondto the idea of homology in algebraic topology. 58 A B First eigenvectors of the graph Laplacian capture (near) connected components, or zeroth-order homology. First eigenvectors of the Hodge Laplacian capture (near) topological holes, or first-order homology A good reference is Hodge Laplacians on Graphs by Lek-Heng Lim.
  • 59.
    We also havesimplicial Personalized PageRank. 59 <latexit sha1_base64="G87jKzztl/ZL8VW+UOYg28uFc2o=">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</latexit> ( I + L1)x = v for (asymmetric) normalized Hodge 1-Laplacian L1.
  • 60.
    Abstract simplicial complexesare another way to model and analyze higher-order network data. 60 • Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian.Michael T.Schaub,Austin R. Benson,Paul Horn,Gabor Lippner,and Ali Jadbabaie.SIAM Review,2020. • Graph-based Semi-Supervised & Active Learning for Edge Flows. Junteng Jia,Michael T.Schaub,Santiago Segarra,and Austin R.Benson.Proc.of KDD,2019. github.com/000Justin000/ssl_edge 1. Algebraic topology provides the computational framework. 2. The hard part is getting a normalization scheme that connects the Hodge Laplacian to diffusions and “respects the topology.” 3. We can apply these ideas to graph algorithms based on random walks. BA Edge View initial position initial position Lifted space upper adjacent walk lower adjacent walk forward backward C
  • 61.
    61 THANKS! Austin R.Benson Slides. bit.ly/arb-TAMU-20 http://cs.cornell.edu/~arb @austinbenson arb@cs.cornell.edu Computational frameworks for higher-order data analysis. A B C Supported by ARO MURI, ARO Award W911NF19- 1-0057, NSF Award DMS-1830274, and JP Morgan Chase & Co. Lots of data available at https://www.cs.cornell.edu/~arb/data/