# Dynamics on higher-order networks: A review

Soumen Majhi<EMAIL_ADDRESS>Department of Mathematics, Bar-Ilan
University, Ramat-Gan, 5290002, Israel  Matjaž Perc<EMAIL_ADDRESS>Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška
cesta 160, 2000 Maribor, Slovenia  Department of Medical Research, China
Medical University Hospital, China Medical University, 404332 Taichung, Taiwan
Complexity Science Hub Vienna, Josefstödter Str. 39, 1080 Vienna, Austria
Alma Mater Europaea, Slovenska ulica 17, 2000 Maribor, Slovenia  Dibakar Ghosh
<EMAIL_ADDRESS>Physics and Applied Mathematics Unit, Indian Statistical
Institute, 203 B. T. Road, Kolkata 700108, India

###### Abstract

Network science has evolved into an indispensable platform for studying
complex systems. But recent research has identified limits of classical
networks, where links connect pairs of nodes, to comprehensively describe
group interactions. Higher-order networks, where a link can connect more than
two nodes, have therefore emerged as a new frontier in network science. Since
group interactions are common in social, biological, and technological
systems, higher-order networks have recently led to important new discoveries
across many fields of research. We here review these works, focusing in
particular on the novel aspects of the dynamics that emerges on higher-order
networks. We cover a variety of dynamical processes that have thus far been
studied, including different synchronization phenomena, contagion processes,
the evolution of cooperation, and consensus formation. We also outline open
challenges and promising directions for future research.

higher-order networks, synchronization, cooperation, dynamics

##  I Introduction

The theory of complex networks Albert and Barabási (2002); Newman (2003)
provides us with a framework for investigating the structure and dynamics of
interacting systems. Indeed, network science has proven highly efficient in
elucidating the dynamics of complex systems arising from many different
contexts in the physical, biological as well as technological and social
sciences Boccaletti et al. (2006); Barrat et al. (2008). Many key developments
have been made in view of identifying and improving the concepts of
association among the constituents of a network. To illustrate, the necessity
of considering the links of networks that are different in nature has led to
the formulation and detailed analysis of multilayer networks Boccaletti et al.
(2014). Further, time-varying networks Holme and Saramäki (2012); Ghosh et al.
(2022) are investigated in which interactions do not persist for all the
course of time, rather come up or vanish over time. It is unquestionably true
that all these developments have helped us to perceive many scenarios better,
but specifically assumed dyadic or pairwise interactions as the backbone for
connections among the units of the system. However, a complete explanation of
many complex systems requires more to furnish. Indeed, real-world systems
often exhibit higher-order relationships beyond dyadic interactions Battiston
et al. (2020); Bianconi (2021). For instance, group interactions take place
predominantly in systems arising in neurobiology Giusti et al. (2016); Petri
et al. (2014) and social systems Benson et al. (2016); Bianconi (2021) to
ecology Levine et al. (2017); Grilli et al. (2017). Network framework has been
intrinsically limited to explanation through the pairwise interactions which
are sufficient only to model the dyadic relationships and so larger group
interactions do need a better formulation for networked systems Lambiotte et
al. (2019); Johnson (2013); Bick et al. (2021). It has been argued that
higher-order structures, namely hypernetworks and simplicial complexes, are
excellent frameworks to characterize the organization of many social,
biological and other scenarios encoded in group interactions of three or more
constituents Battiston et al. (2020, 2021).

Heretofore, not much attention has been paid to the analysis of networks
exposed to higher-order interactions. However, a significant amount of recent
advances is demonstrating that the incorporation of higher-order architecture
can remarkably improve our understanding and prediction ability of their
dynamics. The studies related to these higher-order structures have thus come
to the forefront of network dynamics research. Among some highly significant
researches on higher-order networks, the one by Benson et al. Benson et al.
(2018) is particularly noteworthy that investigates datasets from different
disciplines ranging from various social networks to biology, and demonstrates
variety of characteristics of the higher-order structures emerging therein.
Besides, the problem of higher-order link prediction is formulated which has
been found to be essentially different from the traditional dyadic link
prediction Liben-Nowell and Kleinberg (2007). This issue of link prediction in
networks subject to the presence of higher-order structures is also studied in
the Ref. Sharma et al. (2021) while dealing with different link prediction
algorithms. The inverse problem of inferring higher-order interactions from
observational data has also been treated in the Ref. Beentjes and Khamseh
(2020), whereas higher-order interactions are inferred from the traditional
dyadic interaction network data through a Bayesian approach Young et al.
(2021). An analytical treatment (Statistically Validated Hypergraphs) is
propounded Musciotto et al. (2021) for the problem of finding the most
important relationships among the constituents of a higher-order network. Tie
strengths are quantified considering higher-order interactions encoded by
groups of three or more individuals in social networks by the measure named as
‘Edge PageRank’ Sarker et al. (2021). This measure is proved to be much more
efficacious than the traditional approaches for detection of tie strength.
Vector centrality measure is proposed for higher-order networks with an aim to
identify the most influential nodes in the system Kovalenko et al. (2021a).

Different models of higher-order networks Bobrowski and Krioukov (2021) have
been developed so far. Detailed analysis of models of growing simplicial
complexes Kovalenko et al. (2021b); Courtney and Bianconi (2017, 2018) is
presented, built upon the concept of ‘network geometry with flavor (NGF)’
Bianconi and Rahmede (2016); Mulder and Bianconi (2018). The models yield
exponential or scale-free generalized degree distribution based on the non-
preferential or preferential attachment rules. ‘Simplicial activity driven
model’ Petri and Barrat (2018) is proposed and analyzed, that captures both
the higher-order structure and the temporal nature of interactions among the
nodes. ‘Simplicial configuration model’ Young et al. (2017); Courtney and
Bianconi (2016) with an uniform Markov chain Monte Carlo sampler is
introduced, even for arbitrary degree and size distributions Yen (2021). In
order to provide formalism for modeling heterogeneous, polyadic network data,
the configuration models of random hypernetworks Chodrow (2020) and the
annotated hypergraph model Chodrow and Mellor (2020) are presented as a
generalization of directed graphs. On the other hand, higher-order network
setups are utilized in order to generalize the formalism of structural
controllability to time-varying networks Zhang et al. (2021a), for which both
synthetic and real-world data sets are examined to illustrate the minimum time
required to control the concerned systems. Group research collaborations of
three or more individuals are illustrated through higher-order interaction
framework and further encoded under multilayer formalism with collaboration
data taken from different research disciplines Vasilyeva et al. (2021).
Further, heterogeneous dynamics of higher-order structures in time-varying
social networks is examined for a number of social datasets Cencetti et al.
(2021).

The perception of higher-order clusters being particularly important, the
concept of higher-order clustering coefficients are introduced in the Ref. Yin
et al. (2018), which quantifies the closure probability of higher-order
cliques. This measure is used to examine the clustering behavior of both model
and real-world networks. The problem of clustering in hypernetworks with
categorical edge labels can be addressed with a procedure based upon
combinatorial objective function Amburg et al. (2020). The efficiency of this
mechanism is validated for edge-label community detection and clustering with
time-stamped data. Simplicial communities are detected from real-world data of
social networks while showing that the spectra of the Hodge Laplacian encodes
the communities Krishnagopal and Bianconi (2021). A stochastic generative
model is introduced to hypernetwork clustering with heterogeneous node degree
and hyperlink size distribution Chodrow et al. (2021), which is shown to be
highly scalable and efficient with the utilization of synthetic and various
real-world data. Tudisco et al. Tudisco and Higham (2021) has recently come up
with their study of a family of spectral centrality measures in order to
recognize important nodes and hyperlinks in hypernetworks, which extends the
existing concepts for dyadic interactions. However, the formalism constructed
by Veldt et al. Veldt et al. (2021) using hypernetworks to measure
‘homophily’, unravels that homophilous group configurations are impossible due
to a combinatorial impossibility of hypernetworks.

Concerning dynamical processes, framework for hypernetwork robustness and
analysis of higher-order percolation process Sun and Bianconi (2021); Bianconi
et al. (2019); Lee et al. (2021) are put forward for multiplex hypernetworks.
Analogous to the largest eigenvalue of the matrix representing interaction
structure of a network built upon dyadic connections, the concept of
‘expansion eigenvalue’ for hypernetwork dynamical processes is proposed and
approximated through a mean-field approach Landry and Restrepo (2021). Quite
interestingly, in case of random interactions in ecological communities, the
presence of higher-order species interactions can verily alter the traditional
relation between diversity and stability Bairey et al. (2016). For instance,
even though dyadic interactions cause sensitivity to the species addition,
four-way interactions result in sensitivity to the removal of species. Also,
the merger of the dyadic and higher-order interaction induces both upper and
lower bounds on the number of species. Moreover, there exists evidence of
higher-order interactions stabilizing the dynamics in ecological communities
Grilli et al. (2017) where interaction between species is influenced by other
species. In both open and closed ecological communities, higher-order
interactions have noticeable impacts that stabilize the dynamics for
competitive models. Stochastic models under higher-order interactions help
further in sustained coexistence of species (see also the review work Levine
et al. (2017) and the references therein).

