The topography of gravitational graphs

Transcription

1 The topography of gravitational graphs Fernand Meyer Centre de Morphologie Mathématique 20 February 2017 ernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

2 Reminder on graphs ernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

3 Reminder on graphs A non oriented graph G = [N, E]: N = vertices or nodes ; E = of edges ; an edge being a pair of vertices. The nodes are designated with small letters: p, q, r... The edge linking the nodes p and q is designated by e pq. The subgraph spanning a set A N is the graph G A = [A, E A ], where E A are the edges linking two nodes of A. The partial graph G 0 = [N, E 0 ] has the same set of nodes as G but only a subset E 0 E of edges. A path, π, is a sequence of vertices and edges: π starts with a vertex, say p, followed by an edge e pq, incident to p, followed by the other endpoint q of e pq, and so on. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

4 Weighted graphs Let T be a totally ordered set. Nodes and/or edges may be weighted with the functions: F n = Fun(N,T ) : the functions de ned on the nodes N with value in T ; the function ν 2 F n takes the weight ν p on the node p F e = Fun(E,T ) : the functions de ned on the edges E with value in T ; the function η 2 F e takes the value η pq on the edge e pq Notations : A graphs G = [N, E] holding - only node weights ν 2 F n is designated by G (ν, nil) - only edge weights η 2 F e is designated by G (nil, η) - node and edge weights is designated by G (ν, η) Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

5 The topography of weighted graphs Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

6 Flowing edges and owing paths De nition G (ν, nil) : An edge (p, q) e pq is a owing edge of its extremity p i ν p ν q.! p, q, s are 3 successive nodes on a owing path, then ν p ν q ν s De nition G (nil, η) : An edge (p, q) is a owing edge of the node p, if it is one of the lowest adjacent edges of p.! the weights of the successive edges in a owing path are never increasing De nition A path p e pq q e qs s... is a owing path if each node except the last one is followed by one of its owing edges. ernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

7 Flowing paths De nition A path p e pq q e qs s... is a owing path if each node except the last one is followed by one of its owing edges. For a node weighted graph: the weights of the nodes along a owing path are never increasing: if p, q, s are 3 successive nodes on a owing path, then ν p ν q ν s. For an edge weighted graph: the weights of the successive edges in a owing path are never increasing (if p, q and r are three successive nodes of a owing path, then e qr is one of the lowest edges of q and thus e pq e qr. ernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

8 Flowing edges in node weighted graphs In a node weighted graph: each edge is a owing edge of at least one extremity of this edge. For the edge e st. either ν s ν t and e st is a owing edge of s, or ν t ν s and e st is a owing edge of t. The edge e st is a owing edge of both its extremities if ν s = ν t. The nodes without lower neighbors are not the origin of a owing edge, they are isolated regional minima. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

9 Editing node weighted graphs -> owing graphs A sel oop is added to each isolated regional minimum. A drop of water arriving at such a node cycles around it, with no possibility to leave it. We obtain the owing graph G (ν, nil), in which each node is the origin of a owing edge and each edge is a owing edge of one of its extremities. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

10 Flowing edges in edge weighted graphs In a connected graph, each node has one or several neighboring edges ; those with minimal weight are the owing edges for this node. Thus, each node is the extremity of at least one owing edge. However, not each edge is a owing edge: If the extremities of an edge e st have adjacent edges with weights lower than η st, then e st is not a owing edge. Such an edge never belongs to the trajectory of a drop of water freely circulating from node to node. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

11 Editing edge weighted graphs -> owing graphs If an edge of G (nil, η) is not the lowest edge of one of its extremities, it may be cut without disturbing the trajectories of a drop of water. We obtain the owing graph # G (nil, η), in which each node is the origin of a owing edge and each edge is a owing edge of one of its extremities Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

12 From owing graphs to owing digraphs Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

13 Associating a owing digraph to a owing graph Flowing graph G : each node is the origin of a owing edge and each edge is a owing edge of one of its extremities G!! G : G and! G have the same nodes ; the arcs of! G correspond to the owing edges of G : n for G (ν, nil) : p! q in! o G, fν p ν q in G g n for G (nil, η) : p! q in! o G, fe pq is a owing edge of p in G (nil, η)g, fe pq is a owing edge of p in # G (nil, η)g Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

