Topology and biology: Persistent homology of aggregation models

Transcription

1 Topology and biology: Persistent homology of aggregation models Chad Topaz, Lori Ziegelmeier, Tom Halverson Macalester College Ghrist (2008)

2 If I can do applied topology, YOU can do applied topology Chad Topaz, Lori Ziegelmeier, Tom Halverson Macalester College Ghrist (2008)

3 The big picture Questions about biological aggregations (and maybe agent-based systems in general?). How does each individual behave? 2. How does the group behave? 3. How are individual and group behavior linked? Demonstrate utility of computational persistent homology in addressing #2 above

4 What is this group doing?

5 What is this group doing? r = 0.2 m

6 What is this group doing?

7 Aggregation Models

8 Example : Vicsek Model

9 Example : Vicsek Model Averaging + Noise

10 Example : Vicsek Model i N X x i x j appler j + U( /2, /2) v i v 0 (cos i, sin i ) x i x i + v i t L

11 Example : Vicsek Model '(t) = Nv 0 NX i= v i (t) Order Parameter ϕ (A)

12 Example : Vicsek Model 0 '(t) = Nv 0 NX 0 7 i= v i (t) 5 (C) 0 0 5

13 Example : Vicsek Model! =! = 0

14 Example 2: D Orsogna Model

15 Example 2: D Orsogna Model ẋ i = v i m v i = v i 2 v i r i Q i Q i = X j6=i C r e x i x j /L r C a e x i x j /L a.

16 Example 2: D Orsogna Model Social potential Q(r) r = interorganism distance

17 Example 2: D Orsogna Model ẋ i = v i m v i = v i 2 v i r i Q i Q i = X j6=i C r e x i x j /L r C a e x i x j /L a.

18 Example 2: D Orsogna Model.00 (A) Order Params Simulation Time Order Parameter P Polarization Ang. Momentum M M abs Abs. Ang. Mom.

19 Persistent Homology

20 The big picture: Study data via topology. Envision data as high dimensional point cloud e.g., position-velocity for one simulation snapshot 2. Create connections between proximate points build simplicial complex 3. Determine topological structure of complex calculate Betti numbers (measure # holes) 4. Vary proximity parameter to asses different scales calculate persistent homology 5. Evolve in time CROCKER plots

21 The big picture: Study data via topology. Computing persistent homology A. Zomorodian, G. Carlsson. Disc. & Comp. Geom. (2005) 2. Barcodes: The persistent topology of data R. Ghrist. Bull. Am. Math. Soc. (2008) 3. Persistent homology: A Survey H. Edelsbrunner, J. Harer. Contemp. Math. (2008) 4. Topology and Data G. Carlsson. Bull. Am. Math. Soc. (2009)

22 Step : Envision data as point cloud

23 Step 2: Build simplicial complex

24 Step 3: Calculate Betti numbers b0 = 4 b = b2 = 0 b3 = 0 etc.

25 Step 3: Calculate Betti numbers Circle Two-Torus Two-Sphere b=(,,0,0, ) b=(,2,,0, ) b=(,0,,0, )

26 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

27 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

28 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

29 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

30 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

31 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

32 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

33 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

34 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

35 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

36 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

37 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

38 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

39 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

40 Step 4: Find persistent homology b0 b 0 bb Proximity Parameter " Proximity Parameter ε

41 Step 4: Find persistent homology b0 b Proximity Parameter ε

42 Step 4: Find persistent homology Proximity Parameter ε

43 Step 4: Find persistent homology Proximity Parameter ε

44 Step 5: Evolve in time (CROCKER) Contour Realization Of Computed K-dimensional-hole Evolution in the Rips complex Proximity Parameter ε Contour Plot of b0(ε,t) b 0 = (B) b 0 = 2 b level

45 Step 5: Evolve in time (CROCKER) Contour Realization Of Computed K-dimensional-hole Evolution in the Rips complex Proximity Parameter ε Contour Plot of b0(ε,t) b 0 = (B) b 0 = 2 b level