In the next section (Sec. II), we briefly recall the basic definition of
relevant terminology in higher-order interactions. We then start discussing
about the phenomenon of synchronization emerging in higher-order networked
systems (cf. Sec. III). Then, in Sec. IV, we move on to explore various social
dynamics evolving over higher-order structures. Specifically, we investigate
the processes of contagion dynamics, consensus formation and evolutionary game
dynamics in Secs. IV.1, IV.2 and IV.3 respectively. Section V deals with the
dynamics of random walk and diffusion. Finally, Sec. VI offers the summary and
discussion about potential research in the future.

##  II Basic Concepts

Hyperlink: Hyperlink is the fundamental backbone of a higher-order network,
which instead of joining only two nodes (for the traditional networks of
pairwise interactions), can connect any number of nodes.
Hypernetwork: Hypernetwork is a generalization of the notion of network, and
is composed of hyperlinks. This implies that a hypernetwork $\mathcal{H}$ can
be considered as the pair $(V,E)$ in which $V$ is the set of nodes and $E$ (a
subset of the power set $P(V)$ of $V$) is the set of hyperlinks.
Simplex: A $d$-dimensional simplex (or a $d$-simplex) is simply a set of $d+1$
fully interacting nodes. Essentially, a $0$-simplex is a node, a $1$-simplex
is a link, a $2$-simplex is a triangle, a $3$-simplex is a tetrahedron and so
on.
Simplicial complex: Similar to the networks as a collection of links, a
simplicial complex comprises of simplices. From the context of hypernetworks,
a simplicial complex is a particular type of hypernetwork that accommodates
each subset of all the hyperlinks. This means that a simplicial complex
$\mathcal{S}$ is a hypernetwork which fulfills the criteria that for each
$e\in E$ and $\forall e^{\prime}\subseteq e~{}(e^{\prime}\neq\emptyset)$, one
also has $e^{\prime}\in E$.
In Fig. 1, examples of simplices (in upper panel) and hyperlinks (in lower
panel) of dimension from $1$ to $3$ are depicted that clarify about the
higher-order building blocks based upon which the higher-order networks are
generally built.

Figure 1: The higher-order building blocks, namely the simplices (upper panel)
and the hyperlinks (lower panel) of dimension $1$, $2$ and $3$. Figure
reproduced from the Ref. Battiston et al. (2020).

In the next, we focus on elaborating the principal findings and the novel
effects in the dynamical processes that the higher-order interactions bring
about, and hence can be of interest for the perception of a number of natural
occurrences. We, however, mention here that we do not really distinguish
between the dynamics on hypernetworks and dynamics on simplicial complexes,
rather we present an excerpt of diverse dynamics on top of higher-order
networks, in general.

##  III Synchronization

The phenomenon of synchronization Pikovsky et al. (2003); Boccaletti et al.
(2002); Wang et al. (2009); Majhi et al. (2019a); Arenas et al. (2008); Majhi
and Ghosh (2017) corresponds to a process wherein interacting dynamical
systems adjust certain properties of their motion to a common dynamics, and
this interaction pattern plays decisive roles for the emergence of synchrony.
Synchronization is considered to be one of the most important phenomenon in
complex dynamical network theory, having crucial applications in several
physical, biological and technological systems. Hence, there has been a strong
urge in exploring different aspects of synchronization in coupled systems in
the last two decades. However, only recently the investigation tends towards
higher-order interactions.

Specifically, three-body interactions in an ensemble of phase oscillators can
give rise to infinite number of multistable synchronized attractors beyond a
critical interaction strength Tanaka and Aoyagi (2011). Simplicial complexes
of large ensembles of interacting oscillators are considered while modeling
the three-way interactions on top of a multilayer framework Skardal and Arenas
(2019). A continuum of abrupt transitions to desynchronization is observed
therein as a result of multistability comprising of infinite number of stable
partially synchronized states. An analytical treatment is provided based upon
dimensionality reduction through a variation of the Ott-Antonsen ansatz.
Besides, synergistic effects of higher-order interactions of different order
(namely, $1$-, $2$\- and $3$-simplex) on synchrony arising in heterogeneous
Kuramoto phase oscillators is analyzed Skardal and Arenas (2020). Here, it has
been shown that the interplay of these simplicial structures can yield abrupt
transitions to both desynchronization and synchronization, and can stabilize
strong synchrony even under repulsive pairwise interaction. The authors also
enunciate the phenomena while dealing with UK power grid and Macaque brain
networks. Gambuzza et al. Gambuzza et al. (2021) have recently presented a
general framework for the stability of synchronization in networks subject to
higher-order interactions of any order. The authors demonstrate the existence
of complete synchrony as an invariant solution and provided the necessary
conditions for the synchronous solution. The generality of the proposed
formalism is elucidated considering paradigmatic chaotic Rössler system and
model systems pertinent to neurodynamics. Besides, cluster synchronization is
studied in a model of simplicial complex of Rössler oscillators.

Analysis of $D~{}(\geq 2)$-dimensional Kuramoto dynamics on top of simplicial
structures ($1$\- and $2$-simplices predominantly) is presented in the Ref.
Dai et al. (2021). Theoretical analysis together with extensive numerical
simulations are put forward wherein reasoning behind different synchronization
patterns resulting from odd and even dimensions is explained Chandra et al.
(2019). Interestingly, discontinuous transition to desynchronization for any
dimension at positive interaction strength, discontinuous transition for odd
dimensions at zero coupling strength, and the state of partial synchronization
for all odd $D$ (along with $D=2$) at negative interaction strength are
witnessed. Further, a globally coupled ensemble of the $D$-dimensional
Kuramoto oscillators consisting of only contrarians possesses collective
synchrony in absence of any conformists, if the underlying connection among
the units goes beyond dyadic interactions Kovalenko et al. (2021c). This
result, in particular, is forbidden in networks with only pairwise
communications. A notably interesting formulation of the higher-order Kuramoto
dynamics that designates interactions among oscillators placed not only on the
nodes but also on the higher-dimensional simplices like links, triangles etc.
permits one to describe a topologically projected dynamics on lower- and
higher-dimensional faces Millán et al. (2020). It has been shown that besides
a simple continuous transition to synchronization, with an adaptive coupling
between the dynamics projected on the lower- and higher- dynamical phases the
networked system exhibits explosive transition to synchrony.

Furthermore, the interaction between dynamical signals defined on nodes and
links yields explosive topological synchronization wherein the phases ascribed
on the nodes synchronize to the phases defined on the links at a discontinuous
transition Ghorbanchian et al. (2021). Detailed analytical treatment is
provided exploring this scenario and the associated closed hysteresis loop in
the limit of large size of the networks. Besides dealing with simplicial
complexes, the model has been tested on the human and C. elegans connectomes.
To be precise, a simplicial complex of $N_{[n]}$ simplices of dimension $n$
(i.e., $N_{[0]}$ nodes, $N_{[1]}$ number of links, $N_{[2]}$ triangles etc.)
with ${\boldsymbol{B}}_{[n]}$ as the $n$-th incidence matrix for the $n$-th
boundary operator, are assumed. Then the phase vector
$\boldsymbol{\theta}=(\theta_{1},\theta_{2},\cdots,\theta_{N_{[0]}})$
associated to the nodes obeys the following dynamical equation:

$\begin{array}[]{lcl}\par\dot{\boldsymbol{\theta}}=\boldsymbol{\omega}-\sigma{\boldsymbol{B}}_{[1]}\mbox{sin}\Big{(}{{\boldsymbol{B}}^{T}_{[1]}}\boldsymbol{\theta}\Big{)},\end{array}$
(1)

where $\sigma$ is the interaction strength, with each $\omega_{k}$ chosen from
a given distribution, say a normal distribution
$\omega_{i}\sim\mathcal{N}(\Omega_{0},1/\tau_{0})$. Consequently, the
associated order parameter can be written as

$\begin{array}[]{lcl}\par
R_{0}=\dfrac{1}{N_{[0]}}\Big{|}\sum\limits_{k=1}^{N_{[0]}}e^{i\theta_{k}}\Big{|}.\end{array}$
(2)

