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Final_Presentation

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Final_Presentation

  1. 1. 2
  2. 2. 3
  3. 3. 4
  4. 4. 5 Community Detection
  5. 5. Social Networks can be represented in graphs Nodes correspond to individuals Edges represent interaction among them A community can be defined as a group of entities that share similar properties 6
  6. 6. 7
  7. 7. 8
  8. 8. 10
  9. 9. Compute the distance between all vertices and communities Choose two communities based on their similarity Update the distance between communities Merge these two communities into a new community Walk-Trap 11
  10. 10. 12
  11. 11. 13 Modularity is based on the idea that a random graph is not expected to have a community structure π‘€π‘œπ‘‘π‘’π‘™π‘Žπ‘Ÿπ‘–π‘‘π‘¦ = 𝑄 = 1 2π‘š 𝑖𝑗 (𝐴𝑖𝑗 βˆ’ 𝑑𝑖 𝑑𝑗 2π‘š )𝛿(𝐢𝑖, 𝐢𝑗) A: Adjacency Matrix m: the total number of edges in the network 𝑑𝑖: degree of node i 𝛿(𝐢𝑖, 𝐢𝑗) = 1, 𝐢𝑖 = 𝐢𝑗 0, 𝐢𝑖 β‰  𝐢𝑗 The choice of null model is in principle arbitrary, and several possibilities exist
  12. 12. 𝑸 π’Žπ’‚π’™ = 𝐦𝐚𝐱 𝒑 𝒄=𝟏 𝒏 𝒄 𝑰 𝒄 π’Ž βˆ’ 𝒅 𝒄 πŸπ’Ž 𝟐 = 𝟏 π’Ž 𝐦𝐚𝐱 𝒑 𝒄=𝟏 𝒏 𝒄 𝑰(π‘ͺ) βˆ’ 𝑬𝒙(𝑰 𝒄 ) =- 𝟏 π’Ž π¦π’Šπ’ 𝒑 βˆ’ 𝒄=𝟏 𝒏 𝒄 𝑰(π‘ͺ) βˆ’ 𝑬𝒙(𝑰 𝒄 ) =- 𝟏 π’Ž π¦π’Šπ’ 𝒑 π’Ž βˆ’ 𝒄=𝟏 𝒏 𝒄 𝑰(𝒄) βˆ’ π’Ž βˆ’ 𝒄=𝟏 𝒏 𝒄 𝑬𝒙(𝑰 𝒄 ) =- 𝟏 π’Ž π¦π’Šπ’ 𝒑 ( π‘ͺ𝒖𝒕 𝒑 βˆ’ 𝑬𝒙π‘ͺ𝒖𝒕 𝒑) Intra-community edges 14 Inter-community edges
  13. 13. 15 Each node is assigned to its own community The algorithm repeatedly merges pairs of communities together Repeat the procedure until only one community remains Choose the merger for which the resulting modularity is the largest. FastQ
  14. 14. 16 Proposed Method
  15. 15. 17 A majority of community detection methods try to optimize a global metric Several of methods need initial parameters to find out the problems A centralized decision maker has been proposed by most of the algorithms
  16. 16. A distributed framework has been proposed to detect social networks communities Each community acts as a selfish agent to maximize its utility function We use local utility maximization Modularity has been chosen as the community utility function 18
  17. 17. Each community just uses local information to maximize its utility function Each community has some pre-defined actions Each community chooses the best action in order to have maximum utility Our distributed framework can perform as well as the existing centralized approaches 19
