# Superdiffusive to Ballistic Transports in Nonintegrable Rydberg Chains

Chun Chen<EMAIL_ADDRESS>Key Laboratory of Artificial Structures and
Quantum Control (Ministry of Education), School of Physics and Astronomy,
Shenyang National Laboratory for Materials Science, Shanghai Jiao Tong
University, Shanghai 200240, China  Yan Chen<EMAIL_ADDRESS>Department of Physics and State Key Laboratory of Surface Physics, Fudan
University, Shanghai 200433, China  Xiaoqun Wang<EMAIL_ADDRESS>Key
Laboratory of Artificial Structures and Quantum Control (Ministry of
Education), School of Physics and Astronomy, Shenyang National Laboratory for
Materials Science, Shanghai Jiao Tong University, Shanghai 200240, China
School of Physics, Zhejiang University, Hangzhou 310058, Zhejiang, China
Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China
Collaborative Innovation Center of Advanced Microstructures, Nanjing
University, Nanjing 210093, China

###### Abstract

A common wisdom posits that transports of conserved quantities across clean
nonintegrable quantum systems at high temperatures are diffusive when probed
from the emergent hydrodynamic regime. We show that this empirical paradigm
may alter if the strong interaction limit is taken. Using Krylov-typicality
and purification matrix-product-state methods, we establish the following
observations for the strongly interacting version of the mixed-field Ising
chain, a nonintegrable lattice model imitating the experimental Rydberg
blockade array. Given the strict projection owing to the infinite density-
density repulsion $V$, the chain’s energy transport in the presence of a
transverse field $g$ is superdiffusive at infinite temperature featured by an
anomalous scaling exponent $\frac{3}{4}$, indicating the existence of a novel
dynamical universality class. Imposing, in addition, a growing longitudinal
field $h$ causes a drastic factorization of the whole Hilbert space into
smaller subsectors, evidenced by the spectral parsing of the eigenstate
entanglement. Being a consequence of this approximate symmetry, a
superdiffusion-to-ballistic transport transition arises at $h\approx g$.
Interestingly, all the above results persist for large but finite interactions
and temperatures, provided that the strongly interacting condition $g,h\ll
k_{\textrm{B}}T\ll V$ is fulfilled. Our predictions are verifiable by current
experimental facilities.

Introduction.—In quantum dynamics, states of matter can be classified by their
universal transport properties, e.g., normal diffusion for nonintegrable
systems and ballistic propagation for integrable models [1, 2, 3]. In this
regard, one question central to nonequilibrium many-body physics, quantum
simulation, and statistical mechanics concerns the modifications to this
standard paradigm stemming from the interplay among distinct energy scales of
the problem. For closed setups, there are $3$ such quantities, the free and
the interaction parts of the Hamiltonian and the temperature. Most previous
literature focused on the influence of the moderate interactions on the
system’s low-temperature transport behaviors [4, 5]. Intriguingly, a recent
advance in this context takes a different angle that leads to the finding of
the Kardar-Parisi-Zhang (KPZ) [6] spin superdiffusion at infinite temperature
in the integrable Heisenberg chain with a fine-tuned non-Abelian symmetry [7,
8, 9, 10]. It is noteworthy that here temperature sets the maximum scale that
is far beyond the scale of the Hamiltonian [11, 12].

In this Letter, we consider the scenario where the interaction part of the
Hamiltonian $H_{\textrm{int}}$ is first taken to infinity and the subsequent
measurements and evaluations are then performed in the infinite-temperature
limit while the free part of the Hamiltonian $H_{0}$ is maintained as the
reference point. Physically, such an arrangement corresponds to the strongly
interacting condition, $H_{0}\ll k_{\textrm{B}}T\ll H_{\textrm{int}}$. Here we
are chiefly inspired by the recent progress on the Rydberg blockade
experiments [13], where the above construction appears to be achievable. The
anomalous energy transports in the clean Rydberg blockade chain, as will be
reported in detail in the current work, were first hinted in Refs. [14, 15]
where we extensively discussed the randomness-induced dynamical effects in
Rydberg array when subject to this strong interaction limit.

Clean constrained model.—Neutral atoms in optical tweezers with large
principal quantum numbers can be photon driven into Rydberg qubits. The
generated electric dipole-dipole interactions entangle atoms within a blockade
sphere, accommodating no more than one Rydberg excitation. Lasers are then
used to address individual atoms for the onsite qubit operations. Under the
correspondence $b_{i}^{\dagger}+b_{i}=\sigma_{i}^{x},\
b_{i}^{\dagger}b_{i}=n_{i}=\frac{1}{2}(1-\sigma_{i}^{z})$ upon basis $|{\sf
g}\rangle=|\\!\\!\uparrow\rangle,\ |{\sf r}\rangle=|\\!\\!\downarrow\rangle$,
one canonical hard-core boson model realizable in such Rydberg analog
simulators imitates the nonintegrable quantum Ising chain in the transverse
and longitudinal fields [13],

$H=\sum_{i}[g(b^{\dagger}_{i}+b_{i})+h(1-2n_{i})+Vn_{i}n_{i+1}],$ (1)

where Rydberg blockade means $V=\infty$, then specifically the constrained
model [16] reads

$\widetilde{H}=\sum_{i}[gP(b^{\dagger}_{i}+b_{i})P+hP(1-2n_{i})P]$ (2)

with projector $P=\prod_{i}(1-n_{i}n_{i+1})$, thereby preventing $2$
simultaneous excitations for any pair of neighboring atoms. Rabi frequency and
detuning of the coherent laser beam are designated by the uniform transverse
and longitudinal field strengths $g$ and $h$, respectively. Throughout this
paper, we use hard-core boson and spin-$\frac{1}{2}$ representations
interchangeably, choose $g=1$ as the energy unit, and keep
$\hbar=k_{\textrm{B}}=a=1$.

Figure 1:  Finite-size scaling of mean level-spacing ratio $r$ as a function
of ${\big{|}\frac{h}{g}\big{|}}$ in the zero-momentum, odd-parity $(k=0,p=-1)$
sector of the clean periodic Rydberg blockade chain [model (2)] up to maximum
length $L=36$. The inset illustrates the accompanying data collapse after
adjusting the horizontal axis to $(h-1.4)L^{1.8}/g$ while maintaining the
vertical axis to $r$, wherein a dip of $r$, dropping below $0.49$, is
surrounded by two GOE plateaus.

