text:
\section{Introduction}
\par In the last decade, the study on multilayer networks \cite{boccaletti2014structure,bianconi2018multilayer} has become a {prosperous} research area due to its ability in describing many real-world systems \cite{mobility,power,air,social,ecology} with heterogeneous interacting layers. Another interesting generalized  network structure is hypernetworks \cite{sorrentino_njp,sorrentino2007network,DBLP:conf/iclr/HaDL17} where an ensemble of nodes are simultaneously interconnected through various independent connection topologies. For instance, in neuronal network, the neurons are interconnected via chemical and electrical pathways \cite{rakshit2018emergence}. The precise illustrative power of these generalized structures naturally comes up with increasing analytical complication to scrutinize stability of collective behaviors emerge on them. Synchronization \cite{synchronization1,synchronization2,synchronization3,synchronization4,skardal2014optimal}, one such collective phenomena whose stability analysis is very significant as it generally does not emerge if this is not stable.

\par To execute systematic stability analysis of {synchronization} state in large networks \cite{restrepo2006synchronization,restrepo2005onset}, the crucial step is to split up whole space of the variational equation into parallel and transverse subspaces, and evaluate the maximum transverse Lyapunov exponent to find out if perturbations associated with transverse subspace will die out or not. For stability of complete synchronization in large scale network of identical oscillators, the master stability function (MSF) \cite{pecora1998master} approach provides the essential tool, which further broaden in various ways \cite{sun2009master,restrepo2005onset,nishikawa2006synchronization}. In case of hypernetworks, the stability of synchronization has been analyzed by dimensionality reduction \cite{sorrentino_njp} of master stability equation through simultaneous block diagonalization of coupling matrices \cite{sbd_sorrentino}. To scrutinize stability of coherent state in multilayer networks a few new procedures has been provided recently \cite{belykh2019synchronization,multiplex_decom,rakshit2021relay}. However, all of these studies have been performed for static network structures, while most of the real-world systems \cite{onnela2007structure,pastor2004evolution,skufca2004communication,wasserman1994social,motter2013spontaneous} are time-varying \cite{ghosh2022synchronized}, meaning that the connection topologies between generic agents vary over time \cite{holme2012temporal,taylor2010spontaneous}. In {Ref.} \cite{stilwell2006sufficient}, the authors analyzed stability of synchronous state in temporal networks through fast switching stability procedure. Other than fast switching approach, the stability of coherent state in time-varying networks can be performed by connection graph stability method \cite{belykh2004blinking,belykh2004connection}. Very recently, Zhang et al. \cite{zhang2020symmetry} have proposed an all-round simultaneous block diagonalization framework that enables the stability analysis of synchronization patterns for monolayer, multilayer, temporal network structures \cite{zhang2021unified}. The main theme of present work is to analyze the stability of synchronization in generic temporal multilayer hypernetworks, particularly intralayer synchronization \cite{intra1,rakshit2020intralayer}. Until now, the study of synchronization on temporal multilayer networks have been done mainly on case-by-case basis, either by taking multiplex network structure \cite{rakshit2018synchronization,rakshit2020intralayer}, or with assumption of certain type of intralayer, and interlayer coupling functions (mostly diffusive) \cite{rakshit2017time,rakshit2020intralayer}.

\par Here we propose a general mathematical framework to investigate intralayer synchronization phenomena on temporal multilayer hypernetwork. In particular, we consider a group of generic units, represented by nodes of the multilayer hypernetwork, interacting within and between the layers through completely arbitrary linear or nonlinear coupling functions with only consideration of static interlayer links. In such broad circumstances, we uncover that the intralayer synchronous state emerges as a stable invariant solution. Besides, we derive the necessary condition for emergence of stable intralayer coherent state using fast switching stability method and simultaneous block diagonalization technique. We show that the temporal multilayer hypernetwork achieves stable intralayer synchronization state for adequately fast switching whenever the corresponding time-averaged network does. Further, we show the master stability equation optimally decouples into independent transverse modes if one time-average Laplacian matrix commutes with all other time-average Laplacians and interlayer adjacencies. On the other hand, through the simultaneous block diagonalization technique, we show that the parallel and transverse modes can be separated without considering time-average network structure, and furthermore the dimensionality reduction of master stability equation can be possible even if the coupling matrices are not commutative. Lastly, all of our analytical results are verified with the numerical simulations of paradigmatic and real-world neuronal networked systems.

\par The rest of this article is organized as follows. In Sec. \ref{model}, we propose the generalized model for time-varying multilayer hypernetwork. The analytical results associated with intralayer synchronization state is illustrated in Sec. \ref{Analytical Results}. In Sec. \ref{invariance}, the invariance condition for the coherent state is derived. Subsequently, in Sec. \ref{Stability}, we establish the condition for stability of synchronous solution. The corresponding numerical results are provided in the subsequent Sec. \ref{numerical}. Finally, we sum up all our results and conclude in Sec. \ref{conclusion}.

