# Orientation dynamics of two-dimensional concavo-convex bodies

S. Ravichandran<EMAIL_ADDRESS>Nordita, KTH Royal Institute of
Technology and Stockholm University, Stockholm, SE 10691  Interdisciplinary
Programme in Climate Studies, Indian Institute of Technology Bombay, Mumbai
400076  J. S. Wettlaufer<EMAIL_ADDRESS>Yale University, New Haven, CT
06520-8109, USA  Nordita, KTH Royal Institute of Technology and Stockholm
University, Stockholm, SE 10691

###### Abstract

We study the orientation dynamics of two-dimensional concavo-convex solid
bodies more dense than the fluid through which they fall under gravity. We
show that the orientation dynamics of the body, quantified in terms of the
angle $\phi$ relative to the horizontal, undergoes a transcritical bifurcation
at a Reynolds number $Re_{c}^{(1)}$, and a subcritical pitchfork bifurcation
at a Reynolds number $Re_{c}^{(2)}$. For $Re<Re_{c}^{(1)}$, the concave-
downwards orientation of $\phi=0$ is unstable and bodies overturn into the
$\phi=\pi$ orientation. For $Re_{c}^{(1)}<Re<Re_{c}^{(2)}$, the falling body
has two stable equilibria at $\phi=0\text{ and }\phi=\pi$ for steady descent.
For $Re>Re_{c}^{(2)}$, the concave-downwards orientation of $\phi=0$ is again
unstable, and bodies that start concave-downwards exhibit overstable
oscillations about the unstable fixed point, eventually tumbling into the
stable $\phi=\pi$ orientation. The $Re_{c}^{(2)}\approx 15$ at which the
subcritical pitchfork bifurcation occurs is distinct from the $Re$ for the
onset of vortex shedding, which causes the $\phi=\pi$ equilibrium to also
become unstable, with bodies fluttering about $\phi=\pi$. The complex
orientation dynamics of irregularly shaped bodies evidenced here are relevant
in a wide range of settings, from the tumbling of hydrometeors to settling of
mollusk shells.

##  I Introduction

Solid bodies falling under gravity through lighter fluids are ubiquitous, from
tree leaves/seeds fluttering as they fall to the ground (e.g. Varshney2012, ),
to the dynamics of ammonia mushballs in the Jovian atmosphere (Guillot2020, ).
Such objects are rarely spherical, typically exhibiting surfaces of non-
uniform curvature which influence phenomena across the biological and physical
sciences. Biota, from plant to animal, are typically ‘concavo-convex’ and, for
example, irregularly shaped ice crystals can be crucial in determining the
effective albedo, and hence the radiative balance, of the atmosphere
(Yang2015, ). However, the relatively few systematic studies of such concavo-
convex bodies are at high Reynolds numbers. Allen (ALLEN1984, ) studied the
settling behavior of mollusk shells in sea water, finding that the shells
universally settle with the concave side up. This behavior, along with the
observation that sand grains may be trapped in the vortex bubble of the body’s
wake, is used to rationalize observations of concave-up shells on the sea bed.
However, because a concave-up shell on a substrate can re-orient in a shear
flow, whereas a concave-down shell is more stable, of relevance to eggshell
taphonomy (Hayward2011, ), post depositional re-orientation is an important
mechanism in a variety of settings.

Concavo-convex bodies exhibit compelling dynamical phenomena across a wide
range of flow conditions and Reynolds numbers, $Re=VL/\nu\gg 1$, where $V$,
$L$ and $\nu$ are body velocity, the body length scales and the fluid
viscosity respectively. At low $Re$ symmetric large aspect ratio bodies settle
with a horizontal long-axis, perpendicular to gravity (e.g., Khayat1989, ),
whereas more complex bodies, such as dumbbells with different sized spheres,
settle with a vertical long-axis and the large sphere at the bottom
(Candelier2016, ). At high $Re$, such bodies have been used as passive models
to explain hovering flight. For example, Childress and colleagues
(Childress2006, ; Weathers2010, ; Liu2012, ) showed that concave
_-downwards_($\wedge$) hollow pyramid-shaped bodies hover in oscillatory
flows, whereas bodies that either start concave-upwards $\left(\vee\right)$,
or are perturbed into this orientation, can no longer hover. The asymmetry of
the body, and the associated drag asymmetry, underlie the steady hovering
observed in the $\wedge$ orientation.

We use well-resolved numerical simulations to study the orientation dynamics
of falling two-dimensional concavo-convex solid bodies. Two horizontal
orientations, $\phi=0$ (defined as the concave-downwards) and $\phi=\pi$, are
possible for steady descent. At low $Re$, in analogy with (Candelier2016, ),
we expect that a body will settle concave-upwards. Whereas experiments
(ALLEN1984, ; Chan2020, ) suggest that only the concave-upward orientation is
stable at moderate to large $Re$, we find that for a finite range of $Re$, the
$\phi=0$ orientation is stable, indicating a stability boundary. In
particular, we find a transcritical bifurcation at $Re_{c}^{(1)}\approx 2.5$,
where the unstable $\phi=0$ orientation becomes a stable spiral, and a
subcritical pitchfork bifurcation at $Re_{c}^{(2)}\approx 15$, where the
stable spiral at $\phi=0$ becomes an unstable spiral. Finally, we show that
the critical Reynolds number $Re_{c}^{(2)}$ is distinct from and smaller than
the Reynolds number at which vortices begin to be shed from the body.

Our body has density $\rho_{s}$, falls in the $-z$ direction through a fluid
of density $\rho$, and is made up of elliptical or circular segments as shown
in Fig. 1. The perimeter of the body is discretized with marker points that,
in addition to their normal orientations, are translated and rotated with the
body velocity. At each time step the body is “reconstructed” from these marker
points and normals, as described by Engels et al. (Engels2016, ). The fluid
velocity matches the local velocity of the body everywhere on the solid-fluid
interface and thus the body is frictionally coupled to the flow field. Solving
the associated equations of motion and boundary conditions challenge
traditional numerical methods. An alternative is to model the solid body as a
porous medium with a vanishing porosity, which is achieved by introducing a
term that forces the fluid velocity to relax to the local solid velocity
exponentially, using a tunable penalization time constant $\eta$
(Kevlahan2001, ). This volume-penalization can be incorporated into most
existing Navier-Stokes solvers.

We nondimensionalize the governing equations with the semi-major axis $a$ as
the length scale, $U_{b}=\left[\left(\rho_{s}/\rho-1\right)ag\right]^{1/2}$ as
the velocity scale and the fluid density $\rho$ as the density scale. Thus,
the equations of motion for the velocity field $\boldsymbol{u}=(u,w)$, are

$\displaystyle\frac{D\boldsymbol{u}}{Dt}$ $\displaystyle=-\nabla
p+\frac{1}{Re}\nabla^{2}\boldsymbol{u}+\chi\frac{\boldsymbol{v-u}}{\eta},\text{
and }$ (1) $\displaystyle\nabla\cdot\boldsymbol{u}$ $\displaystyle=0,$ (2)

where $Re=U_{b}a/\nu$ is the Reynolds number,
$\boldsymbol{v}\left(\boldsymbol{x},t\right)$ is the local velocity of the
solid body, and the mask function is $\chi=1$ in the region of space occupied
by the solid and vanishes elsewhere. The coordinates of the body center of
mass, $\boldsymbol{x}_{g}$, and the angle the major axis makes with the
horizontal, $\phi$, obey

$\displaystyle m_{b}\frac{d^{2}\boldsymbol{x}_{g}}{dt^{2}}$
$\displaystyle=\boldsymbol{f}_{b}+\boldsymbol{g}\qquad\text{and}$ (3)
$\displaystyle I_{b}\frac{d^{2}\phi}{dt^{2}}$ $\displaystyle=\tau_{b},$ (4)

where the buoyancy-corrected mass and moment of inertia of the body are
$m_{b}$ and $I_{b}$ respectively, and the forces $\boldsymbol{f}_{b}$ and
torque $\tau_{b}$ exerted by the fluid on the body are calculated as volume
integrals (see e.g., Kolomenskiy2009, ).

Figure 1: The idealized concavo-convex body used here. The four elliptical
sections are shown in different colors and correspond to (a) blue: a semi-
ellipse with semi-major axis $a=1+\delta$ and unit semi-minor axis; (b)
orange: a semi-ellipse with semi-major axis $a=1-\delta$ and semi-minor axis
$b=0.5$; (c) red and green: semi-circular sections with radius $\delta$. Here,
$\delta=0.05$. The $\times$ denotes the location of the center of mass of the
body.