Static spectral & entanglement diagnostics.—Because time-reversal symmetry is
preserved, for generic nonzero parameters, clean mixed-field Ising chain
including its infinitely interacting version is anticipated to obey eigenstate
thermalization hypothesis (ETH) [17, 18] whose eigenspectrum shall be
describable by Gaussian orthogonal ensemble (GOE) [19]. Using exact
diagonalization, we scrutinize the spectral properties of the blockade model
(2) via computing its level statistics. The level-spacing ratio is defined by
$r_{n}=\frac{\min\\{\delta_{n},\delta_{n-1}\\}}{\max\\{\delta_{n},\delta_{n-1}\\}}$
whose mean $r\approx 0.53$ if GOE is assumed where $\delta_{n}=E_{n}-E_{n-1}$
with $\\{E_{n}\\}$ an ascending set of all available eigenvalues [20].

Figure 1 shows the evolution of the mean ratio $r$ under the increase of
${\big{|}\frac{h}{g}\big{|}}$ for the periodic Rydberg chain in the blockade
regime. By exploiting translation and reflection symmetries, we pick the zero-
momentum, odd-parity sector of the model and push the maximum system size to
$L=36$. The calculated spectral statistics exhibit several peculiarities. (i)
Within $0\leqslant{\big{|}\frac{h}{g}\big{|}}\lesssim 1$ (relative sign
between $g,h$ is immaterial), the system’s energy-level distribution is well
captured by GOE. (ii) However, once beyond ${\big{|}\frac{h}{g}\big{|}}\gtrsim
1.5$, this trend slows down and longer chains are entailed to witness the full
convergence toward GOE further. (iii) In between, these two GOE plateaus are
separated by a sudden dip where $r$ deviates from $0.53$ appreciably and drops
below $0.49$, indicating a potential eigenstate phase transition between these
two thermal regions. As illustrated by the inset of Fig. 1, the companion
finite-size data collapse of $r$ supports the finding of the transition where
a critical ${\big{|}\frac{h}{g}\big{|}}\approx 1.4$ can be identified.

Consequently, what are the fundamental distinctions between the thermal phases
at small and large ${\big{|}\frac{h}{g}\big{|}}$? To this end, we proceed to
investigate the energy-resolved entanglement entropy for the entirety of the
eigenstates of model (2). For each eigenfunction with eigenenergy $E$, we
compute its half-chain von Neumann entropy $S_{\textrm{vN}}=-{\sf
Tr}\rho_{R}\log_{2}\rho_{R}$ via reduced density matrix $\rho_{R}$ by tracing
out the degrees of freedom on left half-chain and then plot $S_{\textrm{vN}}$
as a function of $E$.

Figure 2:  Spectral structure of entanglement entropy for periodic Rydberg
blockade chain of length $L=32$ within zero-momentum, odd-parity sector. For
every eigenstate, a point is drawn whose coordinates are marked by its
eigenenergy and entanglement entropy. (a)-(d) correspond to
${\big{|}\frac{h}{g}\big{|}}=0,0.5,1,1.5$, respectively.

Figure 2 depicts the spectral parsing of $S_{\textrm{vN}}$ for the periodic
Rydberg blockade chain at $4$ representative parametric points across regions
of small, intermediate, to large values of ${\big{|}\frac{h}{g}\big{|}}$. In
accord with the signature of a transition in Fig. 1, the distribution of
$S_{\textrm{vN}}$ along the eigenenergy axis displays $2$ distinctive
patterns. As shown by Figs. 2(a),(b), for thermal phase at small
$0\leqslant{\big{|}\frac{h}{g}\big{|}}<1$, the entanglement data tend to form
a single, smoothing curve following the prediction of ETH [21]. This tendency
alters dramatically once $|h|$ exceeds $|g|$. Figs. 2(c),(d) demonstrate that
in this circumstance, the whole Hilbert space of the constrained chain appears
to factorize into smaller subsectors, each resembling the line shapes of Figs.
2(a),(b) but with negligible interference.

Physically, these subsectors emerging once ${\big{|}\frac{h}{g}\big{|}}\gtrsim
1$ can be roughly tagged by the total number of hard-core bosons
$N_{b}=\sum_{i}n_{i}$ in the Rydberg chain. Besides the constraint, the
selection of the allowed values for $N_{b}$ is also subject to the
compatibility condition with the additional symmetry indices, for instance,
the momentum and/or parity quantum numbers. If without using auxiliary
symmetries, there typically exist about $\frac{L}{2}$ such subsectors
corresponding to the $\frac{L}{2}$ possible values of $N_{b}$ for a blockade
chain of length $L$. Since $|g|$ is in the same order as $|h|$, $N_{b}$ is not
a good quantum number. Moreover, it is noticeable from Fig. 1 that the model’s
level statistics within this parameter region approach GOE. In this regard,
such an approximate symmetry of $N_{b}$ steered by the longitudinal field
$|h|$ is not expected, which probably implies a hidden constraint-facilitated
effectiveness of $|h|$ in suppressing the inter-subsector mixture induced by
the transverse flip term $|g|$. As the total number of subsectors is
proportional to $L$, the above observations of Fig. 2 are different from the
Hilbert-space fragmentation where this growth is an exponential function of
$L$ [22, 23].

Figure 3:  Time evolution of the return probability at infinite temperature
using OBCs. The upper (lower) row addresses the ${XX}\ ({ZZ})$ component. For
model (2) at $T=\infty$, the ${XZ}$ component vanishes. A range of parameters
${\big{|}\frac{h}{g}\big{|}}=0,0.5,1,1.5,2$ is organized into $5$ columns. The
Krylov-typicality technique solves system sizes ranging from $L=21$ to $31$.
The results converge onto a power law of time, whose exponent from fitting is
summarized in the bottom-left panel as a function of
${\big{|}\frac{h}{g}\big{|}}$.

Dynamical scaling exponent from return probability.—Contrast between thermal
phases at small and large ${\big{|}\frac{h}{g}\big{|}}$ becomes reinforced
when inspecting how conserved energy is transported across the blockade chain
under the unitary time evolution. At high temperatures, transportation of
conversed quantity in generic nonintegrable isolated systems is obedient to
the diffusive behavior, which is particularly commonplace for energy
propagation.

