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 3 2 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 1 3 2 4 B 1 3 2 4
Can one guarantee that distance between any pair of agents will be preserved ? C

4. Rigidity RIGID ! Ã NOT 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) 2 E.
Realization of d: repres. p s.t. ||p(i)-p(j)|| = dij 8 (i,j) 2 E (d is realizable if there exists a realization p of d. d is then induced 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)|| 8 i,j 2 V) A graph is RIGID if almost all its representations are rigid

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

6. Rigidity not sufficient1 3 2 4
B is rigid. But, if 3 moves, 4 is unable to react A
NOT RIGID
 Rigidity insufficient because
Essentially undirected notion (although definition OK for directed graphs)
Considers all constraints globally (as if guaranteed by external observer) 1 3 2 4 B ??
So, need to take directions and localization of the constraints into account 1 3 2 4 C

7. Fitting representations
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) 1
Example: d41=d42=d43=c
Continuous edges active 2 3 4 c 1 2 3 4 c
p’(4) fitting
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
A representation p is PERSISTENT if there exists  > 0 s.t. every representation p’2 B(p,) fitting for the distance set induced by p is congruent to p A graph is PERSISTENT if almost all its representations are persistent
p’(3)
=p’(2)
p’(4)=
=p’(1)
p(2)
p(1)
p’ fitting but not congruent to p
Example:
 p not persistent
(although p rigid)
p(4)
p(3)
What is the difference between Persistence and Rigidity ?

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

10. Persistence $ Rigidity + C. Consistence
Summary
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” 1 3 2 4
Rig. NO
C.C. YES A 1 3 2 4
Rig. YES
C.C. NO B 1 3 2 4
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
Application of the criterion:
Subgraph not rigid
Graph not persistent 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
Persistence is preserved after addition/deletion of vertex with 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
A cycle-free graph is persistent iff there exists L,F 2 V s.t.
d+(L) = 0 (Leader)
d+(F) = 1, (F,L) 2 E (First Follower)
d+(i) ¸ 2 for every other i 2 V

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