2. GEOLOGÍA REGIONAL
4.4. CUENCA DE LUDO SIGSIG
Relabeling the nodes of a given networkNX = (X, AX) results in a networkNY = (Y, AY)
capture this notion formally, we say thatNX and NY areisomorphic whenever there exists
a bijective mapφ:X →Y such that for all points x, x0∈X we have
AX(x, x0) =AY(φ(x), φ(x0)). (5.78)
When networks NX and NY are isomorphic we write NX ∼= NY. The space where all
isomorphic networks are represented by a single point is called the space of networks modulo isomorphism and denoted as ˜N mod ∼=.
To motivate the definition of a distance on the space ˜N mod ∼= of networks modulo isomorphism, we start considering networks NX and NY with the same number of nodes and assume that a bijective transformationφ:X →Y is given. It is then natural to define thedistortion dis(φ) of the mapφas
dis(φ) := max x,x0∈X AX(x, x0)−AY(φ(x), φ(x0) . (5.79)
Since different maps φ:X →Y are possible, we further focus on those maps φthat make the networksNX andNY as similar as possible and define the distanced∞between networks
NX and NY with the same cardinality as
d∞(NX, NY) :=
1
2minφ dis(φ), (5.80)
where the factor 1/2 is added for consistency with the definition of the Gromov-Hausdorff distance for metric spaces [9, Chapter 7.3]. To generalize (5.80) to networks that may have different number of nodes we consider the notion of correspondence between node sets to take the role of the bijective transformation φ in (5.79) and (5.80). More specifically, for node setsX andY consider subsetsR⊆X×Y of the Cartesian product spaceX×Y with elements (x, y)∈R. The set R is a correspondence between X and Y if for all x0 ∈X we
have at least one element (x0, y) ∈R whose first component is x0, and for all y0 ∈ Y we
have at least one element (x, y0)∈R whose second component isy0. The distortion of the
correspondenceR is defined as dis(R) := max (x,y),(x0,y0)∈R AX(x, x0)−AY(y, y0) . (5.81)
In a correspondenceRall the elements ofXare paired with some point inY and, conversely, all the elements of Y are paired with some point in X. We can then think of R as a mechanism to superimpose the node spaces on top of each other so that no points are orphaned in either X or Y. As we did in going from (5.79) to (5.80) we now define the distance between networksNX andNY as the distortion associated with the correspondence
R that makes NX and NY as close as possible, dN˜(NX, NY) :=
1
2minR dis(R) =
1
2minR (x,y)max,(x0,y0)∈R
AX(x, x0)−AY(y, y0)
. (5.82)
Notice that (5.82) does not necessarily reduce to (5.80) when the networks have the same number of nodes. Since for networks NX, NY with |X| = |Y|, correspondences are more
general than bijective maps there may be a correspondence R that results in a distance
dN˜(NX, NY) smaller than the distance d∞(NX, NY).
The definition in (5.82) is a verbatim generalization of the Gromov-Hausdorff distance in [9, Theorem 7.3.25] except that the dissimilarity functionsAX andAY are not restricted to be metrics. It is legitimate to ask whether the relaxation of this condition renders
dN˜(NX, NY) in (5.82) an invalid metric. We prove in the following theorem that this is
not the case since dN˜(NX, NY) becomes a legitimate metric in the space ˜N mod ∼= of
networks modulo isomorphism.
Theorem 13 The function dN˜ : ˜N ×N →˜ R+ defined in (5.82) is a metric on the space
˜
N mod ∼= of networks modulo isomorphism. I.e., for all networks NX, NY, NZ ∈N˜, dN˜
satisfies the following properties: Nonnegativity: dN˜(NX, NY)≥0.
Symmetry: dN˜(NX, NY) =dN˜(NY, NX).
Identity: dN˜(NX, NY) = 0 if and only if NX ∼=NY.
Triangle ineq.:dN˜(NX, NY)≤dN˜(NX, NZ) +dN˜(NZ, NY).
Proof : Proof of nonnegativity and symmetry statements: That the distance
dN˜(NX, NY)≥0 is nonnegative follows from the absolute value in the definition of (5.82).
The symmetry dN˜(NX, NY) =dN˜(NY, NX) follows because a correspondenceR ⊆X×Y
with elementsri = (xi, yi) results in the same associations as the correspondenceS⊆Y×X
with elementssi= (yi, xi). This proves the first two statements.
Proof of identity statement: In order to show the identity statement, assume thatNX
and NY are isomorphic and let φ:X→Y be a bijection proving this isomorphism. Then,
consider the particular correspondence Rφ ={(x, φ(x)), x ∈ X}. By construction, for all
x0 ∈ X there is an element r = (x0, y) ∈ Rφ and since φ is surjective – indeed, bijective
– for all y0 ∈Y there is an element s= (x, y0) ∈Rφ. Thus, Rφ is a valid correspondence
betweenX and Y, which, according to (5.78), satisfies
for all (x, y),(x0, y0)∈Rφ. SinceRφis a particular correspondence while in definition (5.82)
we minimize over all possible correspondences it must be that
dN˜(NX, NY)≤
1
2(x,y),max(x0,y0)∈Rφ|AX(x, x 0)−A
Y(y, y0)|= 0, (5.84)
where the equality follows because AX(x, x0)−AY(y, y0) = 0 for all (x, y),(x0, y0)∈Rφ by
(5.83). Since we already argued that dN˜(NX, NY) ≥ 0 it must be that dN˜(NX, NY) = 0
when the networks NX ∼=NY are isomorphic.