  18. 18. Local information is used to identify communities Every community only utilizes the knowledge obtained from its neighbors Nodes belonging to a community fall into two types: 1-Core Set(C): no node in C is linked to the outside of the community 2-Boundary Set(B): every node in B has at least one connection to the outside of the community 20
  19. 19. 𝑸 = π‘ͺ=𝟏 𝒏 π‘ͺ 𝑰(π‘ͺ) π’Ž βˆ’ 𝑫 π‘ͺ πŸπ’Ž 𝟐 𝐃 𝐂 = 𝐝 𝟏 + 𝐝 𝟐 + β‹― + 𝐝 𝐧 𝟐 = 𝐝 𝟏 𝟐 + 𝐝 𝟐 𝟐 + β‹― + 𝟐𝐝 𝟏 𝐝 𝟐 + β‹― + 𝐝 𝐧 𝟐𝐈 𝐂 = 𝐒𝐣 𝐀 𝐒𝐣 π‘ͺ 𝑰(π‘ͺ) 𝑫 π‘ͺ π’Ž
  20. 20. 22 C1 C2 C C C1 C2
  21. 21. There exist 2 Possible Merge for C1 𝑼 𝟏 = 𝑰(π‘ͺ 𝟏) π’Ž βˆ’ 𝑫 π‘ͺ 𝟏 πŸπ’Ž 𝟐 = πŸ‘ 𝟏𝟏 βˆ’ 𝟏𝟎 𝟐𝟐 𝟐 = 𝟎. πŸŽπŸ”πŸ” 𝑼 πŸ‘ = 𝑰(π‘ͺ πŸ‘) π’Ž βˆ’ 𝑫 π‘ͺ πŸ‘ πŸπ’Ž 𝟐 = 𝟏 𝟏𝟏 βˆ’ πŸ“ 𝟐𝟐 𝟐 = 𝟎. πŸŽπŸ‘πŸ— 𝑼 = 𝑰 π‘ͺ 𝟏 + 𝑰 π‘ͺ πŸ‘ + 𝒙 π’Ž βˆ’ 𝑫 π‘ͺ 𝟏 + 𝑫 π‘ͺ πŸ‘ πŸπ’Ž 𝟐 = πŸ• 𝟏𝟏 βˆ’ πŸπŸ“ 𝟐𝟐 𝟐 = 𝟎. πŸπŸ•πŸ Suppose C1 is a player Merging between C1 & C3 occurs If and only if 𝐔 > 𝐔 𝟏 + 𝐔 πŸ‘ 𝒙 > 𝑫 π‘ͺ 𝟏 𝑫 π‘ͺ πŸ‘ πŸπ’Ž = πŸ‘ > πŸ“πŸŽ 𝟐𝟐 = 𝟐. πŸπŸ• 23
  22. 22. 25 Our goal is to find a division in which modularity has been maximized 𝐬𝐒 = βˆ’πŸ C C1 C2 𝐒𝐒 = +𝟏 𝐐 = 𝟏 𝟐𝐦 𝐒𝐣 (𝐀𝐒𝐣 βˆ’ 𝐝𝐒 𝐝𝐣 𝟐𝐦 ) 𝐬𝐒 𝐬𝐣 𝑸 = 𝟏 πŸπ’Ž 𝒔 𝑻 𝑩𝒔 S is a vector whose elements are π’”π’Š 𝑺 = π’Š=𝟏 𝒏 πœΆπ’Š π’–π’Š π’–π’Š is ith Eigen vector of B B is a modularity matrix whose elements are: π‘©π’Šπ’‹ = (π‘¨π’Šπ’‹ βˆ’ π’…π’Š 𝒅𝒋 πŸπ’Ž ))
  23. 23. 26 𝒙 > 𝑫 π‘ͺ 𝟏 𝑫 π‘ͺ 𝟐 πŸπ’Ž , 𝒙 < 𝑫 π‘ͺ 𝟏 𝑫 π‘ͺ 𝟐 πŸπ’Ž ,
  24. 24. The proposed method may get stuck at a local modularity It may be possible that no community can improve itself and also modularity is not maximized 27
  25. 25. 28 U1 U2 U 𝑼 𝟏 β€² 𝑼 𝟐 β€² 𝐔 < 𝐔 𝟏 + 𝐔 𝟐 Merging between C1 and C2 is irrational 𝐔 𝟏 + 𝐔 𝟐 < 𝐔′ 𝟏 + 𝐔′ 𝟐 But splitting of C is rational
  26. 26. 29 U1 U2 x C 𝑼 𝟐 β€² x’ Irrational Merge Split Condition 𝐔 𝟏 + 𝐔 𝟐 < 𝐔′ 𝟏 + 𝐔′ 𝟐 (π’™β€²βˆ’π’™) π’Ž > πŸπ‘« 𝒄 𝟏 𝑫 𝒄 𝟐 βˆ’ πŸπ‘« 𝒄 𝟏 β€² 𝑫(𝒄 𝟐 β€² ) πŸ’π’Ž 𝟐 𝑫 π‘ͺ 𝟏 + 𝑫 π‘ͺ 𝟐 = 𝑫 π‘ͺ 𝟏 β€² + 𝐃(𝐂 𝟐 β€² ) 𝑰 π‘ͺ 𝟏 + 𝑰 π‘ͺ 𝟐 + 𝒙 = 𝑰 π‘ͺ 𝟏 β€² + 𝐈 𝐂 𝟐 β€² + 𝐱′ 𝑼 𝟏 β€²
  27. 27. 30
  28. 28. 31 𝒙 > 𝑫 π‘ͺ 𝟏 𝑫 π‘ͺ 𝟐 πŸπ’Ž , 𝒙 < 𝑫 π‘ͺ 𝟏 𝑫 π‘ͺ 𝟐 πŸπ’Ž (π’™β€²βˆ’π’™) π’Ž > πŸπ‘« 𝒄 𝟏 𝑫 𝒄 𝟐 βˆ’πŸπ‘« 𝒄 𝟏 β€² 𝑫(𝒄 𝟐 β€² ) πŸ’π’Ž 𝟐
  29. 29. 32 Experimental Results
  30. 30. DataSet Number of Nodes Number of edges Karate 34 77 Risk 42 83 Dolphin 62 159 Politics 105 441 AdjNoun 112 425 Football 115 613 Jazz 198 2742 USAir97 332 2126 Email 1133 5452 Power 4941 6594 Internet 22960 48436 33
  31. 31. 34 The community structures of the ground truth communities and those detected by 1st proposed Method and 2nd proposed method on Zachary’s karate club network.