We solve Eqs (1)–(2) in two-dimensions using a Fourier-pseudospectral method
as described in (Ravichandran2017b, ) (validation studies are presented in the
supplementary material). We use an $x-z$ domain of $40\times 80$ (or $40\times
160$ for long-time simulations) and find that periodic boundary conditions do
not affect the dynamics beyond this domain size. The $x-z$ domain is
discretized with $2048\times 4096$ (or $2048\times 8192$) gridpoints, so that
there are $>50$ points per unit length. We have verified that the results are
insensitive at this grid resolution, and small variations thereof, with a
volume penalization parameter of $\eta=10^{-3}$. The concavo-convex bodies are
released from rest with the major axis tilted at an initial angle
$\phi\left(t=0\right)\equiv\phi_{0}$ relative to the horizontal as shown in
Fig. 1. Thus, for a given $Re$, finite initial angles act as a perturbations
to a fixed point, which we observe to grow or decay as described presently.

In Fig. 2 we show contour plots of the $y$ component of the vorticity,
$\omega_{y}=\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}$,
around the body for $Re=10,12,17$ and $25$. The following are our four key
observations.

(1): $Re<Re_{c}^{(1)}$. The $\phi=0$ orientation is unstable, while $\phi=\pi$
is stable. Bodies with all initial orientations overturn into the $\phi=\pi$
orientation. The trajectories $\dot{\phi}\left(\phi\right)$ _do not_ spiral
outwards from zero, smoothly converging to $\phi=\pi$ (see Figs. 3 and 4),

(2): $Re_{c}^{(1)}<Re<Re_{c}^{(2)}$. Orientations $\phi=0$ and $\phi=\pi$ are
both _stable spirals_ , with $Re$ dependent basins of attraction. As shown in
Fig. 3, the initial orientation, $\phi_{0}$, determines which equilibrium is
attained. In particular, the basin of attraction, which is the set of values
$\phi_{0}$ that converge to the $\phi=0$ fixed point, shrinks to zero as $Re$
increases to approximately $15$. For $\phi_{0}$ within the basin of attraction
of $\phi=\pi$ but concave-downwards (for example see $Re=12,\phi_{0}=\pi/12$
in Fig. 3), the oscillations of the body are associated with the significant
meandering of the streamline separating the $\omega_{y}<0$ and $\omega_{y}>0$
regions (see Fig. 2b).

Figure 2:  The vorticity, $\omega_{y}$, shown in color and the outline of the
concavo-convex body shown as the solid black line. (a) For
$Re=10<Re_{c}^{(2)}$, the $\phi=0$ orientation is a stable fixed point. (b)
For $Re=12$, the initial orientation $\phi_{0}=\pi/12$ is outside the basin of
attraction of the fixed point at $\phi=0$. As the oscillations increase in
amplitude they are reflected in the deviations of the dividing streamline from
the vertical, and vortices are shed. (c) For $Re=17>Re_{c}^{(2)}$, the
$\phi=0$ orientation is unstable and all initial orientations $\phi_{0}$ lead
to steady descent at $\phi=\pi$. Note, however, this $Re$ is smaller than
required for the onset of vortex shedding. (d) For $Re=25$, in the concave-up
orientation, an incipient Karman vortex street is seen and the system
oscillates with a finite amplitude. Figure 3:  The orientation evolution
$\phi\left(t\right)$ of the bodies (see Fig. 1). The final orientation is
determined by the combination of the Reynolds number $Re$ and the initial
orientation $\phi_{0}$. For small Reynolds number, $Re=2.5<Re_{c}^{(1)}$,
$\phi$ increases monotonically with $t$ for all initial orientations
$\phi_{0}$, overshooting $\phi=\pi$ slightly before reaching $\phi=\pi$. For
$Re_{c}^{(1)}<Re=12<Re_{c}^{(2)}$ the trajectories are oscillatory; for small
$\phi_{0}=0.07,$ $\phi(t)$ decays to zero as the body reaches a state of
steady descent, while for $\phi_{0}=\pi/12$ the body tumbles into the
$\phi=\pi$ orientation before descending steadily. For $Re>Re_{c}^{(2)}\approx
15$, the body flips to the $\phi=\pi$ orientation for all $\phi_{0}$, and the
time in flight before the body tumbles decreases with increasing $Re$ and
increasing $\phi_{0}$.

(3): $Re_{c}^{(2)}<Re\lesssim 20$. The orientation $\phi=\pi$ is _stable_ ,
with the amplitude of oscillations in $\phi$ decaying with time. This is seen
in Fig. 4, showing that trajectories in the phase space
$\left(\phi,\dot{\phi}\right)$ spiral inwards to the point
$\phi=\pi,\dot{\phi}=0$. Figure 2(c) also shows that $Re=17$ is too small to
initiate vortex shedding in the $\phi=\pi$ orientation which, since the convex
outer boundary has semi-major and semi-minor axes $b=a=1$, should occur at
$Re\approx 20$.

(4): $Re\gtrsim 20$. The point $\phi=\pi$ is weakly unstable, due to the
periodic shedding of vortices behind the body as shown in Fig. 2(d). The
trajectories in $\left(\phi,\dot{\phi}\right)$ space reach limit cycles whose
amplitudes increase with increasing $Re$, as seen in Fig. 4. The periodic
oscillations about $\phi=\pi$ are consistent with the experimental
observations of curved bodies by Chan et al. (Chan2020, ). The weak
instability at $\phi=\pi$ is also reminiscent of the behavior of passively
hovering bodies in oscillatory flow (e.g., Childress2006, ) where the concave-
downwards orientation is stable.

Fig. 5 summarizes the points (1)-(4) as follows. The point
$\left(\phi,\dot{\phi}\right)=\left(0,0\right)$ undergoes a transcritical
bifurcation at $Re\approx 2.5$ and a subcritical pitchfork bifurcation at
$Re\approx 15$. The point $\left(\phi,\dot{\phi}\right)=\left(\pi,0\right)$
undergoes a supercritical Hopf bifurcation at a higher $Re\gtrsim 20$.

Figure 4:  Phase space plots of the angular velocity $\dot{\phi}$ versus the
angle $\phi$ for Reynolds numbers $Re=2.5,10,17,25,100$. For $Re=2.5$, the
orientation $\phi=0$ is unstable and the trajectory
$\dot{\phi}\left(\phi\right)$ towards $\phi=\pi$ is non-oscillatory. For
$Re=10$, the orientation $\phi=0$ is a stable spiral and the trajectory
spirals inwards. For $Re=17$, perturbations about $\phi=\pi$ decay to zero and
the trajectory is a decaying spiral around
$\left(\phi=\pi,\dot{\phi}=0\right)$. For $Re=25$, the system is overstable
about the $\phi=0$ orientation, and the trajectory spirals outwards, reaching
the $\phi=\pi$ orientation where it becomes a limit cycle of finite radius. A
limit cycle of larger radius is reached for $Re=100$.  Figure 5:  Bifurcation
diagram for $\left(\phi,\dot{\phi}\right)=\left(0,0\right)$ as the initial
orientation, $\phi_{0}\geq 0$, and the Reynolds number $Re$ are varied. Blue
circles denote initial conditions that are attracted to
$\left(\phi,\dot{\phi}\right)=\left(0,0\right)$, while black squares denote
initial conditions that are attracted to
$\left(\phi,\dot{\phi}\right)=\left(\pi,0\right)$. The dot-dashed lines
separating the circles from the squares are separatrices drawn by hand. For
$Re_{c}^{(1)}<Re<Re_{c}^{(2)}$, two stable fixed points, $\phi=0$ and
$\phi=\pi$ exist. At the subcritical pitchfork bifurcation at
$Re=Re_{c}^{(2)}$, the $\phi=0$ ($\phi=\pi$) fixed point becomes unstable
(remains stable). At a higher $Re$ associated with vortex shedding, the
$\phi=\pi$ fixed point becomes unstable and a limit cycle in
$\left(\phi,\dot{\phi}\right)$ appears (Fig. 4).

The dynamics of the body are controlled by the torques exerted by the fluid
upon it, which are related to the circulation $\Gamma_{d}$ on a contour at a
normal distance $d$ from the surface, encompassing the velocity boundary
layer. In Fig. 6, we plot $\Gamma_{0}-\Gamma_{0.05}$ for $Re=10$ and $Re=20$,
where $\Gamma_{0}=\Gamma_{d=0}$ is the circulation at the surface of the body
and, by the Stokes theorem, is equal to the angular velocity of the body times
its area. The initial orientation is close to the equilibrium $\phi=0$, and we
see that $\Gamma_{d}$ decays with time for $Re=10$. For $Re=20$, the initial
(concave-down) orientation is unstable, and $\Gamma_{0}-\Gamma_{d}$ shows the
oscillatory increase associated with the instability (see Fig. 3). We also
find that whilst $\Gamma_{0}-\Gamma_{d}$ and $\dot{\phi}$ are in phase for
$Re=10$, $\Gamma_{0}-\Gamma_{d}$ _lags behind_ $\dot{\phi}$ by a few flow time
units. Thus, as the body reaches an extremity in its oscillation, the opposing
torque continues to be nonzero, thus providing an impulse for the reverse
motion and leading to an increase in the oscillation amplitude. We note that
this oscillatory instability is not seen in lumped-mass approximations for the
concavo-convex bodies as discussed in the Supplementary Material Sec. II.