\section{Generalized Mathematical Model: Multilayer Hypernetworks}\label{model}
\par We consider a multilayer network consists of two layers, each composed of $N$ generic units. In each layer, the nodes are simultaneously interconnected through two or more topologically different connections, called tiers. In particular, we assume $N$ nodes in each layer are collaborated via $M$ distinct tiers characterizing various types of connections between the dynamical units. We assume that the equations of motion defining the dynamics of our multilayer hypernetwork can be represented by the following sets of equations,

\begin{equation}\label{eq.1}
\begin{array}{lcl}
\dot{\bf x}_i(t) = F_1({\bf x}_i)+\sum\limits_{\beta=1}^{M}\epsilon_\beta\sum\limits_{j=1}^{N}{\mathscr{A}_{ij}^{[1,\beta]}}(t) G_{\beta}^{[1]}({\bf x}_i,{\bf x}_j) \\ ~~~~~~~~~~~~~~~~~~~~ + \lambda \sum\limits_{j=1}^{N}{\mathscr{B}_{ij}^{[1]}}H_1({\bf x}_i,{\bf y}_j), \\
\dot{\bf y}_i(t) = F_2({\bf y}_i)+\sum\limits_{\beta=1}^{M}\epsilon_\beta\sum\limits_{j=1}^{N}{\mathscr{A}_{ij}^{[2,\beta]}}(t) G_{\beta}^{[2]}({\bf y}_i,{\bf y}_j) \\~~~~~~~~~~~~~~~~~~~~ + \lambda \sum\limits_{j=1}^{N}{\mathscr{B}_{ij}^{[2]}}H_2({\bf y}_i,{\bf x}_j), \hspace{10 pt} i=1,2,...,N.
\end{array}
\end{equation}
Here $\mathbf{x}_i(t)$ and $\mathbf{y}_i(t)$ are the $d$-dimensional state vectors illustrating the dynamics of generic node $i$, $F_{1,2} : \mathbb{R}^d \rightarrow  \mathbb{R}^d$ represent the isolated node dynamics presumed to be identical for each generic agent of a particular layer, but different for different layers.  $G_{\beta}^{[l]}  : \mathbb{R}^d\times \mathbb{R}^d \rightarrow  \mathbb{R}^d$ and $H_{1,2} : \mathbb{R}^d\times \mathbb{R}^d \rightarrow  \mathbb{R}^d$ are continuously differentiable functions, representing the  output vectorial functions within each layer for tier-$\beta$ and  between the layers, respectively. Here, we assume the coupling functions to be arbitrary nonlinear functions. Further, the intralayer coupling functions conforming to tier-$\beta$ can be different for different layers and the interlayer functions are non-symmetric, i.e.,  $H_l(x, y)$ is different from $H_l(y, x)$, $l=1,2$. The real-valued parameters $\epsilon_ \beta$ and $\lambda$ describe the intralayer coupling strength for the tier-$\beta$ and interlayer coupling strength, respectively.

\par Moreover, the adjacency matrix $\mathscr{A}^ {[l, \beta]}(t)$ recounts the intralayer network structure associated with the tier-$\beta$ of layer-$l$ at a time instance $t$, where $\mathscr{A}_{ij}^{[l,\beta ]} (t) = 1$ if the $i^{\mbox{th}}$ and $j^{\mbox{th}}$ nodes are interconnected in tier-$\beta$ of layer-$l$ at time t and zero otherwise. The layer-wise connections of the multilayer hypernetwork are varying in time through the probabilistic modification of the whole intralayer network with a rewiring frequency $f$. Smaller $f$ indicates that the intralayer links are almost static, while large $f$ corresponds to very fast swapping of the links over time. Here, we assume that the network structure of tier-$\beta$ for both the layers are equivalent. Although, their corresponding adjacency matrices will not usually be the same because of the {kinematic} behavior of the links of each individual tier. The adjacency matrices $\mathscr{A}^{[l,\beta ]}(t)$ define {the associated} zero-row sum Laplacian matrices {$\mathscr{L}^ {[l, \beta]}(t)$}, whose diagonal entries {are $\mathscr{L}_{ii}^{[l,\beta ]} (t)=\sum_{j=1}^{N}\mathscr{A}_{ij}^{[l,\beta ]}(t)$,} and off-diagonal entities are obtained by just considering negatives of the non-diagonal entries in $\mathscr{A}^{[l,\beta ]}(t)${, i.e., $\mathscr{L}_{ij}^{[l,\beta ]}(t)=-\mathscr{A}_{ij}^{[l,\beta ]}(t)$ for $i\ne j$}. On the other hand, the interlayer connections between the nodes of two layers are characterized by the interlayer adjacency {matrices} $\mathscr{B}^{[l]}$, $(l = 1, 2)$. $\mathscr{B}_{ij}^{[l]}= 1 $ when there is a pairwise link between the $i^{\mbox{th}}$ node of one layer and the $j^{\mbox{th}}$ node of the other layer, and zero otherwise. The interlayer links are {considered to be} static over time. We define the intralayer degree of the $i^{\mbox{th}}$ node of tier-$\beta$ in layer-$l$ as $d_i^{[l,\beta]}(t)=\sum_{j=1}^{N}\mathscr{A}_{ij}^{[l,\beta]}(t)$ and the interlayer node degree in layer-$l$ is denoted by $e_i^{[l]} =\sum_{j=1}^{N}\mathscr{B}_{ij}^{[l]}$. This is the most comprehensive system for a temporal multilayer hypernetwork we can consider, as the coupling functions and the adjacency matrices does not fall under extra limitations.

\begin{figure}[ht]
\centerline{\includegraphics[scale=0.225]{fig1.png}}
\caption{{\bf Schematic representation of temporal multilayer hypernetwork.} The left and right panel, respectively describes the structure of multilayer hypernetwork at two time instances $t=t_1$ and $t=t_2$. Solid black and yellow lines in each layer represents the interconnection between nodes through two distinct tiers. Dotted black lines represents static interlayer connections.}\label{fig.1}
\end{figure}

\par Figure \ref{fig.1} represents a schematic sketch of temporal multilayer hypernetwork consisting of two layers, each composed of $N=8$ units and $M=2$ distinct tiers. The connection between nodes through different tiers are shown in two different colors, solid black lines corresponds to one tier, and solid yellow for another one. The left and right panels delineates the structure of the multilayer network for two different time instances $t=t_1$ and $t=t_2$. Here, the connections between nodes of two different layers are static over time, which are shown in dotted black line.

