Parallel Algorithms for Generating Random Networks with Given Degree - PowerPoint PPT Presentation

. . . . . . . . . . . . . . . Parallel Algorithms for Generating Random Networks with Given Degree Sequences Maleq Khan 2 1 Department of Computer Science Virginia Tech 2 Network Dynamics and Simulation Science Laboratory Virginia Bioinformatics Institute September 17, 2015 Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 34 Maksudul Alam 1 , 2

. Massive Random Network . . . . . . . . . . Outline 1 Emergence of Massive Real-World Networks Real-World Networks . 2 Generating Networks with a Given Expected Degree Sequence Chung–Lu Model Generative Algorithm for the Chung–Lu Model 3 Parallel Generation of Networks using Chung–Lu Model Challenges Parallel Algorithm Load Balancing and Partitioning Results Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 34

. . . . . . . . . . . . . . . Emergence of Massive Real-World Networks Massive Random Network Emergence of Massive Networks Random graphs are used to model complex real-world systems We need massive networks for realistic analysis Many patterns emerge only in massive datasets A smaller network may not exhibit the same behavior as that of a larger network [Leskovec 2008] Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 / 34

. . . . . . . . . . . . . Emergence of Massive Real-World Networks . Real-World Networks Real-World Networks Real-world networks are massive in size Many networks exhibits power-law degree distribution Many networks have no well-defined degree distribution How to generate networks with these diverse distributions? Twitter 316 M active users Facebook 1.49 B active users Degree Distribution Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 34

. . . . . . . . . . . . . . . Emergence of Massive Real-World Networks Real-World Networks Diverse Degree Distributions Twitter Friendstar Blackout Miami Earth Quake Fires Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 / 34

. Generating Networks with a Given Expected Degree Sequence . . . . . . . . . . . . Chung–Lu Model . Chung–Lu Model Chung–Lu model generates networks from any expected degree seq. w i w j If no self-loop is allowed, expected degree of node i is: w i w j w 2 i w 2 i Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . . 6 / 34 . . . . . . . . . . . . . . Given n nodes V = 0 , 1 , 2 , . . . , n − 1 with corresponding weights w = w 0 , w 1 , . . . , w n − 1 An edge between nodes i and j is added with probability p i , j = ∑ k w k ∑ j w j − w i ∑ E [ d i ] = = w i = w i − ∑ ∑ ∑ i ̸ = j k w k k w k k w k In large graph typically: E [ d i ] ≈ w i as k w k → 0 [Miller 2011] ∑

. 3: . . . . . . . . Generating Networks with a Given Expected Degree Sequence Generative Algorithm for the Chung–Lu Model Naïve Sequential Algorithm for the CL Model Algorithm 1.1: Sequential Algorithm 2: 5: . 6: 7: 8: w i w j S 9: return E For n nodes there are 2 2 How to improve it? Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . . . . . . 7 / 34 . . . . . . . . . . . . . . 1: procedure Serial–Chung–Lu( w , V ) S ← ∑ k ∈ V w k E ← Create–Edges( w , S , V ) 4: procedure Create–Edges( w , S , V ) E ← ∅ for all i ∈ [ 0 , n − 2 ] do for all j ∈ [ i + 1 , n − 1 ] do Add edge ( i , j ) to E with probability = n ( n − 1 ) ( n ) = O ( n 2 ) possible edges Tierefore the runtime complexity of naïve algorithm is O ( n 2 )

. 3: . . . . . . . . Generating Networks with a Given Expected Degree Sequence Generative Algorithm for the Chung–Lu Model Naïve Sequential Algorithm for the CL Model Algorithm 1.1: Sequential Algorithm 2: 5: . 6: 7: 8: w i w j S 9: return E For n nodes there are 2 2 How to improve it? Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . . . . . . 7 / 34 . . . . . . . . . . . . . . 1: procedure Serial–Chung–Lu( w , V ) S ← ∑ k ∈ V w k E ← Create–Edges( w , S , V ) 4: procedure Create–Edges( w , S , V ) E ← ∅ for all i ∈ [ 0 , n − 2 ] do for all j ∈ [ i + 1 , n − 1 ] do Add edge ( i , j ) to E with probability = n ( n − 1 ) ( n ) = O ( n 2 ) possible edges Tierefore the runtime complexity of naïve algorithm is O ( n 2 )