Figure 6: The angular velocity (blue, left axis) and the circulation
$\Gamma_{0}-\Gamma_{0.05}$ (red, right axis) for $Re=10$ and $Re=20$ with
initial angles $\phi_{0}$ to the vertical. The latter curves are displaced
vertically for clarity. The former (latter) initial condition is stable
(unstable) in the concave-downwards orientation.

Our results show that over a finite range of $Re$, the $\phi=0$ fixed point
becomes a stable spiral. The $\phi=0$ orientation becomes a stable spiral for
$Re>Re_{c}^{(1)}$ due to the transient torque’s discussed above. For
sufficiently small $Re$, a falling concavo-convex body may be expected to
settle into the $\phi=\pi$ orientation because of inertial torques
(Candelier2016, ). We explain the subcritical pitchfork bifurcation at
$Re_{c}^{(2)}$ in analogy to the fluttering-to-tumbling transition seen in the
dynamics of thin plates and disks, which have been more widely studied at
higher $Re$. These studies find steady, oscillatory and chaotic dynamics as
$Re$ is increased, wherein bodies of sufficiently large (nondimensional)
moments of inertia exhibit tumbling (Tanabe1994, ; Field1997, ; Belmonte1998,
; Mahadevan1999, ; Andersen2005, ; Kanso2014, ). Of particular note are the
experiments of Belmonte et al., (Belmonte1998, ), with two-dimensional thin
flat plates of length $L$, thickness $d$, and width $w$ (in the third,
homogeneous, dimension). The time scale for “fluttering”, or side-to-side
oscillation, is given by that of a buoyant pendulum
$\tau_{p}=\left[\left(\rho_{s}/\rho-1\right)L/g\right]^{1/2}$, assuming that
the drag on the plates is $F_{d}=\rho C_{d}V^{2}A$, with $V$ the terminal
velocity, $A$ the cross sectional area, and $C_{d}$ a constant of order unity.
Thus $\tau_{v}=L/V$ is the timescale for vertical descent and hence when the
body descends by a body length before it has completed a full oscillation, the
plate tumbles instead of fluttering, characterized by the Froude number,
$Fr=\tau_{p}/\tau_{v}>Fr_{c}={O}\left(1\right)$. Mahadevan et al.,
(Mahadevan1999, ) show that this behavior may also be interpreted using the
dimensional argument that balances the drag against gravity. All other things
being equal, narrower (in the third dimension) plates tumble end over end more
readily than wider plates. Belmonte et al. (Belmonte1998, ) study a range of
$Re$ from about $3\times 10^{3}\text{ to }4\times 10^{4}$ and note that their
experiments are in agreement with previous observations that the motion is
$Re$ independent, so long as $Re$ is sufficiently large. Here, however, we
examine $Re$ much smaller than did Belmonte et al. (Belmonte1998, ). Thus, if
we modify the drag law and assume that $C\sim Re^{-1}\sim\nu/\left(LV\right)$,
then $V\sim\left(\rho_{s}/\rho-1\right)gLd/\nu$, leading to
$Fr=\tau_{p}/\tau_{v}\sim Re^{1/2}$. Therefore, whilst Belmonte et al.,
(Belmonte1998, ) find a critical value of $Fr_{c}={O}\left(1\right)$
corresponding to a pendulum driven at resonance, their $Fr$ scaling shows that
the fluttering is independent of viscosity. This suggests that there must be a
critical value for the Reynolds number above which the (steady) oscillation
amplitude is large enough for the body to flip into the more stable concave-up
orientation.

The smaller Reynolds numbers studied here are characteristic of systems with
smaller length scales, such as water droplets or ice particles in clouds
pruppacher1980  or in Saturn’s rings goldreich1978 ; Esposito2010 , and a wide
range of bio-particles settling in the ocean (Guasto2012, ; Wheeler2019, ). Of
particular relevance is the dependence of the stability of the $\phi=0,\pi$
equilibria as the dimensions of the body are altered. We note, for instance,
that the concave-upward orientation is more resistant to tumbling than a flat
plate. In organisms where passive stability in a certain orientation is
beneficial, particular physiological dimensions or shapes may confer fitness,
and thus be evolutionarily favored (see, e.g. chirat2021 ). It would therefore
be of interest to know if and how the stability boundary
$\left(Re_{c}^{(1)},Re_{c}^{(2)}\right)$ varies with the thickness of the
body. Furthermore, the trapping of small particles in the body’s wake,
hypothesized by Allen (ALLEN1984, ), suggests that particle segregation by
size may be achievable by simply dropping suitable shaped concavo-convex
bodies into a suspension. The trapping of small heavy particles in such flows
is of considerable general interest in particle transport (see, e.g.
(Ravichandran2014, ; Angilella2017, )), thereby focusing attention on the role
of particle shape, size and their distribution. Finally, it is clear from this
work that extension to three dimensions, where a body may precess about the
vertical in addition to fluttering as described here, is of significant basic
and practical interest.

## Acknowledgements

Support from Swedish Research Council under grant no. 638-2013-9243 and
computational resources from the Swedish National Infrastructure for Computing
(SNIC) under grants SNIC/2019-3-386, SNIC/2020-5-471, SNIC/2021-5-449 are
gratefully acknowledged. Computations were performed on Tetralith. We thank
Bernhard Mehlig for an interesting discussions on the topic.

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# Event Prediction using Case-Based Reasoning over Knowledge Graphs

Sola Shirai  0000-0001-6913-3598 Rensselaer Polytechnic
InstituteTroyNYUSA12180<EMAIL_ADDRESS>,  Debarun Bhattacharjya
0000-0002-9125-1336 IBM ResearchYorktown HeightsNYUSA<EMAIL_ADDRESS>and
Oktie Hassanzadeh  0000-0001-5307-9857 IBM ResearchYorktown HeightsNYUSA
<EMAIL_ADDRESS>
###### Abstract.

Applying link prediction (LP) methods over knowledge graphs (KG) for tasks
such as causal event prediction presents an exciting opportunity. However,
typical LP models are ill-suited for this task as they are incapable of
performing inductive link prediction for new, unseen event entities and they
require retraining as knowledge is added or changed in the underlying KG. We
introduce a case-based reasoning model, EvCBR, to predict properties about new
consequent events based on similar cause-effect events present in the KG.
EvCBR uses statistical measures to identify similar events and performs path-
based predictions, requiring no training step. To generalize our methods
beyond the domain of event prediction, we frame our task as a 2-hop LP task,
where the first hop is a causal relation connecting a cause event to a new
effect event and the second hop is a property about the new event which we
wish to predict. The effectiveness of our method is demonstrated using a novel
dataset of newsworthy events with causal relations curated from Wikidata,
where EvCBR outperforms baselines including translational-distance-based, GNN-
based, and rule-based LP models.

Knowledge Graph Completion, Case-Based Reasoning, Event Prediction

††journalyear: 2023††copyright: acmlicensed††conference: Proceedings of the
ACM Web Conference 2023; May 1–5, 2023; Austin, TX, USA††booktitle:
Proceedings of the ACM Web Conference 2023 (WWW ’23), May 1–5, 2023, Austin,
TX, USA††price: 15.00††doi: 10.1145/3543507.3583201††isbn:
978-1-4503-9416-1/23/04††ccs: Computing methodologies Knowledge representation
and reasoning

##  1\. Introduction

In recent years, the use of knowledge graphs (KG) in various applications has
become increasingly common. The coverage and scope of domain-specific KGs
continues to grow, and more general-purpose KGs such as Wikidata are capable
of representing enormous amounts of knowledge. While a variety of methods
exist that can leverage KGs to perform tasks, one of the most prevalent
methods is to perform KG completion – and more specifically, link prediction
(LP) – to glean new knowledge. Since we can expect KGs to not have complete
coverage of all available knowledge (Paulheim, 2017), LP is a valuable tool to
apply and refine the contents of a KG.

Figure 1. A running example depicting a snippet of Wikidata containing causal
events as well as a new cause event and effect event for which we wish to make
predictions.

In this work we aim to apply a KG-driven approach to the task of event
prediction for news events as a motivating example. Consider Figure 1, which
depicts a snippet of Wikidata (Vrandecic and Krötzsch, 2014) surrounding
several events. In addition to properties about events such as their
instanceOf and country, we can also observe a number of causal relations
between events. For example, the Kanto Earthquake is connected by the
hasEffect111https://www.wikidata.org/wiki/Help:Modeling_causes relation to the
Nebukawa Rail Accident, indicating that the earthquake caused the train wreck
event.

As a running example, suppose we wish to predict the effect of a new
Megathrust Earthquake event that occurred in Japan. We might represent this
earthquake as a new entity, New Cause, which has a causal relation to another
new entity, New Effect, as depicted in Figure 1. In order to predict
properties about New Effect, such as its instanceOf and country, we can apply
LP methods to connect New Effect to the appropriate entities in the KG.