\par Throughout the following section, our primary target is to derive analytically the necessary conditions for which the intralayer coherent state of the multilayer hypernetwork \eqref{eq.1} is stable.

\section{Analytical Results}\label{Analytical Results}
\subsection{Invariance of Intralayer Synchronization }\label{invariance}
\par Intralayer synchronization state emerges in the multilayer network \eqref{eq.1} if each node in individual layers evolves synchronously with other nodes of the same layer. Mathematically, there exists two solutions $\mathbf{x}_s(t),\mathbf{y}_s(t) \in \mathbb{R}^d$ such that,
\begin{equation*}
\begin{array}{l}
\norm{\mathbf{x}_i(t)-\mathbf{x}_s(t)} \to 0, ~~ \mbox{and} \\  \norm{\mathbf{y}_i(t)-\mathbf{y}_s(t)} \to 0 ~~ \mbox{as}~~ t \to 0 ~~ \mbox{for} ~~ i=1,2,\dots,N.
\end{array}
\end{equation*}
Then the associated synchronization manifold is defined by,
\begin{equation*}
\begin{array}{l}
\mathcal{S}=\{(\mathbf{x}_s(t),\mathbf{y}_s(t)) \subset \mathbb{R}^{2d} : \mathbf{x}_i(t)=\mathbf{x}_s(t), ~~ \mbox{and} \\ \mathbf{y}_i(t)=\mathbf{y}_s(t) ~~ \mbox{for} ~~ i=1,2,\dots,N ~~ \mbox{and} ~~ t \in \mathbb{R}^+ \}.
\end{array}
\end{equation*}
If the coupling functions are confined to synchronization solutions, i.e.,
\begin{equation*}
	\begin{array}{l}
	G_{\beta}^{[1]}({\bf x}_s,{\bf x}_s)=0, G_{\beta}^{[2]}({\bf y}_s,{\bf y}_s)=0, \; \mbox{and}\; \\ H_1({\bf x}_s,{\bf y}_s)=0,  H_2({\bf y}_s,{\bf x}_s)=0,
	\end{array}
\end{equation*}
then the existence and invariance of an intralayer coherent state will be guaranteed. Furthermore we consider arbitrary sort of coupling functions. In this case, as the network structure varies with respect to time, the existence and invariance condition on the network topologies for the emergence of intralayer synchronization is a challenging problem.

\par Suppose the multilayer hypernetwork begins to evolve with the intralayer coherence solution at a time instance $t = t_0$, then $\mathbf{x}_i(t_{0})=\mathbf{x}_s{(t_{0})}$ and $\mathbf{y}_i(t_{0})=\mathbf{y}_s{(t_{0})}$, for $i=1,2,\dots,N$. Then the evolution of the $i^{\mbox{th}}$ node for the two layers at $t = t_0$ can be written from Eq.\eqref{eq.1} in terms of node degrees as,
\begin{equation}\label{eq.2}
\begin{array}{l}
\dot{\bf x}_i(t_{0})=F_1({\bf x}_s)+\sum\limits_{\beta=1}^{M}\epsilon_\beta d_i^{[1,\beta]}(t) G_{\beta}^{[1]}({\bf x}_s,{\bf x}_s)  \\ ~~~~~~~~~~~~~+ \lambda e_{i}^{[1]} H_1({\bf x}_s,{\bf y}_s), \\
\dot{\bf y}_i(t_{0})=F_2({\bf y}_s)+\sum\limits_{\beta=1}^{M}\epsilon_\beta d_i^{[2,\beta]}(t) G_{\beta}^{[2]}({\bf y}_s,{\bf y}_s)  \\ ~~~~~~~~~~~~~+ \lambda e_{i}^{[2]}H_2({\bf y}_s,{\bf x}_s).
\end{array}
\end{equation}
To sustain intralayer synchronization solution, both the layers should advance on the same time evolution. Therefore, the velocities of any two distinct nodes in layer-1 should be identical, i.e.,
\begin{equation*}
\begin{array}{l}
\dot{\mathbf{x}}_i(t_{0}) = \dot{\mathbf{x}}_k(t_{0}), \;\; {\mbox{for any}~i\ne k,~\mbox{and}}~i,k=1,2,\dots,N.
\end{array}
\end{equation*}
Since the intralayer and interlayer coupling functions are arbitrary, this yields for layer-1,
\begin{equation*}
\begin{array}{l}
d_{i}^{[1,\beta]}(t)=d_{k}^{[1,\beta]}(t)$ \; \mbox{and} \; $e_{i}^{[1]}=e_{k}^{[1]}.
\end{array}
\end{equation*}
Similarly for layer-2,
\begin{equation*}
\begin{array}{l}
d_{i}^{[2,\beta]}(t)=d_{k}^{[2,\beta]}(t)$ \; \mbox{and} \; $e_{i}^{[2]}=e_{k}^{[2]}.
\end{array}
\end{equation*}
{\it Hence for the synchronization solution to be invariant the node degree of each dynamical unit for tier-$\beta $ in individual layers are identical and also the interlayer degree of all the nodes is equal for each layer.}

\par We further consider that the in-degree of nodes associated with tier-$\beta$ is constant in time, i.e.,
\begin{equation}{\label{eq.3}}
\begin{array}{l}
\sum\limits_{j=1}^{N}{\mathscr{A}}_{ij}^{[l,\beta]}(t)=d^{[\beta]}, l=1,2 \; \;\mbox{and} \;\; \beta=1,2,...,M.
\end{array}
\end{equation}