14 Associating a owing digraph to a node weighted owing graph A: a node weighted graph G (ν, nil) B: the associated owing digraph! G (ν, nil) Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

15 The owing paths of a owing graph or digraph A: a node weighted graph G (ν, nil) B: the associated owing digraph! G (ν, nil) To each owing path in the owing graph corresponds an oriented path in the owing digraph. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

16 The owing paths of a node weighted owing digraph Two oriented paths with the same origin towards 2 distinct regional minima. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

17 Breaking the isolation of isolated regional minima An isolated regional minimum has no lower or equal neighbor. For the elegance of the theoretical developments below we add a loop edge linking the isolated regional minimum with itself. However, in the algorithms developed for implementing the theory, this step is often not needed. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

18 Associating a owing digraph to an edge weighted owing graph A: an edge weighted owing graph # G (nil, η) B: the associated owing digraph! # G (nil, η) Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

19 The owing paths of a owing graph or digraph A: an edge weighted owing graph G (nil, η) B: the associated owing digraph! G (nil, η) To each owing path in the owing graph corresponds an oriented path in the owing digraph. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

20 The owing paths of an edge weighted owing digraph Two oriented paths with the same origin towards 2 distinct regional minima. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

21 Flat zones ernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

22 Notations and relations p! q designates an oriented path between p and q in a digraph; the relation p & q indicates that there exists an oriented path p! q connecting p and q. p $ q designates a path between p and q in a digraph in which all arcs are bidirectional arcs; the relation p $ $ q indicates that there exists a path p $ q, connecting p and q. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

23 Oriented paths and bi-directional paths If we suppose that the relations & and $ are re exive, i.e. p & p and p $ p, then: the relation p & q, i.e. there exists a path p! q connecting p and q, is a preorder relation. the relation fp & q and q & pg is an equivalence relation. Its equivalence classes are the at zones of the oriented graph! G. the relation p $ q i.e. there exists a path p $ q, connecting p and q and containing only bidirectional arcs, is also an equivalence relation. Its equivalence classes are the super at zones of the non oriented graph G. n o As fp & q and q & pg ( p $ q, the super at zones are sub-sets of the at zones. They are identical only in the case of gravitational digraphs. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

24 A counter example We de ne: for p, q neighbors, fp! qg, fν p + λ ν q g, implying fp $ qg, fν p λ ν q ν p + λg. A: a digraph, which is not a gravitational graph (λ = 1) B: The equivalence classes for the relation prq = fp & q and q & pg : at zones C: The partial non oriented graph G, obtained by replacing (p $ q) by (p q). D: The. equivalence classes of! G for the relation (p $ $ q) : super at zones Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

25 Upstream and downstream propagation ernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

26 Upstream propagation in the digraph associated to a node weighted graph A node weighted graph and its associated gravitational graph. A red is highlighted and its upstream (in blue) extracted. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

27 Upstream propagation in the digraph associated to an edge weighted graph An edge weighted graph and its associated gravitational graph. A red node is highlighted and its upstream (in blue) extracted. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

28 Downstream propagation in the digraph associated to a node weighted graph Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

29 Downstream propagation in the digraph associated to an edge weighted graph Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

30 Gravitational digraphs Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

31 Perpetuum mobile Ever descending stairs are impossible in node or edge weighted digraphs Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

32 No perpetuum mobile in node weighted digraphs Ever descending stairs are impossible in node weighted digraphs: Consider a owing path, i.e. a never ascending path! π pq = p! q between p and q, implying ν p ν q. If there exists a second never ascending path! σ qp = q! p between q and p, then ν q ν p. Together, these two relations show that ν p ν q ν p, implying that all nodes along both paths p! q and q! p have the same weight. These paths are in fact bidrectional graphs! π pq =! π pq and! σ pq =! σ pq. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

33 No perpetuum mobile in edge weighted digraphs Ever descending stairs are impossible in edge weighted digraphs: If there exist two owing paths! π pq = p q between p and q, and σ qp = q p between q and p,.the concatenation between both paths! π pq B! σ qp is a loop along which the edge weights are never increasing. The rst and the last edge of the loop verify : η pt η sp p! t!...! }! q " # s }... } } On the other hand, the path s! p! t also is a non increasing paths and the rst and last edge verify: η sp η pt. Hence η sp = η pt and the weights of all edges along the loop! π qp B! σ qp are equal. These paths are in fact bidrectional graphs! π pq =! π pq and! σ pq =! σ pq. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

