Rigidity and Persistence of Directed Graphs Julien Hendrickx

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

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

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

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’’)| - 3  Not rigid

Rigidity not sufficient 1 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

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”

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 ?

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

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

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

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

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…

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

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)

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

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

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

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

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

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

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 ?

“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

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

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

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