\par Hence the synchronous solutions $(\mathbf{x}_s,\mathbf{y}_s)$ evolves according to the following equations,
\begin{equation}\label{eq.4}
\begin{array}{lcl}
\dot{\bf x}_s = F_1({\bf x}_s)+\sum\limits_{\beta=1}^{M}\epsilon_\beta d^{[\beta]} G_{\beta}^{[1]}({\bf x}_s,{\bf x}_s) + \lambda e^{[1]}H_1({\bf x}_s,{\bf y}_s),\\
\dot{\bf y}_s = F_2({\bf y}_s)+\sum\limits_{\beta=1}^{M}\epsilon_\beta d^{[\beta]} G_{\beta}^{[2]}({\bf y}_s,{\bf y}_s) + \lambda e^{[2]}H_2({\bf y}_s,{\bf x}_s).
\end{array}
\end{equation}

\subsection{Stability Analysis}\label{Stability}
\par The intralayer synchronization occurs when the synchronous manifold $\mathcal{S}$ is stable under small perturbation in the transverse subspace. We consider small perturbation of the $i^{\mbox{th}}$ node around synchronous state, i.e.,
\begin{equation*}
\begin{array}{l}
\delta\mathbf{x}_i = \mathbf{x}_{i} - \mathbf{x}_{s} \;\mbox{and}\; \delta\mathbf{y}_i = \mathbf{y}_{i} - \mathbf{y}_{s},
\end{array}
\end{equation*}
and execute linear stability analysis of Eq.\eqref{eq.1}. Therefore, the linearized equation can be expressed in terms of stack vectors as,
\begin{equation}\label{eq.5}
\begin{array}{l}
\delta\dot{\mathbf{x}}=I_{N}{\otimes}JF_1(\mathbf{x}_s)\delta\mathbf{x}+\sum\limits_{\beta=1}^{M}\epsilon_{\beta} d^{[\beta]}[I_N{\otimes}JG_{\beta}^{[1]}(\mathbf{x}_s,\mathbf{x}_s)+\\~ I_N{\otimes}DG_{\beta}^{[1]}(\mathbf{x}_s,\mathbf{x}_s)]\delta\mathbf{x} -\sum\limits_{\beta=1}^{M}\epsilon_{\beta}\mathscr{L}^{[1,\beta]}(t){\otimes}DG_{\beta}^{[1]}({\bf x}_s,{\bf x}_s)\delta\mathbf{x}\\~~~~ +\lambda[e^{[1]}I_{N}{\otimes}JH_1(\mathbf{x}_s,\mathbf{y}_s)\delta\mathbf{x}+\mathscr{B}^{[1]}{\otimes}DH_1(\mathbf{x}_s,\mathbf{y}_s)\delta\mathbf{y}], \\\\

\delta\dot{\mathbf{y}}=I_{N}{\otimes}JF_2(\mathbf{y}_s)\delta\mathbf{y}+\sum\limits_{\beta=1}^{M}\epsilon_{\beta} d^{[\beta]}[I_N{\otimes}JG_{\beta}^{[2]}(\mathbf{y}_s,\mathbf{y}_s)+\\~ I_N{\otimes}DG_{\beta}^{[2]}(\mathbf{y}_s,\mathbf{y}_s)]\delta\mathbf{y} -\sum\limits_{\beta=1}^{M}\epsilon_{\beta}\mathscr{L}^{[2,\beta]}(t){\otimes}DG_{\beta}^{[1]}({\bf y}_s,{\bf y}_s)\delta\mathbf{y} \\~~~~ +\lambda[e^{[2]}I_{N}{\otimes}JH_2(\mathbf{y}_s,\mathbf{x}_s)\delta\mathbf{y}+\mathscr{B}^{[2]}{\otimes}DH_2(\mathbf{y}_s,\mathbf{x}_s)\delta\mathbf{x}],
\end{array}
\end{equation}
where,
\begin{equation*}
\begin{array}{l}
\delta\mathbf{x}(t)=[\delta\mathbf{x}_{1}(t)^{tr}, \delta\mathbf{x}_{2}(t)^{tr},\dots,\delta\mathbf{x}_{N}(t)^{tr}]^{tr}, \; \mbox{and} \\\\ \delta\mathbf{y}(t)=[\delta\mathbf{y}_{1}(t)^{tr}, \delta\mathbf{y}_{2}(t)^{tr}, ..., \delta\mathbf{y}_{N}(t)^{tr}]^{tr}.
\end{array}
\end{equation*}
$J$ and $D$ are the Jacobian derivative operators corresponding to the first and second variables, respectively.

\par These linearized set of variational equations contains two components: one attributing for the motion along intralayer synchronization manifold, called parallel modes, and other describing the modes transverse to the manifold, called transverse modes. The necessary condition for synchronization state to be stable requires all the transverse modes must converge to zero in time. To perform the linear stability analysis one then needs to decouple the variational equation into parallel and transverse mode and examine whether the latter ones die out or not.  To accomplish stable synchronization state for static network, the classical MSF approach appropriately decouples the high-dimensional variational equation to perturbation mode independent low-dimensional equations through a suitable coordinate transformation which completely diagonalizes the coupling matrices. But for the temporal network scenario, as the Laplacian matrices changes with time, it accounts for many non-commuting matrices. Since, non-commuting matrices are not simultaneously diagonalizable, we can not directly apply the classical MSF formalism to these circumstances. For this reason, we review two different well-known techniques for stability analysis of our temporal multilayer hypernetwork. One is based on the fast switching stability technique, for which time-averaged networks are considered to bring down the problem of stability analysis in MSF form and the other one follows simultaneous block diagonal (SBD) approach, which simultaneously decouples an arbitrary set of symmetry matrices into finest block diagonalized form and as a result decouples the variational equation into optimal form without considering further conditions like time-averaged network formation.

