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Hypergraph Cuts with General Splitting Functions

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Talk at PANNG mini-symposium, SIAM MDS 2020.

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Hypergraph Cuts with General Splitting Functions

  1. 1. 1 Joint work with Nate Veldt & Jon Kleinberg (Cornell) Hypergraph Cuts with General Splitting Functions Austin R. Benson · Cornell University Pattern Analysis for Networks and Network Generalizations SIAM MDS 2020 · June 16, 2020 Slides. bit.ly/arb-PANNG-20
  2. 2. Graph minimum s-t cuts are fundamental. 2 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] • Computer vision [Bokykov-Kolmogorov 01; Kolmogorov-Zabih 04] • Densest subgraph [Goldberg 84; Shang+ 18] • First graph-based semi-supervised learning algorithms [Blum-Chawla 01] • Local graph clustering [Andersen-Lang 08; Oreccchia-Zhu 14; Veldt+ 16] Also see any undergraduate algorithms class poly-time algorithms!
  3. 3. Real-world systems are composed of“higher-order” interactions that we can model with hypergraphs. 3 Physical proximity • nodes are students • hyperedges are students in the same class Drug compounds • nodes are substances • hyperedges are substances combined in a drug linear-algebra discrete-mathematics math-software combinatorics category-theory logic terminology algebraic-graph-theory combinatorial-designs hypergraphs graph-theory cayley-graphs group-theory finite-groups Categorical information • nodes are tags • hyperedges are groups of tags (e.g.,for the same question on mathoverflow.com) Networks beyond pairwise interactions: structure and dynamics. Battiston et al., 2020. The why, how, and when of representations for complex systems. Torres et al., 2020.
  4. 4. Real-world systems are composed of“higher-order” interactions that we can model with hypergraphs. 4 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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  5. 5. 5 1. What is a hypergraph minimum s-t cut? 2. If we know what they are, can we find them efficiently? 3. If we can find them efficiently, what can we use them for? We should have a foundation for hypergraph minimum s-t cuts,but…
  6. 6. What is a hypergraph minimum s-t cut? 6 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] 1 3 2 4 5 6 7 8 s t There is only one way to split an edge (1/1).
  7. 7. We model hypergraph cuts with splitting functions. 7 s t Non-negativity we(U) 0 for all U ⇢ e. Symmetry we(U) = we(eU) for all U ⇢ e. Non-split ignoring we(e) = we(;) = 0.<latexit 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Splitting function for separating edge e into U and U e. For each edge e, we have a function we with minimizeS⇢V P e2E we(e S) ⌘ cutH(S) subject to s 2 S, t /2 S.<latexit 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Hypergraph minimum s-t cut problem. 1. Anonymity. A node’s identity doesn’t affect the function. 2. Heterogeneity. Same splitting function at each edge. Cardinality-based splitting functions. 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cutH(S) = f (2) + f (1)<latexit 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we(U) = f (min(|U|, |Ue|))<latexit 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  8. 8. Cardinality-based splitting functions appear throughout the literature. 8 [Lawler 73; Ihler+ 93; Yin+ 17] [Hu-Moerder 85; Heuer+ 18] [Agarwal+ 06; Zhou+ 06; Benson+ 16] [Yaros- Imielinski 13] [Li-Milenkovic 18] All-or-nothing we(U) = ( 0 if U 2 {e, ;} 1 otherwise Linear penalty we(U) = min{|U|, |eU|} Quadratic penalty we(U) = |U| · |eU| Discount cut we(U) = min{|U|↵ , |eU|↵ } L-M submodular we(U) = 1 2 + 1 2 · min n 1, |U| b↵|e|c , |eU| b↵|e|c o <latexit 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  9. 9. Cardinality-based splitting functions are easy to specify. 9 Cardinality-based splitting functions. minimizeS⇢V P e2E we(e S) ⌘ cutH(S) subject to s 2 S, t /2 S.<latexit 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sha1_base64="dNi2W8uQiA5FM9ge7UsagYj+LZU=">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</latexit> s t One extra scaling DOF, so set w1 = 1. Specify w2, ... , wbr/2c.<latexit 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sha1_base64="SMjjx0KffHfUKRIVd6aJj9NDt0M=">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</latexit><latexit sha1_base64="SMjjx0KffHfUKRIVd6aJj9NDt0M=">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</latexit> Non-negativity we(U) 0 for all U ⇢ e. Non-split ignoring we(e) = we(;) = 0. C-B we(U) = f (min(|U|, |Ue|)).<latexit 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cutH(S) = f (2) + f (1) = w2 + 1<latexit 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Only need to specify f(1), f(2), …, f(⌊r / 2⌋), where r = max hyperedge size. Just scalars. f(i) = wi.
  10. 10. Cardinality-based splitting functions are easy to specify. 10 Just need to specify w2, ... , wbr/2c and assume w1 = 1.<latexit 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r = 2 (graphs) r = 3 (3-uniform hypergraph) “Only one way to split a triangle” [Benson+ 16; Li-Milenkovic 17; Yin+ 17] s t s t s t r = 4 w2 = 0.5 solution w2 = 1.5 solution w3 = 1.5 solution
  11. 11. 1.0 1.25 1.5 1.75 2.0 fusion- systems topological- stacks graph- invariants adjacency- matrix signed- graph gorenstein cohen- macaulay topological- k- theory difference- sets pushforward regular- rings graph- connectivity block- matrices directed- graphs eulerian- path central- extensions group- extensions semidirect- product wreath- product graded- algebras supergeometry geometric- complexity soliton- theory matrix- congruences teichmueller- theory superalgebra string- theory riemann- surfaces group- cohomology dglas celestial- mechanics s- seed = symplectic- linear- algebra t- seed = bernoulli- numbers Different weights lead to different min cuts in practice. 11 1.00 1.25 1.50 1.75 2.00 0.7 0.8 0.9 1.0 JaccardSimilarity
  12. 12. 12 1. What is a hypergraph minimum s-t cut? 2. If we know what they are, can we find them efficiently? 3. If we can find them efficiently, what can we use them for? We should have a foundation for hypergraph minimum s-t cuts,but…
  13. 13. We solve hypergraph cut problems with graph reductions. 13 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.
  14. 14. s t s t s t s t Existing gadgets model cardinality-based splitting functions. 14 1/21/2 1/2 1 1 1 1 ∞ ∞ ∞ ∞ ∞∞ clique expansion star expansion Lawler gadget [1973]hyperedge Quadratic penalty wi = i ( k – i ) k = hyperedge size Linear penalty wi = i All-or-nothing wi = 1
  15. 15. s t Existing gadgets model cardinality-based splitting functions. 15 1 ∞ ∞ ∞ ∞ ∞∞s t 1 ∞ ∞ ∞ ∞ ∞∞with s with t with t must go with s must go with t ⟶ penalty = 1 1 ∞ ∞ ∞ ∞ ∞∞with s with s with s must go with s must go with s ⟶ penalty = 0 Directed min s-t graph cut

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