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

•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

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Can one guarantee that distance between any pair of agents will be preserved ?

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B

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Rigidity

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

à NOT RIGID

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

Not rigid

|E’| = 2 |V| - 3

Rigid

|E’| = 2 |V| - 3

But, |E’’| > 2 |V(E’’)| - 3 Not rigid

Rigidity not sufficient

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4NOT RIGID

So, need to take directions and localization of the constraints into account

B is rigid. But, if 3 moves, 4 is unable to react

??

Rigidity insufficient because

•Essentially undirected notion (although definition OK for directed graphs)

•Considers all constraints globally (as if guaranteed by external observer)

Fitting representationsDistance 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)

Repres. p’ is fitting (for d): positions of all vertices are fitting

1Example:d41=d42=d43=cContinuous edges active

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cc

“fitting if every agent tries to satisfy all its constraints”

p’(4) Not fitting

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p’(4) fitting

Persistence

What is the difference between Persistence and Rigidity ?

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

Example:p(1)

p(3)

p(2)

p(4)

p’ fitting but not congruent to p

p not persistent(although p rigid)

p’(3)

=p’(2)

p’(4)=

=p’(1)

Constraint ConsistenceA 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

Examples:

A graph having no vertex with an out-degree > 2 is always constraint consistent

p(2) p’(2)

p’ fitting but not a realization Not C.C

C.C.

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”

Persistence $

Rigidity + C. Consistence

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Rig. NO C.C. YES

Rig. YESC.C. NO

Rig. YESC.C. YES

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Outline

• Problem Description and Modelisation

• Characterization of persistent graphs

• Minimal persistence

• Persistence for Cycle-free graphs

• Further works and open questions

CharacterizationA persistent graph remains persistent after deletion of an edge leaving a vertex with out-degree ¸ 3

Examples:

Graph remains persistent

Obtained graph not rigid not 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

Persistent

Addition of an edge

Subgraph not rigid

Graph not persistent

So, one can lose persistence by adding edges,

Still open…

Question: when can one add edges ?

“because of unfortunate selections among possible information architectures”

Application of the criterion:

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Outline

• Problem Description and Modelisation

• Characterization of persistent graphs

• Minimal persistence

• Persistence for Cycle-free graphs

• Further works and open questions

Minimal Rigidity

G=(V,E) is minimally rigid iff rigid and |E|=2|V|-3

G is minimally rigid if it is rigid and if no single edge can be removed without losing rigidity.

(directions have no importance)

Vertex addition: Edge splitting:

Minimal rigidity preserved by:

Henneberg sequences

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

Example:

K2

Minimal PersistenceA 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

Vertex addition: Edge splitting:

But, not all min. persistent graphs can be obtained using these operations on smaller min. persistent graphs.

Examples:

Three vertices with d+ = 1

One v. with d+ = 0One v. with d+ = 1Others have d+ = 2

Minimal persistence preserved by:

Outline

• Problem Description and Modelisation

• Characterization of persistent graphs

• Minimal persistence

• Persistence for Cycle-free graphs

• Further works and open questions

Cycle Free GraphsPersistence 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