S PACETIME R EPRESENTATION L EARNING
Anonymous authors
Paper under double-blind review
A BSTRACT
Much of the data we encounter in the real world can be represented as directed graphs.
In this work, we introduce a general family of representations for directed graphs through connected time-oriented Lorentz manifolds, called “spacetimes” in general relativity.
Spacetimes intrinsically contain a causal structure that indicates whether or not there exists a causal or even chronological order between points of the manifold, called events.
This chronological order allows us to naturally represent directed edges via imposing the correct ordering when the nodes are embedded as events in the spacetime.
Previous work in machine learning only considers embeddings lying on the simplest Lorentz manifold or does not exploit the connection between Lorentzian pre-length spaces and directed graphs.
We introduce a well-deﬁned approach to map data onto a general family of spacetimes.
We empirically evaluate our framework in the tasks of hierarchy extraction of undirected graphs, directed link prediction and representation of directed graphs.
I NTRODUCTION
Most of the machine learning literature has focused on learning representations that lie on a Riemannian manifold such as the Euclidean space, the d -sphere (e.g., (cid:96) 2 -normalized representations) (Wang et al., 2017), hyperbolic geometry to represent graphs without cycles (Nickel & Kiela, 2017), or a statistical manifold in information geometry (Amari, 1998).
Concepts of Euclidean geometry, such as distances, are naturally generalized to Riemannian geometry which remains easy to interpret.
In contrast, recent approaches have considered learning representations that lie on a pseudo-Riemannian manifold to extract hierarchies in graphs with cycles (Law & Stam, 2020; Law, 2021) or represent directed graphs (Clough & Evans, 2017; Sim et al., 2021).
Pseudo-Riemannian manifolds are generalizations of Riemannian manifolds where the constraint of positive deﬁniteness of the nondegenerate metric tensor is relaxed.
The machine learning literature on pseudo-Riemannian manifolds can be divided into two categories.
The ﬁrst category focuses on how to optimize a given function whose domain is a pseudo-Riemannian manifold without accounting for possible (time-)orientations of the manifold (Law & Stam, 2020; Law, 2021).
The second category exploits the interpretation of a speciﬁc family of pseudo-Riemannian manifolds called “spacetimes” in general relativity (Clough & Evans, 2017; Sim et al., 2021).
More speciﬁcally, spacetimes are connected time-oriented Lorentz manifolds.
They intrinsically contain a causal structure that indicates whether or not there exists a causal order between points of the manifold, called events.
This causal structure has been utilized to represent directed graphs where each node is an event and the existence of an arc (i.e., directed edge) between two nodes depends on the causal character of the curves joining them (Bombelli et al., 1987; Kunzinger & Sämann, 2018).
In particular, Clough & Evans (2017) consider learning representations via the Minkowski spacetime which is the simplest such manifold.
On the other hand, Sim et al.
(2021) use three types of spacetimes and propose an ad hoc method based on the sign of a time coordinate difference function to determine the orientation of edges.
The sign of such a function is not always meaningful as it for instance alternates periodically when the manifold is non-chronological and does not generalize to all spacetimes.
Moreover, the distance function that they optimize is constant when two points cannot be joined by a geodesic.
Contributions.
We propose a framework inspired by the Lorentzian causality theory (Kronheimer & Penrose, 1967), and in particular Lorentzian pre-length spaces (Kunzinger & Sämann, 2018), to learn directed graph representations lying on a larger family of spacetimes than in Sim et al.
To this end, we present tools to account for time-orientation and exploit distances speciﬁc to
Lorentz geometry.
In particular, we propose to restrict the existence of edges to pairs of nodes whose representations lie in an open globally hyperbolic convex normal neighborhood.
Such a neighborhood can be deﬁned for any spacetime (see Theorem 2.7 of Minguzzi (2019)) and admits simple distance and time separation functions whose sign determines the direction of edges.
We experimentally show that spacetimes can extract hierarchies in social networks better than standard approaches.
Our framework also outperforms existing methods in link prediction on graphs with directed cycles.
S PACETIME D IFFERENTIAL G EOMETRY
We introduce some differential geometry background about spacetimes.
Spacetimes have been widely studied, and we refer the reader to Hawking & Ellis (1973); Uhlenbeck (1975); O’Neill (1995); Beem et al.
; Wolf (2011); Gourgoulhon (2016), Chapter 5-8 & 14 of O’Neill (1983).
Pseudo-Riemannian Manifold.
A d -dimensional pseudo-Riemannian manifold ( M , g ) is a smooth manifold such that every point x ∈ M has a d -dimensional tangent space T x M
whose metric tensor g x : T x M × T x M
→ R is a nondegenerate symmetric bilinear form (called a scalar product ).
Nondegeneracy means that ∀ v ∈ T x M , g x ( u , v ) =
= ⇒ u
= 0 .
When the context is clear, we write (cid:104)· , ·(cid:105) :
=
g x ( · , · ) to deﬁne the metric tensor at x to simplify the notation.
We also write M instead of ( M , g ) .
We write points x ∈ M of the manifold in bold serif font, and tangent vectors u ∈ T x M in bold sans-serif font when we want to distinguish them from points.
Lorentz manifold.
Every tangent space T x M of a d -dimensional pseudo-Riemannian manifold M admits an orthonormal basis { e 1 , . .
. , e d } that satisﬁes ∀ i, (cid:104) e i , e i (cid:105) = ± 1 and ∀ i (cid:54) =
j, (cid:104) e i , e j (cid:105) = 0 .
The index ν ≤ d of M is the number of orthonormal vectors e i that satisfy (cid:104)
e i , e i (cid:105) = − 1 .
If ν = 0 ,
M is Riemannian and its metric tensor is positive deﬁnite (i.e., ∀ x ∈ M , ∀ u ∈ T x M , (cid:104) u , u (cid:105) ≥ 0 and (cid:104) u , u (cid:105) =
⇐⇒ u = 0 ).
If ν = 1 , M is a Lorentz manifold and T x M is a Lorentz vector space.
Future timecone.
A nonzero tangent vector u is called timelike (or chronological ), null, spacelike or non-spacelike (or causal ) if (cid:104) u , u (cid:105) is negative, zero, positive or nonpositive, respectively.
The type into which u falls is called its causal character .
Every Lorentz tangent space contains two timecones.
Some timelike tangent vector t ∈ T x M can arbitrarily be used to deﬁne the future timecone as the following set of timelike tangent vectors: C + x ( t ) :
= { v ∈ T x M : (cid:104) v , v (cid:105) < 0 , (cid:104) t , v (cid:105) < 0 } whereas − t deﬁnes the past timecone C − x ( t ) :
=
C + x ( − t ) .
Two timelike tangent vectors u and v are in the same timecone iff (cid:104) u , v (cid:105) < 0 ; they belong to different timecones if v = − u .
Time-orientability and time-orientation.
A continuous vector ﬁeld X is a function that assigns to each point x ∈ M a tangent vector of M at x denoted by X ( x ) ∈ T x M .
X and − X are timelike if ∀ x ∈ M , (cid:104) X
( x ) , X ( x ) (cid:105) < 0 .
A Lorentz manifold is time-orientable iff there exists a timelike vector ﬁeld.
If M is assigned such a timelike vector ﬁeld X , it is time-oriented by X .
In this case, non-spacelike tangent vectors u at each point x can be divided into two separate classes: future (resp.
past ) directed if (cid:104) X ( x ) , u (cid:105) < 0 (resp.
(cid:104) X ( x ) , u (cid:105) > 0 ).
A curve γ x → u : I → M where I ⊆ R is deﬁned such that its initial point is γ x → u (0) =
x and its initial velocity is γ (cid:48) x
→ u (0) =
u ∈ T x M .
We denote it by γ when its initial conditions are clear from the context.
If its acceleration is zero, then it is called a geodesic.
The curve γ x → u is called timelike, null or spacelike if its velocity on the whole domain I is timelike, null or spacelike, respectively.
It is called future-directed (or future-pointing) if its velocity is future-directed (and causal) on I .
Completeness.
The manifolds that we consider in this paper are geodesically complete (i.e., I = R ), and the exponential map exp x : T x M → M of M at x is deﬁned as exp x ( u ) :
=
γ x → u (1) where γ x → u is a geodesic.
The maximal normal neighborhood of x is the maximal subset U x ⊆ M where the logarithmic map log x := exp − 1 x : U x → T x M is a diffeomorphism, it satisﬁes U x := { y ∈ M : exp x (exp − 1 x ( y ))
=
y } .
To simplify the notation, we also write −→ xy :=
log x ( y ) where y ∈
U x .
U x is convex if ∀ y ∈
U x , there exists a unique geodesic totally contained in U x from x to y .
We now present some pseudo-Riemannian manifolds that will be relevant.
Their differential geometry tools (e.g., exponential/logarithmic map and parallel transport) are provided in Appendix B.
• Pseudo-Euclidean space R dν .
The ﬂat d -dimensional pseudo-Riemannian manifold of index ν is denoted by R dν .
In particular, R d 0 is the Euclidean space and R d 1 is the Minkowski space.
Since it is a vector space, we can identify its tangent space to the space itself by means of the natural isomorphism R dν
≈ T
x R dν .
R dν is equipped with the following scalar product:
[Equation]
The maximal normal neighborhood for all x ∈ R dν is U x
=
R dν .
In special relativity, the ﬁrst ν elements of x ∈ R dν are called time coordinates and the other ones are called space coordinates.