34 No perpetuum mobile in node or edge weighted digraphs The existence of a never ascending path p q in G (ν, nil) is expressed by p & q. in the digraph. Thus for nboth node o and edge weighted digraphs we have : fp & q and q & p g ) p $ q. The inverse implication is always true. We call "gravitational digraphs", n o the digraphs verifying fp & q and q & p g, p $ q We now study the properties of the gravitational digraphs, independently of any node or edge weights. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

35 Black holes and catchment zones Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

36 Black holes of gravitational digraphs The gravitational attraction of a black hole is so high that it catches everything around it and not even the light gets out. A node without any outgoing arc is an isolated black hole. Non isolated black holes are at zones without outgoing arcs: De nition A black hole M of a gravitational graph is de ned by the following equivalence : p 2 M, f8q : p & q ) q & pg If there exists no node q such that p & q, then p is an isolated black hole. If on the contrary, p & q for a given node q, then q & p, implying p $ $ q, i.e. p and q belong to the same at zone: there is no way to escape from this at zone through an outgoing arc. Suppose that for p 2 M : p! s for a particular node s, then: p! s ) p & s ) s & p ) p $ $ s, showing that s also belongs to the same at zone. ernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

37 Catchment zones Each black hole is surrounded by a catchment zone. De nition The catchment zone of a black hole M is the set of nodes at the origin of an oriented path leading inside M : CZ(M) = fp 2 N j p & q, q 2 Mg If there exists q 2 M, such that p & q, then p & s for each s 2 M, as q $ $ s. The following theorem shows that each node of a gravitational graph belongs to the attraction zone of at least one black hole. These attraction zones may overlap and a node belong to the attraction zones of several black holes. Theorem Each node of a gravitational graph belongs to the attraction zone of at least one black hole. ernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

38 Proof of the theorem If p, a node of a gravitational graph! G, belongs to a black hole, the theorem is veri ed. If not: 9 q 1 : p & q 1 and q 1 & p Again there are two possibilities : a) q 1 belongs to a black hole, the theorem is veri ed b) q 1 does not belong to a black hole and there exists q 2, such that q 1 & q 2 and q 2 & q 1. Necessarily q 2 & p otherwise there exists a cycle $ verifying p & q 1 & q 2 & p, implying q 1 q2, and in particular q 2 & q 1, which is contrary to the hypothesis. The relation q 2 & p also shows that q 2 6= p. The same reasoning may be applied again and again, producing a series of nodes q i, verifying p & q 1 & q 2 &...q k and q k & q k 1 &...q 1 & p all distinct from each other. As the number of nodes of N is nite, the series q i converges to a node belonging to a black hole of the graph. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

39 An algorithm for detecting the black holes If p does not belong to a black hole, there exists an oriented path p! q connecting p with a node q inside a black hole. There exists a couple of nodes s, u, along this path, verifying s! u and u 9 s otherwise we have p $ q, and p would belong to the same black hole as the node q. We may suppose that (s, u) is the rst couple of nodes if one follows this path upstream, verifying s! u and u 9 s. Then p belongs to the upstream of the node s, i.e. to ft 2 N j t & sg. Lemma The non black holes are the upstream of all nodes s having a neighbor u verifying s! u and u 9 s. The black holes are obtained by taking their complement. ernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

40 An algorithm for detecting the black holes The red nodes : s! u and u 9 s. The blue nodes : upstream of the red nodes. Together, red and blue nodes are the non black holes. The black holes,their complement are labeled in g.d Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

41 An algorithm for detecting the black holes on an initially edge weighted graph Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

42 Extracting the catchment zone containing a marked node. A: A gravitational graph with its labeled regional minima. B: marking a node p. C: Downstream of the marked node p. D: Upstream of the downstream CZ (p) =%& p Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

43 Extracting overlappings of catchment zones C,D: 2 overlapping catchment zones. E: Upstream propagation of the cyan and violet label. The violet node pointing towards a cyan label belongs to both catchment zones. F: its upstrem constitutes the overlapping of both catchment zones. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

