Rigidity and Persistence of Directed Graphs

1. Rigidity and Persistence of Directed Graphs Julien Hendrickx

2. Outline • Problem Description and Modelisation • Characterization of persistent graphs • Minimal persistence • Persistence for Cycle-free graphs • Further works and open questions

3. Problem description 1 1 1 2 2 2 3 3 3 4 4 4 • Set of autonomous agents (possibly) moving continuously in <2, represented by vertices • Edge from i to j if i has to maintain its distance from j constant • No other hypothesis made about the agents movement  if only one constraint, agent can move freely on a circle centered on its neighbor A B Can one guarantee that distance between any pair of agents will be preserved ? C

4. Rigidity RIGID ! ÃNOT RIGID Representation of G=(V,E):p: V!<2(d(p1,p2) = maxi2 V||p1(i)-p2(i)||) Distances set d:dij>0 8 (i,j) 2E. Realization of d:repres. p s.t. ||p(i)-p(j)|| = dij8 (i,j) 2E(d is realizable if there exists a realization p of d. d is theninduced by p ) A representation p is RIGID if there exists  > 0 s.t. every realization p’ 2 B(p,) of the distance set induced by p is congruent to p. (i.e. , ||p’(i)-p’(j)|| = ||p(i)-p(j)|| 8i,j2V)A graph is RIGIDif almost all its representations are rigid

5. Laman’s criterion • G=(V,E) is rigid (in <2) iff there exists E’µE s.t. • |E’| = 2|V| - 3 • 8E’’ µE’, |E’’| · 2|V(E’’)| - 3 Examples: |E’| = 2 |V| - 3 |E’| = 2 |V| - 3 |E| = 4 < 2 |V| - 3 = 5 But, |E’’| > 2 |V(E’’)| - 3 Not rigid  Not rigid Rigid

6. Rigidity not sufficient 1 1 1 2 2 2 3 3 3 ?? 4 4 4 B is rigid. But, if 3 moves, 4 is unable to react  Rigidity insufficient because A NOT RIGID • Essentially undirected notion (although definition OK for directed graphs) • Considers all constraints globally (as if guaranteed by external observer) B So, need to take directions and localization of the constraints into account C

7. Fitting representations 1 1 c c Example:d41=d42=d43=c Continuous edges active c c 2 2 3 3 4 p’(4) fitting 4 Distance set d on G=(V,E) and representation p’ of G Edge (i,j) is active: ||p’(i)-p’(j)|| = dij Position p’(i) is fitting (for d): impossible to increase set of active edges by modifying only p’(i). (increase set ≠ increase number) p’(4) Not fitting Repres. p’ is fitting (for d): positions of all vertices are fitting “fitting if every agent tries to satisfy all its constraints”

8. Persistence =p’(1) =p’(2) p’(3) p’(4)= A representation p is PERSISTENT if there exists  > 0 s.t. every representation p’2B(p,) fitting for the distance set induced by p is congruent to pA graph is PERSISTENTif almost all its representations are persistent p(1) p(2) p’ fitting but not congruent to p Example:  p not persistent (although p rigid) p(3) p(4) What is the difference between Persistence and Rigidity ?

9. Constraint Consistence p’(2) p’ fitting but not a realization  Not C.C A representation p is CONSTRAINT CONSISTENT if there exists  > 0 s.t. every representation p’2B(p,) fitting for the distance set d induced by p is a realization of dA graph is CONSTRAINT CONSISTENTif almost all its representations are constraint consistent p(2) Examples: C.C. A graph having no vertex with an out-degree > 2 is always constraint consistent

10. Summary 1 1 1 2 2 2 3 3 3 4 4 4 • Rigidity:“All constraints satisfied  structure preserved” • Constraint Consistence: “Every agent tries to satisfy all its constraints  all the constraints are satisfied” • Persistence:“Every agent tries to satisfy all its constraints  structure preserved” Rig. NO C.C. YES A Rig. YES C.C. NO B Persistence $Rigidity + C. Consistence Rig. YES C.C. YES C