We find that this paradigm fails in Rydberg blockade chain. To examine the
transport properties of model (2) where energy is the only conserved quantity,
we utilize Krylov-typicality technique [24, 25] to numerically evaluate the
connected equal-site time-dependent return probability defined by [12],

$C^{\mathcal{OO}}_{L/2,\beta}(t)=\langle\delta\rho_{\mathcal{O}}(L/2,t)\delta\mathcal{O}(L/2,0)\rangle\propto
t^{-\alpha},$ (3)

where $\langle\cdots\rangle\coloneqq{\sf
Tr}[e^{-\beta\widetilde{H}}\cdots]/{\sf Tr}[e^{-\beta\widetilde{H}}]$,
$\rho_{\mathcal{O}}(L/2,t)=e^{i\widetilde{H}t}P_{\frac{L}{2}-1}\mathcal{O}(L/2)P_{\frac{L}{2}+1}e^{-i\widetilde{H}t}$
with
$\delta\rho_{\mathcal{O}}(L/2,t)\coloneqq\rho_{\mathcal{O}}(L/2,t)-\langle\rho_{\mathcal{O}}(L/2,t)\rangle$
is the density matrix associated to the component $\mathcal{O}$ of the
Hamiltonian density on site $\frac{L}{2}$, $P_{i}=1-n_{i}$ is the local
projector to render the initial energy disturbance concentrated around the
chain center thus allowing for a convenient inspection on how this
inhomogeneity gets smeared, and $\beta=\frac{1}{k_{\textrm{B}}T}$ is the
inverse temperature. At long wavelengths, hydrodynamic fluctuations dominate
the collective motion involving many particles, giving rise to a universal
power-law scaling $t^{-\alpha}$ for the later-time dependence of return
probability. This dynamical scaling exponent $\alpha$ provides a scheme to
classify the transport behaviors into diffusive when $\alpha=\frac{1}{2}$ or
ballistic when $\alpha=1$. In between, the transport is subdiffusive if
$0<\alpha<\frac{1}{2}$ or superdiffusive if $\frac{1}{2}<\alpha<1$.

In Fig. 3, we focus attention on $C^{\mathcal{OO}}_{L/2}(t)$ at infinite
temperature $(\beta=0)$ and choose
$\mathcal{O}=\widetilde{X},\widetilde{Z}=P\sigma^{x,z}P$. Deploying the
Krylov-typicality approximation by averaging over $1000$ independent pure
states, we evolve a spectrum of open chains with length from $L=21$ up to
$31$. After the initial transient oscillations, all curves of different sizes
converge to one steady power-law decay of time. Intriguingly, rather than
diffusion, the scaling exponent of the energy transport throughout the
small-${\big{|}\frac{h}{g}\big{|}}$ regime is close to $\frac{3}{4}$,
indicating a superdiffusion. As no long-range interactions exist, Lévy flight
[26] is not immediately relevant. Moreover, this exponent of value
$\frac{3}{4}$ does not belong to the Fibonacci family of dynamical
universality classes [27], of which the Gaussian diffusion with exponent
$\frac{1}{2}$ and the KPZ superdiffusion with exponent $\frac{2}{3}$ are $2$
members.

By contrast, the large-${\big{|}\frac{h}{g}\big{|}}$ regime is featured by a
ballistic energy transport typical for integrable systems. On the one hand,
this ballistic behavior is unexpected because from Fig. 1 the level statistics
inside this thermal phase exhibit a mounting tendency toward GOE, thereby
being nonintegrable and shall predict a diffusive behavior. On the other hand,
an indication of the emergent approximate $N_{b}$ symmetry evidenced by Fig. 2
hints at the consistency between this ballistic transport and the quasi-
integrals of motion associated to $N_{b}$. However, although the number of the
permitted values of $N_{b}$ is linear in $L$, it is far from exponentially
many compared to the Hilbert-space dimension. This partly explains why, in
large-${\big{|}\frac{h}{g}\big{|}}$ region, $r$ is GOE rather than Poisson.
Accordingly, reasonings assumed integrability may not be applicable to the
case at hand.

Figure 4:  Purified TEBD results for the finite-$V$ and finite-$T$ energy
transports of model (1) using an open Rydberg chain of length $L=65$. Panels
(a)-(d) focus on demonstrating the anomalous superdiffusion in the pure
transverse field limit $(\frac{h}{g}=0)$, while panels (e)-(h) switch to
illustrate the realization of a ballistic energy transport after enhancing the
longitudinal field strength to $h=3g$. For more explanations, see text.

The proposed eigenstate transition between small- and
large-${\big{|}\frac{h}{g}\big{|}}$ phases is now signaled by a crossover
between superdiffusive to ballistic energy transports. In Fig. 3 we show that
close to criticality the scaling exponent $\alpha\approx 0.9$ when
${\big{|}\frac{h}{g}\big{|}}=1$. Physically, the $x,z$ quantization axes are
different due to the infinite-interaction projection. But Fig. 3 reveals that
their respective return probabilities are nevertheless governed by the same
scaling exponent, suggesting energy transport is likely isotropic in the spin
space, consistent with the expectations from GOE and ETH. Furthermore, we
check that the entanglement spreadings in both thermal regions are ballistic,
similar to the conventional ETH phases [12]. Therefore, the computational
resource for capturing the long-time dynamics of longer Rydberg chains should
grow promptly.

Finite-$V$ & finite-$T$ results.—We have exclusively focused on model (2)
which is derived from the more realistic model (1) by first sending
$V\rightarrow\infty$ and then $T\rightarrow\infty$. Now we switch to the
purification TEBD method [28, 29, 30] to show that the energy-transport
properties found so far are quantitatively preserved in the model (1) at
finite $V$ and $T$, thus being experimentally measurable. The essence is to
achieve the large-$V$ and high-$T$ condition, $g,h\ll k_{\textrm{B}}T\ll V$.
To this end, we fix $V=100g$ and tune $h\in[0,3g]$ and
$k_{\textrm{B}}T\in[10g,50g]$ in the following matrix-product-state
simulation.

Figures 4(a)-(d) target the case of superdiffusion at $h=0$. Via computing the
${XX}$ component of the return probability at $k_{\textrm{B}}T=20g$ by
replacing $\widetilde{H}$ with $H$ and setting $\mathcal{O}=\sigma^{x}$ in Eq.
(3), we illustrate in Fig. 4(c) up to the maximum bond dimension $\chi=1024$
that in accord with the top-left panel of Fig. 3, the same type of energy
superdiffusion characterized by a fitted dynamical exponent $\frac{3}{4}$
arises also on a longer open Rydberg chain of length $L=65$ described by model
(1). (Finite-size analyses covering $L=33,49,65$ yield the same value of
$\alpha$.) Furthermore, Fig. 4(d) unveils that with fixed $\chi=1024$, the
just-claimed superdiffusion occurs at $k_{\textrm{B}}T=10g$ as well. Thus, a
superdiffusive phase is stabilized in a temperature range of
$k_{\textrm{B}}T\in[10g,20g]$. Increasing the temperature to
$k_{\textrm{B}}T=40g$, however, restores the system’s transport to a normal
diffusion characterized by $\alpha=\frac{1}{2}$. In this sense, a super-to-
normal diffusion crossover exists in the model (1) triggered solely by
temperature.