Higher-order topological Kuramoto model defined on phases
$\boldsymbol{\phi}=(\phi_{l_{1}},\phi_{l_{2}},\dots,\phi_{l_{N_{[1]}}})$
associated to the links is described as

$\begin{array}[]{lcl}\par\dot{\boldsymbol{\phi}}=\boldsymbol{\tilde{\omega}}-\sigma{{\boldsymbol{B}}^{T}_{[1]}}\mbox{sin}\Big{(}{{\boldsymbol{B}}_{[1]}}\boldsymbol{\phi}\Big{)}-\sigma{{\boldsymbol{B}}_{[2]}}\mbox{sin}\Big{(}{{\boldsymbol{B}}^{T}_{[2]}}\boldsymbol{\phi}\Big{)},\end{array}$
(3)

with ${\tilde{\omega}}_{l}\sim\mathcal{N}(\Omega_{1},1/\tau_{1})$ as the
internal frequencies for the links. With the synchronization dynamics defined
on the higher-order $n(=1)$-dimensional signals, the projections
${\boldsymbol{\phi}}^{[-]}$ and ${\boldsymbol{\phi}}^{[+]}$ on the $n-1$
simplices (i.e., nodes) and $n-2$ simplices (i.e., triangles) are
${\boldsymbol{\phi}}^{[-]}={{\boldsymbol{B}}_{[1]}}{\boldsymbol{\phi}}$ and
${\boldsymbol{\phi}}^{[+]}={{\boldsymbol{B}}^{T}_{[2]}}{\boldsymbol{\phi}}$,
which respectively follow the following dynamics

$\begin{array}[]{lcl}\par\dot{{\boldsymbol{\phi}}}^{[-]}={{\boldsymbol{B}}_{[1]}}\boldsymbol{\tilde{\omega}}-\sigma{{\boldsymbol{L}}_{[0]}}\mbox{sin}({{\phi}}^{[-]}),\\\
\dot{{\boldsymbol{\phi}}}^{[+]}={{\boldsymbol{B}}^{T}_{[2]}}\boldsymbol{\tilde{\omega}}-\sigma{{\boldsymbol{L}}^{\mbox{\tiny
down}}_{[2]}}\mbox{sin}({{\phi}}^{[+]}),\end{array}$ (4)

where
${{\boldsymbol{L}}_{[0]}}={{\boldsymbol{B}}_{[1]}}{{\boldsymbol{B}}^{T}_{[1]}}$
and ${{\boldsymbol{L}}^{\mbox{\tiny
down}}_{[2]}}={{\boldsymbol{B}}^{T}_{[2]}}{{\boldsymbol{B}}_{[2]}}$, and the
corresponding order parameters obtain the forms:

$\begin{array}[]{lcl}\par{R}^{\mbox{\tiny
down}}_{1}=\dfrac{1}{N_{[0]}}\Big{|}\sum\limits_{k=1}^{N_{[0]}}e^{i{{\phi}}^{[-]}_{k}}\Big{|},\\\
~{}~{}~{}{R}^{\mbox{\tiny
up}}_{1}=\dfrac{1}{N_{[2]}}\Big{|}\sum\limits_{k=1}^{N_{[2]}}e^{i{{\phi}}^{[+]}_{k}}\Big{|}.\end{array}$
(5)

Then, unlike the adaptive coupling between these two dynamics of the same
dimension as in the Ref. Millán et al. (2020), here signals of different
dimensions are coupled through the order parameters of the node and link
dynamics (i.e., Eqs. (2) and (5)). In particular, two models named as Nodes-
Links (NL) and Nodes-Links-Triangles (NLT) are considered. The former one is
described as

$\begin{array}[]{lcl}\dot{\boldsymbol{\theta}}=\boldsymbol{\omega}-\sigma{R}^{\mbox{\tiny
down}}_{1}{\boldsymbol{B}}_{[1]}\mbox{sin}\Big{(}{{\boldsymbol{B}}^{T}_{[1]}}\boldsymbol{\theta}\Big{)},\\\
\dot{\boldsymbol{\phi}}=\boldsymbol{\tilde{\omega}}-\sigma
R_{0}{{\boldsymbol{B}}^{T}_{[1]}}\mbox{sin}\Big{(}{{\boldsymbol{B}}_{[1]}}\boldsymbol{\phi}\Big{)}-\sigma{{\boldsymbol{B}}_{[2]}}\mbox{sin}\Big{(}{{\boldsymbol{B}}^{T}_{[2]}}\boldsymbol{\phi}\Big{)},\end{array}$
(6)

with the projected dynamics following

$\begin{array}[]{lcl}\par\dot{{\boldsymbol{\phi}}}^{[-]}={{\boldsymbol{B}}_{[1]}}\boldsymbol{\tilde{\omega}}-\sigma
R_{0}{{\boldsymbol{L}}_{[0]}}\mbox{sin}({{\phi}}^{[-]}),\\\
\dot{{\boldsymbol{\phi}}}^{[+]}={{\boldsymbol{B}}^{T}_{[2]}}\boldsymbol{\tilde{\omega}}-\sigma{{\boldsymbol{L}}^{\mbox{\tiny
down}}_{[2]}}\mbox{sin}({{\phi}}^{[+]}).\end{array}$ (7)

The latter one is defined as

$\begin{array}[]{lcl}\dot{\boldsymbol{\theta}}=\boldsymbol{\omega}-\sigma{R}^{\mbox{\tiny
down}}_{1}{\boldsymbol{B}}_{[1]}\mbox{sin}\Big{(}{{\boldsymbol{B}}^{T}_{[1]}}\boldsymbol{\theta}\Big{)},\\\
\dot{\boldsymbol{\phi}}=\boldsymbol{\tilde{\omega}}-\sigma
R_{0}{R}^{\mbox{\tiny
up}}_{1}{{\boldsymbol{B}}^{T}_{[1]}}\mbox{sin}\Big{(}{{\boldsymbol{B}}_{[1]}}\boldsymbol{\phi}\Big{)}-\sigma{R}^{\mbox{\tiny
down}}_{1}{{\boldsymbol{B}}_{[2]}}\mbox{sin}\Big{(}{{\boldsymbol{B}}^{T}_{[2]}}\boldsymbol{\phi}\Big{)},\end{array}$
(8)

where the projected dynamics obeys

$\begin{array}[]{lcl}\par\dot{{\boldsymbol{\phi}}}^{[-]}={{\boldsymbol{B}}_{[1]}}\boldsymbol{\tilde{\omega}}-\sigma
R_{0}{R}^{\mbox{\tiny
up}}_{1}{{\boldsymbol{L}}_{[0]}}\mbox{sin}({{\phi}}^{[-]}),\\\
\dot{{\boldsymbol{\phi}}}^{[+]}={{\boldsymbol{B}}^{T}_{[2]}}\boldsymbol{\tilde{\omega}}-\sigma{R}^{\mbox{\tiny
down}}_{1}{{\boldsymbol{L}}^{\mbox{\tiny
down}}_{[2]}}\mbox{sin}({{\phi}}^{[+]}).\end{array}$ (9)

Figure 2:  The evolution of the order parameters $R_{0}$, ${R}^{\mbox{\tiny
down}}_{1}$ and ${R}^{\mbox{\tiny up}}_{1}$ with respect to the coupling
strength $\sigma$. The first two columns (from left) correspond to the NGF
model, with the last two columns representing the configuration model. Also,
the second and fourth columns (from left) represent the Nodes-Links (NL) model
whereas the first and third columns (from left) correspond to the Nodes-Links-
Triangles (NLT) model. The cyan lines designate the forward transitions and
the green lines indicate the backward transitions. For both the network
models, $N_{[0]}=500$. Further, $\Omega_{0}=\Omega_{1}=2$ and
$\tau_{0}=\tau_{1}=1$ are chosen. Figure reproduced from the Ref. Ghorbanchian
et al. (2021).