• The pseudo-sphere S dν ( r ) of radius r > 0 is called the de Sitter space when ν = 1 and deﬁned as:
[Equation]
It is not time-orientable if d − ν is odd, and U x = { y ∈ S dν ( r
[Equation]
: (cid:104) x
[Equation]
• The pseudo-hyperbolic space H dν ( r ) is called the anti-de Sitter space when ν = 1 and deﬁned as:
[Equation]
The anti-de Sitter space is time-orientable for all d , and
U x = { y
∈
H
dν
[Equation]
r
[Equation]
y
[Equation] [Equation]
• The cylindrical Minkowski space L d 1 ( C ) :
= R d 1 / ∼ is a quotient set deﬁned such that x ∈ R d 1 and y ∈
R d 1 are equivalent (i.e., x ∼ y )
iff ∀ i > 0 , y i
=
x i and ∃ k ∈ Z , y
= x 0 + kC where C > 0 is a circumference hyperparameter.
See page 148 of O’Neill (1983) for other types of Lorentz cylinders.
We have
U x = { y =
[Equation]
+ C/ 2) } .
S PACETIME G RAPH R EPRESENTATION
A spacetime is a connected time-oriented Lorentz manifold M whose points are called events.
Informally, time-oriented is often weakened to time-orientable , and Hawking & Ellis (1973) ignore the time-orientability criterion to deﬁne spacetimes although it is required to study their causal structure .
Our main contribution is to exploit this causal structure via Lorentzian pre-length spaces to represent graphs.
In the following, we always assume that M is a spacetime unless stated otherwise.
A ﬁnite directed graph can be given the structure of a Lorentzian pre-length space (Kunzinger &
Sämann, 2018).
To deﬁne our graphs, we then choose a special type of Lorentzian pre-length space that is easy to optimize.
We now provide and follow the deﬁnitions of Kunzinger & Sämann (2018).
Causal space.
Let X be a set with a reﬂexive and transitive relation ≤ , and (cid:28) a transitive relation contained in ≤ (i.e., (cid:28)⊆≤ , so x (cid:28) y =
⇒ x ≤ y ).
Then ( X , (cid:28) , ≤ ) is called a causal space.
This deﬁnition is more general than the one given in the seminal work of Kronheimer & Penrose (1967).
Following general relativity, the event x ∈ M causally (resp.
chronologically) precedes the event y ∈ M , and we write x < y (resp.
x (cid:28) y )
iff there exists a future-directed causal (resp. timelike) curve from x to y .
This condition might be difﬁcult to verify in general.
However, if y is in a convex normal neighborhood of x denoted by V x ⊆
U x
(and we always assume in the following that x ∈ V x ), we have x < y (resp.
x (cid:28) y )
iff there exists a nonconstant future-directed causal
(resp. timelike) geodesic from x to y (see Proposition 4.5.1 of (Hawking & Ellis, 1973)).
We note x ≤ y ⇐⇒ x = y or x < y .
Any open subset W ⊆ M of a spacetime is a spacetime in its own right, and the intrinsic causal relations of W imply the corresponding ones in M .
In the case where
W ⊆ M is a convex open set, the intrinsic causality of W is as simple as that of Minkowski space (see page 403 of O’Neill (1983)).
Moreover, both ( M , (cid:28) , ≤ ) and ( W , (cid:28) , ≤ ) are causal spaces.
Lorentzian pre-length space.
Let ( X , (cid:28) , ≤ ) be a causal space and d a metric on X .
Let τ :
X × X → [0 , ∞ ] be a lower semicontinuous map ( w.r.t.
the metric topology induced by d ) that satisﬁes the reverse triangle inequality: ∀ x , y , z ∈ X with x ≤ y ≤ z , τ ( x , z ) ≥ τ ( x , y )
+ τ ( y , z ) .
Suppose that τ ( x , y )
=
if x (cid:54)≤ y and τ ( x , y ) >
iff x (cid:28) y .
Then ( X , d , ≤ , (cid:28) , τ ) is a Lorentzian pre-length space and τ is called the time separation function .
See details about τ in Section 3.3.
3. G RAPH CONSTRUCTION VIA L ORENTZIAN PRE - LENGTH SPACES
We now explain our methodological contribution, which is the construction of directed graphs from the properties mentioned above.
Let us deﬁne a directed graph G =
( V, E ) where V = { v i } ni
= is the node set and E is the set of arcs.
Our goal is to represent each node v i by a point x i ∈ M so that there exists an arc from v i to v j (i.e., ( v i , v j ) ∈ E ) only if x i (cid:28) x j .
We propose to consider subgraphs G i =
( V, E i ) deﬁned such that E
= (cid:83) ni
=1 E i .
Each subgraph G i is given by the structure of the Lorentzian pre-length space ( V x i , d , ≤ , (cid:28) , τ ) where V x i ⊆
U x
i is an open subset of the maximal normal neighborhood U x i .
More precisely, we consider the chronological future
I + ( x i , V x i ) :
=
{ y ∈ V x
i : x i (cid:28) y } of the point x i relative to some given set V x i ⊆
U x i (see page 402 of O’Neill (1983)), and we draw an arc from v i to v j iff x j ∈
I + ( x i , V x i ) .
Assuming V x
i is a convex normal neighborhood and x j ∈ V x
i , we have x j ∈
I
+ ( x i , V x i ) only if −−→ x i x j is deﬁned and future-directed timelike (see Proposition 4.12 of Beem et al.
Our framework produces a subgraph G =
(cid:83) ni
=1 G i of the graph G described by the chronological relations of M , and the causal relations between events depend on the choice of M .
It is worth noting that for any spacetime, chronological order is transitive (i.e., x i (cid:28) x j and x j (cid:28) x k = ⇒ x i (cid:28) x k (i.e., x k ∈
I + ( x i , M ) ), see Chapter 3.2 of Beem et al.
v i is then connected by an arc to all the successors of v j in G .
We avoid this degenerate case by drawing an arc from v i to v k in G iff x k ∈ I + ( x i , V x i ) .
The choice of M and V x i ⊆ M for all i is then crucial to construct G ⊆ G .
Existence of directed cycles.
Our framework can represent graphs with directed cycles only if M is non-chronological (i.e., there exists at least one closed timelike curve : ∃ x ∈ M , x (cid:28) x , see Fig. 1), which is the case if M is the anti-de Sitter space
H d 1 ( r ) or the Cylindical Minkowski space L d 1 ( C ) .
Spacetimes that do not contain closed timelike curves are called chronological (i.e., (cid:64) x ∈ M , x (cid:28) x ) and can represent only Directed Acyclic Graphs (DAGs) in our framework.
Some chronological spacetimes such as S d 1 ( r ) (when d ≥ 3 is odd) or R d 1 are called globally hyperbolic and satisfy: x ≤ y = ⇒ x
and y can be joined by a (longest) causal geodesic that is not necessarily unique (see page 66 of Beem et al.
Nonetheless, if M is S d 1 ( r ) or R d 1 , we also have y ∈
I + ( x , M ) iff y ∈
U x and −→ xy is future-directed timelike (see page 411 of (O’Neill, 1983) for details on conditions).
Although DAGs do not contain directed cycles, they can contain undirected cycles.
For simplicity, we consider that V x ⊆ U x is the convex normal neighborhood of x that contains points y such that the arc length d γ of the geodesic γ from x = γ (0) to y = γ (1) is smaller than some arbitrary threshold ε ∈ (0 , ∞ ] and can be formulated as d γ ( x , y ) :
= (cid:112) |(cid:104)−→ xy , −→ xy (cid:105)| .
We have:
[Equation]
where C + x ( t ) is the future timecone parametrized by some arbitrary timelike tangent vector t ∈ T x M .
The motivation is to choose or learn ε small enough that v i is not connected to undesired successors of v j .
However, in most of our experiments we simply consider that V x = U x .
3. E XAMPLES OF SPACETIMES
We ﬁrst illustrate how to represent directed graphs with the simplest spacetime, which is the (ﬂat)
Minkowski space R d 1 .
It is used in special relativity which is a special case of the general relativity of a spacetime isometric to R 41 .
It is also the geometry induced on each ﬁxed tangent space of an arbitrary Lorentz manifold.
It was used in Clough & Evans (2017) to represent DAGs due to its global hyperbolicity and the fact that any pair of points of R d 1 can be joined by a geodesic.
It is worth noting that the Hopf-Rinow theorem does not hold for non-Riemannian manifolds such as spacetimes.
Therefore for many spacetimes, there exist pairs of points that cannot be joined by a geodesic even if the spacetime is geodesically complete.
Working with convex normal neighborhoods
V x
i allows us to constrain chronological order between points only via timelike geodesics as explained in Section 3.1.
• Minkowski spacetime R d 1 .
We recall that ∀ ν, R d ν ≈ T x
R d ν .
Its geodesic γ x → y : R → R d ν is γ x → y ( t ) :
= x + t y .
The exponential map at x is exp x ( y ) :
= x + y and its inverse is −→ xy :=
y − x .
R d 1 is time-oriented by the
vector
ﬁeld
∂/∂x
(e
.g.
, ∀ x ,
y
, τ ( x , y
:= y
− x
, and we should in theory deﬁne τ ( x , y ) :
=
if x (cid:2) y to properly follow the deﬁnition of Lorentzian pre-length spaces but we ignore this criterion during training for optimization purpose).
Let us deﬁne t := (1 , 0 , . .
. , 0) (cid:62) ,
[Equation]
According to equation 4, we have y ∈
I + ( x , V x ) iff −→ xy is future-directed timelike (i.e., β < 0 and (cid:104)−→ xy , t (cid:105) < 0 , or equivalently α < 0 ) and the arc length of −→ xy is smaller than ε (i.e., − ε 2 < β ).
There might exist a path but no arc between x (cid:28) y (i.e., y ∈
I + ( x , M ) \I + ( x ,
V x ) ) iff (cid:104)−→ xy , t (cid:105) <and β ≤ − ε 2 .