However, a major limitation of most LP methods is that relations can only be
predicted among entities that already exist in the KG. A more practical and
challenging perspective for this task is to consider LP for entirely new
entities (i.e., inductive link prediction). This perspective also better
reflects the needs of an event prediction system, which would aim to make
predictions about entirely new events. The inductive setting introduces an
additional level of difficulty for LP models, as many of them (especially
embedding-based ones) operate under the closed world assumption – that is,
they can only perform predictions for entities that have been seen during
training, and facts that are not seen during training are assumed to be false.
Additionally, even for models that are capable of this task, some KGs (such as
Wikidata) constantly undergo updates and changes, which can negatively affect
a model’s performance as its training data becomes outdated.

Towards our goal of applying KGs to perform event prediction, we develop a
case-based reasoning approach, EvCBR, which leverages examples of past events
in the KG to perform LP for the properties of unseen event entities. We frame
our problem as a 2-hop link prediction task – e.g., starting from a cause
event, we assume the existence of a causal relation to a new effect event and
predict its properties. This approach allows us to perform inductive link
prediction for the properties of the unseen effect without the need for
external data. Importantly, EvCBR requires no training, relying instead on
statistical measures and subclass hierarchies to identify similar entities and
follow paths through the KG to perform predictions. This makes EvCBR well-
suited to KGs such as Wikidata that frequently experience changes in their
structure and content. More generally, we can also apply EvCBR to perform
2-hop LP for properties of any type of unseen entity that is connected to a
known entity by some relation.

Our contributions are as follows:

  1. (1)

We introduce a case-based reasoning model, EvCBR,222We publicly release our
code at https://github.com/solashirai/WWW-EvCBR for event prediction. Our
model considers this task as a LP task in a 2-hop setting, leveraging
knowledge about similar cause-effect events to make predictions about the
unseen effect. EvCBR requires no training, making it well-suited to handle new
events and changes to the underlying KG.

  2. (2)

Compared to similar work, we introduce novel similarity metrics to identify
similar cause-effect event cases as well as a refinement step to improve the
precision of predictions.

  3. (3)

We curate and release a novel dataset surrounding causal events in Wikidata,
extracting news events that are connected by causal relations as well as their
local connections.333Dataset available at
https://doi.org/10.5281/zenodo.7196049.

  4. (4)

In our 2-hop inductive link prediction task, our model shows superior
performance on our event dataset as well as competitive performance on a
modified evaluation dataset based on the FB15k-237 dataset.

##  2\. Related Work

Zhao (Zhao, 2022) presents a comprehensive survey of different kinds of event
prediction methods across different domains. Under their taxonomy of event
prediction methods (Zhao, 2022, Fig. 3), our method is most closely related to
the “causality-based” methods under “semantic prediction” techniques. Other
notable work that falls under this class of methods includes Radinsky et al’s
Pundit algorithm (Radinsky et al., 2012) which is based on automated
construction of a causality graph through extraction of causal patterns and
generalization. Zhao et al. (Zhao et al., 2017) also construct a causal graph
from text documents and build embeddings to perform a simple (one-hop)
prediction. While prior work has explored applying LP methods to KGs of causal
events (Khatiwada et al., 2022; Shirai et al., 2022), to our knowledge, our
work is the first to apply 2-hop LP over a knowledge graph for event
prediction.

In the space of LP in KGs, a significant number of embedding-based methods
have been developed as detailed in recent surveys (Wang et al., 2017, 2021).
Such methods often show trade-offs in performance based on the dataset and how
well the model hyperparameters are tuned (Ruffinelli et al., 2020). While GNN-
based models (Zhang, 2022) have often shown recent state-of-the-art
performance, there has been discussion surrounding what semantics are actually
captured in such models (Jain et al., 2021) as well as re-evaluations of how
one evaluates such LP models (Tiwari et al., 2021; Akrami et al., 2020; Sun et
al., 2020). Furthermore, most embedding-based models are trained and evaluated
in the closed world setting, with a relatively sparse number of recent works
such as (Daza et al., 2021; Ali et al., 2021) considering inductive link
prediction using additional knowledge sources such as text or hyper-relational
facts.

While less prevalent than embedding-based models, there are also a variety of
rule-based LP models (Galárraga et al., 2013; Lajus et al., 2020; Meilicke et
al., 2018; Ortona et al., 2018; Sadeghian et al., 2019). Methods such as
AnyBURL (Meilicke et al., 2019) have shown similar performance to state-of-
the-art embedding models while requiring significantly less training time
(Rossi et al., 2021). Works such as (Das et al., 2020b, a) present case-based
reasoning models which, like our method, require no training and make path-
based predictions. Our method differs from such similar works in our method of
computing entity similarity, approach to apply and refine prediction paths,
and task formulation of performing 2-hop LP.

##  3\. Problem formulation

Our work considers the application of LP methods over a knowledge graph for
the task of event prediction. We define a knowledge graph $G=(E,R,T)$ as
consisting of a set of entities $E$, a set of relations $R$, and a set of
triples $T$. $T$ consists of triples of the form ($h$, $r$, $t$), where
$h,t\in E$ and $r\in R$.

The task of LP is then to correctly predict the missing tail entity given a
partial triple ($h$, $r$, $?t$).444We note the related LP tasks of predicting
a missing head $?h$ or relation $?r$, but in this paper we focus on LP for
predicting $?t$ as it aligns best with our event prediction task. The standard
assumption in this task is that the head entity $h$ and the missing tail
entity $?t$ are known entities in $E$. To perform inductive link prediction,
it is necessary to predict the missing $?t$ where $?t\notin E$. Since LP
models can not predict such an entity $?t$ solely from the content of $G$,
inductive link prediction methods often also rely on additional background
data (such as textual descriptions) to extract and represent new entities.

###  3.1. Event Prediction Task

Extending the task of LP to event prediction, our goal is to make predictions
about new events based on a causal relation between two events. We represent
our task as a query triple $(c,r,e)$, where $c$, $r$, and $e$ denote the cause
event, causal relation, and effect event, respectively. Furthermore, the point
of the event prediction task is to predict properties about the new effect
event – i.e., predicting outgoing properties about $e$ in the form of
$(e,r_{e},?z)\in Out_{e}$, where $r_{e}\in R$ and $?z\in E$. In this task, we
assume that $c\in E$ while $e\notin E$ – if we wish to make predictions about
a cause $c$ which was not in the original KG, we assume that triples
indicating the properties of $c$ are supplied as input and added to the KG.
Additionally, we assume that we are given the relations $r_{e}$ to predict.

##### Example.

In Figure 1, our task is to perform predictions for New Effect based on the
triple (New Cause, hasEffect, New Effect). Further, we make predictions about
each of New Effect’s properties, as (New Effect, instanceOf, $?z$) and (New
Effect, country, $?z$). ∎

To make predictions under these assumptions, we must consider two points:
first, we must make a prediction for an unseen effect event $e$ given a cause
$c$ of interest and a relation $r$; second, from the unseen event $e$, we must
make predictions about its properties by performing LP to tail entities $?z\in
E$.

Given this problem formulation for event prediction, we can then perform
predictions for properties of $e$ without the need for any background
knowledge about $e$ by performing a 2-hop LP task starting from $c$ – that is,
predicting the tail entity for the 2-hop link ($c$, $r$, $r_{e}$, $?z$) for
each triple in $Out_{e}$. By omitting explicit consideration of $e$ in the
prediction task, we are able to overcome the limitation that LP models must
have a way of extracting or representing the entity. From a practical
perspective, when performing event predictions, one need not explicitly
represent $e$ as the primary focus is on predicting properties about the
effect event.

Figure 2. The task setup for our running example.

We additionally denote the set of outgoing triples from $c$ as $Out_{c}$ – we
assume that outgoing properties of the cause event $c$ are provided or already
present in the KG, and our method only makes use of such outgoing triples555We
are motivated to only consider outgoing edges because we can expect a new
event entered into the KG to have little to no incoming relations present.
from $c$. A graphical depiction of this problem setup for our running example
is shown in Figure 2.

###  3.2. A General 2-Hop Prediction Task

We pose a generalization of the event prediction task, which could be useful
in other applications: to predict properties about a novel entity based on a
known relation between that entity and some entity in $E$. Given some
representative relation $r\in R$ that connects an entity $h\in E$ to a novel
entity $e_{n}\notin E$, we can make predictions about its properties ($e_{n}$,
$r_{n}$, $t_{n}$) using the 2-hop link ($h$, $r$, $r_{n}$, $?t$), where $?t\in
E$. This generalization operates under the assumption that the relation
connecting $h$ and $e_{n}$ is semantically meaningful such that we can make
useful predictions about $e_{n}$ based on knowledge about $h$. The validity of
this assumption may vary for different types of relations.