Now, with these two dynamical models NL and NLT in hand, extensive numerical
simulations are carried out on two models of simplicial complexes, namely the
configuration model Courtney and Bianconi (2016) and the NGF model Bianconi
and Rahmede (2016). Figure 2 displays the values of the order parameters
$R_{0}$, ${R}^{\mbox{\tiny down}}_{1}$ and ${R}^{\mbox{\tiny up}}_{1}$ as
functions of the coupling strength $\sigma$ in top, middle and bottom panels,
respectively. The first two columns correspond to the NGF model for flavor
$s=-1$ and $d=3$-dimensional simplicial complexes with an underlying power-law
network (exponent $\gamma=3$), whereas the last two columns are for the
configuration model for power-law (exponent $\gamma=2.8$) generalized degree
distribution, with $N_{[0]}=500$ nodes for both the cases. Moreover, the first
and third (second and fourth) columns depict the outcomes for the NLT (NL)
model. As confirmed, in both the scenarios, explosive transitions to the state
of synchronization take place. The transitions occurs along with the
hysteresis loop formed by the forward and backward transitions. For the NLT
model, all the order parameters $R_{0}$, ${R}^{\mbox{\tiny up}}_{1}$ and
${R}^{\mbox{\tiny down}}_{1}$ show discontinuous transition to synchrony at
the same coupling strength. But in case of the NL model, although $R_{0}$ and
${R}^{\mbox{\tiny down}}_{1}$ demonstrate discontinuous transition for some
critical interaction strength, ${R}^{\mbox{\tiny up}}_{1}$ ensues an
independent transition at the zero coupling strength for both the network
models. This is because in the NL model the adaptive interaction couples only
the phases ${{\boldsymbol{\phi}}}^{[-]}$ and ${\boldsymbol{\theta}}$, and not
the phase ${{\boldsymbol{\phi}}}^{[+]}$. For the further details of the nice
analytical treatment alongside the numerical results, please see the Ref.
Ghorbanchian et al. (2021).

Synchronization in an ensemble of Kuramoto phase oscillators subject to the
interplay of interactions built upon $1$-simplex (i.e., the links) and the
$2$-simplex (i.e., the triangles) faces of homogeneous $4$-dimensional
simplicial complexes is reported Chutani et al. (2021). In presence of only
dyadic interactions, increasing positive coupling strength leads to a
continuous transition to complete synchrony whereas negative coupling results
in a partial synchronization. It needs to be mentioned here that no synchrony
is observed for negative coupling in scale-free networks. Moreover,
introduction of the higher-order ($2$-simplex) coupling impedes the synchrony
induced by the pairwise interaction, and causes the hysteresis loop with
abrupt transition to desynchronization for negative pairwise coupling. Also,
in a very recent work Kachhvah and Jalan (2021), the authors assume an
adaptive higher-order (triadic) interaction formalism relying on the Hebbian
learning mechanism in networks of Kuramoto phase oscillators and showed that
such a coupling can induce first-order transition to desynchronization. The
presented scenario is further explained through a detailed mean-field
analysis. Partial loss of synchronization can also be witnessed in a
generalized Sakaguchi-Kuramoto model formed through the inclusion of linear
and nonlinear frustrations into the simplicial Kuramoto model Arnaudon et al.
(2021) and weights on the simplices. A precise mathematical framework is
presented in the article apart from the computational results for this
frustrated model on links while dealing with measures like the order parameter
and Hodge decomposition.

Stability of synchronization in ensembles of oscillators subject to higher-
order interactions of any order and built upon any complex underlying
hypernetwork structure with arbitrary coupling functions can be analyzed
through a general formalism of multiorder Laplacian Lucas et al. (2020).
Different network setups are investigated including the one that deals with
both attractive and repulsive interactions Majhi et al. (2020). From synthetic
to empirical datasets are studied under this framework. Higher-order
interactions embedded in clique complexes can optimize the collective
synchronization in Kuramoto model subject to equitable increase in the
strength of the higher-order connections relative to the pairwise interactions
Skardal et al. (2021). Synchronization dynamics and formation of Turing
patterns in nonlinear chaotic systems are studied in higher-order networks
while opting the master stability function (MSF) framework and analyzing an
appropriate combinatorial Laplacian Carletti et al. (2020a). The processes are
examined for general hypernetworks with heterogeneous distribution of the
hyperlinks, and not restricting to any specific forms of the coupling
function. Different synchronization patterns from cluster synchrony to
chimeras are realized in generalized networks, including multilayer networks,
hypernetworks and time-varying networks through the simultaneous block
diagonalization (SBD) approach Zhang et al. (2021b). MSF formalism has also
been generalized for hypernetworks in the Ref. Mulas et al. (2020) to treat
the linear stability of the phenomenon of synchronization, where the special
class of Laplace-type interactions have also been examined. The dynamical
systems known as coupled map lattices are extended to the scenario of higher-
order interaction, namely to the concept of coupled hypergraph maps Böhle et
al. (2021). The process of cluster synchronization is investigated in such a
system through the analysis of hypernetwork Laplacian for different chaotic
discrete dynamical systems. Very recently, in the Ref. Parastesh et al. (2022)
the authors consider three-body interactions along with the dyadic couplings
for interacting Hindmarsh-Rose neuronal model while deriving the necessary
conditions for the emergence of synchrony by means of linear stability
analysis.

##  IV Social dynamics

Diverse social processes have always been a major area of research in
complexity science. A wide list of scenarios ranging from opinion, cultural
and language dynamics to crowd behavior, hierarchy formation, human dynamics,
evolution of cooperation and social spreading is considerably influenced by
peer-to-peer interaction among individuals embedded in social networks. Such
contagion effects directs researchers to explore the dynamics from
mathematical point of view. For this purpose, network science has emerged to
play the most significant role. In the past decade, we have witnessed a golden
age in the study of social dynamics over networks, from different
perspectives. The research communities have long been interested in the
interactions between individuals leading to diverse emerging behavior.
However, as noted in the review by Castellano et al. Castellano et al. (2009),
and the Ref. Malmgren et al. (2009), there are significant aspects of real
social contagion phenomena that needs to be captured with much more
sophisticated approaches than before, from the perspective of both dynamics on
networks and dynamics of networks. In the following, we go through different
genres of studies of social dynamics exposed to higher-order interactions.

###  IV.1 Contagion processes

In order to take into account the group interactions of different sizes,
Iacopini et al. Iacopini et al. (2019) formulated a higher-order simplicial
model of social contagion. The model incorporates both pairwise and higher-
order contacts, and thus combines the essences of both simple and complex
contagion processes. Simplicial structure leads to a discontinuous transition
to the endemic state and bistability emerges in which endemic and healthy
states coexist. The former scenario has been demonstrated analytically along
with numerical demonstrations on random Erdős-Rényi model and empirical
higher-order networks. This model is then extended to the framework of
temporal networks Chowdhary et al. (2021), in which dyadic and higher-order
interactions can be formed and destroyed temporally. Going through a
microscopic Markov chain approach it has been shown that the same number of
infectious seed may or may not generate endemic state, which actually depends
on the temporal properties of the underlying network. The impact of degree
heterogeneity on the simplicial contagion over time-varying higher-order
networks is also investigated in the article thereafter. Social contagion
dynamics is further investigated on hypergraphs in the Ref. de Arruda et al.
(2020). The authors particularly embodied the critical-mass dynamics into the
previously framed model of the Ref. Iacopini et al. (2019). Analytical and
numerical results are presented to show the emergence of continuous and
discontinuous transitions together with bistable regimes and hysteresis.

In addition to depicting that the standard network generating algorithms with
tunable clustering characteristic can yield diverse higher-order structures so
that dynamics can differ on the networks with same clustering and degree
distribution profiles, the Ref. Ritchie et al. (2014) formulated a new metric
for measuring order-four structures. The authors emphasize on how the higher-
order structural differences (arising in networks possessing same clustering)
have consequences on epidemic prevalence and epidemic threshold while dealing
with Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Recovered
(SIR) dynamical models. Landry et al. Landry and Restrepo (2020) studied the
dynamics of SIS model on hypergraphs by means of hyperdegree-based mean-field
analysis on networks with higher-order interactions. Both degree-correlated
and uncorrelated cases are analyzed while it has been unraveled that the
abrupt first-order transitions can be suppressed through heterogeneous degree
distribution of the dyadic interactions under certain assumptions on degree
correlations. Besides inferring the conditions for bistability and hysteresis,
the issues related to higher-order healing and the ‘hipster effect’ have been
further discussed in their article.

Figure 3: (a) Infection kernel with $\beta=0.1$ where the infection
probability ${\theta}_{m}$ has a power law scaling
${\theta}_{m}(\rho)\propto{\rho}^{\alpha}$. (b) Supralinear kernels $\nu>1$
result in a superexponential growth of the prevalence $I(t)$.
$\beta=0.0005,~{}0.025$ and $0.077$ for $\nu=0.5,~{}1.0$ and $1.5$ are chosen
respectively, where $\bar{\tau}$ is the median exposure period. (c) Stationary
prevalence $I^{*}$ with respect to $\beta$, where continuous phase transition
appears for sublinear and linear kernels $\nu\leq 1$ while a discontinuous
phase transition arises with a bistable solution for supralinear kernels
$\nu>1$. Poisson distributions are opted for both the distribution
$\bar{P}(m)$ of the size $m$ of the hyperlinks and the distribution
$\tilde{P}(k)$ of hyperdegree $k$ of the individuals with $\langle
m\rangle=10$ and $\langle k\rangle=5$ respectively, and $\mu=0.05$. Figure
reproduced from the Ref. St-Onge et al. (2021a).