\subsubsection{Fast switching approach} \label{fast_switching}
\par Fast switching \cite{stilwell2006sufficient,belykh2004blinking,belykh2004connection} suggests that the change of network structure with time is much faster as compared to the evolution of oscillators coupled to each other. To perform the stability analysis we convert the variational equation \eqref{eq.5} to a suitable time-averaged form where the intralayer topologies corresponding to each tier are converted into time-averaged structures through the transformation of time-varying Laplacians into static time-averaged ones. In this regard, we recall the subsequent Lemma proposed by Stillwell et al. \cite{stilwell2006sufficient} regarding fast switching stability convention.
\begin{lemma}\label{lemma_1}
If there exits a time average matrix $\bar{M}$ of the matrix valued function $M(t)$ such that
\begin{equation*}
\begin{array}{l}
\bar{M}=\frac{1}{T} \int_{t}^{t+T}{M(z)} dz,~\forall~t \in \mathbb{R}^{+},
\end{array}
\end{equation*}
and for some constant T, then for fairly fast switching the system,
\begin{equation}{\label{eq.6}}
\begin{array}{l}
\dot{\mathbf{w}}(t)=[A(t)+M(t)] \mathbf{w}(t),\hspace{10pt}\mathbf{w}(t_0)=\mathbf{w}_0,\hspace{10pt}t\ge t_0,
\end{array}
\end{equation}
will be uniformly asymptotically stable whenever the time average system,
\begin{equation}{\label{eq.7}}
\begin{array}{l}
\dot{\bar{\mathbf{w}}}(t)=[A(t)+\bar{M}] \bar{\mathbf{w}}(t), \hspace{10pt}\bar{\mathbf{x}}(t_0)=\bar{\mathbf{x}}_0, \hspace{10pt}t \ge t_0,
\end{array}
\end{equation}
is also uniformly asymptotically stable.
\end{lemma}
\par Using the above convention, we establish that the time-varying multilayer hypernetwork \eqref{eq.1} obtains stable intralayer synchronization state whenever the corresponding time-averaged system shows asymptotically stable {intralayer synchronization} state (The proof is illustrated in Appendix \ref{time_average}). As the network structures of tier $\beta$ for both the layers are equivalent, their corresponding time-averaged adjacency and Laplacian matrices are identical. Then for an arbitrary real constant T,
\begin{equation*}
\begin{array}{l}
\frac{1}{T} \int_{t}^{t+T} \mathscr{A}^{[l,\beta]}(z) dz = \bar{\mathscr{A}^{[\beta]}}, \;\; \mbox{and} \\
\frac{1}{T} \int_{t}^{t+T} \mathscr{L}^{[l,\beta]}(z) dz = \bar{\mathscr{L}^{[\beta]}}, \;l=1,2\; \mbox{and}\; \beta=1,2,...,P.
\end{array}
\end{equation*}
Taking the time-average intralayer network topologies, the time-averaged multilayer hypernetwork can be expressed in terms of the following sets of equations as:
\begin{equation}\label{eq.8}
\begin{array}{l}
\dot{{ \bar{\mathbf{x}}}}_i = F_1({\bf {\bar{x}}}_i)+\sum\limits_{\beta=1}^{M}\epsilon_\beta\sum\limits_{j=1}^{N}{\bar{\mathscr{A}}_{ij}^{[\beta]}} G_{\beta}^{[1]}({\bf \bar{x}}_i,{\bf \bar{x}}_j)  \\ ~~~~~~~~~~~~~+ \lambda \sum\limits_{j=1}^{N}{\mathscr{B}_{ij}^{[1]}}H_1({\bf \bar{x}}_i,{\bf \bar{y}}_j),\\
\dot{ \bar{\mathbf{y}}}_i = F_2({\bf \bar{y}}_i)+\sum\limits_{\beta=1}^{M}\epsilon_\beta\sum\limits_{j=1}^{N}{\bar{\mathscr{A}}_{ij}^{[\beta]}} G_{\beta}^{[2]}({\bf \bar{y}}_i,{\bf \bar{y}}_j)  \\ ~~~~~~~~~~~~~+ \lambda \sum\limits_{j=1}^{N}{\mathscr{B}_{ij}^{[2]}}H_2({\bf \bar{y}}_i,{\bf \bar{x}}_j),
\end{array}
\end{equation}
where $\mathbf{\bar{x}}_i(\mathbf{\bar{y}}_i)$ represents the states of the $i$-th node in layer-1 (layer-2) for time-averaged network.

\par As, the stability of intralayer synchronization for time-average multilayer hypernetwork \eqref{eq.8} guarantees the stability of intralayer synchronization in time-varying network \eqref{eq.1}, the extinction of transverse modes corresponding to time-averages system will then necessary implies the achievement of stable coherent state. Through a series of theoretical steps detailed in Appendix \ref{time_average}, we derive the dynamics of transverse modes for time-average system as following:
\begin{equation}\label{eq.9}
\begin{array}{l}
\dot{\eta}_T^{(\mathbf{\bar{x}})}=I_{N-1}{\otimes}JF_1(\mathbf{x}_s)\eta_T^{(\mathbf{\bar{x}})} +\sum\limits_{\beta=1}^{M}\epsilon_{\beta} d^{[\beta]}[I_{N-1}{\otimes}JG_{\beta}^{[1]}(\mathbf{x}_s,\mathbf{x}_s) \\~+I_{N-1}{\otimes}DG_{\beta}^{[1]}(\mathbf{x}_s,\mathbf{x}_s)]\eta_T^{(\mathbf{\bar{x}})}   -\sum\limits_{\beta=1}^{M}\epsilon_{\beta}\bar{W}_2^{[\beta]}{\otimes}DG_{\beta}^{[1]}({\bf x}_s,{\bf x}_s)\eta_T^{(\mathbf{\bar{x}})}\\~~ +\lambda[e^{[1]}I_{N-1}{\otimes}JH_1(\mathbf{x}_s,\mathbf{y}_s)\eta_T^{(\mathbf{\bar{x}})}+U_3^{[1]}{\otimes}DH_1(\mathbf{x}_s,\mathbf{y}_s)\eta_T^{(\mathbf{\bar{y}})}],  \\\\