44 Extracting overlappings of catchment zones We consider a labeled owing digraph! G (nil, nil, λ) in which the labels form a partial partition, result of the upstream label propagation of the labels of some marked nodes. If two node p, q verifying (p! q) hold distinct labels, the node p belongs to two distinct catchment zones. The label of q could not be propagated to the node p, as the label of q has a lower priority than the label of p. Such nodes are called "overlapping seeds", forming a set S de ned by: S = fp j p, q 2 N : (p! q) and (λ p 6= λ q )g Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

45 Extracting overlappings of catchment zones C,D: 2 overlapping catchment zones. E: Upstream propagation of the yellow and violet labels. A violet node pointing towards a yellow label belongs to both catchment zones. F: its upstrem constitutes the overlapping of both catchment zones. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

46 The restricted catchment zones. A: A gravitational digraph; B: The catchment zone of the black hole labeled in violet. C: The violet nodes pointing towards a node with a distinct label also belong to other catchment zones, as does their upstream in C D:The remaining violet nodes constitute a restricted catchment zone. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

47 Authorized prunings Each node belongs to the attraction zone of at least a black hole, sometimes of several. In order to reduce the attraction zone of a black hole, it is su cient to reduce the number of oriented paths leading to it. This may be done by suppressing arcs in the graph, without modifying the black holes themselves. If e! pq is an arc which is suppressed, then all paths passing through this arc are broken. Such an operation suppressing arcs is called pruning the graph. By pruning the graph the attraction zone of the black holes will be reduced. Authorized prunings leave the black holes unchanged but narrow their catchment zones. De nition In an authorized pruning, the arc p! q may be suppressed if after pruning there remains a node u such that p! u and u 9 p. Lemma An authorized pruning preserves the black holes. ernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

48 Authorized prunings A: a black hole in violet ; B: its catchment zone C: authorized pruning suppressing an arrow, leaving the black holes unchanged ; D: restricted catchment zone Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

49 Back to the node or edge weighted graphs Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

50 Gravittational graphs derived from node weighted graphs Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

51 Gravitational digraphs derived from node weighted graphs It is de ned by : for p, q neighbors fp! qg, fν p ν q g. Thus fp $ qg, fν p = ν q g. Its at zones are maximal connected components of the graph, surrounded by nodes with a di erent weight. If no neighboring node has a lower altitude, a at zone is a regional minimum. The black holes are thus the regional minima. They are either isolated regional minima, i.e. nodes with no lower or equal neighbors. Or they are extended at zones without lower neighbors.. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

52 Gravitational digraphs derived from node weighted graphs The node weighted graph and the associated gravitational digraph have the same regional minima. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

53 A pruning keeping only the arcs towards the lowest neighbbors In an authorized pruning, the arc p! q may be suppressed if after pruning there remains a node u such that p! u and u 9 p. The pruning which leaves for each node only the arrows towards its lowest neighbors is an authorized pruning. If the node q is not one of the smallest neighbors of p, then there exists at least a lowest neighbor u such that p! u and u 9 p and the arc p! q may be suppressed. Submitted to an authorized pruning, the resulting graph has the same regional minima and smaller catchment zones. The at zones also are smaller. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

54 A pruning keeping only the arcs towards the lowest neighbbors Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

55 Gravitational graphs derived from edge weighted graphs Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

56 Gravitational digraphs derived from edge weighted graphs In G (nil, η), an edge (p, q) is a owing edge of the node p, if it is one of the lowest adjacent edges of p. The arcs of the digraph are de ned by: for p, q neighbors fp! qg, e pq is a owing edge of p. The at zones are the equivalence classes of the equivalence relation q $ $ s. The at zones are thus maximal connected components of nodes such that two neighboring nodes are connected by an edge which is the lowest edge of both its extremities. The regional minima are at zones without outgoing arcs. Isolated regional minima do not exist. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57

57 Flat zones and black holes of a gravitational digraph A,B: an edge weighted graph and its owing digraph. C,D: the at zones obtained by replacing the double arrows by edges. E,F: the at zones without outgoing arcs are the black holes of the digraph and the regional minima of the edge weighted graph.. Fernand Meyer (Centre de Morphologie Mathématique) The topography of gravitational graphs 20 February / 57