There exists no path between x and y (i.e., x (cid:54)(cid:28) y and y (cid:54)(cid:28) x ) iff β ≥ 0 .
• de Sitter spacetime
S
d 1 ( r ) .
The original de Sitter spacetime S 41 ( r ) is not time-orientable (see Corollary 11.2.6 of Wolf (2011)).
However, when d ≥ 3 , ν > 0 and d − ν is even, S dν ( r ) is orientable and time-orientable, the projective space S dν ( r ) / ± 1 used in Law (2021) is orientable but not timeorientable (see page 247 of O’Neill (1983)).
We assume in the following that conditions hold so that S dν ( r ) is a spacetime and we refer the reader to the supp.
material for details
For any pair of points x ∈ S dν ( r ) and y ∈ S dν ( r ) , −→ xy is deﬁned iff y ∈
U x
(i.e., (cid:104) x , y (cid:105) 1 > − r 2 ), and −→ xy is timelike iff (cid:104) x , y (cid:105) 1 > r 2 .
Let p :=
, 0 , r ) (cid:62)
∈ S d 1 ( r ) denote the positive pole, and Γ xp : T
p S d 1 ( r ) → T x S d 1 ( r ) the parallel transport from T p S d 1 ( r ) to T x S d 1 ( r ) .
Since the parallel transport preserves the causal character of any tangent vector v , we can deﬁne the future timecone
C + x ( −→ xy ) with respect to the timelike tangent vector t :=
(1 , 0 , . . . , 0) (cid:62) ∈ T ± p S d 1 ( r ) as follows:
Lemma
−→ xy is future-directed iff Γ xp ( t ) ∈ C
+ x ( −→ xy ) if Γ xp is deﬁned (i.e., p ∈
U x ), and Γ x − p ( t )
∈ C + x ( −→ xy ) otherwise (i.e., − p ∈
U x ).
See proof in Appendix C.1.2.
• The anti-de Sitter spacetime H d 1 ( r ) and its projective version P d 1 ( r ) :
= H d 1 ( r ) / ± 1 are nonchronological and satisfy x (cid:28) y =
⇒ y
(cid:28) x , which is convenient to represent graphs with directed cycles.
Sim et al.
(2021) use H d 1 ( r ) to represent directed graphs but also promote arcs (i.e., causal relation) between pairs of nodes that are not connected by any geodesic, which makes their problem hard to optimize.
In this work, we only consider the existence of arcs if there exists a timelike geodesic in the convex
normal neighborhood V x joining two events.
See Appendix C for details.
• A warped product is a manifold ( M 1 × M 2 , g 1 ⊕ fg 2 ) denoted by M 1 × f M 2 where ( M 1 , g 1 ) , ( M 2 , g 2 ) and f : M 1 → (0 , ∞ ) is a smooth function called the warping function (e.g., f = 1 ).
Let M
=
B
×
f F be a Lorentz warped product where B is Lorentz and F is a complete Riemannian manifold.
M is time-orientable iff B is (see page 417 of (O’Neill, 1983)).
M satisﬁes the chronology, causality or strong causality iff B does.
Our framework can then be extended to warped products.
3. L ORENTZIAN DISTANCE AND L ORENTZIAN LENGTH SPACES
The Lorentzian distance indicates chronological order (hence causality) between events when it is positive, satisﬁes the reverse triangle inequality and its squared function is of class C 2 on a normal neighborhood.
These properties make it ideal for optimization, and it can be used as time separation function.
We refer the reader to Chapter 4 of Beem et al.
and Section 5 of Minguzzi (2019).
Proposition 4.5.3 of Hawking & Ellis (1973): Let x and y lie in a convex normal neighborhood
V x ⊆
U x .
If x and y can be joined by a causal curve in V x , the longest such curve is the unique causal geodesic in V x from x to y .
This is in contrast with Riemannian geometry where the (spacelike) geodesic corresponds to the shortest curve joining points.
Since V x is a convex normal neighborhood, we can deﬁne the arc length of such a curve by χ V ( x , y ) :
= (cid:112) −(cid:104)−→
xy , −→ xy (cid:105) ≥ 0 .
If x (cid:28) y , χ V ( x , y ) is called the Lorentzian distance from x to y on V x , it corresponds to the elapsed proper time between the events x and y (i.e., as measured by a clock along the geodesic γ x →−→ xy ) (Gourgoulhon, 2016).
If we deﬁne τ := χ V , then ( V x , d , ≤ , (cid:28) , τ ) is called a Lorentzian length space .
In this paper, we consider the squared Lorentzian distance which is deﬁned as y ∈
I + ( x , V x ) =
⇒ χ
2 V ( x , y )
= −(cid:104) −→ xy , −→ xy (cid:105) .
In the literature (Beem et al., 1996; Hawking & Ellis, 1973), the (squared) Lorentzian distance is deﬁned such that χ 2 V ( x , y )
=
if x and y are not joined by a timelike curve due to lack of causality.
When evaluating learned representations (i.e., at test time), we therefore consider that χ 2 V ( x , y )
=if this is the case.
However, during training and for optimization purpose, we propose to consider:
If −→ xy is undeﬁned or not timelike, χ 2 V ( x , y ) =
[Equation]
This makes the function χ 2 V differentiable everywhere (except when (cid:104) x , y (cid:105) ν
+
=
if M = P dν ( r ) ), equal to −(cid:104)−→ xy , −→ xy (cid:105) > 0 if −→ xy is timelike, and non-positive otherwise.
χ 2 V is also deﬁned for any pair of points x and y on M by using extrinsic geometry, whether −→ xy is deﬁned or not.
R ELATED W ORK
Exploiting spacetimes to represent directed graphs was proposed by Clough & Evans (2017) to extend
MultiDimensional Scaling (MDS) (Kruskal, 1964) to the simplest spacetime R d 1 .
As in standard
MDS, a target distance matrix between nodes is given as input and the goal of the method is to ﬁnd the vector representations of nodes that return the best approximation of the target distance matrix when using the squared Lorentzian distance as the dissimilarity function.
However, as discussed in
Section 3.3, it is difﬁcult to deﬁne target distances between pairs of nodes that are not connected.
Moreover, the formulation in (Clough & Evans, 2017) considers only R d 1 and does not account for future or past-direction between pairs of nodes during training.
We solve this problem in Section 3.
by enforcing future timelike geodesics to have their length smaller than some threshold ε ∈ (0 , ∞ ] .
Sim et al.
(2021) extended Clough & Evans (2017) to the anti-de Sitter space and Lorentz cylinder.
Although our motivation is similar, our contributions are methodological, rely on the intuitions of
Lorentzian pre-length spaces, and provide an easier interpretation of the learned representations as we explain below.
First, Sim et al.
(2021) do not address clearly the case when there is no geodesic between pairs of points, and their optimization framework leads to a distance loss term with a zero gradient in this case, which is difﬁcult to optimize.
Moreover, their prediction of an arc between a pair of nodes is determined via a Triple Fermi-Dirac (TFD) probability function that accounts for the distance between the nodes, the time coordinate difference ∆ t and its opposite value − ∆ t (i.e., TFD accounts for both the chronological future and past of a given node).
In contrast, we restrict the representation of nodes connected by an arc to belong to I + ( x , V x ) where V x is an open convex normal neighborhood, which is simple to interpret and optimize.
In general, the Lorentzian distance function from x to y on M is deﬁned to be inﬁnite if M is non-chronological and there exists a closed timelike curve joining x and y .
Moreover, the Hopf-Rinow theorem does not hold for spacetimes.
Working with convex normal neighborhoods allows us to restrict the existence of arcs between nodes to the existence of geodesics (of ﬁnite length) joining events.
The deﬁnition of time separation functions also becomes straightforward and we use their sign to determine the direction of an edge.
Our framework shares similarities with Sim et al.
(2021) when M = R d 1 = V x because R dis globally hyperbolic.
Otherwise, the way we formulate our time separation function and creation of arcs is different.
Further comparisons with Sim et al.
(2021) are provided in Appendix E.
The negative of the pseudo-Riemannian gradient is generally not a descent direction.
We then use the optimization tools introduced in Gao et al.
(2018); Law & Stam (2020); Law (2021) as described in
Appendix F to learn our representations.
Our approach could be extended to time-oriented pseudoRiemannian manifolds of index ν > 1 (see pages 240-242 of (O’Neill, 1983)).
We also have x (cid:28) y if there exists a future-directed timelike curve joining them.
Lemma 3.1 that exploits parallel transport can for instance be generalized to timelike tangent vectors t 1 , . . .
, t ν to deﬁne different types of arcs.
Some Riemannian approaches (Bordes et al., 2013; Ganea et al., 2018; Vilnis et al., 2018) have been used to represent DAGs.
Ganea et al.
(2018) consider entailment relations where x (cid:28) y means that y is a subconcept of x by constructing cones for each node representation in hyperbolic geometry.
The
DAGs they consider are directed trees and their underlying undirected graphs are trees (i.e., graphs without cycles).
We propose to consider the chronological future and past that are intrinsic to our manifolds and have been studied for decades to deﬁne the causal structure.
For instance, it is known that causal relations induce partially ordered sets (called causal sets ) on causal spacetimes and can then represent DAGs (Bombelli et al., 1987).
Our framework can represent DAGs that are not directed trees (see example in Appendix G.1) and is not limited to DAGs (see examples in Appendix G.2).
[Table]
E XPERIMENTS
We show how our framework can represent graphs with directed cycles and predict links effectively in Section 5.1.
We also show how the causal interpretation of our model can be used to represent hierarchical graphs with cycles in Section 5.2.