The dynamical universality class for the anomalous transport phenomenon is
dictated by the scaling exponent and the scaling function. By changing the
$2$nd operator position in Eq. (3) from the chain center to the end, we plot
the corresponding spatiotemporal evolution contour of the connected
correlation function $|C^{{XX}}_{\beta=0.05}|$ in Fig. 4(a). Physically, it is
reasonable to assume that the correlation function in the hydrodynamic regime
satisfies the scaling relation $|C^{{XX}}_{\beta=0.05}|\propto
t^{-\alpha}f[(i-i_{c})/t^{\alpha}]$ where $\alpha$ is the scaling exponent and
the unknown function $f[\cdots]$ is the scaling function. Although the system
sizes and the time scales achieved in the present work do not allow for a
reliable investigation of the detailed form of the scaling function, in Fig.
4(b), we proceed to show the preliminary data collapse of the
$|C^{{XX}}_{\beta=0.05}|$ profiles collected at $3$ different moments from
Fig. 4(a) and compare it to a Gaussian fit. With $\alpha=\frac{3}{4}$ fixed,
we find that these various data points do fall onto a single curve at large
$i,t$, and this line decays rapidly, thus becomes progressively deviating from
the Gaussian fit. This trend is consistent with the prediction of the
superdiffusive behavior by the scaling exponent $\frac{3}{4}$.

Figures 4(e)-(h) target the case of ballistic energy transport at $h=3g$. The
signature for this ballistic diffusion is most transparent from Fig. 4(g)
where the ${ZZ}$ component of the return probability exhibits a clean
convergent tendency toward the characteristic linear power-law decay with
scaling exponent $1$ at $k_{\textrm{B}}T=20g$ under the enhancement of $\chi$.
Further, despite the extra complication from the more significant
oscillations, the accompanying ${XX}$ component of the return probability
converges to the same ballistic transport as displayed by Fig. 4(e).
Therefore, threading a growing longitudinal field while maintaining other
parameters intact induces a field-tuned crossover between the anomalous
superdiffusion at small $h$ [Fig. 4(c)] and the ballistic diffusion at
moderate $h$ [Figs. 4(e),(g)]. In addition, by varying $k_{\textrm{B}}T$ at
fixed $\chi=1024$, Figs. 4(f),(h) show that this ballistic behavior persists
down to $k_{\textrm{B}}T=10g$. Conversely, lifting the temperature to
$k_{\textrm{B}}T=50g$ turns the system’s energy propagation to the normal
diffusion. Accordingly, at medium $h$, a temperature-tuned crossover exists
from the ballistic to the diffusive energy transport.

Conclusion & outlook.—Based on large-scale numerical studies of $1$D Rydberg
blockade array, we predict that under strong interaction conditions, there
exist two distinct types of constrained thermal phases. The first arises in
the transverse-field dominated regime whose energy transport is superdiffusive
featured by a dynamical exponent $\frac{3}{4}$. The second is stabilized after
adding a growing longitudinal field whose energy diffusion is ballistic.
Complementary spectral and dynamical analyses suggest that there might exist
an eigenstate transition between the two constrained thermal phases. An
analytical understanding of these numerical findings is lacking. Future works
can be oriented toward determining the scaling function for the superdiffusive
phase where the nonlinear fluctuating hydrodynamics [31] could be a promising
direction.

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†† This study was supported by JSPS KAKENHI(21K13800).
2010 Mathematics Subject Classification. Primary 91A12; Secondary 91C43.
Keywords; Myerson value, directed hypergraph, stability, safety.

# Myerson value of directed hypergraphs

Taiki Yamada  Interdisciplinary Faculty of Science and Engineering, Shimane
University, Shimane 690-8504, Japan<EMAIL_ADDRESS>
###### Abstract.

This paper presents directed hypergraphs as cooperative networks and defines
their Myerson value. The axiomatization of the Myerson value, i.e., strong
component efficiency and fairness, has been demonstrated. Furthermore, the
concept of safety, as defined by Li and Shan, has been modified, and the
hyperedge safety condition is proven in terms of the Myerson value of directed
hypergraphs.

##  1\. Introduction

Game theory can be classified into types—cooperative and noncooperative. In
cooperative game theory, all players cooperate with each other to increase
their utility, thereby forming a coalition and generating worth. The
cooperative game theory is employed in several analyses, including those of
markets in economics, bill votes by political parties, and cost sharing.
Recently, it has also found application in market design by way of auction and
matching theories. The primary focus in the application of these theories
concerns the sharing or distribution of the worth gained from a cooperation.
To this end, we usually consider a transferable utility or TU game. In 1953,
Shapley introduced the Shapley value [Sh] based on the results of one of the
most famous investigations concerning this topic. Although it corresponds to a
method of fairly distributing the worth gained from a cooperation, it is
assumed there exist no restrictions on the cooperation possibilities of the
players. To solve this problem, in 1977, Myerson introduced a graph-restricted
game by modifying the original game and defined the corresponding Shapley
value, namely the Myerson value [My]. Subsequently, several studies have been
conducted concerning the Myerson value. An interesting topic, which has
hitherto been seldom investigated in available literature, concerns the type
of cooperative network. Initially, Myerson considered an undirected graph as a
cooperative network. Therefore, Li–Shan considered the Myerson value for a
directed graph [LS2], and Nouweland–Borm–Tijs considered the Myerson value for
a hypergraph [VDN]. Another topic of research concerns the characterization of
the Myerson value. In 2001, Algaba–Bilbao–Borm–Lopez characterized the Myerson
value for union stable structures [ABBL]. In 2004,
Gomez—Gonzalez–Manuel—Owen—Pozo–Tejada calculated the Myerson value of
splitting graphs for pure overhead games [GGMO]. In 2012, Kim–Hee introduced
various concepts of betweenness and centrality. Moreover, they deduced the
relationship between them and the Myerson value [Kim]. In 2016,
Beal—Casajus–Huettner studied the efficient egalitarian Myerson value
citeBA. In 2021, Wang–Shan proved the decomposition property for the weighted
Myerson value [WE].