  32. 32. 35 The community structures of the ground truth communities and those detected by 1st proposed Method and 2nd proposed method on Dolphin Network.
  33. 33. 36 The community structures of the ground truth communities and those detected by 1st proposed Method and 2nd proposed method on NCCA Football Network.
  34. 34. 37 The community structures of the 1st proposed Method and 2nd proposed method on Risk Network.
  35. 35. 38 The community structures of the 1st proposed Method and 2nd proposed method on Politics Network.
  36. 36. 0 1 2 3 4 5 6 Karate Risk Dolphin Politics AdjNoun Football Jazz USAir97 Email Power Internet Rank Dataset Rank of Modularity per Dataset 39
  37. 37. DataSet FastQ walktrap Laplacian SLAP 1st Proposed Method 2nd Proposed Method Karate 0.252 0.36 0.255 0.399 0.4197 0.4197 Risk 0.624 0.624 0.624 0.626 0.631 0.637 Dolphin 0.341 0.517 0.365 0.511 0.509 0.529 Politics 0.447 0.524 0.527 0.494 0.52 0.527 AdjNoun 0.1845 0.229 0.259 0.286 0.272 0.306 Football 0.577 0.604 0.604 0.6045 0.6043 0.6045 Jazz 0.403 0.437 0.441 0.428 0.425 0.444 USAir97 0.29 0.315 0.363 0.351 0.356 0.366 Email 0.506 0.534 0.543 0.47 0.548 0.566 Power 0.447 0.886 0.932 0.64 0.933 0.939 Internet 0.472 0.647 0.646 0.574 0.588 0.6489 Modularity Obtained From Several Popular Approaches And Our Proposed Method On Real World Networks 40
  38. 38. 0 1 2 3 4 5 6 Karate Risk Dolphin Politics AdjNoun Football Jazz USAir97 Email Power Rank Dataset Rank of Execution Time per Dataset FastQ walktrap SLAP 1st Prposed Method 2nd Proposed Method 41
  39. 39. DataSet FastQ walktrap SLAP 1st Proposed Method 2nd Proposed Method Karate 77 77 45 31 39 Risk 93 84 38 18 90 Dolphin 211 117 63 54 141 Politics 414 197 88 107 314 AdjNoun 426 194 82 126 379 Football 350 190 100 152 380 Jazz 740 314 295 625 1100 USAir97 3600 497 211 1020 4200 Email 5452 1833 458 2042 8201 Power 31458 7153 762 9472 39763 42
  40. 40. 43 0 1 2 3 4 5 6 100 200 300 400 500 600 700 800 900 1000 Rank Dataset Rank of Modularity per Dataset(MU=0.3) FastQ walktrap Laplacian SLAP 1st Prposed Method 2nd Proposed Method
  41. 41. DataSet FastQ walktrap Laplacian SLAP 1st Proposed Method 2nd Proposed Method 100 0.35 0.365 0.365 0.365 0.324 0.365 200 0.500 0.549 0.549 0.549 0.523 0.549 300 0.541 0.593 0.593 0.563 0.549 0.593 400 0.562 0.606 0.606 0.6058 0.58 0.606 500 0.574 0.613 0.613 0.613 0.602 0.613 600 0.597 0.608 0.608 0.587 0.589 0.608 700 0.591 0.612 0.612 0.612 0.604 0.612 800 0.59 0.613 0.613 0.613 0.596 0.613 900 0.595 0.611 0.611 0.610 0.579 0.613 1000 0.59 0.609 0.609 0.609 0.586 0.609 MODULARITY OBTAINED FROM SEVERAL POPULAR APPROACHES AND OUR PROPOSED METHOD ON SYNTHETIC NETWORK(MU=0.3) 44
  42. 42. 45 0 1 2 3 4 5 6 100 200 300 400 500 600 700 800 900 1000
  43. 43. DataSet FastQ walktrap SLAP 1st Proposed Method 2nd Proposed Method 100 332 196 123 116 240 200 649 364 212 485 731 300 1162 480 390 780 1340 400 1284 682 486 992 2210 500 1555 873 685 1240 3406 600 2060 1041 1047 1570 4210 700 2776 1289 1358 1743 5378 800 2745 1580 1637 2020 6421 900 3980 1860 2438 2320 8745 1000 3565 2179 2599 2610 10255 The Execution Time From Several Popular Approaches And Our Proposed Method On Synthetic Network(mu=0.3) 46