. . . . . . . . . . . . . . . . Generating Networks with a Given Expected Degree Sequence Generative Algorithm for the Chung–Lu Model Recall Erdos–Renyi Model Probability of an edge between nodes i and j is p (constant) For every possible node pairs, add an edge with probability p Generating an ER network with 5 nodes Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . 8 / 34 . . . . . . . . . . . . . A set of n nodes V = { 0 , 1 , 2 , . . . , n − 1 } 0 4 1 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 (0,1) (0,2) (0,3) (0,4) (1,2) (1,3) (1,4) (2,3) (2,4) (3,4) 3 2 Graph List of all possible edges Runtime complexity of O ( n 2 ) Batagelj and Brandes proposed an effjcient O ( n + m ) algorithm

. . . . . . . . . . . . . . . . Generating Networks with a Given Expected Degree Sequence Generative Algorithm for the Chung–Lu Model Recall Erdos–Renyi Model Probability of an edge between nodes i and j is p (constant) For every possible node pairs, add an edge with probability p Generating an ER network with 5 nodes Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . 8 / 34 . . . . . . . . . . . . . A set of n nodes V = { 0 , 1 , 2 , . . . , n − 1 } 0 4 1 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 (0,1) (0,2) (0,3) (0,4) (1,2) (1,3) (1,4) (2,3) (2,4) (3,4) 3 2 Graph List of all possible edges Runtime complexity of O ( n 2 ) Batagelj and Brandes proposed an effjcient O ( n + m ) algorithm

. . . . . . . . . . . . . . . . Generating Networks with a Given Expected Degree Sequence Generative Algorithm for the Chung–Lu Model Recall Erdos–Renyi Model Probability of an edge between nodes i and j is p (constant) For every possible node pairs, add an edge with probability p Generating an ER network with 5 nodes Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . 8 / 34 . . . . . . . . . . . . . A set of n nodes V = { 0 , 1 , 2 , . . . , n − 1 } 0 4 1 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 (0,1) (0,2) (0,3) (0,4) (1,2) (1,3) (1,4) (2,3) (2,4) (3,4) 3 2 Graph List of all possible edges Runtime complexity of O ( n 2 ) Batagelj and Brandes proposed an effjcient O ( n + m ) algorithm

. . . . . . . . . . . . . . . . Generating Networks with a Given Expected Degree Sequence Generative Algorithm for the Chung–Lu Model Recall Erdos–Renyi Model Probability of an edge between nodes i and j is p (constant) For every possible node pairs, add an edge with probability p Generating an ER network with 5 nodes Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . 8 / 34 . . . . . . . . . . . . . A set of n nodes V = { 0 , 1 , 2 , . . . , n − 1 } 0 4 1 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 (0,1) (0,2) (0,3) (0,4) (1,2) (1,3) (1,4) (2,3) (2,4) (3,4) 3 2 Graph List of all possible edges Runtime complexity of O ( n 2 ) Batagelj and Brandes proposed an effjcient O ( n + m ) algorithm

. . . . . . . . . . . . . . . Generating Networks with a Given Expected Degree Sequence Generative Algorithm for the Chung–Lu Model Discarded Potential Edges A sequence of potential edges are discarded before creating an edge What if we could count the number of discarded edges beforehand and add the next potential edge to the graph? Tie only challenge is how to determine the number of discarded edges and still maintain the same edge creation probability p Maksudul Alam (Virginia Tech) Generating Rand. Net. with Given Deg. Seq. September 17, 2015 . . . . . . . . . . . . . . . . 9 / 34 . . . . . . . . . . . . . It would take O ( m ) time (instead of O ( n 2 ) ), where m is the number of edges 2 edges discarded 4 edges discarded 0 4 1 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 (0,1) (0,2) (0,3) (0,4) (1,2) (1,3) (1,4) (2,3) (2,4) (3,4) 3 2 Graph List of all possible edges