Notably, the stability of an edge is an important concept in the study of the
Myerson value. Myerson suggested that an edge is stable if none of the
connected players benefit from breaking it under the Myerson value. Moreover,
he proved that all edges in a super additive game are stable. However, there
nonetheless exist games wherein one’s payoff might affect those of others. To
solve this problem, in 2020, Li—Shan defined the safety of a link and deduced
the appropriate safety conditions in terms of the Myerson value [LS1]. Their
findings imply that all bridges pertaining to a convex game are safe. In
addition, they demonstrated that if a game is strictly convex, the
corresponding graph link is safe if and only if it represents a bridge.
However, the strictly convex of the game is a bit strong. Therefore, the
motivation of this study is to improve their results.

The proposed study considers a directed hypergraph, a natural generalization
of an undirected graph, a directed graph, and undirected hypergraph, as a
cooperative network. Therefore, directed hypergraphs represent the most common
discrete-object types. However, the number of vertices containing an edge on a
directed hypergraph usually exceeds 2. Moreover, each edge is oriented.
Therefore, studies involving directed hypergraphs are challenging compared to
those involving graphs. In fact, no research concerning the Myerson values for
directed hypergraphs has been undertaken to date. Moreover, in this study, we
identified hypergraphs considering a simplicial complex. Therefore, the
findings of this study are applicable to topological homology.

The remainder of this paper is organized as follows. Section 2 presents the
basic notations and properties of directed hypergraphs and the Myerson value.
Section 3 describes the modifications made in this study to the concept of
safety, as defined by Li—Shan, as well as the condition of hyperedge safety in
terms of the Myerson value for directed hypergraphs. Notably, the statement of
the proposed result is more comprehensive compared to that proposed by
Li–Shan, while the corresponding proof is comparatively straightforward.
Section 4 describes the calculation of the Myerson value for each player
within a specific graph. Finally, Section 5 lists the major conclusions drawn
from this study and identifies the scope for future research.

##  2\. Preliminaries

Let $H=(N,E)$ represent a hypergraph, wherein $N$ denotes a set of players
$N=\left\\{1,\cdots,n\right\\}$ and $E$ is a subset of $2^{N}$. Let $v_{i}$
and $e_{i}$ denote individual elements of $N$ and $E$, respectively. Moreover,
$v_{i}$ and $e_{i}$, can be referred to as player and hyperedge, respectively.
Because the object of this study corresponds to a directed hypergraph, every
hyperedge $e\in E$ represents a directional relation between two non-empty
subsets $A_{e}$ and $B_{e}$ of $N$. Accordingly, $A_{e}$ and $B_{e}$ represent
the tail and head sets of $e$, respectively.

Some notations pertinent to graph theory can be described as follows.

######  Definition 2.1.

For any two players $s$ and $t$, a path from $s$ to $t$ represents a sequence
of players and hyperedges
$(s=v_{0},e_{0},v_{1},e_{1},\cdots,e_{l-1},v_{l}=t)$, where

$\displaystyle s$ $\displaystyle\in A_{e_{0}},\ t\in B_{e_{l-1}},$
$\displaystyle v_{i}$ $\displaystyle\in A_{e_{i}}\cap B_{e_{i+1}}$

for any $i\in\left\\{1,\cdots,l-1\right\\}$.

######  Definition 2.2.

For any two players $s$ and $t$, $P(s,t)$ denotes the set of paths from $s$ to
$t$. Player $i$ is critical for $s$ and $t$ if $i\in p$ for any $p\in P(s,t)$.

If there exist different paths from $s$ to $t$ and $t$ to $s$, then $s$ and
$t$ are assumed to be connected. Moreover, if any player pair in $N$ is
connected, then $H=(N,E)$ can be assumed to be connected. However, it is
noteworthy that if there exists a path from $s$ to $t$, a path from $t$ to $s$
need not exist necessarily.

######  Definition 2.3.

An induced subgraph $H^{\prime}$ of $H=(N,E)$ represents a pair of
$(N_{H^{\prime}},E_{H^{\prime}})$ that satisfy $N_{H^{\prime}}\subseteq N$ and

$\displaystyle E_{H^{\prime}}=\left\\{e\mid e\in E,\ e\subseteq
N_{H^{\prime}}\right\\}.$

A maximal connected induced subgraph is called the strong component of $H$,
and $N/E$ denotes the set of all strong components in $H$. Therefore, the
union of all strong components in $H$ equals $N$. A hyperedge $e$ is referred
to as a bridge of $H$ if the strong component containing $e$ splits into
disconnected sets upon the deletion of $e$.

The TU game represents a pair $(N,v)$, where $N$ denotes a set of players and
$v:2^{N}\to\mathbb{R}$ is a characteristic function with $v(\emptyset)=0$.
Each $S\in 2^{N}$ is referred to as a coalition, and $v(S)$ represents the
worth of $S$. $TU(N)$ represents the set of all TU games. Given $(N,v)\in
TU(N)$, $v$ can be referred to as super additive if $v(S\cup T)\geq v(S)+v(T)$
and convex if $v(S\cup T)+v(S\cap T)\geq v(S)+v(T)$. Moreover, $v$ can be
referred to as strictly super additive (strictly convex) if $\geq$ is replaced
with $>$ in the above-described inequalities.

Now, let us allocate each player to the worth, thereby defining an allocation
rule, which can be expressed as a function $f:TU(N)\to\mathbb{R}^{n}$, wherein
$f_{i}(N,v)$ represents the payoff for player $i\in N$. A popular allocation
rule is the Shapley value $Sh(N,v)$ that can be described as follows for any
$i\in N$.

$\displaystyle Sh_{i}(N,v)=\sum_{S\subseteq
N\setminus\left\\{i\right\\}}\cfrac{s!(n-s-1)!}{n!}(v(S\cup\left\\{i\right\\})-v(S)).$

In the above expression, $s=|S|$. An important theorem concerning the Shapley
value can be stated as follows.

######  Theorem 2.4 ([Ne]).

The Shapley value can be uniquely determined as efficient, additive,
symmetric, and conforming to the null-player property subject to the following
conditions.

  1. (1)

An allocation rule $f$ on $TU(N)$ is efficient if $\sum_{i\in
N}f_{i}(N,v)=v(N)$ for each $(N,v)\in TU(N)$.