We provide experiments on DAGs in Appendix D.1.
5. L INK PREDICTION ON GRAPHS WITH DIRECTED CYCLES
We now consider the link prediction task on the Saccharomyces Cerevisiae , in silico and Escherichia coli DREAM5 datasets (Marbach et al., 2012)
(see values of hyperparameters and results on Escherichia coli with standard deviations in Appendix D.2).
We follow the experimental protocol of
Sim et al.
by considering only the positive-regulatory nodes from the networks mentioned above while omitting the gene-expression data itself.
Each network is randomly split into train and test sets, following 85/15 splits, and a part of the training set is used for validation.
Sim et al.
(2021) use the Minkowski space R d 1 , the anti de-Sitter space H d 1 ( r ) , and the Cylindrical
Minkowski space L d 1 ( C ) .
Both H d 1 ( r ) and L d 1 ( C ) are non-chronological and can then represent graphs with directed cycles.
Sim et al.
(2021) design a probability function that they call Triple
Fermi-Dirac (TFD) speciﬁcally for their Cylindrical Minkowski space.
Instead, we propose to deﬁne the chronological future I + ( x , V x ) of x ∈ L d 1 ( C ) such that if −→ xy is timelike, we have y ∈
I + ( x , V x ) if ∃ k ∈ Z , y 0 + kC ∈ ( x 0 , x 0 + C/ 2) .
Similarly, the chronological past I − ( x , V x ) is deﬁned such that if −→ xy is timelike, we have y ∈ I − ( x , V x ) if ∃ k ∈ Z , y 0 +
kC ∈
( x 0 − C/ 2 , x 0 ) .
In detail, we deﬁne the time separation function for L d 1 ( C ) as follows:
τ ( x , y ) :
= (cid:0)(cid:0) ( y 0 − x 0 + C 2 ) mod C (cid:1) − C 2 (cid:1) ∈ [ − C 2 , C 2 ) where we use the modulo operation for real values which can be written as follows: a mod b :=
a − b · (cid:98) ab (cid:99) , and (cid:98)·(cid:99) is the ﬂoor function.
Following our spacetime interpretation, we propose to learn our representations by optimizing:
[Equation]
where F ( x i , x j ) :
= σ θ 1 ( χ 2 V ( x i , x j )) · σ θ 2 ( τ ( x i , x j )) , θ 1 and θ 2 > 0 are temperature hyperparameters, and σ θ ( x ) :
= 1 / (1 + e − x/θ ) is the sigmoid function.
This promotes future-directed timelike geodesics only between pairs of nodes connected by an arc.
We report in Table 1 the Average Precision scores of our method (i.e., using equation 6) that outperforms baselines reported in Sim et al.
The non-chronological cylindrical Minkowski space obtains a higher performance gap in low-dimensional space.
This suggests it manages to account for directed cycles better than chronological spacetimes.
On the other hand, the gap is reduced in higher dimension except on Escherichia coli where the gap increases (see Table 4).
5. H IERARCHY EXTRACTION ON A SOCIAL NETWORK DATASET
Pseudo-Riemannian manifolds were introduced in the machine learning literature to extract hierarchies in graphs with cycles (Law & Stam, 2020).
Spacetimes are a special family of pseudo-Riemannian manifolds whose time-orientation was not taken into account in Law & Stam (2020).
In this paper, we found that spacetimes are able to consistently outperform other pseudo-Riemannian manifold approaches by exploiting the causality interpretation implicit in the squared Lorentzian distance.
↓ the lower the metric, the better.
↑ the larger the metric (in absolute value), the better
[Table] [Table]
Graph dataset.
We consider an undirected graph G =
( V, E ) where V = { v i } ni
=1 is the node set and E is the edge set such that ( v i , v j
) ∈
E indicates that v i and v j are connected by an edge.
Zachary’s karate club dataset (Zachary, 1977) is a social network of n = 34 members of the karate club, each represented by a node v i .
The club was split in two factions due to a conﬂict between the instructor (node v 1 ) and the administrator (node v 34 ) that are the two most important members (i.e., leaders) of the dataset.
The other members had to decide between joining the new club created by v 1 or stay with v 34 .
Two nodes are joined by an edge ( v i , v
j ) ∈
E if the members are friends.
In the original ultrahyperbolic approaches (Law & Stam, 2020; Law, 2021), each node v i is represented by a point x i ∈ M on some pseudo-Riemannian manifold M .
The goal is to learn embeddings { x i } ni =1 so that pairs of nodes joined by an edge ( v i , v j ) ∈ E are closer to each other than pairs of nodes not joined by an edge.
To this end, the embeddings are learned in Law & Stam (2020); Law (2021) by minimizing the following problem:
[Equation]
where the dissimilarity function d ( x i , x j ) is the arc length d γ ( x i , x j ) :
= (cid:112) |(cid:104)−−→ x i x j , −−→ x i x j (cid:105)| of the geodesic joining x i and x j .
If M is Riemannian, d γ is the Riemannian distance.
In this paper, we propose instead to consider in equation 7 that d is the squared Lorentzian distance χ 2 U , as deﬁned in
Section 3.3, where U x is the (convex) maximal normal neighborhood of x ∈ M .
We also propose to simply enforce the embeddings of v i and v j to be joined by a timelike geodesic iff ( v i , v j ) ∈ E .
Since G is an undirected graph, the future- or past-direction is not provided so we do not constrain the future-direction of the geodesics during training.
Our problem formulation is then:
[Equation]
where λ > 0 is a regularization parameter and we set d ( x i , x j )
=
χ 2 U ( x i , x j ) .
The main goal of equation 8 is to enforce pairs of nodes ( v i , v j ) ∈ E to be joined by a timelike geodesic and pairs ( v a , v b ) / ∈ E to be joined by a spacelike geodesic since there is no causality between them.
1.5 x 1 ( space ) 1.
x 0 ( t i m e )     5  9 12 15 21  16 17 202426 27
Faction 1 Faction
Evaluation metrics.
We report in Table 2 the results obtained when optimizing equation 7 or equation 8 using different dissimilarity functions.
At test time, a score δ i =
(cid:80) nj =1 d ( x i , x j ) summing pairwise distances is assigned to each node v i and used as an indicator of importance in the hierarchical graph.
Following Section 3.3, when the dissimilarity function is the arc length d γ (resp.
squared
Lorentzian distance χ 2 U ), we sort the scores δ 1 , . . .
, δ n is ascending (resp.
descending) order and report the ranks of the two leaders v 1 and v n (in no particular order) in the ﬁrst two columns of Table 2 averaged over 10 different initializations.
We also report in Table 2 the Spearmans rank correlation coefﬁcient ρ (Spearman, 1904) between the ordered δ i scores and the order of the 5 and 10 most important nodes which are 34, 1, 33, 3, 2, 32, 24, 4, 9, 14 (in that order, see Law & Stam (2020)).
Results.
In general, the squared Lorentzian distance χ 2 U returns better performance than the arc length d γ .
Moreover, the optimization problem in equation 8 enforcing timelike geodesics between connected nodes (i.e., causality) and spacelike geodesics between unconnected nodes (i.e., noncausality) extracts the most important nodes in low-dimensional spacetimes more effectively.
The predicted ρ also shows higher rank correlation with χ 2 U .
It is worth noting that hyperbolic geometry was shown relevant to represent graphs without cycles (Gromov, 1987)
but our hierarchical graph contains cycles, which explains why geometries other than hyperbolic may be more relevant.
The performance gap with non-Riemannian baselines (Law & Stam, 2020; Law, 2021) can be explained by the fact that those baselines compare the arc lengths of different types of geodesics (i.e., timelike or spacelike) that are not necessarily comparable.
On the other hand, our framework sets non-causal distances to zero at test time, and compares only the lengths of causal geodesics.
Figure 2 illustrates spacetime diagrams of the learned representations for R 21 and S 31 ( r ) .
The most important nodes tend to be further away from the other nodes with respect to the Lorentzian distance.
Most ground truth edges are timelike geodesics and belong to the chronological past or future of an edge representation.
We report in Appendix D.3 the same kind of hierarchy extraction experiment on a dataset that describes co-authorship information from papers published at NIPS from 1988 to 2003 (Globerson et al., 2007).
The number of nodes/authors is | V | = 2715 and the number of edges is | E | = 4733 .
C ONCLUSION
We have proposed a general framework to represent data, and in particular nodes of a directed graph, in a spacetime.
Compared to previous work, our framework is properly endowed with the structure of a Lorentzian pre-length space, which makes it easy to optimize and applicable to a large family of spacetimes.
In the case of hierarchical undirected graphs, we show that we can enforce causality between nodes connected by an edge to extract the most important nodes in the hierarchy.
A S UMMARY OF S UPPLEMENTARY M ATERIAL
The supplementary material is structured as follows:
• Section B gives the differential geometry tools that are general to pseudo-Riemannian manifolds of constant curvature (called space forms ).
• Section C gives the differential geometry tools that are speciﬁc to our spacetimes such as the time separation function and the squared Lorentzian distance.
• Section D presents additional results and provides further experimental details such as hyperparameter values.
• Section F gives the details of the optimizers we use.
• Section G gives examples of directed graphs described with our model.
B D IFFERENTIAL G EOMETRY OF P SEUDO -R IEMANNIAN S PACE F ORMS
We provide here the necessary differential geometry tools to work on the pseudo-Euclidean space R d ν , the pseudo-sphere S dν ( r ) and the quotient manifold P d 1 ( r ) .
Most of them are explained in (Law, 2021;
Law & Stam, 2020).
We also refer the reader to (Hawking & Ellis, 1973; O’Neill, 1983; Wolf, 2011).