##  4\. Methods

EvCBR is a case-based reasoning approach which aims to make predictions about
new events based on background knowledge about similar cause-effect events.
While our methods can be generalized to make predictions about new entities in
other domains (conceptually replacing the “cause event” and “effect event”
with different types of entities and connecting them by different relations as
mentioned in the previous section), we describe our approach from the
perspective of event prediction in this section.

At a high level, case-based reasoning aims to solve a new problem based on
experience and knowledge about similar cases in the system’s case base. In our
task, the target problem refers to a new cause-effect event pair query for
which we wish to make predictions, and a case refers to an event pair present
in the KG. This approach can then be broken down as (1) retrieve cases from
the KG which are similar to the new problem query, (2) identify and score
paths through the KG that can be used to predict properties of the effect
events in each case, and (3) reuse the learned paths to make predictions for
our new problem.

###  4.1. Case Retrieval

We define a case $s$ as a triple ($c_{s}$, $r$, $e_{s}$) connecting the cause
$c_{s}$ to the effect $e_{s}$ by the causal relation $r$, where
$(c_{s},r,e_{s})\in T$. In our running example KG, there are three example
cases of cause and effect events connected by the hasEffect relation.

The first step of EvCBR is to retrieve cases from the KG that are similar to
our problem, which we break down into two subtasks. First, we compute a
similarity measure among entities in $E$ based on their subclass hierarchy and
outgoing connections in the KG. Second, we compute the similarity of cases to
the query $(c,r,e)$.

####  4.1.1. Entity Similarity

To compute entity similarity, we apply a relatively simple similarity metric
which can be computed based on count statistics and vector multiplication. Our
goal in this step is to acquire a very rough sense of similarity between
individual entities, which will be utilized in our subsequent steps.

For each entity $h_{i}\in E$, we form a vector representation
$v_{h_{i}}=[a_{1},a_{2},...,a_{N_{h}}]$, where $N_{h}$ is the total number of
entities in $E$. Each value $a_{j}=1$ if there is a triple $(h_{i},r,t_{j})\in
T$ for any relation $r$ or if $h_{i}$ is a subclass of $a_{j}$, and otherwise
$a_{j}=0$.

Intuitively, this vector allows us to capture the idea that similar entities
are likely to be connected to the same entities.

##### Example.

The Kanto Earthquake and Tohoku Earthquake both have an outgoing connection to
Japan, which might indicate that they are more similar than other earthquake
events that occurred in different countries. Additionally, while these two
earthquakes do not have any other outgoing connections that are identical, the
subclassOf relation between Megathrust Earthquake and Earthquake indicates
that both the Kanto and Tohoku Earthquakes are a type of Earthquake. ∎

We also weight the importance of each entity in this vector based on how
frequently it occurs as an outgoing neighbor or superclass. Our intuition here
is to ensure that sharing outgoing connections to a less common entity is more
meaningful than sharing connections to a common one. We compute this weight as
an inverse document frequency (IDF) measure, $IDF(h)=\log(N_{h}/count(h))$,
where $count(h)$ is the number of times $h$ has an incoming edge or is a
superclass of an entity, and form the vector containing the IDF weighting of
all entities as $v_{IDF}=[IDF(h_{1}),...,IDF(h_{N_{h}})]$. Additionally, we
apply normalization to $v_{IDF}$.

Lastly, we apply the IDF weighting to each vector by elementwise
multiplication, and compute the similarity between two entities based on a
weighted Jaccard similarity between those vectors. The entity similarity $ES$
is then computed as:

(1) $wJaccard(x,y)=\frac{x\cdot
y}{\sum_{i=1}^{N_{h}}x_{i}+\sum_{i=1}^{N_{h}}y_{i}-x\cdot y}$ (2)
$ES(h_{i},h_{j})=wJaccard(v_{h_{i}}\odot v_{IDF},v_{h_{j}}\odot v_{IDF})$

##### Example.

The entity that is the most similar to the Tohoku Earthquake would be the
Indian Ocean Earthquake, since both of them have outgoing connections to
Megathrust Earthquake, Earthquake (through the subclass hierarchy), and
Tsunami. While an entity like the Kanto Earthquake shares the outgoing
connection to Japan, this link has a lower weighting since Japan has more
connections in the KG compared to Megathrust Earthquake and Earthquake. ∎

####  4.1.2. Case Head Similarity

Next, we begin to determine the similarity of a case $(c_{s},r,e_{s})$ to our
query $(c,r,e)$.

Our goal for case head similarity is to determine how similar the case’s cause
is to c. Rather than directly using entity similarity from Equation 2, we
instead compute similarity based on the set of $c$’s outgoing triples, denoted
as $(c,r_{c},t_{c})\in Out_{c}$.

For each triple in $Out_{c}$, we define the importance of that triple based on
(1) the probability that any triple in $T$ containing the tail entity $t_{c}$
also contains the relation $r_{c}$, denoted $P(r_{c}|t_{c})$, and (2) the
probability that the tail entity $t_{c}$ occurs in any triple in $T$, denoted
$P(t_{c})$. We posit that triples containing uncommon relations or that lead
to uncommon entities should be considered more important.

The importance $I$ of a triple is then computed as:

(3) $I(c,r_{c},t_{c})=\log\left(\frac{P(r_{c}|t_{c})}{P(t_{c})}\right)$

##### Example.

The importance of ($c$, instanceOf, Megathrust Earthquake) will be greater
than the importance of ($c$, country, Japan). The $P(r_{c}|t_{c})$ terms for
both triples will be equal to 1 in our KG snippet, while the $P(\text{Japan})$
is greater than $P(\text{Megathrust Earthquake})$. ∎

After computing the importance of each triple in $Out_{c}$, we also normalize
the values (denoted $nI(c,r_{c},t_{c})$) by dividing the importance of each
triple by the sum of all importance values.

Case head similarity of $c_{s}$ to the cause $c$ is then calculated as the
weighted sum of similarities between each outgoing triple from $c_{s}$
(denoted $cOut_{s}$) and $Out_{c}$, using the most similar triple from each
set for each unique relation. Denoting $ROut_{c}$ as the set of relations that
occur in $Out_{c}$, we compute the case head similarity $CS_{h}$ as:

(4) $CS_{h}(c,c_{s})=\sum_{r_{c}\in
ROut_{c}}\max_{\begin{subarray}{c}(c_{s},r_{c},t_{s})\in cOut_{s}\\\
(c,r_{c},t_{c})\in Out_{c}\end{subarray}}(nI(c,r_{c},t_{c})*ES(t_{c},t_{s}))$

##### Example.

Tohoku Earthquake has the greatest $CS_{h}$ to our cause $c$, given that both
the instanceOf and country have an exact match. Kanto Earthquake has the
second highest $CS_{h}$, because it has an exact match with $c$ for the
country relation leading to Japan and in its instanceOf relation, the entity
similarity $ES$ between Earthquake and Megathrust earthquake is high. ∎

####  4.1.3. Case Tail Similarity

Besides judging how similar a case’s cause event is to $c$, we also want some
notion of how similar the case’s effect is to the effect e. Since $e$ is
unseen in the KG and we do not actually know what the tail entities are for
triples in $Out_{e}$, we determine similarity based on the types of outgoing
relations from each entity. This similarity metric closely resembles those of
works such as (Das et al., 2020a, b), which form a one-hot vector for each
outgoing relation of entities to determine similarity.

We compute case tail similarity, $CS_{t}$, as a simple Jaccard similarity
between the prediction relations, denoted $ROut_{e}$, and the outgoing
relations for the case’s effect, denoted $ROut_{es}$.

(5) $CS_{t}(e_{s})=\frac{|ROut_{e}\cap ROut_{es}|}{|ROut_{e}\cup ROut_{es}|}$

Lastly, we compute the coverage of the outgoing relations for the case tail,
denoted $CC_{t}$. While $CS_{t}$ identifies case effects with similar
relations to our effect event $e$, it also penalizes effects with a large
number of relations. This second measure aims to ensure that we can identify
cases whose tail entity can be used to make predictions about $e$ without
applying such a penalty – this measure might result in some entities that are
less “similar” but more capable of performing reasoning through the KG.

(6) $CC_{t}(e_{s})=\frac{|ROut_{e}\cap ROut_{es}|}{|ROut_{e}|}$

####  4.1.4. Case Selection

Finally, we select a number of similar cases to retrieve from the KG which we
will use for our case-based reasoning. After identifying the initial set of
candidate cases, we score the similarity of the case to our query using
Equation 7 below. The cases are then sorted and we select the top $N_{h}$
cases. When selecting these top $N_{h}$ cases, we only select cases with a
unique cause entity $c_{s}$ – if another case with the same cause ranks among
the top $N_{h}$ cases, it is disregarded. We follow this procedure to ensure a
level of diversity in the cases over which we perform our reasoning.