  2. (2)

An allocation rule $f$ on $TU(N)$ is additive if $f(N,v+w)=f(N,v)+f(N,w)$ for
all $(N,v),(N,w)\in TU(N)$.

  3. (3)

An allocation rule $f$ on $TU(N)$ is symmetric if $f_{i}(N,v)=f_{j}(N,v)$ for
each $(N,v)\in TU(N)$ and any two players $i,j\in N$ with
$v(S\cup\left\\{i\right\\})=v(S\cup\left\\{j\right\\})$ for all $S\subseteq
N\setminus\left\\{i,j\right\\}$.

  4. (4)

An allocation rule $f$ on $TU(N)$ satisfies the null-player property if
$f_{i}(N,v)=0$ for each $(N,v)\in TU(N)$ and any $i\in N$ with
$v(S)=v(S\cap\left\\{i\right\\})$ for all $S\subseteq N$.

To define the Myerson value, let us consider a graph game.

######  Definition 2.5.

  1. (1)

A directed hypergraph game on $N$ represents a triplet $(N,v,E)$ such that
$(N,v)$ is a TU game and $(N,E)$ is a directed hypergraph. The set of all
directed hypergraph games are denoted by $DHG(N)$

  2. (2)

Given a directed hypergraph game $(N,v,E)$, the directed hypergraph-restricted
game $(N,v^{E})$ is defined by for all $S\subseteq N$; therefore,

$\displaystyle v^{E}(S)=\sum_{T\in S/E_{S}}v(T).$

  3. (3)

The Myerson value $\mu(N,v,E)$ can be expressed as

$\displaystyle\mu(N,v,E)=Sh(M,v^{E})=\sum_{S\subseteq
N\setminus\left\\{i\right\\}}\cfrac{s!(n-s-1)!}{n!}(v^{E}(S\cup\left\\{i\right\\})-v^{E}(S)).$

##  3\. Characterization of Myerson value for directed hypergraph games

In accordance with extant research, the properties of the Myerson value can be
modified as follows.

  1. (1)

An allocation rule $f$ on $DHG(N)$ is called strong component efficient if

$\displaystyle\sum_{i\in T}f_{i}(N,v,E)=v(T)$

for any $(N,v,E)\in DHG(N)$ and any $T\in N/E$.

  2. (2)

An allocation rule $f$ on $DHG(N)$ is called fair if

$\displaystyle f_{i}(N,v,E)-f_{i}(N,v,E\setminus
e)=f_{j}(N,v,E)-f_{j}(N,v,E\setminus e)$

for any $(N,v,E)\in DHG(N)$ and any $e\in E$ with $i\in A_{e}$ and $j\in
B_{e}$.

The Myerson value can be characterized using these properties.

######  Theorem 3.1.

For any $(N,v,E)\in DHG(N)$, the Myerson value represents a unique allocation
rule that satisfies the conditions of strong component efficiency and
fairness.

###### Proof.

Consider two distinct allocation rules $h(N,v,E)$ and $g(N,v,E)$ that satisfy
the strong component efficiency and fairness conditions. Next, let us consider
a directed hypergraph $H=(N,E)$ comprising a minimal number of hyperedges such
that $h(N,v,E)\neq g(N,v,E)$. Therefore, for any $e\in E$, it follows that

$\displaystyle h(N,v,E\setminus e)=g(N,v,E\setminus e).$

In accordance with the fairness condition, for any $i\in A_{e}$ and $j\in
B_{e}$, it follows that

$\displaystyle h_{i}(N,v,E)-h_{i}(N,v,E\setminus
e)=h_{j}(N,v,E)-h_{j}(N,v,E\setminus e),$

which implies

$\displaystyle h_{i}(N,v,E)-h_{j}(N,v,E)$ $\displaystyle=$ $\displaystyle
h_{i}(N,v,E\setminus e)-h_{j}(N,v,E\setminus e),$ $\displaystyle=$
$\displaystyle g_{i}(N,v,E\setminus e)-g_{j}(N,v,E\setminus e),$
$\displaystyle=$ $\displaystyle g_{i}(N,v,E)-g_{j}(N,v,E).$

Considering any other hyperedge $e^{\prime}\in E$ with $j\in A_{e^{\prime}}$
and $k\in B_{e^{\prime}}$, we get

$\displaystyle h_{j}(N,v,E)-h_{k}(N,v,E)=g_{j}(N,v,E)-g_{k}(N,v,E).$

Therefore, there exists a constant $C_{T}$ such that for any $T\in N/E$ and
any $i\in T$ we have

$\displaystyle h_{i}(N,v,E)-g_{i}(N,v,E)=C_{T}.$

In accordance with the strong component efficiency condition,

$\displaystyle 0=\sum_{i\in T}(h_{i}(N,v,E)-g_{i}(N,v,E))=|T|C_{T}.$

Therefore, $C_{T}=0$. This is obviously a contradiction.

Next, we prove that the Myerson value satisfies the strong component
efficiency and fairness conditions. Let $T_{1},\cdots,T_{m}$ represent a
strong component in $H=(N,E)$. Therefore,

$\displaystyle\bigsqcup_{i=1}^{m}T_{i}=N.$

For any induced subgraph $H^{\prime}=(S,E_{S})\subseteq H$, because it holds
that

$\displaystyle S/E_{S}=\left\\{S\cap T_{1},S\cap T_{2},\cdots,S\cap
T_{m}\right\\},$

we have

$\displaystyle v^{E}(S)=\sum_{i=1}^{m}v^{E}(S\cap T_{i})$

In accordance with the additivity condition of the Shapley value,

$\displaystyle Sh(N,v^{E})=\sum_{i=1}^{m}Sh(N,v^{E}(S\cap T_{i})).$

Clearly, for each $S\subseteq N\setminus\left\\{i\right\\}$, if $i\in T_{j}$,

$\displaystyle v^{E}((S\cup\left\\{i\right\\})\cap T_{j})-v^{E}(S\cap
T_{j})=0.$

Therefore, if $i\in T_{j}$

$\displaystyle Sh_{i}(N,v^{E}(S\cap T_{j}))=0$

For each fixed $T_{k}\in N/E$, the efficiency condition of the Shapley value
requires that

$\displaystyle\sum_{i\in T_{k}}Sh_{i}(N,v^{E}(S\cap T_{k}))$ $\displaystyle=$
$\displaystyle v^{E}(T_{k}\cap T_{k})$ $\displaystyle=$ $\displaystyle
v^{E}(T_{k})=v(T_{k}).$

Therefore,

$\displaystyle\sum_{i\in T_{k}}\mu_{i}(N,v,E)$ $\displaystyle=$
$\displaystyle\sum_{i\in T_{k}}Sh_{i}(N,v^{E})$ $\displaystyle=$
$\displaystyle\sum_{i\in T_{k}}\sum_{j=1}^{m}Sh_{i}(N,v^{E}(S\cap T_{j}))$
$\displaystyle=$ $\displaystyle\sum_{i\in T_{k}}Sh_{i}(N,v^{E}(S\cap
T_{k}))=v(T_{k}),$

This confirms that the Myerson value satisfies the strong component efficiency
condition.