We now argue that the converse is also true, i.e., dN˜(NX, NY) = 0 implies that X and Y are isomorphic. IfdN˜(NX, NY) = 0 there is a correspondence R0 such thatAX(x, x0) =
AY(y, y0) for all (x, y),(x0, y0)∈R0. Define then the function φ:X→Y that associates to
x any valuey among those that form a pair withx in the correspondence R0,
φ(x) =y0∈
y(x, y)∈R0 . (5.85)
SinceR0 is a correspondence the set
y(x, y)∈R0 6=∅is nonempty implying that (5.85) is defined for all x ∈ X. Moreover, since we know that (x, φ(x)) ∈ R0 we must have
AX(x, x0) =AY(φ(x), φ(x0)) for all x, x0. From this observation it follows that the function
φ must be injective. If it were not, there would be a pair of points x 6= x0 for which
φ(x) =φ(x0). For this pair of points we can then write,
AX(x, x0) =AY(φ(x), φ(x0)) = 0, (5.86)
where the first equality follows from the definition of φ and the second equality from the fact thatφ(x) =φ(x0) and that dissimilarity functions are such thatAY(y, y) = 0. However, (5.86) is inconsistent with x6=x0 because the dissimilarity function is AX(x, x0) = 0 if and
only x = x0. It then must be φ(x) = φ(x0) if and only if x = x0 implying that φ is an injection.
Likewise, define the function ψ:Y →X that associates toy any valuex among those that form a pair with y in the correspondenceR0,
ψ(y) =x0 ∈
x(x, y)∈R0 (5.87)
Since R0 is a correspondence the set
x(x, y)∈R0 6= ∅ is nonempty implying that (5.87) is defined for all y ∈ Y and since we know that (ψ(y), y) ∈ R0 we must have
AX(ψ(y), ψ(y0)) =AY(y, y0) for all y, y0 from where it follows that the function ψmust be
injective.
We have then constructed reciprocal injectionsφ:X →Y andψ:Y →X. The Cantor- Bernstein-Schroeder theorem [48, Chapter 2.6] applies and guarantees that there exists a
bijection between X and Y. This forces X and Y to have the same cardinality and, as a consequence, it forces φ and ψ to be bijections. Pick the bijection φ and recall that since (x, φ(x))∈R0 we must haveAX(x, x0) =AY(φ(x), φ(x0)) for all x, x0 from where it follows
thatNX ∼=NY.
Proof of triangle inequality: To show the triangle inequality let correspondences R∗
between X and Z and S∗ between Z and Y be the minimizing correspondences in (5.82) so that we can write
dN˜(NX, NZ) = 1 2(x,z),max(x0,z0)∈R∗ AX(x, x0)−AZ(z, z0) . dN˜(NZ, NY) = 1 2(z,y),max(z0,y0)∈S∗ AZ(z, z0)−AY(y, y0) . (5.88)
Define now the correspondence T betweenX and Y as the one induced by pairs (x, z) and (z, y) sharing a common pointz∈Z,
T :=(x, y)∃ z∈Z with (x, z)∈R∗,(z, y)∈S∗ . (5.89)
To show that T is a correspondence we have to prove that for every x ∈ X there exists
y0 ∈ Y such that (x, y0) ∈ T and that for every y ∈ Y there exists x0 ∈ X such that
(x0, y) ∈T. To see this pick arbitrary x ∈ X. Because R is a correspondence there must
existz0 ∈Z such that (x, z0)∈R. SinceSis also a correspondence, there must existy0∈Y
such that (z0, y0)∈S. Hence, there exists (x, y0)∈T for everyx∈X. Conversely, pick an
arbitrary y ∈Y. Since S and R are correspondences there must exist z0 ∈Z and x0 ∈X
such that (z0, y) ∈ S and (x0, z0) ∈ R. Thus, there exists (x0, y) ∈ T for every y ∈ Y.
Therefore, T is a well defined correspondence.
The correspondence T need not be the minimizing correspondence for the distance
dN˜(NX, NY), but since it is a valid correspondence we can write [cf. (5.82)]
dN˜(NX, NY)≤
1
2(x,y)max,(x0,y0)∈T|AX(x, x 0
)−AY(y, y0)|. (5.90)
According to the definition of T in (5.89) the requirement (x, y),(x0, y0) ∈ T is equivalent to requiring (x, z),(x0, z0) ∈ R∗ and (z, y),(z0, y0) ∈ S∗. Further adding and subtracting
AZ(z, z0) from the maximand and using the triangle inequality on the absolute value yields
dN˜(NX, NY)≤
1
2 (x,z),max(x0,z0)∈R∗ (z,y),(z0,y0)∈S∗
We can further bound (5.91) by maximizing each summand independently so as to write dN˜(NX, NY)≤ 1 2(x,z),max(x0,z0)∈R∗|AX(x, x 0)−A Z(z, z0)| +1 2(z,y),max(z0,y0)∈S∗|AZ(z, z 0)−A Y(y, y0)|. (5.92)
Sunbstituting the equalities in (5.88) for the summands on the right hand side of (5.92)
yields the triangle inequality.
Having shown the four statements in Theorem 13, the global proof concludes.
The guarantee offered by Theorem 13 entails that the space ˜N mod ∼= of networks modulo isomorphism endowed with the distance defined in (5.82) is a metric space. Re- striction of (5.82) to symmetric networks shows that the space N mod ∼= of symmetric networks [cf. Section 3.2.1] modulo isomorphism is also a metric space. Further restriction to metric spaces shows that the space of finite metric spaces modulo isomorphism is met- ric [9, Chapter 7.3]. A final restriction of (5.82) to finite ultrametric spaces shows that the space U mod ∼= of ultrametrics modulo isomorphism is a metric space. Having a prop- erly defined metric to measure distances between networks ˜N and therefore also between ultrametrics U ⊂ N˜ permits the study of stability of hierarchical clustering methods for asymmetric networks that we undertake in the following section.