Lately, a higher-order model is developed addressing a number of issues that
has been mostly neglected in epidemic modeling. Heterogeneity that arises in
environments like offices and households etc., and the temporal heterogeneity
in participation of the individuals in these environments are analyzed St-Onge
et al. (2021a). This heterogeneous exposure subject to minimal infective dose
yields a universal nonlinear relation between the risk of infection and the
infected contacts, challenging the prevalent assumption of linear relationship
between these two. As a result, discontinuous transition to epidemic outbreak
takes place and bistable regime emerges as well in which outbreak and healthy
states coexist.
Let us consider the interaction framework to be a hypernetwork in which the
environments are defined by hyperlinks of $m$ individuals where each
individual is incident to $k$ hyperlinks. A discrete-time process
$(t=1,2,...)$ is then assumed where for each environment a participation time
$\tau\in[1,~{}{\tau}_{\mbox{max}}]$ is chosen for each individual. Then if a
susceptible individual participating in an $m$-sized environment for duration
$\tau$ under the presence of a fraction $\rho$ of the other infected
participants, it receives an infective dose $l\in[0,\infty)$ from the infected
individuals, according to the distribution $f(l;\lambda)$, where
$\lambda\equiv\langle l\rangle$. Similar to the threshold models, it is then
considered that someone develops the disease when $l>K$ and
${\theta}_{m}(\rho)$ is the infection kernel that represents the he
probability of getting the infection in an $m$-sized environment subject to a
fraction $\rho$ of other infected participants. Also,
$w\leq{\tau}_{\mbox{max}}$ is the clearing window which represents the
characteristic time for the immune system to be free of any dose $l<K$. Then,
for heterogeneous exposure periods described by a Pareto distribution
$P(\tau)\propto{\tau}^{-\alpha-1}$, $\alpha>0$ and the characteristic time to
be infected as ${\tau}_{c}\equiv\frac{K}{\beta g(m)\rho}$, ($g(m)$ governs the
number of contacts frequented by $m$ nodes and $\beta$ is a dose accumulation
rate), ${\theta}_{m}(\rho)$ takes the form ${\theta}_{m}(\rho)\sim
D_{\alpha}{{\tau}_{c}}^{-\alpha}\propto{\rho}^{\alpha}$ St-Onge et al.
(2021a), with $D_{\alpha}$ being a constant. This is demonstrated in Fig. 3(a)
for an exponential dose distribution $f(l;\lambda)=e^{-l/\lambda}$, $g(m)=1$
and $w\rightarrow\infty$ for different values of $\alpha$.
Further, the effects of nonlinear infection kernels are discussed while
considering a SIS model with recovery rate being $\mu$. With a degree-based
mean-field approximation, for the marginal probability ${\rho}_{k}(t)$ of an
individual to be infected at time $t$ and $\tilde{P}(k)$ as the distribution
of hyperdegree $k$ of the individuals, the global prevalence is
$I(t)=\sum_{k}{\rho}_{k}(t)\tilde{P}(k)$. The temporal evolution of this
prevalence is portrayed in Fig. 3(b) that mirrors the impact of the nonlinear
infection kernel. Specifically, superexponential growth is observed whenever
$\nu>1,~{}({\theta}_{m}(\rho)\propto{\rho}^{\nu})$ whereas the growth is
regular exponential until saturation if $\nu\leq 1$. Finally, Fig. 3(c)
depicts the stationary prevalence $I^{*}$ as a function of $\beta$. The
absorbing state $I^{*}=0$ remains unstable (cf. the dashed line in Fig. 3(c))
whenever $\beta>{\beta}_{u}$ (the invasion threshold). On the other hand,
$I^{*}=0$ is globally stable (cf. the dotted line in Fig. 3(c)) for
$\beta<{\beta}_{s}$ (the persistence threshold). It is conspicuous from the
figure that the transition of $I^{*}$ can be either continuous
(${\beta}_{s}={\beta}_{u}$) or discontinuous (${\beta}_{s}<{\beta}_{u}$) with
a bistable solution.

A simplicial complex environment of interaction can again result in a
discontinuous transition to the endemic state Li et al. (2021). In particular,
here two different facets of contagion has been encountered, at the initial
stage governed by the dyadic interactions whereas the later stage is
controlled by the higher-order interactions. Theoretical analysis is provided
in the homogeneous mixing limit along with rigorous computation in order to
explain the associated bistable regime. By now, we all are aware of the fact
that in case of contagions over standard network models built upon dyadic
interactions, hubs play quite crucial roles. However, in higher-order
networks, not only the individuals but also the groups play decisive roles.
So, with the view of exploring the roles of set of groups on the hypergraph
contagion dynamics, considering heterogeneity in both hyper-degree and
hyperlink cardinality, in the Ref. St-Onge et al. (2021b) the authors
construct a framework based upon approximate master equations analyzing
contagion dynamics on top of random higher-order networks. Assuming the rate
of infection as a nonlinear function of the number of infectious individuals
in groups, it is shown how influential groups can govern the initial dynamics
as well as the final stationary state of the contagion. A mathematical
formulation has been provided with the view of analyzing the linear stability
of general dynamical processes on arbitrary hypernetworks on the basis of
weighted-graph projection of the hypernetwork de Arruda et al. (2021). In
particular, the processes of social contagion and diffusion dynamics are dealt
with. Apart from these, the study in the Ref. Li et al. (2022) investigated
two competing SIS epidemic dynamics on a higher-order networked system
composed of $1$\- and $2$-simplex. Rigorous computations and the analysis of
mean-field equations depicted a repertoire of dynamical features owing to the
higher-order interaction. From the absolute dominance of the epidemics for
weak triadic infection strength to the alternative dominance for higher
triadic infection strength depending on the initial seed of infection are
observed.

###  IV.2 Consensus formation

Consensus dynamics on higher-order networked systems ($3$-body systems,
mainly) is analyzed analytically and numerically Neuhäuser et al. (2020) in
which it was disclosed that the dynamical consequences of multibody
interactions can be effective only when interaction function is nonlinear. As
a result of bringing in a nonlinear function herein, the emerged dynamics
causes shifts off the state of average system, depending on the underlying
network and the initial configuration. Consensus dynamics in higher-order
networks of any order is further studied in the Ref. Sahasrabuddhe et al.
(2021) while contemplating with a number of social processes like homophily
and peer-pressure for modeling the interactions. Apart from the hypergraph
models like block hypergraphs, analysis has been performed on real-world
networks as well. In Hickok et al. (2021), the authors formulated a hypergraph
bounded confidence model and showed the appearance of a scenario named as
‘opinion jumping’ in which individuals’ opinion can jump from one cluster of
opinions to another, which one doesn’t observe in dyadic connectivity.
Moreover, echo chambers are witnessed to emerge on hypergraphs with community
structure. Large hyperlinks are found to be playing decisive roles for the
consensus more than the small hyperlinks. Besides the computational
demonstrations, the scenarios are treated mathematically. Consensus dynamics
over higher-order networked systems can be investigated through the concept of
generalized Hodge Laplacians for the instances in which the weights for lower-
and higher-order interactions between simplices are different Ziegler et al.
(2021). Using the Hodge Decomposition, convergence can be analyzed and
thereafter with the techniques of algebraic topology the role of simplicial
complex homology can be studied. In fact, lower- and higher-order interactions
can be balanced to optimize consensus dynamics.

Figure 4: (a) Consensus value with respect to the parameter $p$ corresponding
to the scenarios of first-mover A, first-mover B and the aggregated dynamics,
in which the error bars reflect one standard deviation. (b) Time taken to
reach consensus as a function of the fraction $p$ that shows faster rate of
convergence for asymmetric configuration. (c) Time-scale $\tau$ dependence of
the consensus value for $p=0.5$. Decreasing $\tau$ helps in making the
convergence to the aggregated dynamics. Figure reproduced from the Ref.
Neuhäuser et al. (2021).