(7) $CaseScore(c_{s},r,e_{s})=CS_{h}(c,c_{s})*CS_{t}(e_{s})$

We select an additional $N_{t}$ cases, where $N_{t}<N_{h}$, in which we
prioritize the coverage of outgoing edges from the case’s effect $e_{s}$. The
motivation of this second set of cases is to ensure that we select some cases
in which the case’s effect contains as many relations in $ROut_{e}$ as
possible. Therefore, we formulate the scoring equation to give $CC_{t}$ a
greater influence than $CS_{h}$. We follow the same ranking and selection
procedure as for the first $N_{h}$ cases, using Equation 8 to rank the cases.

(8) $CaseScore_{cov}(c_{s},r,e_{s})=(1+CS_{h}(c,c_{s}))*CC_{t}(e_{s})$

We denote the set of cases selected through this procedure as $S$, containing
$N_{s}=N_{h}+N_{t}$ total cases.

##### Example.

In our KG snippet, we find that all of the cases’ effect events have the same
properties as those we wish to predict for $e$, and so our selection of the
best cases will be judged by their head similarity $CS_{h}$ – out of our three
example cases, the Tohoku Earthquake case will have the highest $CaseScore$. ∎

###  4.2. Prediction Path Enumeration and Scoring

We next use the retrieved cases to enumerate paths through the KG that can be
used to make predictions about $e$.

####  4.2.1. Path Enumeration

Our first step is to enumerate a set of paths, using cases retrieved from the
KG, which can be used to connect cause events to properties of their effect
events. We define a path as a sequence of triples through the KG that can
connect two entities. Given a start entity $x$ and end entity $y$, we can
express a path of length $n$ connecting them as a sequence of triples,
$p=[(x,r_{1},e_{1}),...,(e_{n-1},r_{n},y)]$. We define the relation path as
the sequence of relations used in each triple of a path, denoted
$rel(p)=(r_{1},...,r_{n})$. We denote the list of entities that can be reached
starting from entity $x$ following the relation path $rel(p)$ as
$\mathbb{E}_{p,x}$. Note that $\mathbb{E}_{p,x}$ is not a set, and may contain
repeated entries of an entity that can be reached by different paths through
the KG.

For each case $(c_{s},r,e_{s})$ in our set of retrieved cases $S$ and relation
$r_{e}\in ROut_{e}$, let $E_{r_{e},e_{s}}$ denote the set of entities
connected to $e_{s}$ by the outgoing relation $r_{e}$, i.e.,
$(e_{s},r_{e},t)\in T$ for all $t\in E_{r_{e},e_{s}}$. We then randomly sample
up to $N_{p}$ unique paths of length $\leq 3$ connecting $c_{s}$ to any entity
in $E_{r_{e},e_{s}}$.

Figure 3. Example paths for the Tohoku Earthquake case.

Here, we restrict the paths that are sampled such that (1) the first relation
of $rel(p)$ must be an outgoing relation of our cause event $c$, and (2) the
path does not traverse through the case’s effect event $e_{s}$. Our first
restriction ensures that we identify paths that are relevant to our query
event – since our goal is to follow relation paths starting from the event
$c$, if the first relation is not an outgoing relation of $c$ it will not be
possible to utilize the path. Also note that the triple $(c,r,e)$ is not
present in the KG, so a path starting with $r$ would only be valid if $c$ has
some other outgoing triple with the relation $r$. Similarly, our second
restriction aims to prevent us from sampling paths that traverse through the
causal relation connecting the cause and effect event. Again, since $e$ is
unseen in the KG, such a path would not provide us with any useful information
to apply to our event query.

##### Example.

Figure 3 shows two example paths, highlighted in red (light) and blue (dark),
connecting the Tohoku Earthquake event to its effect’s instanceOf property.
$rel(p_{red})$ consists of the three relations [instanceOf, subclassOf,
hasCause-1] while $rel(p_{blue})$ is the single relation [instanceOf]. We do
not sample paths that traverse through the Aftermath event. ∎

The path sampling steps are repeated for each case in $S$, and the set of all
paths sampled from each case, connecting the case’s cause to its effect’s
$r_{e}$ properties, is denoted as $\mathcal{P}_{S,r_{e}}$.

####  4.2.2. Path Scoring

Next, we compute confidence scores for each unique relation path present in
$\mathcal{P}_{S,r_{t_{q}}}$. This confidence score will correspond to the
notion of how confident we are that a given path leads to the correct entity
prediction.

We base our scoring on a simple precision measure, aiming to score how well a
given relation path leads to the correct entities for the target relation.
Additionally, following from the path confidence measure implemented in
(Meilicke et al., 2019), we add a smoothing constant $\epsilon$ (set to
$\epsilon=5$ in our experiments) to the denominator when calculating the
precision – this allows for relation paths with the same precision and a
greater number of samples to have a higher score than relation paths with
fewer samples. The relation path score for a target relation to predict,
$r_{e}$, and relation path $rel(p)$ is given as:

(9) $PScore(r_{e},rel(p))=\frac{\sum_{(c_{s},r,e_{s})\in
S}\sum_{t^{\prime}\in\mathbb{E}_{p,c_{s}}}\mathds{1}[t^{\prime}\in
E_{r_{e},e_{s}}]}{\epsilon+\sum_{(c_{s},r,e_{s})\in S}|\mathbb{E}_{p,c_{s}}|}$

where $\mathds{1}[t^{\prime}\in E_{r_{e},e_{s}}]=1$ if $t^{\prime}\in
E_{r_{e},e_{s}}$ is true and 0 otherwise.

The $PScore$ for a particular $rel(p)$ may be influenced by sampling multiple
different paths that have the same relation path. This allows us to implicitly
add a weighting to each relation path based on how frequently they occur,
which corresponds to how likely they are to be randomly sampled.

##### Example.

To make a prediction about the effect’s country in our three cases, we can
sample the relation path $rel(p)$ = [country]. For all three of our cases,
following the relation path [country] from the cause leads to the correct
country property of its effect. The $PScore$ of this path would then be
$3/(\epsilon+3)$. On the other hand, for a relation path like $rel(p)$ =
[instanceOf] from our example in Figure 3, which aims to predict the
instanceOf relation of the effect, the correct entity is only reached in 1 out
of 2 results for the Tohoku Earthquake, and 0 out of 3 total results for the
other two cases. The $PScore$ for this path would then be $1/(\epsilon+5)$. ∎

Based on these path scores, produced for the set of distinct relation paths in
$\mathcal{P}_{S,r_{e}}$, we select $N_{p}$ relation paths with the highest
scores, denoted $\mathcal{P}_{r_{e}}$, to make predictions for $(e,r_{e},?z)$.

###  4.3. Applying Prediction Paths

Given our set of relation paths $\mathcal{P}_{r_{e}}$, selected and scored
using our set of retrieved cases, we can now apply these paths to our cause
event c to make predictions about property $r_{e}$ of the effect event. We
perform these predictions by following each relation path, starting from our
cause $c$, and using the $PScore$ for each path to produce a total confidence
score for each predicted entity.

Formally, for our query event $(c,r,e)$ and a single property for which we
want to predict $(e,r_{e},?z)$, we score each candidate prediction entity $z$
as follows:

(10)
$EScore(c,r,r_{e},z)=\sum_{\begin{subarray}{c}rel(p)\in\mathcal{P}_{r_{e}}\\\
z^{\prime}\in\mathbb{E}_{p,c}\end{subarray}}PScore(r_{e},rel(p))[z^{\prime}=z]$

In Equation 10, we choose to calculate the score for a prediction $z$ as a
summation of path $PScore$s so that the score is increased for (1) entities
that are frequently reached by a given relation path, and (2) entities that
are reached by a variety of different relation paths.

For the target relation $r_{e}$ we compute all $EScore$ values for all
entities for which a path $rel(p)\in\mathcal{P}_{r_{e}}$ exists between it and
$c$, denoted $E_{\mathcal{P}_{r_{e}},c}$. We use these scores to sort and rank
our predictions for the given 2-hop link. To make predictions for all
properties $Out_{e}$ of that we wish to predict, we repeat the procedures in
Section 4.2 for each $r_{e}\in ROut_{e}$ and the aforementioned scoring
procedure.

##### Example.

Let us consider applying paths to make predictions for $e$’s instanceOf
relation. There are only two valid paths that may be sampled to make this
prediction, $rel(p_{1})$ = [instanceOf] and $rel(p_{2})$ = [instanceOf,
subclassOf, hasCause-1], both coming from the Tohoku Earthquake case. Setting
$\epsilon=0$ for simplicity, the $PScore$ for these two path are
$PScore(p_{1})=1/5$ and $PScore(p_{2})=1$. Applying these two paths to our
cause event $c$ results in predicting $e$ to be an instance of
$MegathrustEarthquake$ with a score of $1/5$ and $Tsunami$ with a score of 1.
The best prediction in our minimal running example is that the new effect
event will be a tsunami. ∎

Algorithm 1 provides a high-level overview of our model’s inputs and the steps
that are performed to produce predictions for each property of the unseen
effect event up to this point.