B. P SEUDO -E UCLIDEAN SPACE
We recall that ∀ ν, R dν ≈ T x R dν .
Its geodesic γ x → y : R → R dν is γ x → y ( t ) :
= x + t y .
The exponential map at x is deﬁned as exp x ( y ) :
=
γ x → y (1) =
x + y and its inverse is −→ xy :=
exp − 1 x ( y ) =
y − x .
Moreover, ∀ x ∈ R d 1 , y ∈ R d 1 , the parallel transport Γ yx : T x R dν
→
T y R dν is the identity function.
B. P SEUDO - SPHERE S dν ( r )
We give here the differential geometry tools speciﬁc to the pseudo-sphere which is deﬁned as the following hypersurface: S
[Equation]
Geodesic.
The geodesic γ x → u : R → S dν ( r ) satisfying γ x → u (0) = x and γ (cid:48)
x → u (0) =
u ∈ T x S dν
( r ) is formulated for all t ∈ R :
[Equation]
Exponential map.
The exponential map exp x : T x S dν ( r ) → S dν ( r ) is deﬁned such that ∀ u ∈
T x S dν ( r ) , exp x ( u ) =
γ x → u (1) .
We then have:
[Equation]
Logarithmic map.
The logarithmic map log x is deﬁned as the inverse of the exponential map exp x on a normal neighborhood of x ∈ S dν ( r ) denoted by U x = { y ∈ S
dν ( r ) : (cid:104) x , y (cid:105)
ν r> − 1 } .
It is then formulated:
[Equation]
One can verify from equation 10 and equation 11 that −→ xy is timelike (i.e., (cid:104)−→ xy , −→ xy (cid:105) < 0)
iff (cid:104) x , y (cid:105) ν
> r 2 .
Geodesic “distance”.
As explained in (Law & Stam, 2020) and Chapter 5 of (O’Neill, 1983), when the logarithmic map log exists for some pseudo-Riemannian manifold M , the arc length of the tangent vector joining x ∈ M and y ∈ M corresponds to the radius function: (cid:112) | g x (log x ( y ) , log x ( y ))
| where g x : T x M × T x M
→
R is the metric tensor at x and log x is the logarithmic map.
In the case of the pseudo-sphere, we have g x ( · , · ) = (cid:104)· [Equation]
ν . The geodesic distance d γ : S
[Equation]
ν
[Equation]
r ) × S
[Equation]
r ) → R is then:
[Equation]
d γ is not a “distance metric” but a symmetric premetric: it satisﬁes (i) d [Equation]
( x , y ) = d γ ( y , x ) ≥ 0 and (ii) d γ ( x , x ) = 0 .
In (O’Neill, 1983), the “minimizing geodesic” is deﬁned by its arc length and then also corresponds to the geodesic distance mentioned above.
Given the minimizing geodesic γ connecting x to y , the parallel transport Γ yx : T x S dν ( r ) → T y S dν ( r ) is a linear isometry such that ∀ u , v , (cid:104) u , v (cid:105)
ν = (cid:104) Γ yx ( u ) , Γ yx ( v ) (cid:105) ν .
The parallel transport along γ from x = γ (0) to y = γ (1) (where x and y satisfy (cid:104) x , y (cid:105) ν > − r 2 ) is:
[Equation]
Theorem B.1 (Diffeomorphism (Wolf, 2011)) .
There exists a diffeomorphism ψ : S dν ( r ) → R ν ×
S d − ν( r ) .
Let us note x = (cid:18) ts (cid:19)
∈ S dν ( r ) with t ∈ R ν and s ∈ R d − ν +1 ∗ .
Let us note z = (cid:18) tv (cid:19) ∈
R ν × S d − ν( r ) where v ∈ S d − ν( r ) .
The mapping ψ and its inverse ψ − 1 can be formulated:
[Equation]
and
[Equation]
B. P SEUDO - HYPERBOLOID H dν ( r )
We recall here the differential geometry tools (from (Law & Stam, 2020)) speciﬁc to the pseudohyperboloid which is deﬁned as the following set:
H dν ( r ) :
= { x ∈ R d +1 ν
+1 : (cid:104) x , x (cid:105)
[Equation]
+1 = − r 2 } .
Geodesic.
The geodesic γ x → u : R → H dν ( r ) satisfying γ x → u (0) = x and γ (cid:48)
x → u (0) =
u ∈ T
[Equation]
H dν [Equation] r ) is formulated for all t ∈ R :
[Equation]
Exponential map.
The exponential map exp x : T x H dν ( r ) → H dν ( r ) is deﬁned such that ∀ u ∈
T x H dν ( r ) , exp x ( u ) =
γ x → u (1) .
We then have:
[Equation]
Logarithmic map.
The logarithmic map log x is deﬁned as the inverse of the exponential map exp x on a normal neighborhood of x ∈ H dν ( r ) denoted by U x =
{ y ∈ H dν ( r ) : (cid:104) x , y (cid:105) ν
+1 r< 1 } .
It is then formulated:
[Equation]
One can verify that a geodesic from x to y is timelike iff (cid:104)
x , y (cid:105) ν ∈ ( − r 2 , r 2 ) or y =
− x .
Geodesic “distance”.
The geodesic distance d γ : H d ν ( r ) × H d ν ( r ) → R is then:
[Equation]
Parallel transport on H dν ( r ) .
The parallel transport connecting x ∈ H dν ( r ) to y ∈ H dν ( r ) is:
[Equation]
where (cid:104) x , y (cid:105) ν
+1 < r(19)
Theorem B.2 (Diffeomorphism (Wolf, 2011)) .
There exists a diffeomorphism ψ : H dν ( r ) → S ν 0 ( r ) ×
R d − ν .
Let us note x = (cid:18) ts (cid:19) ∈
H dν ( r ) with t ∈ R ν +1 ∗ and s ∈ R d − ν .
Let us note z = (cid:18) uv (cid:19) ∈
S ν 0 ( r ) × R d − ν where u ∈ S ν 0 ( r ) and v ∈ R d − ν .
The mapping ψ and its inverse ψ − 1 can be formulated:
[Equation]
and
[Equation]
The anti-de Sitter spacetime H d 1 ( r ) is non-chronological and satisﬁes x (cid:28) y =
⇒ y
(cid:28) x , which is convenient to represent graphs with directed cycles.
It was used in Sim et al.
(2021) to represent directed graphs.
Nonetheless, it is worth noting that the problem formulation of Sim et al.
(2021) for H d 1 ( r ) also promotes arcs (i.e., causal relation) between pairs of nodes that are not connected by any geodesic, which makes their problem hard to optimize.
In this work, following general relativity, we only consider the existence of arcs if there exists a timelike geodesic in V x joining two events x and y .
See Appendix C for details.
B. P ROJECTIVE SPACE P d 1 ( r )
The manifold P d 1 ( r ) :
= H d 1 ( r ) / ± 1 used in Law (2021) is time-orientable for all d ≥ 2 (see page 214 of O’Neill (1983)).
We refer the reader to (Law, 2021) for details about P d 1 ( r ) :
= H d 1 ( r ) / ± 1 .
C F UTURE D IRECTION , TIME SEPARATION FUNCTION AND L ORENTZIAN D ISTANCE C. F UTURE DIRECTION
We discuss in this section how to constrain future direction for our spacetimes.
C.1. M INKOWSKI SPACE
The explanation can be found in Section 3.2.
We recall that we deﬁned: t :=
(1 , 0 , 0 , 0) (cid:62) , u :=
( u 0 , . . .
, u d − 1 ) (cid:62) =
−→ xy =
( y 0 − x 0 , y 1 − x 1 , . . .
, y d − 1 − x d − 1 ) (cid:62) and α = − u
+ (cid:113)(cid:80) d − 1 i =
u 2 i .
If α is negative, then −→ xy is timelike (i.e., (cid:104)−→ xy , −→ xy (cid:105) < 0 ) and −→ xy ∈ C
+ x ( t ) .
By deﬁnition, −→ xy is then future-directed timelike.
C.1. D E S ITTER SPACE
We provide here the proof of Lemma 3.1 which is written as follows: −→ xy is future-directed iff Γ xp ( t ) ∈ C + x ( −→ xy ) if Γ xp is deﬁned (i.e., p ∈ U x ), and Γ x − p ( t ) ∈ C
+ x ( −→ xy ) otherwise (i.e., − p ∈
U x ).
We ﬁrst recall a property from page 146 of (O’Neill, 1983): If M is a Lorentz manifold, if a piecewise smooth curve α is timelike, this means not only that α (cid:48) ( t ) is timelike, but at each break t i of α :
[Equation]
Here the ﬁrst vector derives from α | [ t i − 1 , t i ] and the second from α | [ t i , t i +1 ] .
Thus α (cid:48) does not switch timecones at the break.
Let us note the poles p = (0 , . . . , 0 , r ) (cid:62)
∈ S
d 1 ( r ) , − p =
, 0 , − r ) (cid:62)
∈ S d 1 ( r ) and t = (1 , 0 . . . , 0) (cid:62) where t ∈ T
p S d 1 ( r ) and t ∈ T − p S d 1 ( r ) .
We also note x :=
( x 0 , x 1 , . . .
, x d ) (cid:62) ∈ S d 1 ( r ) .
If x d ∈ ( − r, r ) , we show below that equation 21 can be satisﬁed by deﬁning α such that α ( t 0 )
=
p , α ( t 1 ) =
x , α (
[Equation]
) = − p ,
[Equation]
=
[Equation] [Equation]
=
Γ
[Equation] [Equation]
+ 1 )
=
Γ x − p ( t ) , α (cid:48) ( t 2 ) =
t .