In the above, we have already discussed how temporal higher-order interaction
patterns modulate the discrete dynamics of social contagion Chowdhary et al.
(2021). Let us now elaborate how temporality in network connectivity affects
the continuous dynamics of consensus process developing in higher-order
networks Neuhäuser et al. (2021).
The nodal dynamics is described by the following set of equations:

$\begin{array}[]{lcl}\par\dot{x}_{i}=\displaystyle\sum_{j,k}A_{ijk}\mbox{exp}(l|x_{k}-x_{j}|)\Big{[}(x_{j}-x_{i})+(x_{k}-x_{i})\Big{]},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}i=1,2,\cdots,N.\par\end{array}$
(10)

where $A_{ijk}\in{\mathbb{R}}^{N\times N\times N}$ is the adjacency tensor
representing the interaction structure of the $3$-hypernetwork and the term
$\mbox{exp}(l|x_{k}-x_{j}|)$ is the scaling function that regulates the
impacts of $j$-th and $k$-th nodes on the $i$-th node. Then the temporal
network model of $3$-regular hypergraphs is constructed by defining a sequence
of adjacency tensors $A^{[1]},~{}A^{[2]},...$ representing the network
structures at different times with $\tau$ being the length of the time-periods
between any two successive adjacency tensors.

Let us then assume a network set-up with two (individually globally connected)
clusters (say, cluster-‘A’ and ‘B’) of same size ($N=10$ nodes) in which both
intra-cluster and inter-cluster hyperlink connections exist with the nodes in
cluster A (B) having the initial state $x_{A}(0)=1$ ($x_{B}(0)=0$). Further,
the clusters are connected via $20$ randomly placed hyper $3$-links in such a
way that $p\in[0,~{}1]$-fraction of $3$-links are oriented towards the cluster
A (i.e., lesser number of hyperlink nodes are part of the cluster A than of
the cluster B) and the rest of the $3$-links are oriented towards B. Then
three different schemes are studied, namely the first-mover A, first-mover B
and the aggregated scenario. To be precise, in case of first-mover A (first-
mover B), firstly for certain time all the A-majority (B-majority) subgroups
interact, then all the B-majority (A-majority) subgroups and then the entire
hypernetwork interacts. For the aggregated scenario, the hypernetwork remains
static and all the interactions take place concurrently.
Figure 4(a) portrays the value of consensus in terms of the node state
$x_{i}(t)$ as a function of the fraction $p$, averaged over $10$ simulations.
The consensus evolving in the hypernetwork tends towards the initial opinion
in cluster A or B for the aggregated scenario. Whenever $p=0$ ($p=1$), the
connecting $3$-links are oriented towards cluster B (A) making the initial
opinion of cluster A (B) prevail. This outcome turns out to be qualitatively
same in the first mover cases. As far as the convergence speed is concerned,
convergence is effectively faster for asymmetric initial opinion and when
orientation of the $3$-links and the first-mover group line up (cf. Fig.
4(b)). Further, Fig. 4(c) demonstrates the consensus dynamics with respect to
the time-scale $\tau$ for $p=0.5$. It is clear from the figure that the system
is more prone to converge to the aggregated dynamics whenever $\tau$ is small.

In the Ref. Horstmeyer and Kuehn (2020), the authors propose and study an
adaptive voter model under higher-order interactions, specifically on a
simplicial complex, that incorporates the influence of the important social
factor of peer-pressure. The rewiring rule of linking to agreeing nodes is
adopted while focusing mainly on the $2$-simplex framework. Peer-pressure
speeds up the transitions to both single-opinion and two-opinion states. Also,
this higher-order model can exhibit multiple time-scales in which the
$2$-simplices vanish before the exhaustion of the active links. In a recent
work DeVille (2021), a generalized dynamical model on simplicial complex of
several consensus and synchronization processes is proposed and analyzed. Many
behaviors are detected here for consensus dynamics that occur for dyadic
interactions and also the emergence of multistability in the steady states due
to this model is put forward.

###  IV.3 Evolutionary game dynamics

Figure 5:  $\rho-T_{2}$ phase diagrams depicting the frequency $F_{C}$ of
cooperators, for fixed values of $S_{2}=0.5$ for the first row and
$S_{2}=-0.5$ for the second row, with (a,b)
$T_{1}=T_{3}=0.8,~{}S_{1}=S_{3}=0.2$ that correspond to the Harmony game,
(c,d) $T_{1}=T_{3}=0.8,~{}S_{1}=S_{3}=-0.2$ that represent the Stag Hunt game,
(e,f) $T_{1}=T_{3}=1.2,~{}S_{1}=S_{3}=0.2$ that define the Snowdrift game, and
(g,h) $T_{1}=T_{3}=1.2,~{}S_{1}=S_{3}=-0.2$ associated to the Prisoners
Dilemma game. In the first row, Game2 is SD for $T_{2}\geq 1$ and H dilemma
when $T_{2}\leq 1$, while in the second row, Game2 is PD if $T_{2}\geq 1$ and
a SH dilemma for $T_{2}\leq 1$. Figure reproduced from the Ref. Guo et al.
(2021).

Cooperation Nowak (2006); Perc et al. (2017); Perc and Szolnoki (2010);
Szolnoki et al. (2014); Hilbe et al. (2014); Perc et al. (2013); Perc and
Szolnoki (2008) is the process in which individuals function together in
groups for mutual benefits, which is observed in diverse real systems
including microorganisms and human society. Significant attempts have been
made earlier in order to explore the evolutionary game dynamics in populations
subject to group interactions (cf. Perc et al. (2013) and references therein).
In evolutionary game theory, the higher-order interactions differ from
pairwise interactions in the derivation of payoffs. If one’s payoff in a
higher-order interaction, to some degree, is equivalent to the sum of payoffs
in interactions with each individual opponent, both higher-order and pairwise
interactions essentially are the same. Otherwise, if one’s payoff in his
neighborhood is nonlinear to the sum of payoffs in all pairwise interactions,
higher-order interactions lead to different dynamical processes. An approach
to capture the higher-order interaction is general multiplayer game, where one
player’s payoff is a function of his and all neighbors’ strategies (the Refs.
Li et al. (2014); Peña et al. (2016); Li et al. (2016); Zhou et al. (2018); Su
et al. (2019a, 2018, b) furnish nice strict analytical results on multiplayer
games). When the payoff function is nonlinear to the number of cooperative
neighbors, it presents the higher-order effects.

In particular, the review by Perc et al. Perc et al. (2013) clarifies how
larger group sizes can help in preserving cooperation in networks formed upon
dyadic interactions which are often insufficient to explain all the essence of
group interactions. Keeping this in mind, Burgio et al. Burgio et al. (2020)
came up with their work on diverse hypernetworks in pursuance of having
clearer perception of the development of cooperation in networked groups while
examining the evolution of cooperation in public goods game (PGG), and
demonstrated that group interactions can, indeed, enhance cooperation. The
method adopted to generate the hypernetworks preserve the dyadic projection
and the authors, in particular, deal with hypernetworks formed from the Holme-
Kim and the Dorogovtsev-Mendes models. Besides mean-field approximation for
homogeneous interactions, invasion analysis is presented for heterogeneous
structures explaining how increasing order of connections can cause higher
reciprocity. The developed reciprocity specifically owes to the adopted
mechanism that replaces some first-order $3$-cliques with second-order
triangles. The article also discusses how cooperative and non-cooperative
states can coexist subject to the modality of interaction structures.

Evolutionary dynamics of public goods game is also investigated in social
networks built upon higher-order interactions Alvarez-Rodriguez et al. (2021).
The study reveals that this game on uniform hypernetworks in which there is no
hyperdegree-hyperdegree correlations is consistent with the replicator
dynamics in the well-mixed regime. The article further incorporates
heterogeneity in both order and hyperdegree, and demonstrates how these
characteristic features affect the evolutionary game dynamics. This higher-
order network framework being capable of appropriately describing the group
structures, the study actually has been able to depict how synergy factors
depending on the group size result in critical scaling in the defection to
cooperation transition. Hierarchical hypernetworks are observed to impede
cooperation in a structured population. The network set-up is further employed
to collaboration datasets as well. Higher-order group ($3$-player with
$2$-player) interactions along with adaption are taken into account in order
to propose the adaptive simplicial Snow-drift game Schlager et al. (2021).
Adaptation in the network topology and the state of the system is assumed
which explores for both mathematical and numerical treatments that even under
the higher-order structural framework, the stability of the equilibrium points
remains unaltered. An evolutionary model of group choice dilemmas is proposed
and analyzed on hypernetworks where the decisions between a safe alternative
and a risky one are taken in different sized groups, which is capable of
explaining how opinion diffuses following an imitation process Civilini et al.
(2021). Further, an organized study for a different form of strategic
interaction of signaling games in populations subject to higher-order
structures, namely the dynamical evolution of honesty in the sender-receiver
game is presented very recently Kumar et al. (2021). Unlike the instance of
the sole presence of dyadic interactions, honesty has been witnessed to be
existent even under the temptation to lie. Also, moral strategy persists even
if lies favor the receiver at a cost to the sender. The evolutionary dynamics
is investigated in populations based upon the assumption of well-mixed
setting, in hyper-ring as well as in real-world hypernetworks.