Algorithm 1 EvCBR Event Prediction Overview

1:Inputs: $(c,r,e),ROut_{e},N_{h},N_{t},N_{p}$

2:$(c,r,e)\leftarrow$ the cause $c$, causal relation $r$, and unseen effect
$e$

3:$ROut_{e}\leftarrow$ the set of relations about the effect to predict

4:$N_{h}\leftarrow$ number of cases to retrieve using $CaseScore$

5:$N_{t}\leftarrow$ number of cases to retrieve using $CaseScore_{cov}$

6:$N_{p}\leftarrow$ number of paths to sample

7:Retrieve cases from the KG

8:for $r_{e}\in ROut_{e}$ do $\triangleright$ Repeat for each prediction
relation

9: Sample $N_{p}$ paths from each case

10: Compute each path’s score using $PScore$

11: Follow each path starting from $c$ to generate predictions

12: Score each prediction using $EScore$

13: Rank and report top predictions for $r_{e}$

###  4.4. Prediction Score Refinement

Optionally, after making predictions for all of our event prediction relations
$ROut_{e}$, we introduce an additional step to refine our prediction rankings
produced by Equation 10. Given our task setup, in which we perform a
prediction based on some properties about a cause event, our refinement step
aims to apply our prediction methods in the opposite direction to make
“predictions” about the cause event’s properties starting from the effect
event.

The key intuition behind this step is that if we have chosen the correct
entity $z$ for the prediction triple $(e,e_{r},?z)$, we would expect that
applying our prediction methods starting from the event $e$ to predict
properties of $c$ should yield good results. Furthermore, because we know the
true properties of $c$, we can refine the score of the prediction
$(c,r,r_{e},?z)$ based on how accurately it can be used to perform this
reverse prediction.

Figure 4. Example paths connecting the effect event (Aftermath of…) to the
Tohoku Earthquake’s instanceOf property.

##### Example.

A visualization of our refinement step is shown in Figure 4, for the case of
the Tohoku Earthquake event. Two paths connecting the effect to the cause’s
instanceOf property are highlighted in orange (light) and blue (dark). Similar
to sampling paths from this case to make predictions about the effect, we now
aim to sample paths to make predictions about the cause. ∎

To perform our refinement step we reuse our set of previously retrieved cases
$S$ and proceed with producing refined scores for one prediction relation
$r_{e}$ at a time. Following through the path enumeration and scoring methods
of Section 4.2, we now produce paths for each unique outgoing relation of the
cause event. Let $ROut_{c}$ denote the set of outgoing relations from our
cause event $c$. For each $r_{c}\in ROut_{c}$, we sample a set of paths
$\mathcal{P}_{S^{-1},r_{c}}$ indicating paths connecting $e_{s}$ to any entity
in $E_{r_{c},c_{s}}$ for each case $(c_{s},r,e_{s})\in S$. We apply similar
restrictions on our path sampling as in Section 4.2.1, sampling up to $N_{p}$
paths, limiting paths’ length to 3 relations, and only selecting paths whose
first relation is in the set of prediction relations $ROut_{e}$.

For each relation path in $rel(p)\in\mathcal{P}_{S^{-1},r_{c}}$ , we produce a
path score $PScoreR$, as:

(11) $PScoreR(r_{c},rel(p))=\frac{\sum_{(c_{s},r,e_{s})\in
S}\sum_{t^{\prime}\in\mathbb{E}_{p,e_{s}}}\mathds{1}[t^{\prime}\in
E_{r_{c},c_{s}}]}{\epsilon+\sum_{(c_{s},r,e_{s})\in S}|\mathbb{E}_{p,e_{s}}|}$

We note that $PScoreR$ is identical to $PScore$ from Equation 9, except that
the starting points of paths are switched to reflect our swap to following
paths from the effect to the cause event.

For each predicted entity $z\in E_{\mathcal{P}_{r_{e}},c}$ which we produced
using the methods up to Section 4.3, we produce a sequence refinement score,
$RS(z,r_{e},(c,r_{c},t_{c}))$, corresponding to how accurately paths that
contain the entity $z$ can predict outgoing triples of $c$, where
$(c,r_{c},t_{c})\in Out_{c}$. Additionally, we temporarily treat the
prediction triple $(e,r_{e},z)$ as being present in the KG so that we can
follow paths starting from our unseen effect event $e$. Since we only
temporarily add this single triple, only paths that traverse through the
prediction entity $(e,r_{e},z)$ will be considered in calculating $RS$.

(12)
$RS(z,r_{e},(c,r_{c},t_{c}))=\sum_{\begin{subarray}{c}rel(p)\in\mathcal{P}_{S^{-1},r_{c}}\\\
t^{\prime}\in\mathbb{E}_{p,e}\end{subarray}}\frac{PScoreR(r_{c},rel(p))[t^{\prime}=t_{c}]}{|\mathbb{E}_{p,e}|}$

We normalize these values by dividing each $RS$ by the maximum $RS$ score
produced for each triple – i.e., for each triple in $Out_{c}$ and entity $z\in
E_{\mathcal{P}_{r_{e}},c}$, the normalized refinement score $nRS$ is
calculated:

(13)
$nRS(z,r_{e},(c,r_{c},t_{c}))=\frac{RS(z,r_{e},(c,r_{c},t_{c}))}{\max_{z_{i}\in
E_{\mathcal{P}_{r_{e}},c}}(RS(z_{i},r_{e},(c,r_{c},t_{c}))}$

$nRS$ now provides us with a score in the range of [0,1] which indicates how
well a given entity $z$ can “predict” a particular cause event, where the
$nRS=1$ for the entity $z$ that produces the highest $RS$ score for a
particular triple $(c,r_{c},t_{c})$. We then compute our final refinement
score as:

(14) $\displaystyle nRS_{max}$
$\displaystyle(z,r_{e})=\max_{(c,r_{c},t_{c})\in
Out_{c}}(nRS(z,r_{e},(c,r_{c},t_{c})))$ $\displaystyle nRS_{avg}$
$\displaystyle(z,r_{e})=\frac{1}{|Out_{c}|}*\sum_{(c,r_{c},t_{c})\in
Out_{c}}(nRS(z,r_{e},(c,r_{c},t_{c})))$ $\displaystyle ReScore$
$\displaystyle(h_{q},r_{q},r_{e},z)=$ $\displaystyle
EScore(h_{q},r_{q},r_{e},z)*(nRS_{max}(z,r_{e})+nRS_{avg}(z,r_{e}))$

In Equation 14, high values of $nRS_{max}$ provides us with evidence that $z$
can be used to predict some triple in $Out_{c}$ well, while high values of
$nRS_{avg}$ indicate that $z$ can predict many triples in $Out_{c}$ well. We
combine these scores so that we can reward prediction entities that can
predict all of the cause’s properties while not over-penalizing entities that
still are able to reach some of the cause’s properties with high accuracy.

##### Example.

Our initial predictions for $e$’s instanceOf relation were Tsunami and
Megathrust Earthquake. To refine the scores of these two predictions, our
method determines how well we can reach $(c,\text{country},\text{Japan})$ and
($c$, instanceOf, Megathrust Earthquake). Using paths such as the two shown in
Figure 4, we find that the Tsunami prediction can more accurately reach $c$’s
instanceOf property compared to the Megathrust Earthquake prediction. This
leads to Tsunami receiving a higher $nRS_{max}(z,instanceOf)$ score, which
subsequently leads to a higher $ReScore$ value. ∎

After producing the refined scores for each entity, we re-rank them to produce
our refined predictions. For each prediction relation $r_{e}$, the above
process is repeated.

##  5\. Experiments

To evaluate our work, we perform a modified evaluation over a commonly used
benchmark dataset, FB15k-237 (Toutanova and Chen, 2015), as well as a novel
dataset curated from Wikidata. Model performance is measured in terms of the
predicted tail entity for a given 2-hop link.

###  5.1. Datasets

Given our problem setup as a 2-hop LP task for properties of unseen entities,
we split our test data based on entities rather than individual triples. We
split both datasets into training triples, validation connections, validation
triples, test connections, and test triples. We refer to any entity contained
in the training triples as part of the training set, while head entities of
triples in the test and validation triples are part of the test and validation
sets, respectively. The test connections indicate relations from an entity in
the training set to an entity in the test set. The test triples then indicate
outgoing triples from an entity in the test set to one in the training set.

####  5.1.1. FB15k-237

For the FB15k-237 dataset, we iteratively select random entities to place in
the test set based on the following conditions: for a candidate test entity
$x$, (1) there must be no triple connecting $x$ to an entity in the test set,
(2) $x$ must have at least one incoming and outgoing triple, and (3) any
entity connected to $x$ must have at least one other triple connecting it to a
different entity in the training set. These conditions ensure that test
triples can plausibly be predicted by models and that all training entities
have training data available. After selecting entities to place in the test
set, we randomly select half of those for our validation set.

From the 14,541 entities in FB15k-237, we select 500 entities each for the
test and validation sets. The connecting and test triples together combine to
form a total of 170,000 2-hop links over which we perform evaluation.