Finally, it can be demonstrated that the Myerson value satisfies the fairness
condition. For any $e\in E$ with $i\in A_{e}$ and $j\in B_{e}$, the following
allocation rule can be defined.

$\displaystyle w(S):=v^{E}(S)-v^{E\setminus e}(S).$

Furthermore, given a coalition $S$, if $i\notin S$ or $j\notin S$, it follows
that $S/E_{S}=S/(E_{S}\setminus e)$ and

$\displaystyle w(S)=v^{E}(S)-v^{E\setminus e}(S)=\sum_{T\in
S/E_{S}}v(T)-\sum_{T\in S/(E_{S}\setminus e)}v(T)=0.$

Therefore, the coalitions that contain both players $i$ and $j$ can be
exclusively considered. In accordance with the symmetry and additivity
conditions of the Shapley value, we have

$\displaystyle\mu_{i}(N,v,E)-\mu_{i}(N,v,E\setminus e)$ $\displaystyle=$
$\displaystyle Sh_{i}(N,v^{E})-Sh_{i}(N,v^{E\setminus e})$ $\displaystyle=$
$\displaystyle Sh_{i}(N,v^{E}-v^{E\setminus e})$ $\displaystyle=$
$\displaystyle Sh_{i}(N,w)=Sh_{j}(N,w)$ $\displaystyle=$
$\displaystyle\mu_{j}(N,v,E)-\mu_{j}(N,v,E\setminus e).$

This completes the proof. ∎

##  4\. Safety of directed hyperedges with respect to the Myerson value

Based on the stability of the allocation rules defined by Myerson, the
stability of the Myerson value for a directed hypergraph game can be defined
as explained hereunder.

######  Definition 4.1.

For any $(N,v,E)\in DHG(N)$, a hyperedge $e$ can be considered stable with
respect to the Myerson value if and only if

$\displaystyle\mu_{k}(N,v,E)\geq\mu_{k}(N,v,E\setminus e)$

for any $k\in e$. Therefore, if a given edge is stable, the stability of the
Myerson value follows.

In this context, stability implies that if a hyperedge $e$ is deleted, any
vertex containing $e$ decreases the profit. Myerson proved that if a TU game
is a super additive, the corresponding Myerson value is stable. Li–Shan
defined safety considering a stronger definition of stability.

######  Definition 4.2.

For any $(N,v,E)\in DHG(N)$, a hyperedge $e$ can be considered safe with
respect to the Myerson value if and only if

$\displaystyle\mu_{k}(N,v,E)\geq\mu_{k}(N,v,E\setminus e)$

for any $k\in N$.

Therefore, safety implies that if a hyperedge $e$ is deleted, all vertices in
$N$ decrease the profit. That is, if a hyperedge is not safe, there exists a
player that increases the benefit from breaking the edge. In this study, we
modify Li–Shan’s result and demonstrate the condition of safety with respect
to the Myerson value. The following theorem can be stated.

######  Theorem 4.3.

For any $(N,v,E)\in DHG(M)$, if the TU game $(N,v)$ is convex, a hyperedge $e$
of the hypergraph $(N,E)$ represents a bridge if and only if it is safe.

###### Proof.

Let $e$ denote a hypergraph bridge. To prove that $e$ is safe, it is
sufficient to demonstrate that for any $i\in N$,

$\displaystyle\sum_{S\subseteq
N\setminus\left\\{i\right\\}}\cfrac{s!(n-s-1)!}{n!}(v^{E}(S\cup\left\\{i\right\\})-v^{E}(S)-v^{E\setminus
e}(S\cup\left\\{i\right\\})+v^{E\setminus e}(S)).$

Therefore, it must be demonstrated that for any $i\in N$ and any $S\subseteq
N\setminus\left\\{i\right\\}$,

$\displaystyle v^{E}(S\cup\left\\{i\right\\})-v^{E\setminus
e}(S\cup\left\\{i\right\\})\geq v^{E}(S)-v^{E\setminus e}(S)\geq 0.$ ( 4.1)

In accordance with the super additivity and convexity definitions, if $(N,v)$
is convex, it follows that it is super additive. Therefore, $v^{E}(S)\geq
v^{E\setminus e}(S)$, which implies $v^{E}(S)-v^{E\setminus e}(S)\geq 0$ for
any subset $S\subseteq N$.

If there exists a player $k\in e$ such that $k\notin S$, it follows that
$e\notin S$. Moreover, it is clear that (4.1) holds because
$v^{E}(S)-v^{E\setminus e}(S)=0$; that is,
$v^{E}(S\cup\left\\{i\right\\})-v^{E\setminus e}(S\cup\left\\{i\right\\})\geq
0$.

Therefore, in this study, we exclusively considered the case wherein all
players of $e$ are contained by a strong component of $S$. Let
$N/E=\left\\{T_{0},\cdots,T_{m}\right\\}$. Therefore, we have
$S/E_{S}=\left\\{T_{0}\cap S,\cdots,T_{m}\cap S\right\\}$. Without loss of
generality, it can be considered that

$\displaystyle e\subseteq T_{0}\cap S,$

and $T_{0}\cap S$ splits the two strong components $T_{0}^{\prime}$ and
$T_{0}^{\prime\prime}$ in $E_{T_{0}\cap S}\setminus e$. Depending on the
strong component player in which $i$ is included, one of the following cases
hold (Figure 1).

Figure 1. Location of player $i$

Case 1. $i\notin T_{0}$. Both sides of (4.1) correspond to $v(T_{0}\cap
S)-v(T_{0}^{\prime})-v(T_{0}^{\prime\prime})$, and this completes the proof.

Case 2. $i\in T_{0}$. This can be further divided into two cases.

