Alan Kuhnle*, Victoria G. Crawford, and My T. Thai Scalable and Adaptive Algorithms for the Triangle Interdiction Problem on Billion-Scale Networks Alan Kuhnle*, Victoria G. Crawford, and My T. Thai Email: kuhnle@ufl.edu November 20, 2017 IEEE ICDM 2017, New Orleans, LA * Presenter

Network Transitivity 3 × Number of Triangles / Number of Connected Triplets Transitivity: 0.571 Figure degrades all triangles in multiplex by removing one edge from each layer. The smaller the objective, the less reliable the network is.

Transitivity and Reliability Transitivity allows redundant paths Bottom two nodes want to communicate

Transitivity and Social Influence Complex propagation Empirical studies (e.g. Centola1) link transitivity and influence 1 Centola, D. (2010). The Spread of Behavior in an Online Social Network Experiment. Science, 329(5996), 1194–1197.

Triangle Interdiction Problem Transitivity degradation by edge removal Triangle Interdiction Problem (TIP) Collection of triangles Triangles after removal of S

Triangle Interdiction Problem Transitivity: 0 Transitivity: 0.571

TIP Challenges TIP is an NP-hard problem Kortsarz1 introduced a 2-approximation algorithm for TIP Requires up to |E| LP solutions Does not scale to large graphs Only for static graphs Primal-dual algorithm List all triangles, iterate through them. If current triangle is unbroken, remove all of its edges. Efficient 3-approximation but performs close to its worst-case guarantee Modern networks are very large and dynamic 1 Guy Kortsarz. Approximating Maximum Subgraphs without Short Cycles. Siam Journal of Discrete Math, 24(1):255–269, 2010.

Our contributions TARL, a 5/2 approximation that solves one LP Provides speed-up of up to |E| over Kortsarz. Better worst-case guarantee than primal-dual DART, fully dynamic 3-approximation Usually within factor of 1.5 of optimal, better on sparse graphs Efficiently update its solution upon edge / vertex insertion / deletion Average update time on the order of a few microseconds Does not need to store triangle list in memory Runs on network with 4 billion triangles in only 37 GB memory

TIP IP IP formulation of TIP:

TARL Description List all triangles in G Construct and solve LP 1 optimally -> 𝑥 ∗ Break triangles containing a “low LP-weight” edge: For each 𝑒∈𝐸 If 𝑥 ∗ 𝑒 <1/5 For each unbroken triangle containing e, add edge with largest 𝑥 ∗ value into the solution. Break all other triangles: Construct large (at least ½ of edges) bipartite subgraph of residual graph Remove the complement of this graph

TARL TARL has performance guarantee of 5/2 It requires the solution of exactly one LP So provides speed-up over 2-approximation of Kortsarz of up to |E| However, the LP solution is still a large bottleneck

DART Overview Scalable algorithm with competitive ratio of 3 Maintains list of disjoint triangles 𝑈 Solution 𝑆⊆𝑈 Efficient dynamic data structure enables fast updates Vertices Edges Triangles (in 𝑈) Pointers from Edges to Triangles tell which triangle in U the edge belongs to. Pointers from Triangles back to edges tell which edges form the triangle. Pointers from Vertices to Edges tell which Edges are incident with that vertex. Easy to update since all three of these are adjacency lists K is number of layers in multiplex

DART Overview Visualization 𝑈 is maximal Disjoint triangles, 𝑈 Solution 𝑆⊆𝑈 (red edges) 𝑊=𝑈\S (green edges) 𝑈 is maximal 𝑊 comprises all edges not forced to be in 𝑆 by adjacent triangles

DART Overview All DART methods rely on the following two procedures: The AUGMENT procedure Takes an input a single edge e Ensures feasibility with respect to triangles containing e Maintains disjoint triangle list U The PRUNE procedure DART DART-BASE DART-ADD DART-REMOVE

DART Overview DART-BASE calls AUGMENT on each edge 𝑒 DART-ADD calls AUGMENT on newly inserted edge 𝑒 DART-REMOVE Determines which triangle 𝑇 in 𝑈 an edge 𝑒 belongs to, if any Removes 𝑇= {𝑒,𝑓,𝑔} from 𝑆∪𝑊 Calls AUGMENT on 𝑓,𝑔 All three DART procedures conclude with a call to PRUNE

DART Overview AUGMENT takes as input an edge e Ensures 𝑆 hits all triangles containing e Avoids adding new triangles to U if possible Otherwise, adds new disjoint triangles to 𝑈

DART Overview Have the following performance guarantee for DART

Results DART returns solution close to OPT (compared on small ER network( 30 nodes)) at much faster running times (here TARL is an alg. We formulated but unscalable).

Visualization on small ER graph

Results on smaller traces

DART Scalability In practice, DART is capable of running on networks with billions of triangles in under 2 hours It can update its solution in microseconds |S| is size of solution. Mem is max. memory used. Dynamic times are averaged over 1 million edge additions /removals. The size increase / decrease is how much solution changed after adding /removing all 1 million edges. Repeated 10 times and averaged over all.

Conclusions Provided two algorithms, TARL and DART, to approximate TIP: find minimum-size set of edges whose removal breaks all triangles. TARL Performance guarantee of 5/2 Provides significant speed-up over Kortsarz (up to 103 times faster) DART Competitive guarantee of 3 Performs close to optimal on sparse networks Fully dynamic Thank you! Questions? Contact: kuhnle@ufl.edu