####  5.1.2. Wikidata Events

To curate our Wikidata causal event dataset, we use an approach similar to
(Hassanzadeh, 2021) to identify 307 event-related classes based on links
between entries in Wikidata and news articles in Wikinews. Following
Wikidata’s guidelines on modeling causes, we select 6 causal
relations666https://www.wikidata.org/wiki/Wikidata:List_of_properties/causality
which we then use to query Wikidata for pairs of event entities that are
connected by a causal relation.

This query yielded a set of 1,953 pairs of events, encompassing 157 unique
event classes. These triples connect 284 unique “cause” events to 311 unique
“effect” events. We then collect the 3-hop neighborhood of outgoing
connections surrounding each of these event entities, removing all literals
(e.g., strings, integers) and any entity for which only one triple existed in
the dataset. Our final dataset consists of 758,857 triples for 123,166
entities.

From our set of cause-effect event pairs, we randomly select 100 effect events
to serve as the test set. After filtering to remove test events related to
each other, we end up with a test set consisting of 89 unique effect events,
with 104 connections to causal events to test over, corresponding to a total
of 1,365 2-hop links to evaluate.

###  5.2. Baseline Models

To compare EvCBR’s performance against baselines, we modify each model’s
scoring function to incorporate the 2-hop link. For instance, for TransE
(Bordes et al., 2013), which aims to learn embeddings for $(h,r,t)$ by
optimizing for $h+r=t$, we can score a 2-hop prediction as $h+r_{1}=t_{1}$,
$t_{1}+r_{2}=t_{2}$ $\xrightarrow{}$ $h+r_{1}+r_{2}=t_{2}$. We follow a
similar procedure for each of our baseline models which rely on learning
embeddings. This modified scoring is only used for testing, and model training
is performed normally using the training triples.

For our baselines, we choose three embedding-based models which have seen
widespread use in recent years – TransE (Bordes et al., 2013), ComplEx
(Trouillon et al., 2016), and RotatE (Sun et al., 2019). For each model, we
test embedding sizes of {100, 200, 300} dimensions and train the model for 100
iterations777We select these hyperparameters to explore based on prior
experience and related work, and consider further fine-tuning (Ruffinelli et
al., 2020) to be beyond the scope of this research. using self-adversarial
negative sampling (Sun et al., 2019).

We also compare against NoGE (Nguyen et al., 2022), a graph neural network
(GNN) model which models both entities and relations as nodes in the graph.
NoGE was developed and evaluated on a Wikidata-based dataset, CoDEx (Safavi
and Koutra, 2020), which we believe makes it a strong candidate for showing
good performance on our own causal event dataset. We train NoGE for 100
iterations using default configurations, and test using embedding sizes of
{64, 128, 256}.

Lastly, we compare against two rule-based baselines: ProbCBR (Das et al.,
2020b) and AnyBURL (Meilicke et al., 2019). ProbCBR is a similar model to
ours, which leverages clustering of entities to better estimate scores of
reasoning paths through the KG. We perform experiments with ProbCBR using
parameter settings of retrieving {5, 10, 20} cases and sampling {60, 80, 100}
paths. AnyBURL on the other hand is a rule learning model which can
efficiently sample the KG to learn and generalize logical rules. Following
from the original publication, we train AnyBURL for 1,000 seconds with default
parameters.

For our own models, we report our results for three variations of our entity
scoring. EvCBRbase is our basic case-based reasoning approach, following the
path sampling and scoring methods up to Section 4.3. EvCBRre denotes the use
of our score refinement method, from Section 4.4. Lastly, EvCBRre+base
indicates the addition of the scores produced by the base and refinement
methods. We evaluate our methods by retrieving $N_{h}=\\{5,10,20\\}$ cases
using $CaseScore$, $N_{t}=\\{1,3,5\\}$ cases using $CaseScore_{cov}$, and
$N_{p}=\\{60,80,100\\}$ sample paths.

###  5.3. Results and Discussion

We apply each model to rank predictions for each tail entity in the test set
of 2-hop links and report the mean reciprocal rank (MRR) and Hits@K metrics
for each model. Experimental results are shown in Table 1; the best performing
model for each dataset is highlighted in bold and the second best is
underlined (for EvCBR’s results we only highlight the best performance of any
one variation).

Table 1. Results for our 2-hop LP experiments. Dataset: FB15k-237
---
Model | MRR | Hits@1 | Hits@10
TransE | 0.355 | 0.253 | 0.553
ComplEx | 0.151 | 0.101 | 0.249
RotatE | 0.290 | 0.191 | 0.477
NoGE | 0.324 | 0.238 | 0.491
AnyBURL | 0.381 | 0.289 | 0.560
ProbCBR | 0.289 | 0.204 | 0.452
EvCBRbase | 0.364 | 0.273 | 0.537
EvCBRre | 0.349 | 0.256 | 0.524
EvCBRre+base | 0.368 | 0.277 | 0.543
Dataset: Wikidata Causal Events
Model | MRR | Hits@1 | Hits@10
TransE | 0.139 | 0.096 | 0.207
ComplEx | 0.030 | 0.024 | 0.050
RotatE | 0.097 | 0.081 | 0.125
NoGE | 0.149 | 0.118 | 0.210
AnyBURL | 0.149 | 0.121 | 0.202
ProbCBR | 0.122 | 0.107 | 0.154
EvCBRbase | 0.156 | 0.118 | 0.222
EvCBRre | 0.158 | 0.130 | 0.212
EvCBRre+base | 0.159 | 0.125 | 0.227

In the causal event data, we find that variations of EvCBR show superior
performance over baselines, while for FB15k-237 EvCBR shows second-best
performance. In particular, we observe that applying our refinement step
EvCBRre leads to the best Hits@1 performance for the event dataset, while
EvCBRre+base shows the highest MRR and Hits@10. EvCBRre’s lower Hits@10 may be
attributed to situations where refinement fails to sample paths connecting a
prediction entity to the input cause’s properties (i.e., leading to
$nRS_{max}(h,r_{e})=0$, which subsequently makes its $ReScore$ equal 0).

Our results suggest that EvCBR shows strong performance for the task of event
prediction, and more generally for the task of 2-hop LP for properties about
unseen entities. Our model does not require any training, which further
bolsters its applicability to this task in the open-world setting, where we
might see frequent changes to the underlying KG. In all of our experiments,
the performance of EvCBR is reflective of making predictions for the effect
event if the cause event’s properties were newly added to the KG. In contrast,
all baselines except ProbCBR perform training while including the cause events
in the training set. To compare the impact of this fact, for our AnyBURL
baseline, if we remove all learned rules that explicitly refer to the cause
entity when performing predictions, MRR decreases to 0.363 and 0.127 for the
FB15k-237 and causal event datasets, respectively – under this condition,
EvCBR now outperforms AnyBURL on the FB15k-237 dataset.

As an example of applying EvCBR to ongoing events, we performed a prediction
for the effect of a Protest event in Iran. Similar cases retrieved from the KG
included the Bahraini Protests of 2011, the Iranian Revolution, and the
2019-21 Chilean Protests. The top instanceOf predictions for the effect were
Resignation, Demonstration, and Civil Resistance, while the top country
predictions were Iran, Iraq, and Azerbaijan. We find that our method shows
promise in terms of factors such as predicting the possibility of events in
Iraq and Azerbaijan, which both are geographically and politically intertwined
with Iran, and are already dealing with consequences of the events in Iran.
Our method also performs well in retrieving cases of past events that have
similar causes and likely similar consequences.

On the other hand, we also observe some situations in which EvCBR struggles.
One failure pattern was when no particularly similar event pairs were present
in the KG – this was not uncommon due to Wikidata’s variable coverage and
level of detail. Another noticeable issue was the overabundance of COVID-19
related events in Wikidata, which frequently were retrieved by EvCBR due to
matching the target country of a query event.

##  6\. Conclusion

We introduce EvCBR, a case-based reasoning model developed to perform event
prediction between a causal event and its unseen effect event. Framing event
prediction as a 2-hop link prediction task, EvCBR retrieves cases of cause-
effect events from the KG to sample and apply paths through the KG that
connect causes to their respective effect’s properties. A novel refinement
step helps improve the accuracy of predictions. EvCBR does not require
training, making it well suited to our intended application to KGs under the
open-world assumption. We evaluate the effectiveness of EvCBR over the
FB15k-237 dataset as well as a newly curated dataset of causal events from
Wikidata, showing strong performance compared to baselines consisting of three
embedding-based models, a GNN model, and two rule-based models which use
similar scoring and sampling methods as ours. Future work should continue to
explore the practically important application of event prediction using KGs,
as well as make further advances in reasoning techniques over constantly
evolving KGs such as Wikidata.

###### Acknowledgements.

This work was supported by the Rensselaer-IBM AI Research Collaboration, part
of the IBM AI Horizons Network.

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