A different approach is adopted very recently to model evolutionary game
dynamics for higher-order interactions among individuals, where apart from the
strategies of a focal player and one of the neighbors, strategies of other
neighbor(s) coming out of an indirect interactions also influence the game
dynamics Guo et al. (2021). Diverse social dilemmas with different Nash
equilibria being played over $1$\- and $2$-simplices are investigated
demonstrating that such a simplicial framework results in the appearance of
the non-dominant strategies and its coexistence with the dominant strategies.
Further, transition from dominant defection state to the state of cooperation
with respect to the higher-order structure is established.
The two-strategy (cooperation (C) and defection (D)) two-player game
configuration can be described by the following payoff matrix:

$\begin{array}[]{lcl}&C&D\\\ C&R&S\\\ D&T&P\end{array}$ (11)

Each player receives a payoff $R=1$ (Reward) under mutual cooperation and
$P=0$ (Punishment) for mutual defection on the agreement of the strategies.
Instead, if the players’ strategies disagree, the cooperator receives a payoff
$S\in[-1,1]$ (Sucker) whereas the defector receives $T\in[0,2]$ (Temptation).
Concerning the network formulation, initially starting with a fully connected
sub-network of $n_{0}(=5)$ nodes, in the next time-step $m(=1)$ new nodes are
added. These new nodes are linked to the endpoints of randomly chosen $m$
links, and thus $m$ new triangles are created in the sub-network. Reiterating
this step of addition of nodes, the final network of $N$ nodes is constructed.
The network thus formed exhibits power-law degree distribution with
exponential generalized-degree distribution Kovalenko et al. (2021b). A
fraction $\rho\in[0,1]$ of random triangles in the network are chosen to
characterize actual three-body ($2$-simplex) interactions whereas the
remaining fraction ($1-\rho$) of triangles represents three two-body
($1$-simplex) interactions. A strategy matrix $\tilde{S}=\\{s_{ij}\\}$ is also
defined that takes different values based on whether $i$-th and $j$-th nodes
cooperate, defect or do not interact. The accumulated payoff $\Pi_{i}$ of the
$i$-th node is then calculated as $\Pi_{i}=(1/k_{i})\sum_{j\in
N_{i}}\Pi_{i,(ij)}$, where $N_{i},~{}k_{i}$ and $\Pi_{i,(ij)}$ are
respectively the neighbourhood, degree and the total payoff obtained along the
link $(i,j)$ of the $i$-th node. Further,
$\Pi_{i,(ij)}=(1/k_{ij})\sum_{\tau\in\Delta}\Pi_{i,(ij),\tau}$, where the set
$\Delta$ comprises of the $k_{ij}$ triangles constituted by the link $(i,j)$.
Now, if $\tau$ is simply a sum of three $1$-simplices, then
$\Pi_{i,(ij),\tau}$ is obtained from Game1, the payoff values of which are
$S=S_{1},~{}T=T_{1}$ with $R=1,~{}P=0$. On the other hand, if $\tau$
characterizes a $2$-simplex, then assuming the other node to be the $k$-th
node that completes this simplex, $\Pi_{i,(ij),\tau}$ will be calculated from
Game2 ($S=S_{2},~{}T=T_{2}$ with $R=1,~{}P=0$) if $s_{ki}=s_{kj}$. Similarly,
the payoff will be obtained from Game3 ($S=S_{3},~{}T=T_{3}$ with
$R=1,~{}P=0$) if $s_{ki}\neq s_{kj}$. This way each $i$-th node obtains its
payoff $\Pi_{i}$ and subsequently updates (synchronously with others) its
strategy with probability $\mbox{Pr}=1/(1+e^{[(\Pi_{i}-\Pi_{\tilde{j}})/K]})$,
in which $\Pi_{\tilde{j}}$ is the accumulated payoff of the $\tilde{j}$-th
node.

Frequency of cooperation in the $\rho-T_{2}$ parameter plane is shown in Fig.
5 while considering Game1 and Game3 to be the same (i.e., with the same $S$
and $T$ values). Game1,3 are identified respectively by the Harmony (H), Stag
Hunt (SH), Snowdrift (SD) and Prisoner’s Dilemma (PD) games from the left to
the right columns of the figure. Moreover, for the upper (lower) row
$S_{2}=0.5$ ($S_{2}=-0.5$) is assumed so that the Game2 represents H (SH)
dilemma whenever $T_{2}\leq 1$ and SD (PD) game for $T_{2}\geq 1$. For the
left-most Harmony dilemma (cf. Figs. 5(a) and 5(b)), mutual cooperation being
the Nash equilibrium, for small values of $\rho$ cooperation is favored
irrespective of the specifics of Game2. With increasing $\rho$, the number of
three-body interactions increases and the Game2 dynamics starts to matter
which can be any of the considered four game dynamics. For $T_{2}\leq 1$,
Game2 is either H or SH that supports cooperation and hence $\rho$ values do
not matter much. However, whenever $T_{2}\geq 1$, Game2 is SD or PD and hence
the fraction of cooperators decreases. Interesting results start appearing
when Game1 and Game3 correspond to SH, SD and PD (cf. Figs. 5(c-h)). In
absence of simplicial interactions (i.e., $\rho=0$), defection is the dominant
strategy but a transition to cooperation takes place with increasing $\rho$
(implying increasing higher-order interactions) whenever $T_{2}<1$ (i.e.,
Game2 is either H or SH). More than the transition scenario for Game2 playing
the Harmony game (for which CC is the Nash equilibria), the transition to
cooperation (even when Game1 and Game3 represents PD) for the instance of
Game2 playing the SH game is noteworthy as CC and DD are the two pure Nash
equilibria for the SH game.

##  V Random walk and diffusion

With an aim to explore the dynamics of random walks on networks beyond
pairwise interactions, a family of random walks on top of simplicial complexes
is defined by Markov chain Mukherjee and Steenbergen (2016). A relation
between the chain’s stationary distribution and the harmonics of the Hodge
Laplacian is further established. In this context, from the higher-order
homology groups and the role of orientation of the simplexes to the concept of
neighbors of the higher-order simplexes are discussed in detail. Also in the
Ref. Parzanchevski and Rosenthal (2017), the authors have gone through the
concept of random walks on simplicial complexes. Diffusive processes in the
form of a family of random walks on heterogeneous higher-order networks
(hypernetworks) is brought forward while giving an analytical treatment with a
general proposition for the stationary distribution of the walk Carletti et
al. (2020b). A comparison analysis of this distribution to that corresponding
to the traditional random walk over the associated projected network is also
provided. Both model and real-world hypernetworks are treated in order to
explore the proposed random walk dynamics. Specifically, from the applications
in node ranking and centrality measure to classification tasks are explained.
The process of diffusion on simplicial complexes is studied in the Ref. Schaub
et al. (2020) that proposes a normalized Hodge Laplacian matrix and
demonstrates how it is associated to random walk dynamics on simplicial
complexes, specifically on edges. The approach is further utilized in
developing embeddings of edge flows and trajectory data and also the
generalization of personalized PageRank for edges.

Figure 6:  (a) The hypernetwork composed of $16$ nodes and $15$ hyperlinks,
specifically there exists $8$ hyperlinks of size $2$ (blue), $4$ hyperlinks of
size $4$ (red), $2$ hyperlinks of size $4$ (yellow) and $1$ large hyperlink of
size $16$ (grey). (b) The number of communities as a function of the Markov
time for different values of the size bias parameter $\sigma$. (c) Number of
communities for simultaneous variations of Markov time and $\sigma$. (d) $1/Y$
with respect to Markov time and $\sigma$. Figure reproduced from the Ref.
Carletti et al. (2021).