For completeness, we also have ∀ t ∈
[ t 0 , t − 1 ] , α (cid:48) ( t ) =
Γ α ( t ) p ( t ) and ∀ t ∈
[ t + 1 , t 2 ] , α (cid:48) ( t ) =
Γ α ( t ) − p ( t )
Let us assume that t ∈ T p S d 1 ( r ) deﬁnes the future direction of the manifold.
For all x ∈ S
d
1 ( r ) that satisﬁes (cid:104) x , p (cid:105) > − r 2 , we note Γ xp ( t ) the parallel translate of t ∈ T
p S d 1 ( r ) to T x S d 1 ( r ) .
As explained in Appendix B.1.2 of (Law, 2021), it is formulated:
[Equation]
We know from page 66 of (O’Neill, 1983) that parallel transport (also called parallel translation) is a linear isometry that satisﬁes:
[Equation]
[Equation]
and u = Γ xp (Γ px ( u )) .
This implies
[Equation] [Equation]
Similarly, if (cid:104) x , − p (cid:105) >
− r 2 , we have Γ x − p ( t ) ∈ C + x ( u ) ⇐⇒ Γ − px ( u ) ∈ C + − p
( t ) .
Let us now assume that p ∈
U x
(i.e., (cid:104) x , p (cid:105) > − r 2 ) and − p ∈
U x
(i.e., (cid:104) x , p (cid:105) <
r 2 ), which is equivalent to x satisfying x d ∈ ( − r, r ) .
Therefore Γ xp ( t ) ∈ C + x ( u ) , Γ x − p ( t ) ∈ C
+ x ( u ) ⇐⇒ (cid:104) Γ xp ( t ) , Γ x − p ( t ) (cid:105) 1 < 0 .
By deﬁnition, we have:
[Equation]
We have shown that Γ xp ( t ) and Γ x − p ( t ) have the same future direction by showing that (cid:104) Γ xp ( t ) , Γ x − p ( t ) (cid:105) 1 < 0 .
A (piecewise) smooth that preserves causality can then be found by using Γ xp ( t ) or Γ x − p ( t ) .
C.1. A NTI - DE S ITTER SPACE
The proof for H d 1 ( r ) is similar to the proof in Section C.1.2.
Let us note the poles p = ( r, 0 , . .
. , 0) (cid:62) ∈
H d 1 ( r ) , − p = ( − r, 0 , . .
. , 0) (cid:62) ∈
H d 1 ( r ) and t = (0 , 1 , 0 . . . , 0) (cid:62) where t ∈ T
p H d 1 ( r ) and t ∈ T − p H d 1 ( r ) .
Let x = ( x − 1 , x 0 , . . .
, x
d − 1 ) (cid:62)
[Equation]
H d 1 ( r ) where x − 1 ∈ ( − r, r ) .
Let us note Γ xp ( t ) the parallel translate of t ∈ T
p H d 1 ( r ) to T x H d 1 ( r ) .
As explained in AppendixB. of (Law, 2021), it is formulated:
[Equation]
We show below that Γ x p ( t ) and Γ x − p ( − t ) have the same future direction by showing that (cid:104) Γ x p ( t ) , Γ x − p ( − t
[Equation]
1 < 0 .
[Equation]
equation 30 is negative because we have by deﬁnition of H d 1 ( r ) : − x 2 − 1 − x 20 + (cid:80) d − 1 i =1 x 2 i = − r 2 , which implies x 20 ≥ r 2 − x 2 − 1 > 0 , and the denominator is positive because we deﬁned x − 1 ∈ ( − r, r ) .
C. TIME SEPARATION FUNCTION
We now give the formulations we use for the time separation function τ ( x , y ) that is positive (resp.
negative) if y is in the chronological future (resp.
past) of x .
There is no unique way to deﬁne it.
Globally hyperbolic spacetimes.
If the spacetime M is globally hyperbolic, there exists a Cauchy time function c : M → R (i.e., for any t ∈ R , the set Σ t = { y ∈ M : c ( y )
=
t } is a Cauchy hypersurface) that can be used to deﬁne τ ( x , y ) :
= c ( y ) − c ( x ) since it satisﬁes the reverse triangle inequality τ ( x , z ) ≥ τ ( x , y )
+
τ ( y , z ) when x ≤ y ≤ z .
With this formulation, it actually satisﬁes τ ( x , z ) =
τ ( x , y )
+ τ ( y , z ) when x ≤ y ≤ z .
In theory, to have a Lorentzian pre-length space, one could deﬁne τ such that it is 0 if x (cid:2) y , but this is not easy to optimize in practice so we ignore this constraint for optimization purpose.
C.2. M INKOWSKI SPACE R d
If M = R d 1 then we arbitrarily deﬁne:
[Equation]
(31) where t :=
(1 , 0 , . . . , 0) (cid:62) deﬁnes the future cone C
+ x ( t ) .
It is worth noting that the function c : R d 1 → R deﬁned such that ∀ x = ( x 0 , . . .
, x d ) (cid:62) ∈ R d 1 , c ( x ) := x 0 is a Cauchy time function.
C.2. D E S ITTER SPACE S d 1 ( r )
Following our discussion in SectionC.1.2 and using the formulation of the parallel transport in equation 22, we can formulate the time function as:
[Equation]
otherwise.
One limitation of equation 32 is that it assumes that −→ xy is deﬁned, which might not be the case if y / ∈ U x .
From the discussion in Chapter 5.2 and Figure 16 (i) of (Hawking & Ellis, 1973) that uses Cartesian coordinates to formulate space coordinates on the pseudo-sphere, the function c : S d 1 ( r ) → R deﬁned such that ∀ x
= ( x 0 , . .
, x d ) (cid:62) , c ( x ) := x 0 is a Cauchy time function.
One can then also simply deﬁne:
[Equation]
In practice, we found that using equation 33 returned better performance than equation 32 in the experiments of Section D.1.
We then also use it for the experiment of Section 5.1.
See Section D. for details.
C.2. A NTI - DE S ITTER SPACE H d
[Equation]
H dν ( r ) is non-chronological and satisﬁes x (cid:28) y =
⇒ y
(cid:28) x , which is convenient to represent graphs with directed cycles.
There exists a future-directed timelike curve from x to y only if −→ xy is timelike or y = ± x .
Equivalently, x (cid:28) y is equivalent to y = ±
x or (cid:104) x , y (cid:105) ν +1 ∈
( − r 2 , r 2 ) where (cid:104)· , ·(cid:105)
ν
+1 is the metric of R d
+
ν
+1 .
Following our discussion in Section C.1.3 and using the formulation of the parallel transport in equation 27, we can formulate the time function as:
[Equation]
otherwise.
[Equation] [Equation]
P ROJECTIVE VERSION OF THE ANTI - DE S ITTER SPACE P d 1 ( r )
We recall that each point of P d 1 ( r ) is an unordered pair {− x , x } ∈ P d 1 ( r ) where x ∈ H d 1 ( r ) .
By using the formulation of the parallel transport in equation 27, we can then deﬁne:
[Equation]
p
[Equation]
+ if (cid:104) x , p (cid:105) ν +1 > 0 and (cid:104) x , y (cid:105) ν +1 ≤−(cid:104)
Γ px ( −−−−→ x ( − y )) , t (cid:105) ν
+ if (cid:104) x , p (cid:105) ν +1 ≤ 0 and (cid:104) x , y (cid:105) ν +1 >−(cid:104)
Γ − px ( −−−−→ x ( − y )) , − t (cid:105) ν
+ if (cid:104) x , p (cid:105) ν
+1 > 0 and (cid:104) x , y (cid:105) ν +1 >(35)
Let us assume that p =
( r, 0 , . . . , 0) (cid:62) .
We can pick x such that x − 1 ≥
so that (cid:104) x , p (cid:105) 2 ≤ 0 is satisﬁed and y so that (cid:104) x , y (cid:105) 2 ≤ 0 is also satisﬁed.
We can choose deﬁne t :=
(0 , 1 , 0 , . . . , 0) (cid:62) ∈
T p P d 1 ( r ) .
If we deﬁne u = ( u − 1 , u 0 , . .
u d − 1 ) (cid:62) =
Γ px ( −→ xy ) , then equation 35 can be rewritten:
[Equation]
Another possibility is to use the following time separation function:
[Equation]
which is positive iff −→ xy is future-directed timelike.
C.2. C YLINDRICAL M INKOWSKI SPACE L d 1 ( C )
As we explain in the main paper, we propose to deﬁne the chronological future
I + ( x , V x ) of x ∈ L d 1 ( C ) such that if −→ xy is timelike, we have y ∈
I + ( x , V x ) if ∃ k ∈ Z , y 0 + kC ∈ ( x 0 , x 0 + C/ 2) .
Similarly, the chronological past I − ( x , V x ) is deﬁned such that if −→ xy is timelike, we have y ∈
I − ( x , V x ) if ∃ k ∈ Z , y 0 + kC ∈
x 0 − C/ 2 , x 0 ) .
We deﬁne the time separation function for L d 1 ( C ) as follows:
[Equation]
where we use the modulo operation for real values which can be written as follows: a mod b := a − b · (cid:98) ab (cid:99) , and (cid:98)·(cid:99) is the ﬂoor function.
C. S QUARED L ORENTZIAN DISTANCE
We give here the formulation of the squared Lorentzian distance for the different spacetimes that we use in the main paper depending on the nature of M :
If
[Equation] [Equation]
Causality and squared (Lorentzian) distance in Sim et al.
Sim et al.
(2021) acknowledge that there exists no geodesic from x to y if M = H d 1 ( r ) and (cid:104) x , y (cid:105)
[Equation]
1 r> 1 .