46 Step 5: Evolve in time (CROCKER) Contour Realization Of Computed K-dimensional-hole Evolution in the Rips complex Proximity Parameter ε Contour Plot of b0(ε,t) b 0 = (B) b 0 = 2 b level

47 Results

48 Vicsek Model Initial Condition Three-torus T 3 b = (,3,3,,0, ) (A

49 Vicsek Model Long Term 0 Behaviors (A) 7 (B) 5 (C) Clusters? 5 (C) Loose alignment? Strong alignment?

50 (A) 0 25 Intermittent (B) clustering Loss of two topol. circles b = (2-4,, ) 0 7 (C) Alignment! Proximity Parameter ε Proximity Parameter ε (A) b 0 5 b 0 = 2 b 0 = b = b = 0 b = 0 (B) (C) level level

51 (B) 0 7 One group (C) Two persistent topol. circles b = (, 2,, ) 0 5 Proximity Parameter ε Alignment! (A) b 0 5 b 0 = b = 3 b = 0 b 5 b = 2 (B) (C) level level

52 (C) 0 5 One group, one rogue Two persistent topol. circles b = (, 2, 0, ) Hole in the data Order Order Parameter Order Parameter Parameter ϕ ϕ ϕ Alignment! Proximity Proximity Proximity Parameter Parameter Parameter ε ε ε Proximity Parameter ε Proximity Proximity Proximity Parameter Parameter Parameter ε ε ε (A) (A) (A) b 0 5 b 0 5 b 0 5 b 0 = b 0 = b 5 b 0 = b = 2 b 0 = b 5 b 0 = b = 2 (B) (C) (B) (C) b 0 = b 5 b = 2 (B) (C) level level 2 level level level 2 level

53 D Orsogna Model t = 5 t = 23 Simulation 0.6 Snapshots (A) 0.6 (B) 0.6 (C) 0.6 t = t = t = t = t = t = (B) (early) 0.6 (C) (middle) (late)

54 (A) t = 5 (B) (B) t = 23 (C) (A) t = (B) t = (A) t = 5 t = 23 (C) (C) t = 34 t = t = groups topol. circles b = (2, 2, ) Order Order Parameters Order Parameters Order Params. Proximity Proximity Proximity Parameter Parameter Parameter ε ε ε Proximity Parameter ε Proximity Proximity Proximity Parameter Parameter Parameter ε ε ε (A) (A) (A) Simulation Time P Order Parameter M P Order M Parameter abs P M 50 (B) (B) b 0 = b (B) 0 = b 0 = M abs b b 5 Simulation Time M abs b (C) b = 0 Simulation Time (C) b = 0 b = b b= 0 5 b = 2 b 5 b = 2 b = Order Parameter b (C) b = Polarization Ang. Momentum Abs. Ang. Mom. level level 2 level level level 2 level

55 2 3 In four dimensional x - v space, # and #3 are closer than # and #2.

56 (A) t = 5 (B) (B) t = 23 (C) (A) t = (B) t = (A) t = 5 t = 23 (C) (C) t = 34 t = t = groups topol. circles b = (2, 2, ) Order Order Parameters Order Parameters Order Params. Proximity Proximity Proximity Parameter Parameter Parameter ε ε ε Proximity Parameter ε Proximity Proximity Proximity Parameter Parameter Parameter ε ε ε (A) (A) (A) Simulation Time P Order Parameter M P Order M Parameter abs P M 50 (B) (B) b 0 = b (B) 0 = b 0 = M abs b b 5 Simulation Time M abs b (C) b = 0 Simulation Time (C) b = 0 b = b b= 0 5 b = 2 b 5 b = 2 b = Order Parameter b (C) b = Polarization Ang. Momentum Abs. Ang. Mom. level level 2 level level level 2 level

57 Conclusions. Persistent homology computations reveal dynamics missed by order parameters distinguish dynamics missed when OP do not recognize similarity when OP do not useful when manual examination of data is hard 2. Non-topologists can do persistent homology 3. Non-topologists SHOULD do persistent homology