A class of random walks on hypernetworks is defined in such a way that the
random walk process shows propensity towards hyperlinks of high or low size
based upon the variation of a single size bias parameter Carletti et al.
(2021). The resulting dynamics is, in fact, capable of describing diverse
hypernetwork projection on networks for different values of this bias
parameter. These projections can further vary depending on this parameter and
this dissimilation is examined via its effect on community structure while
developing the formulation of Markov stability on hypernetworks. Let us assume
a hypernetwork $H(V,E)$ with $V=\\{V_{1},V_{2},...,V_{N}\\}$ and
$E=\\{E_{1},E_{2},...,E_{M}\\}$ being the sets of $N$ nodes and $M$ hyperlinks
respectively. The incidence matrix associated to the hypernetwork is the
following:

${e}_{i\alpha}=\begin{cases}1,&\text{if}~{}V_{i}\in E_{\alpha},\\\
0,&\text{otherwise}.\end{cases}$

The $M\times M$ hyperlink matrix is defined as $\bf{B}={\bf{e}}^{t}{\bf{e}}$
in which ${\bf{e}}^{t}$ is the transpose of ${\bf{e}}$ and the elements
$B_{\alpha\beta}$ accounts for the number of nodes in $E_{\alpha}\cap
E_{\beta}$. The agents are then placed on the nodes that hop at discrete
times, and the weighted adjacency matrix is described as
${K_{ij}}^{[\sigma]}=\sum_{\alpha}{(B_{\alpha\alpha}-1)}^{\sigma}{e}_{i\alpha}{e}_{j\alpha},~{}\sigma\in\mathbb{R},~{}\forall
i\neq j$ and ${K_{ii}}^{[\sigma]}=0$, from which the transition probabilities
are computed as ${T_{ij}}^{[\sigma]}={K_{ij}}^{[\sigma]}/(\sum_{m\neq
i}{K_{im}}^{[\sigma]}),~{}\forall i\neq j$ and ${T_{ii}}^{[\sigma]}=0$. This
implies how the hyperlinks of large (small) size governs the random walk
dynamics for large (negative) values of the size bias parameter $\sigma$. A
continuous random walk on top of the hypernetwork is then delineated as

$\begin{array}[]{lcl}\par\dot{p}_{i}=\displaystyle\sum_{j}p_{j}{T_{ji}}^{[\sigma]}-\displaystyle\sum_{j}p_{i}{T_{ij}}^{[\sigma]},~{}~{}~{}~{}~{}~{}~{}~{}i=1,2,\cdots,N,\par\end{array}$
(12)

in which $p_{i}~{}(p_{i}(t))$ is the probability of the agent being on the
$i$-th node at time $t$.
A generalization of the formulation of Markov stability Delvenne et al. (2010)
is further adopted in order to find the communities in the hypernetwork, by
assuming a partition of the nodes into $c$ non-overlapping communities,
captured by the indicator matrix ${\bf C}_{N\times c}$ where $C_{ij}$ takes up
the value $1$ when the $i$-th node belongs to the $j$-th community, and $0$
otherwise. The Markov stability $r(t;{\bf C})$ then measures the goodness of
${\bf C}$ as a function of the time horizon of the random walk (please see the
Ref. Carletti et al. (2021) for detailed definitions of Markov Stability).

We then consider a typical hierarchical hypernetwork model that comprises of
$16$ nodes and $15$ hyperlinks (cf. Fig. 6(a)), the projection of which is a
complete network with $16$ nodes. As can be seen, there exists $8$ hyperlinks
each of size $2$, $4$ hyperlinks containing $4$ nodes, $2$ hyperlinks with $8$
nodes and lastly the hyperlink containing all the $16$ nodes. In order to find
the communities, Markov stability is optimized with respect to the Markov time
$t$ for different values of $\sigma$ in Fig. 6(b). The plots demonstrate the
hierarchical structure efficiently while determining all the communities of
decreasing sizes as the Markov time increases. Then having a look at the
entries of ${\bf K}^{[\sigma]}$ one can calculate
$\lim_{\sigma\to+\infty}{T_{ij}}^{[\sigma]}=1/15,~{}\forall
i,j\in\\{1,2,...,16\\}$, with $\lim_{\sigma\to-\infty}{T_{12}}^{[\sigma]}=1$
and $\lim_{\sigma\to-\infty}{T_{1j}}^{[\sigma]}=0$ for all other $j$. The
other values follow from the symmetry in the structure of the hypernetwork.
Figure 6(c) depicts the number of communities as a function of Markov time and
$\sigma$. The algorithm yields the partition of $16$ communities whenever
$\sigma$ is positive and large, and the two communities of size $8$ for
smaller positive $\sigma$. The intermediate communities are realized mainly
for the negative values of $\sigma$, and the algorithm finally identifies the
communities of size $2$. Furthermore, the Simpson diversity index $Y$ is
computed in order to get the size of the communities, where
$Y=\sum_{i=1}^{Q}{S_{i}}^{2}/N^{2}$, in which $S_{i}$ is the number of nodes
present in the $i$-th group. $Y$ varies from $1$ (when all the nodes are in a
single group) to $1/N$ (if there exists $Q=N$ groups, each comprising of a
single node), whereas $Y\sim 1/Q$ whenever the nodes are uniformly distributed
among the $Q$ groups. In Fig. 6(d), the value of $1/Y$ is presented for
simultaneous variations in Markov time and $\sigma$. It is discernible that
$1/Y=2$ is associated to the $Q=2$ communities of size $8$ as here
$Y=2\times{(8/16)}^{2}=1/2$. Similarly, $1/Y=4$ corresponds to the $Q=4$
communities of size $4$, and $1/Y=8$ represents $Q=8$ communities of size $2$.

Spectral properties of higher-order Laplacian associated to the simplicial
complex model known as ‘network geometry with flavor’ (as mentioned above) are
studied in the Ref. Torres and Bianconi (2020). It is demonstrated that these
higher-order up- and down-Laplacians can have a finite spectral dimension that
depends on the order of the Laplacian. Moreover, this higher-order structure
affects the diffusion dynamics taking place on this, with the spectral
dimensions having meaningful influence on the return-time probability of the
concerned diffusion process. Furthermore, relation between the geometry of a
network and diffusion dynamics is unraveled Millán et al. (2021) based on the
investigation of two family of models, namely NGF and ‘Short-Range Triadic
Closure (STC)’. Thus far, many generalizations of different random walk models
for higher-order interactions have been put forward, as discussed above. In
order to explore which combination of model and network representation is best
for resolving different research issues associated with diverse hypernetwork
data, the authors of the Ref. Eriksson et al. (2021) derive unipartite,
bipartite and multilayer network representations of hypernetwork flows with
identical node-visit rates for the same random-walk model. Information-
theoretic and flow-based community detection algorithm Infomap is utilized to
investigate how different hypernetwork models and network representations
alter the number, size, overlap of the detected communities.

##  VI Summary and Future Prospects

The variants of interactions in networked systems essentially regulate the
dynamical processes taking place on them. It has been demonstrated in many
ways that from synchronization to spreading dynamics, the complex interaction
structure strongly decides the destiny of the concerned complex systems.
However, the existing literature predominantly has dealt with pairwise
networked systems, even though the underlying processes are better represented
on top of higher-order structures. Only in the recent times, structural and
dynamical properties of higher-order networks have become a rapidly developing
research field owing to its potential efficacy in describing numerous complex
instances from social processes to neuroscience. In this review article, we
have furnished a review of recent research endeavours that study various
dynamical processes on networks beyond dyadic interactions. We come up with
our investigation with a motive to clarify how diverse can the impact on
different phenomena be while higher-order connections are taken into account.
The fundamental concepts of higher-order networks are briefly discussed in
Section II. In Section III, we started with explaining how the phenomenon of
synchronization gets affected due to the presence of higher-order connections
in the system. Social processes staring from contagion dynamics, consensus
formation to evolution of cooperation are examined in Sections IV A, IV B and
IV C, respectively. The influence of higher-order interactions on random walk
and diffusion dynamics are studied in Section V.

Even though a number of significant developments have been made in view of
analyzing the role of higher-order interactions on dynamical processes, we
would still like to bring forward some of the noteworthy routes of further
research. For instance, there are enough scopes to contribute to the
understanding of the temporal higher-order networks. From its structural
intricacies to the analysis of different dynamics on time-varying higher-order
structures is highly worth of attention. The same applies to the
interdependent network frameworks, specifically the multilayer/multiplex
structures along with higher-order interactions, inspection of which should be
envisaged as a promising research direction. Although there exists important
attempts concerning synchronization in networks beyond pairwise connectivity,
the detailed analysis of cluster synchrony is missing. The specific aspect of
chimera state is mostly untouched so far whereas these patterns have high
resemblance to several neuronal developments Majhi et al. (2019b). So, the
study of chimera states in simplicial networks would be an excellent candidate
for future research. Also, the study of collective behaviors of swarmalator
systems with higher-order interactions could be quite interesting. Moreover,
the dynamical scenarios arising from the increased complexity due to
adaptivity Gross and Blasius (2008) in the higher-order systems require much
more attentive study.

## Acknowledgments

S.M. acknowledges support from the Planning and Budgeting Committee of
Israel’s Council for Higher Education. M.P. acknowledges funding from the
Slovenian Research Agency (Grant nos. P1-0403 and J1-2457).

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