  44. 44. 47 0 1 2 3 4 5 6 100 200 300 400 500 600 700 800 900 1000 Rank Dataset Rank of Modularity per Dataset(Mu=0.5) FastQ walktrap Laplacian SLAP 1st Prposed Method 2nd Proposed Method
  45. 45. DataSet FastQ walktrap Laplacian SLAP 1st Proposed Method 2nd Proposed Method 100 0.233 0.202 0.238 0.229 0.231 0.253 200 0.27 0.356 0.356 0.332 0.288 0.355 300 0.344 0.407 0.402 0.395 0.352 0.407 400 0.363 0.431 0.425 0.406 0.391 0.431 500 0.372 0.433 0.433 0.433 0.406 0.434 600 0.367 0.439 0.426 0.406 0.403 0.44 700 0.377 0.435 0.427 0.425 0.400 0.436 800 0.374 0.428 0.429 0.416 0.396 0.432 900 0.365 0.429 0.43 0.424 0.408 0.43 1000 0.375 0.436 0.431 0.435 0.415 0.436 48 MODULARITY OBTAINED FROM SEVERAL POPULAR APPROACHES AND OUR PROPOSED METHOD ON SYNTHETIC NETWORK(MU=0.5)
  46. 46. 49 0 1 2 3 4 5 6 100 200 300 400 500 600 700 800 900 1000 Rank Dataset Rank of Execution Time per Dataset(Mu=0.5) FastQ walktrap SLAP 1st Proposed Method 2nd Proposed Method
  47. 47. DataSet FastQ walktrap SLAP 1st Proposed Method 2nd Proposed Method 100 354 191 102 91 221 200 723 377 199 463 621 300 886 472 392 720 1420 400 1288 677 544 1009 2451 500 1654 879 734 1120 3231 600 2180 1049 1061 1680 4621 700 2864 1295 1738 1920 5145 800 2848 1583 1896 2007 6352 900 3305 1867 2366 2247 8745 1000 3859 2172 2700 2670 11471 The Execution Time From Several Popular Approaches And Our Proposed Method On Synthetic Network(mu=0.5) 50
  48. 48. 51 0 1 2 3 4 5 6 100 200 300 400 500 600 700 800 900 1000 Rank Dataset Rank of Modularity per Dataset(Mu=0.7) FastQ walktrap Laplacian SLAP 1st Prposed Method 2nd Proposed Method
  49. 49. DataSet FastQ walktrap Laplacian SLAP 1st Proposed Method 2nd Proposed Method 100 0.234 0.196 0.244 0.242 0.23 0.254 200 0.168 0.144 0.178 0.154 0.159 0.179 300 0.155 0.174 0.189 0.166 0.141 0.19 400 0.169 0.239 0.236 0.231 0.177 0.232 500 0.180 0.247 0.245 0.238 0.204 0.247 600 0.181 0.257 0.255 0.202 0.206 0.257 700 0.184 0.26 0.254 0.236 0.229 0.259 800 0.182 0.259 0.255 0.231 0.233 0.259 900 0.185 0.262 0.258 0.252 0.23 0.262 1000 0.180 0.26 0.257 0.23 0.231 0.26 52 Modularity Obtained From Several Popular Approaches And Our Proposed Method On Synthetic Network(mu=0.7)
  50. 50. 53 0 1 2 3 4 5 6 100 200 300 400 500 600 700 800 900 1000 Rank Dataset Rank of Execution Time per Dataset(Mu=0.7) FastQ walktrap SLAP 1st Proposed Method 2nd Proposed Method
  51. 51. DataSet FastQ walktrap SLAP 1st Proposed Method 2nd Proposed Method 100 330 197 111 105 320 200 608 382 259 370 591 300 951 486 383 690 1345 400 1321 693 687 997 2684 500 1569 886 1045 1140 3354 600 2011 1085 1041 1620 4574 700 2198 1374 1492 1749 5354 800 2813 1541 1968 1984 6478 900 2836 1841 2408 2146 8894 1000 3823 2200 2618 2541 12577 The Execution Time From Several Popular Approaches And Our Proposed Method On Synthetic Network(mu=0.7) 54
  52. 52. 55
  53. 53. 56
  54. 54.              57
  55. 55. 58
  56. 56. 59

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