\dot{\eta}_T^{(\mathbf{\bar{y}})}=I_{N-1}{\otimes}JF_2(\mathbf{y}_s)\eta_T^{(\mathbf{\bar{y}})}+\sum\limits_{\beta=1}^{M}\epsilon_{\beta} d^{[\beta]}[I_{N-1}{\otimes}JG_{\beta}^{[2]}(\mathbf{y}_s,\mathbf{y}_s)\\~~~~ +I_{N-1}{\otimes}DG_{\beta}^{[2]}(\mathbf{y}_s,\mathbf{y}_s)]\eta_T^{(\mathbf{\bar{y}})}  \\~~~~~~ -\sum\limits_{\beta=1}^{M}\epsilon_{\beta}\bar{W}_2^{[\beta]}{\otimes}DG_{\beta}^{[2]}({\bf y}_s,{\bf y}_s)\eta_T^{(\mathbf{\bar{y}})} \\~~~~+\lambda[e^{[2]}I_{N-1}{\otimes}JH_2(\mathbf{y}_s,\mathbf{x}_s)\eta_T^{(\mathbf{\bar{y}})}+U_3^{[2]}{\otimes}DH_2(\mathbf{y}_s,\mathbf{x}_s)\eta_T^{(\mathbf{\bar{x}})}],
\end{array}
\end{equation}
where ${\eta}_T^{(\mathbf{\bar{x}})}$ and ${\eta}_T^{(\mathbf{\bar{y}})}$ represents the states of transverse modes of time-average system.

\par Therefore, the problem of stability of intralayer synchronization state for the time-varying multilayer network is then reduced to solving the coupled transverse linear equations \eqref{eq.9} for calculation of maximum Lyapunov exponent (MLE). Stability of coherent state needs MLE associated with the transverse modes to be negative, as a necessary condition. Given node dynamics and coupling functions, MLE is mainly function of coupling strengths, time-averaged Laplacians associated with each tier and interlayer adjacencies, i.e., MLE ($\epsilon_{1}, \epsilon_{2},\dots,\epsilon_{M},\lambda,\bar{\mathscr{L}}^{[1]}, \bar{\mathscr{L}}^{[2]},\dots,\bar{\mathscr{L}}^{[M]},\mathscr{B}^{[1]},\mathscr{B}^{[2]}$). It is notable that, in comparison with the classical MSF scheme, also in time-varying multilayer hypernetwork we are able to separate the motion parallel and transverse to synchronization manifold through fast switching approach. However, more intricacy in network structure in latter circumstance gives a bunch of coupled linear differential equation to analyze the stability, in spite of independent, uncoupled equations as in the case of classical MSF.  Particularly, there remain some terms in the transverse error equation \eqref{eq.9} which are generally not transformable into block diagonal forms and as a result the transverse error dynamics becomes $2d(N-1)$-dimensional coupled equation. Furthermore, there are suitable occasions where the coupled transverse modes can be optimally separated and the $2d(N-1)$-dimensional coupled equation reduce into $(N-1)$, $2d$-dimensional linear differential equations.

\par The relevant instances are as follows:\\
(i) The first instance is when each time average Laplacians $\bar{\mathscr{L}}^{[\beta]}$, interlayer adjacency matrices $\mathscr{B}^{[l]}$, $l=1,2 \;\;\mbox{and}\;\; \beta= 1, 2,\dots,M$ are symmetric, and among them one Laplacian commutes with all the other time average Laplacians as well as with the interlayer adjacencies (The detailed proof is in Appendix \ref{commutative}). In this case the transverse variational equation reduces to $2d$-dimensional $N-1$ systems as,
\begin{equation}\label{eq.10}
\begin{array}{l}
\dot{\eta}_{T_i}^{(\mathbf{\bar{x}})}=JF_1(\mathbf{x}_s)\eta_{T_i}^{(\mathbf{\bar{x}})}+\sum\limits_{\beta=1}^{M}\epsilon_{\beta} d^{[\beta]}[JG_{\beta}^{[1]}(\mathbf{x}_s,\mathbf{x}_s) \\~~~~ +DG_{\beta}^{[1]}(\mathbf{x}_s,\mathbf{x}_s)]\eta_{T_i}^{(\mathbf{\bar{x}})} -\sum\limits_{\beta=1}^{M}\epsilon_{\beta}\bar{\gamma}_{i}^{[\beta]}DG_{\beta}^{[1]}({\bf x}_s,{\bf x}_s)\eta_{T_i}^{(\mathbf{\bar{x}})} \\~~~~ +\lambda[e^{[1]}JH_1(\mathbf{x}_s,\mathbf{y}_s)\eta_{T_i}^{(\mathbf{\bar{x}})}+\varGamma_{i}^{[1]}DH_1(\mathbf{x}_s,\mathbf{y}_s)\eta_{T_i}^{(\mathbf{\bar{y}})}], \\