11. Outline • Problem Description and Modelisation • Characterization of persistent graphs • Minimal persistence • Persistence for Cycle-free graphs • Further works and open questions

12. Characterization A persistent graph remains persistent after deletion of an edge leaving a vertex with out-degree ¸ 3 Obtained graph not rigid  not persistent Examples: Graph remains persistent Initial graph was not persistent A graph is persistent iff all subgraphs obtained by removing edge leaving vertices with d+¸ 3 until all vertices have d+· 2 are rigid

13. Surprising consequence Subgraph not rigid Graph not persistent Application of the criterion: 1 1 Persistent 2 3 2 3 Addition of an edge 4 4 So, one can lose persistence by adding edges, “because of unfortunate selections among possible information architectures”  Question: when can one add edges ? Still open…

14. Outline • Problem Description and Modelisation • Characterization of persistent graphs • Minimal persistence • Persistence for Cycle-free graphs • Further works and open questions

15. Minimal Rigidity G is minimally rigid if it is rigid and if no single edge can be removed without losing rigidity. G=(V,E) is minimally rigid iff rigid and |E|=2|V|-3 Minimal rigidity preserved by: Vertex addition: Edge splitting: (directions have no importance)

16. Henneberg sequences Every minimally rigid graph can be obtained from K2 using these operations (Henneberg sequence) Example: K2

17. Minimal Persistence A graph is minimally persistent if it is persistent and if no single edge can be removed without losing persistence. A graph G=(V,E) is minimally persistent iff it is persistent and minimally rigid, i.e., |E| = 2|V| - 3 • A rigid graph is minimally persistent iff one of the two following conditions is satisfied: • Three vertices have an out-degree 1, the others have an out-degree 2 • One vertex has an out-degree 0, one vertex has an out-degree 1, the others have an out-degree 2

18. Directed sequential operations Minimal persistence preserved by: Vertex addition: Edge splitting: But, not all min. persistent graphs can be obtained using these operations on smaller min. persistent graphs. One v. with d+ = 0 One v. with d+ = 1 Others have d+ = 2 Examples: Three vertices with d+ = 1

19. Outline • Problem Description and Modelisation • Characterization of persistent graphs • Minimal persistence • Persistence for Cycle-free graphs • Further works and open questions

20. Cycle Free Graphs • A cycle-free graph is persistent iff there exists L,F2 V s.t. • d+(L) = 0 (Leader) • d+(F) = 1, (F,L) 2E (First Follower) • d+(i) ¸ 2 for every other i2V Persistence is preserved after addition/deletion of vertexwith d-=0 and d+¸ 2 Example: Leader Follower Every cycle free persistent graph can be obtained by a succession of such additions to initial Leader-Follower seed

21. Outline • Problem Description and Modelisation • Characterization of persistent graphs • Minimal persistence • Persistence for Cycle-free graphs • Further works and open questions

22. Further works and open questions • How to check persistence in polynomial time for the generic case? (polynomial time algorithm exists for cycle-free and minimally rigid graphs) • When can one add edges without losing persistence?  maximally persistent graphs, maximally robust persistent graphs (minimize probability to lose persistence if possible appearance of parasite edges or disappearance of existing links.) • Characterize persistence is other spaces (as <3) • Is there a persistent graph for each rigid graph ?

23. “Almost all” Graph is (generically) rigid, but there exists non-rigid representations. Suppose triangles are congruent, lateral edges are parallel and have the same length: Realization of the same distance set, but no congruence

24. Counterexample for directed sequential operations • If it was obtained by a sequential operation from a smaller minimally persistent graph, then : • Two possibilities for last added vertex • Last operation was edge splitting

25. First possibility Not persistent  This vertex cannot have been the last one added

26. Second possibility Not persistent  This vertex cannot have been the last one added  This minimally persistent graph cannot be obtained from a smaller one by one of the sequential operations