Therefore, they consider that their squared Lorentzian distance is equal to π 2 in this case to keep their function smooth.
This promotes causality between x and y .
However, constraining future-direction between x and y when (cid:104) x , y (cid:105) ν
+ r> 1 becomes problematic without the explicit formulation of a timelike curve.
This is why they formulate their time function based on a different criterion than ours that uses parallel transport (see Section C.2).
Moreover, the fact that their distance function becomes a constant in this case makes it difﬁcult to optimize as the gradient is zero.
Instead, we propose to use the manifold
M = P d 1 ( r ) which can represent the same types of relationships between points as H d 1 ( r ) since P d 1 ( r ) contains elliptic and hyperbolic parts (Law, 2021), and any pair of points of P d 1 ( r ) can be joined by a geodesic.
D A DDITIONAL E XPERIMENTS AND E XPERIMENTAL D ETAILS
We now report additional experiments and provide experimental details.
Setup.
We ran all our experiments on a single desktop with 64 GB of RAM, a 6-core Intel i7-7800X
CPU and a NVIDIA GeForce RTX 3090 GPU.
D. C HRONOLOGICAL ORDER IN DIRECTED GRAPHS
Our goal in this subsection is to represent a directed graph with spacetimes.
As in Clow & Evans (Clough & Evans, 2017), we select the 200 and 1000 most cited papers in the Arxiv High-energy physics theory (HEP-TH) citation network (Gehrke et al., 2003).
HEP-TH is originally a dataset of 27 , 770 papers (each represented by a node) with 352 , 807 edges.
The graph contains an arc from v i to v j if paper i cites paper j .
When selecting the 200 or 1000 most cited papers, the graph is not a Directed Acyclic Graph (DAG) as there exist pairs of papers that cite each other.
We ignore these pairs of arcs.
To simplify the notation, we write v i (cid:28) v j either if there exists a path from v i to v j , or the exists an arc from v i to v j but not from v j to v i (i.e., there can exist a longer path from v j to v i ).
We optimize the problem:
[Equation]
( x i , x j ) (cid:1) + σ θ 2 ( − τ ( x i , x j )) (cid:1) where σ θ ( x ) :
= 1 / (1 + e − x/θ ) is the sigmoid function, θ 1 , θ 2 > 0 are temperature parameters, λ is a regularization parameter and τ ( x , y ) is a time separation function that is positive (resp.
negative) if y is in the chronological future (resp.
past) of x .
For instance, if M = R d 1 then we arbitrarily deﬁne τ ( x , y ) :
= −(cid:104) −→ xy , t (cid:105) =
y 0 − x 0 , where t :=
(1 , 0 , . . . , 0) (cid:62) deﬁnes the future cone C
+ x ( t ) .
In this experiment, our temperature hyperparameter values are θ
1 = θ 2 = 1 , and we ﬁx the radius to r = 1 .
We run our experiments for 10 8 iterations with a step size of 10 − 6 by using the optimization tools of (Law, 2021; Law & Stam, 2020).
The regularization parameter λ is set to λ =
| E c | | E | where | E | is the number of pairs that satisfy v i (cid:28) v j and | E c | is the number of pairs that satisfy v a (cid:54)(cid:28) v b .
We report in Table 3 how well the learned representations manage to preserve chronological order when we select the | V | = 200 or 1000 most cited papers.
For instance if M = R d 1 , we report the
[Table]
percentage of pairs of nodes v i (cid:28) v j represented by x i and x j that satisfy (cid:104)−−→ x i x j , t (cid:105) < 0 .
The chronological manifolds R d 1 and S d 1 ( r ) manage to predict chronological order better than the nonchronological manifold P d 1 ( r ) .
This suggests that chronological spacetimes are more appropriate to represent graphs that are almost DAGs.
The time separation function in equation 33 returns better performance than equation 32, we then use it in the rest of our experiments.
Figure 3 illustrates the embeddings learned when M = R 21 .
Article representations tend to satisfy the chronological order along the time dimension.
D. D IRECTED LINK PREDICTION
We report in Table 4 the link prediction scores on the Saccharomyces Cerevisiae , in silico and
Escherichia coli DREAM5 datasets (Marbach et al., 2012).
Following the evaluation protocol of (Sim et al., 2021), we report the median + standard deviation across 20 random initialization.
We use the same training and test splits as (Sim et al., 2021).
We use the following hyperparameters:
• if M = S d 1 ( r ) , r = 1 , θ 1 = 0 .
15 , θ 2 = 0 .
03 , learning rate = 10 − 5 , number of epochs = 2000 .
• if M = L d 1 ( C ) , C = 8 , θ 1 = 0 .
15 , θ 2 = 0 .
03 , learning rate = 10 − 3 , number of epochs = 2000 .
In conclusion, our approach based on general relativity (i.e., equation 6) returns better performance with the cylindrical Minkowski space L d 1 ( C ) than with the de Sitter space when d is small due to the presence of directed cycles in the graph.
[Figure]
[Table] [Table]
[Table]
equation
[Table] [Table]
equationχ 2 U
[Table]
y )
[Table]
D. H IERARCHY EXTRACTION
For both equation 7 and equation 8, we set θ = 10 − 2 , r = 1 and train the model for 10 5 iterations/epochs.
In equation 8, the regularization parameter λ is set to λ =
| E c | | E | where | E | is the number of pairs that satisfy ( v i , v j ) ∈ E and | E c | is the number of pairs that satisfy ( v a , v b ) / ∈ E .
We report scores for the NIPS dataset in Table 5.
We report the Spearmans rank correlation coefﬁcient ρ (Spearman, 1904) all the the authors least
[Figure] [Figure]
[Figure] [Figure]
[Figure] [Figure]
[Figure]
x 0 ( t i m e )     5  9 12 15 21   17 2024
Faction
Faction
E E XTENDED R ELATED W ORK
To help explain the contributions of our work relative to prior art, we use this section to provide a more detailed comparison of our contributions to that of Sim et al.
Sim et al.
(2021) extended Clough & Evans (2017) to the anti-de Sitter space and Lorentz cylinder.
Although our motivation is similar, our contributions are methodological, rely on a simpler use of the intuitions of general relativity and Lorentzian pre-length spaces, and provide an easier interpretation of the learned representations as we explain below.
First, Sim et al.
(2021) do not address clearly the case when there is no geodesic between pairs of points, and their optimization framework leads to a distance loss term with a zero gradient in this case, which make it difﬁcult to optimize.
Moreover, in Sim et al.
(2021), the prediction of an arc between a pair of nodes is determined via a Triple
Fermi-Dirac (TFD) probability function that accounts for the squared (Lorentzian) distance between the nodes, the time coordinate difference ∆ t and its opposite value − ∆ t .
In other words, TFD accounts for both the chronological future and past (with different weights) of a given node.
The major methodological difference with Sim et al.
(2021) is that we restrict the representation of nodes connected by an arc to belong to I + ( x , V x ) where V x is a convex normal neighborhood.
Although subtle, this difference makes the optimization and interpretation of results easier.
In general, the Lorentzian distance function from x to y on M is deﬁned to be inﬁnite (i.e., χ M ( x , y ) =
+ ∞ ) if M is non-chronological and there exists a closed timelike curve joining x and y .
Moreover, the Hopf-Rinow theorem does not hold for spacetimes.
We propose to work with convex normal neighborhoods, which allows us to restrict the existence of arcs between nodes to the existence of geodesics (of ﬁnite length) joining events.
Our distance function χ V is called a local distance function (see Deﬁnition 4.25 of Beem et al.
when its domain is restricted to a convex normal neighborhood.
Moreover, χ V is continuous and differentiable on V x ×I + ( x , V x ) (see Lemma 4.26 of Beem et al.
In some cases, it might be easier to optimize its squared function χ 2 V which is of class C 2 on V x ×V x
(see Theorem 2.6 of Minguzzi (2019)).
It is worth noting that V x can be deﬁned to be globally hyperbolic for any spacetime (see Theorem 2.7 of Minguzzi (2019)).
This means that V x admits a Cauchy time function that can be used to deﬁne a time separation function τ (see explanation in Section C.2) whose sign deﬁnes the direction of edges.
Our framework shares similarities with Sim et al.
(2021) when M = R d 1 = V x because R d 1 is globally hyperbolic and any pair of points of R d 1 can be joined by a geodesic.
However, the way we deﬁne the time separation τ (instead of using the same ∆ t ) is different when
M = L d 1 ( C ) because we restrict it to be calculated on the maximal convex normal neighborhood.
We construct τ so that it is positive if x j ∈
I + ( x , V x ) and negative if x j ∈ I − ( x , V x ) .
We then enforce τ to be positive if we want an arc from v i to v j .
The fact that we work only with an open convex set instead of the whole manifold M is crucial because x j can belong to both the chronological future
I + ( x i , M ) :=
{ y ∈ M : x i (cid:28) y } and past I − ( x i , M ) := { y ∈ M : y (cid:28) x i } if the M is non-chronological.
Using the entire manifold requires that the sign of ∆ t is not as informative, as in Sim et al.
Our approach allows us to determine the direction of the arc joining v i and v j only via the sign of τ .
We also deﬁne a general way of optimizing τ via the parallel transport (see Appendix C.2) instead of working only with Cartesian coordinates, which is not meaningful for some spacetimes such as P d 1 ( r ) .
Moreover, since we restrict our Lorentzian distances to be calculated in the convex normal neighborhood V x , we also have the nice interpretation that the Lorentzian distance corresponds to the length of the longest causal curve joining points.
One other contribution is the connection of our work with the theory of Lorentzian pre-length spaces (Kunzinger & Sämann, 2018) which does not require notions of differential geometry to be understood and can be applied to discrete topological spaces (see Example 2.16 of Kunzinger & Sämann (2018)).