\dot{\eta}_{T_i}^{(\mathbf{\bar{y}})}=JF_2(\mathbf{y}_s)\eta_{T_i}^{(\mathbf{\bar{y}})}+\sum\limits_{\beta=1}^{M}\epsilon_{\beta} d^{[\beta]}[JG_{\beta}^{[2]}(\mathbf{y}_s,\mathbf{y}_s) \\~~~~ +DG_{\beta}^{[2]}(\mathbf{y}_s,\mathbf{y}_s)]\eta_{T_i}^{(\mathbf{\bar{y}})}  -\sum\limits_{\beta=1}^{M}\epsilon_{\beta}\bar{\gamma}_{i}^{[\beta]}DG_{\beta}^{[2]}({\bf y}_s,{\bf y}_s)\eta_{T_i}^{(\mathbf{\bar{y}})} \\~~~~ +\lambda[e^{[2]}JH_2(\mathbf{y}_s,\mathbf{x}_s)\eta_{T_i}^{(\mathbf{\bar{y}})}+\varGamma_{i}^{[2]}DH_2(\mathbf{y}_s,\mathbf{x}_s)\eta_{T_i}^{(\mathbf{\bar{x}})}],
\end{array}
\end{equation}
where $\bar{\gamma}_{i}^{[\beta]}$ and $\varGamma_{i}^{[l]}$ for $i=2,3,\dots,N$ are the non-zero eigenvalues of time-averaged Laplacian matrices corresponding to tier-$\beta$ and eigenvalues of interlayer adjacency matrices.

\par As for example, if we consider connection topology of one tier of time-varying network as random network with edge rewiring probability $p_{rand}$ and any arbitrary undirected network structure for other tiers with bidirectional interlayer connections, then we can get one such instance where transverse dynamics decouples into independent modes.\\
(ii) Another instance for optimal separation of transverse modes occurs when the number of distinct connection topologies in both layer equals to one, i.e., M = 1 and the time average Laplacian matrices have all the eigenvalues equal except the smallest zero eigenvalue. Further the interlayer adjacencies  $\mathscr{B}^{[1]}$ and $\mathscr{B}^{[2]}$ are identical (see Appendix \ref{one_tier} for full derivation).

\subsubsection{Simultaneous block diagonalization approach} \label{sbd}
\par To separate the parallel and transverse error components the fast switching approach considers the time-averaged network structures. The commutative property of time-average Laplacians and identical eigenvalues are also considered to further reduce the transverse error system into lower dimensional form. To overcome these limitations, here we use the SBD \cite{maehara2011algorithm,sbd_sorrentino,zhang2020symmetry} approach on the set of coupling matrices of time-varying multilayer hypernetwork \eqref{eq.1}.

\par The {SBD} process can be conceptualized as follows: for a prescribed {finite} set of symmetric matrices $\{B^{1}, B^{2},\dots,B^{q}\}$, there exists an orthogonal matrix $P$ such that the matrices $P^{tr}B^{m}P$ possess a common block-diagonal structure for $m=1,2,\dots,q$, leading to an optimal separation of perturbation modes. This common block structure is identified by the largest number of blocks and is unique up to block permutations. In particular, this scheme guarantees the separation of perturbation modes along the synchronization manifold and transverse to the manifold. Further it decouples transverse system into low-dimensional systems through transforming set of matrices to optimal block diagonal forms.

\par To proceed, we rewrite the variation equation \eqref{eq.5} by introducing $\delta \mathbf{ X}(t) =\begin{bmatrix} \delta \mathbf{x}(t) & \delta \mathbf{y}(t) \end{bmatrix}^{tr}$ as,
\begin{equation} \label{eq.11}
\begin{array}{lll}
{\delta\dot{\mathbf X}}(t)=\begin{bmatrix}
\delta\dot{\mathbf{x}}(t) \\ \delta\dot{\mathbf{y}}(t)
\end{bmatrix} = \big[ D_{1}
+\sum\limits_{\beta=1}^{M}\epsilon_{\beta} d^{[\beta]} D_{2} + \sum\limits_{\beta=1}^{M}\epsilon_{\beta} D_{3}\\\hspace{82pt}+\sum\limits_{\beta=1}^{M}\epsilon_{\beta} D_{4}  + \lambda D_{5} +\lambda D_{6} \big] \delta{\mathbf X},
\end{array}
\end{equation}
where \begin{equation*}
		\begin{array}{l}
		D_{1}=\begin{bmatrix}
			I_{N}\otimes JF_{1} & 0 \\ 0  & I_{N} \otimes JF_{2}
		\end{bmatrix},
	  \\\\D_{2}= \begin{bmatrix}
		I_{N}\otimes JG_{\beta}^{[1]}+I_{N}\otimes DG_{\beta}^{[1]} & 0 \\
		0 & I_{N}\otimes JG_{\beta}^{[2]}+I_{N}\otimes DG_{\beta}^{[2]}
	\end{bmatrix}, \\\\D_{3}= \begin{bmatrix}
		\mathscr{L}^{[1,\beta]}(t) \otimes DG_{\beta}^{[1]} & 0 \\
		0 & 0
	\end{bmatrix}, \\\\D_{4}= \begin{bmatrix}
		0 & 0 \\
		0 & \mathscr{L}^{[2,\beta]}(t) \otimes DG_{\beta}^{[2]}
	\end{bmatrix}, \\\\D_{5}= \begin{bmatrix}
		e^{[1]}I_{N}\otimes JH_{1} & 0 \\
		0 & e^{[2]}I_{N}\otimes JH_{2}
	\end{bmatrix}, \\\\ \;\; \mbox{and}\;\; D_{6}= \begin{bmatrix}
		0 & \mathscr{B}^{[1]}\otimes DH_{1}\\
		\mathscr{B}^{[2]}\otimes DH_{2} & 0
	\end{bmatrix}.
    \end{array}
    \end{equation*}