Case 2.1. There exists a player in $T_{0}\cap S$ connecting $i$.
Since $e$ represents a bridge of $T_{0}\cap S$, all players that connect $i$
in $T_{0}\cap S$ are contained within either $T_{0}^{\prime}$ or
$T_{0}^{\prime\prime}$. Therefore, without loss of generality, it can be
considered that all players that connect $i$ in $T_{0}\cap S$ are contained
within $T_{0}^{\prime}$. Therefore,

$\displaystyle v^{E}(S\cup\left\\{i\right\\})-v^{E\setminus
e}(S\cup\left\\{i\right\\})$ $\displaystyle=$ $\displaystyle v((T_{0}\cap
S)\cup\left\\{i\right\\})-v(T_{0}^{\prime}\cup\left\\{i\right\\})-v(T_{0}^{\prime\prime}),$
$\displaystyle v^{E}(S)-v^{E\setminus e}(S)$ $\displaystyle=$ $\displaystyle
v(T_{0}\cap S)-v(T_{0}^{\prime})-v(T_{0}^{\prime\prime}).$

Based on the convexity condition of $v$ for $T_{0}^{\prime}\cup
T_{0}^{\prime\prime}$ and $T_{0}^{\prime}\cup\left\\{i\right\\}$, we have

$\displaystyle v((T_{0}\cap S)\cup\left\\{i\right\\})+v(T_{0}^{\prime})\geq
v(T_{0}\cap S)+v(T_{0}^{\prime}\cup\left\\{i\right\\}).$

This completes the proof.

Case 2.2. There exists no player in $T_{0}\cap S$ that connects $i$. In this
case, both sides of (4.1) correspond to $v(T_{0}\cap
S)-v(T_{0}^{\prime})-v(T_{0}^{\prime\prime})$. This completes the proof.

Conversely, assuming $e$ is safe, it follows that for any $k\in N$,

$\displaystyle\mu_{k}(N,v,E)\geq\mu_{k}(N,v,E\setminus e).$

Let us assume that $e$ does not represent a bridge. Therefore, it follows that
there exists a strong component $T\in N/E$ such that

$\displaystyle T\in N/(E\setminus e),\ e\subset T.$

Based on the component efficiency and fairness conditions of the Myerson
value, for any $i\in A_{e}$ and any $j\in B_{e}$,

$\displaystyle\sum_{k\in
T\setminus\left\\{i,j\right\\}}(\mu_{k}(N,v,E)-\mu_{k}(N,v,E\setminus
e))=2(\mu_{i}(N,v,E\setminus e)-\mu_{i}(N,v,E)).$

Considering the safety condition of $e$, $0\leq\mu_{i}(N,v,E\setminus
e)-\mu_{i}(N,v,E)$, it implies that

$\displaystyle\mu_{i}(N,v,E)\leq\mu_{i}(N,v,E\setminus e).$

This represents a contradiction. ∎

######  Remark 4.4.

Li–Shan demonstrated that if a game is strictly convex, the corresponding
graph link can be considered safe if and only if it represents a bridge.
Therefore, the statement of this theorem is more accurate, and its proof is
more straightforward compared to reported in [LS1].

##  5\. Example

In this study, we calculated the Myerson value for each player depicted in
Figure 2. Considering $H=(N,E)$ denotes the hypergraph in Figure 2, it can be
stated that

$\displaystyle N$ $\displaystyle=$ $\displaystyle\left\\{1,2,3,4,5\right\\},$
$\displaystyle E$ $\displaystyle=$
$\displaystyle\left\\{e_{1}=\left\\{1,2\right\\},e_{2}=\left\\{1,2\right\\},e_{3}=\left\\{2,3,4\right\\},e_{4}=\left\\{1,3,4,5\right\\}\right\\},$

where

$\displaystyle A_{e_{1}}=\left\\{1\right\\},\ B_{e_{1}}=\left\\{2\right\\},$
$\displaystyle A_{e_{2}}=\left\\{2\right\\},\ B_{e_{2}}=\left\\{1\right\\},$
$\displaystyle A_{e_{3}}=\left\\{2,3\right\\},\ B_{e_{3}}=\left\\{4\right\\},$
$\displaystyle A_{e_{4}}=\left\\{3,4,5\right\\},\
B_{e_{4}}=\left\\{1\right\\}.$

Considering the characteristic function $v(S)=|S|$ for any $S\subseteq N$, it
follows that

$\displaystyle\mu(N,v,E)=\left(\cfrac{28}{5},\cfrac{23}{5},\cfrac{23}{5},\cfrac{23}{5},\cfrac{23}{5}\right).$

Because every hyperedge of $H$, except $e_{2}$, represents a bridge, we
verified Theorem 4.3 by deleting $e_{4}$ and calculated the Myerson value for
$(N,v,E\setminus e_{4})$.

$\displaystyle\mu(N,v,E\setminus e_{4})=\left(2,2,1,1,1\right).$

Therefore, for any $k\in N$, it follows that

$\displaystyle\mu_{k}(N,v,E)\geq\mu_{k}(N,v,E\setminus e_{4}).$

Therefore, $e_{4}$ is safe. Similarly, the other edge can be proved to be
safe. Conversely, if we delete $e_{2}$ and calculate the Myerson value for
$(N,v,E\setminus e_{2})$, we get

$\displaystyle\mu(N,v,E\setminus e_{2})=\left(5,5,5,5,5\right).$

Therefore, for any $k\in\left\\{3,4,5\right\\}$, we have
$\mu_{k}(N,v,E)<\mu_{k}(N,v,E\setminus e_{2})$. This proves that $e_{2}$ is
not safe.

Figure 2. Directed hypergraph

##  6\. Conclusions

This paper defines the Myerson value for a directed hypergraph game. Moreover,
it demonstrates the axiomatizations of the Myerson value in terms of the
properties of strong component efficiency and fairness. Accordingly, the
Myerson value can be defined for all discrete objects. Li–Shan had previously
demonstrated that an edge is safe if and only if it represents a bridge with a
strictly convex graph game. Moreover, this safety condition requires that the
above-mentioned strictly convex game condition is accompanied by the strongest
super additivity, convexity, and strict superadditivity. Therefore, a
hyperedge can be considered safe if and only if it represents a bridge with a
directed convex hypergraph game. Other interesting topics for future research
in the same vein include an alternate allocation rule for a directed
hypergraph game as well as another relation between the Myerson value and some
topological notations, such as homology and genus.

## References