The framework proposed by Sim et al.
(2021) is not a Lorentzian pre-length space due to their formulation of their time coordinate difference function that does not satisfy the properties of a time separation function (especially when M is non-chronological).
[Table] [Table]
[Table] [Table]
[Table]
F O PTIMIZATION
The optimizers that we use in the main paper for the different spacetimes can all be optimized as described in Algorithm 1.
The goal is to minimize some function f : M → R by using differential geometry tools described in (Gao et al., 2018; Law, 2021; Law & Stam, 2020).
∇ f ( x ) is the
Euclidean gradient of f at x , and Π x ( z ) is the orthogonal projection of an arbitrary vector z onto
T x M .
We consider in the following that the step size η > 0 is ﬁxed and given.
We give details for the different spacetimes that we consider in the main paper.
F. M INKOWSKI SPACETIME R d
This optimizer was explained in (Gao et al., 2018).
When M = R d 1 , Π x is the identity function.
G is the diagonal matrix with the ﬁrst diagonal element equal to − 1 and the remaining ones equal to 1 .
exp x ( y ) :
= x + y .
Algorithm 1 corresponds to the standard Euclidean gradient descent because χ = ∇ f ( x ) in this case.
[Equation]
C YLINDRICAL M
[Table]
L d
[Equation]
We recall that L d 1 ( C ) =
R d 1 / ∼ , a quotient set deﬁned such that x ∈ R d 1 and y ∈
R d 1 are equivalent (i.e., x ∼ y )
iff ∀ i > 0 , y i
=
x i and ∃ k ∈ Z , y
= x 0 + kC where C > 0 is a circumference hyperparameter.
When M = L d 1 ( C ) , we use the same optimizer as in Section F.1.
Although it is optional, we also reproject the time coordinate of x = ( x 0 , . . .
, x d − 1 ) (cid:62) at the end of each iteration as follows: x 0 ← (cid:0)(cid:0) ( x 0 + C 2 ) mod C (cid:1) − C 2 (cid:1) ∈ [ − C 2 , C 2 ) where we use the modulo operation for real values which can be written as follows: a mod b := a − b · (cid:98) ab (cid:99) , and (cid:98)·(cid:99) is the ﬂoor function.
If the initial value of x 0 is not in [ − C 2 , C 2 ) , this projects x to an equivalent point.
F. D E S ITTER SPACE S d 1 ( r )
This optimizer was introduced in (Law & Stam, 2020).
We recall that S d 1 ( r ) :
= { x ∈ R d +1: (cid:104) x , x (cid:105) 1 = r 2 } .
G is the diagonal matrix with the ﬁrst diagonal element equal to − 1 and the remaining ones equal to 1 .
We have: Π x ( z ) :=
z −
[Equation]
z −
[Equation]
z
[Equation]
r 2 x .
The exponential map is deﬁned in equation 10.
F. A NTI - DE S ITTER SPACE H d 1 ( r )
We recall that H d 1 ( r ) :
= { x ∈ R d + : (cid:104)
x , x (cid:105)
= − r 2 } .
G is the diagonal matrix with the ﬁrst two diagonal elements equal to − 1 and the remaining ones equal to 1 .
We have: Π x ( z ) :
=
z − (cid:104)
z , x (cid:105) 2 (cid:104) x , x (cid:105) 2 x = z +
(cid:104) z , x (cid:105)
2 r 2 x .
The exponential map is deﬁned in equation 15.
F. P ROJECTIVE VERSION OF THE ANTI - DE S ITTER SPACE P d 1 ( r )
The neural network optimizer is given in (Law, 2021).
Otherwise, the embedding optimizer is the same as in Section F.4.
The main difference is how the points are compared to calculate the distance and time separation function.
G E XAMPLES OF GRAPHS
Since our general framework is fairly abstract, we give some explicit examples of directed graphs with or without cycles that can be described by our framework.
[Figure]
We ﬁrst consider a directed acyclic graph that consists of an undirected cycle (Figure 6).
The DAG G =
( V, E ) is deﬁned as V = { v i } 6 i =1 and the set of arcs is E
= { ( v 1 , v 4 ) , ( v 2 , v 4 ) , ( v 2 , v 5 ) , ( v 3 , v 5 ) , ( v 1 , v 6 ) , ( v 3 , v 6 ) } .
One can see that G is not a directed tree since the undirected path v 1 , v 4 , v 2 , v 5 , v 3 , v 6 , v 1 is cyclic.
We recall that if M is globally hyperbolic, M is also chronological and can then only describe
DAGs with our framework.
Let us give an example where M = R d + or M = S d 1 ( r ) ⊂
R d + with d = 3 and r > 0 .
Both R d + and S d 1 ( r ) are globally hyperbolic.
Let us consider any value ε > 0 , a = r + ε and b = √ 2 a 2 − r 2 .
We take the embeddings: x 1 = (0 , r, 0 , 0) (cid:62) , x 2 = (0 , 0 , r, 0) (cid:62) , x 3 = (0 , 0 , 0 , r ) (cid:62) , x 4 = ( b, a, a, 0) (cid:62) , x 5 =
( b, 0 , a, a ) (cid:62) , x 6 = ( b, a, 0 , a ) (cid:62) .
When M = S d 1 ( r ) , we know from Section B.2 that −−→ x i x j is timelike (i.e., (cid:104)−−→ x i x j , −−→ x i x j (cid:105) < 0) iff (cid:104) x i , x j (cid:105) 1 > r 2 .
By using the time separation function in equation 33 and assuming −−→ x i x j is timelike, we only need to compare the ﬁrst coordinate of x i and x j to determine the direction of the edge between v i and v j .
One can verify that ∀ i, j, ( v i , v
j ) ∈
E ⇐⇒ x j ∈
U
x
i and the tangent vector −−→ x i x j is future-directed timelike.
G. G RAPHS WITH DIRECTED CYCLES G
[Figure]
We now illustrate a simple example of graph with directed cycle that can be drawn with L d 1 ( C ) (Figure 7).
The graph G =
( V, E ) is deﬁned as V = { v i }
i =1 and E = { ( v 1 , v 2 ) , ( v 2 , v 3 ) , ( v 3 , v 1 ) } , which is a graph with directed cycle.
Let us consider that C
= 3 and d = 2 .
The maximal normal neighborhood of every point x ∈ R d 1 is
U x = { y = ( y 0 , . .
, y d − 1 ) (cid:62) ∈ R d 1 : y 0 ∈ ( x 0 − 1 .
5 , x 0 + 1 . 5) } .
We also consider that
V x = U x .
Since L d 1 ( C ) is a quotient set, its points are equivalence classes.
We consider three equivalence classes [ x i ] :
= { ( i + 3 k, 0) (cid:62) : k ∈ Z } where i ∈ { 1 , 2 , 3 } and we deﬁne x i :=
( i, 0) (cid:62)
∈
R
d 1 .
In other words, the ﬁve points x 0 = (0 , 0) (cid:62) , x 1 = (1 , 0) (cid:62) , x 2 = (2 , 0) (cid:62) , x 3 = (3 , 0) (cid:62) , x 4 = (4 , 0) (cid:62) in R d 1 actually correspond to three points in L d 1 ( C ) due to the equivalence relation (i.e., x 3 ∼ x 0 and x 4 ∼ x 1 ).
We can then compare those three equivalence classes by comparing x i with x i − 1 and x i + only.
One can verify that ∀ i ∈ { 1 , 2 , 3 } , we have x i +1 ∈
U x
i , x i +1 ∈
I + ( x i , U x i ) , and x i − 1 ∈
U x i .
However, we also have x i − 1 / ∈ I
+ ( x i , U x i ) .
We then obtain the graph illustrated in Figure 7.
[Figure]
P ROJECTIVE VERSION OF THE ANTI - DE S ITTER SPACE
[Figure]
We now consider the graph G =
( V, E ) deﬁned as V = { v i } 4 i =1 and E = { ( v 1 , v 4 ) , ( v 2 , v 1 ) , ( v 3 , v 1 ) , ( v 3 , v 2 ) , ( v 4 , v 2 ) , ( v 4 , v 3 ) } .
This is a graph with directed cycles (e.g., v 1 → v 4 → v 2 → v 1 , see Figure 8).
We recall that every point of P d 1 ( r ) can be written as the unordered pair {− x , x } where x ∈ H d 1 ( r ) ⊂
R d +1.
Taking d = 2 , the maximal normal neighborhood of every point {− x , x } can be written
U {− x , x } = {{− y , y } : y ∈ H d 1 ( r ) , (cid:104) x , y (cid:105) 2 < 0 } .
Let us consider r = 1 , ε 1 = − 0 .
1 , ε 2 =
ε 3 = − 0 .
5 , a = − 1 , b = − (cid:112) ( r + ε 1 ) 2 − r 2 + a 2 , c = − 2 , e = − (cid:112) ( r + ε 2 ) 2 − r 2 + c 2 , f = 1 , g = (cid:112) ( r + ε 3 ) 2 − r
+ f 2 .
We construct the four following points: x 1 = p
=
( r, 0 , 0) (cid:62) , x 2 = ( r + ε 1 , a, b ) (cid:62) , x 3 = ( r + ε 2 , c, e ) (cid:62) , x 4 = ( r + ε 3 , f, g ) (cid:62) .
To deﬁne the future direction, we consider the timelike tangent vector t = (0 , 1 , 0) (cid:62) ∈ T p H d 1 (
In our framework, there exists an edge between x i and x j iff |(cid:104) x i , x j (cid:105) 2 | ∈ (0 , r 2 ) .
The direction of the edge is determined by using Section C.2.4.