	Now, let
	 \begin{equation*}
	 	\begin{array}{l}
	 		U^{[1,\beta]}(t)= \begin{bmatrix}
	 			\mathscr{L}^{[1,\beta]}(t) & 0 \\
	 			0 & 0
	 		\end{bmatrix} ,  \\\\ U^{[2,\beta]}(t)= \begin{bmatrix}
	 			0 & 0 \\
	 			0 & \mathscr{L}^{[2,\beta]}(t)
	 		\end{bmatrix}\;\; \mbox{and} \;\;\mathscr{B}= \begin{bmatrix}
	 			0 & \mathscr{B}^{[1]} \\
	 			\mathscr{B}^{[2]}  & 0
	 		\end{bmatrix}.
	 	\end{array}
	 \end{equation*}  Also, we consider that the interlayer coupling functions hold $JH_{1}(x_0,yo)= JH_{2}(y_0,x_o)$ and $DH_{1}(x_0,y_o)= DH_{2}(y_0,x_o)$.

	Therefore, the variational equation \eqref{eq.11} can be rewritten as,

	\begin{equation} \label{eq.12}
		\begin{array}{l}
			{\delta\dot{\mathbf X}} = \big[ D_{1}
			+ \sum\limits_{\beta=1}^{M}\epsilon_{\beta} d^{[\beta]} D_{2} + \sum\limits_{\beta=1}^{M}\epsilon_{\beta} U^{[1,\beta]}(t) \otimes DG_{\beta}^{[1]}  \\ ~~~~~~~~+
			\sum\limits_{\beta=1}^{M}\epsilon_{\beta} U^{[2,\beta]}(t) \otimes DG_{\beta}^{[2]}  + \lambda D_{5} +\lambda \mathscr{B}\otimes DH_1 \big] \delta{\mathbf X}.
		\end{array}
	\end{equation}
All the terms in this linearized equation are block diagonalized except the terms $U^{[1,\beta]}(t) \otimes DG_{\beta}^{[1]}$, $U^{[2,\beta]}(t) \otimes DG_{\beta}^{[2]}$ and $\mathscr{B}\otimes DH_1$. Now time-varying Laplacian matrices $\mathscr{L}^{[1,\beta]}(t)$, $\mathscr{L}^{[2,\beta]}(t)$ and the interlayer adjacency matrices  are all symmetric matrices. Hence the matrices $U^{[1,\beta]}(t)$, $U^{[2,\beta]}(t)$ and $\mathscr{B}$ are also symmetric. So, we apply the process of SBD on the set of symmetric matrices \{$U^{[1,\beta]}(t),U^{[2,\beta]}(t),\mathscr{B}$\} to obtain a orthogonal matrix $P$ that transforms these matrices to finest block diagonalization form. {Usually the SBD technique works better for a finite collection of symmetric matrices, so we adopt one extra assumption about the time varying intralayer connection topologies. We assume that the collection of time-stamped intralayer adjacency matrices $\{\mathscr{A}^{[l,\beta]}(t):~l=1,2;~\beta=1,2,\dots,M;~\mbox{and}~t\in\mathbb{R}^+\}$ is a finite set, which in turn gives finite number of symmetric matrices $U^{[1,\beta]}(t),U^{[2,\beta]}(t)$}. To obtain the matrix $P$, we follow the algorithm illustrated in \cite{zhang2020symmetry}. Thereafter, we project the state variable $\delta \mathbf{X}$ onto the basis of vectors formed by the columns of $P$ by defining new variable,
\begin{equation*}
\begin{array}{l}
\eta(t)=(P{\otimes}I_{2d})^{T}\delta\mathbf{ X}(t).
\end{array}
\end{equation*} Using this co-ordinate transformation the linearized equation \eqref{eq.12} becomes,
\begin{equation} \label{eq.13}
\begin{array}{l}
\dot{\eta}(t)=\big[ D_{1}+\sum\limits_{\beta=1}^{M}\epsilon_{\beta} d^{[\beta]} D_{2}+\sum\limits_{\beta=1}^{M}\epsilon_{\beta} \tilde{U}^{[1,\beta]}(t) \otimes DG_{\beta}^{[1]}+  \\ ~~~~~~~~ \sum\limits_{\beta=1}^{M}\epsilon_{\beta} \tilde{U}^{[2,\beta]}(t) \otimes DG_{\beta}^{[2]}+ \lambda D_{5}+\lambda \tilde{\mathscr{B}}\otimes DH_1 \big] \eta(t),
\end{array}
\end{equation}
where
\begin{equation*}
\begin{array}{lll}
\tilde{U}^{[1,\beta]}(t) &= P^{T} U^{[1,\beta]}(t) P, \\ \tilde{U}^{[2,\beta]}(t) &= P^{T} U^{[2,\beta]}(t) P, \\ \tilde{\mathscr{B}} &= P^{T} \mathscr{B} P,
\end{array}
\end{equation*}
are the finest block diagonalized forms of $U^{[1,\beta]}(t)$,$U^{[2,\beta]}(t)$ \& $\mathscr{B}$. These block diagonalized matrices have two $1\times1$ blocks which are associated with the parallel modes of synchronization solutions corresponding to two layers and the other blocks are associated with the transverse modes of synchronization manifold. Thus, we can perfectly decouple the perturbation modes parallel and transverse to the synchronization manifold without implementing time-average network structure. Apart from this, the finest block-diagonal forms of the matrices also explains that the transverse modes are not fully coupled. The high-dimensional coupled transverse system reduces to many lower-dimensional linear systems by means of sizes of blocks in the transformed block-diagonal matrices. Hence, the further reduction of transverse error systems is possible through this approach even without considering any limitations like fast switching mechanism. The stability condition of the synchronization manifold is then reduced to solving the linear equation \eqref{eq.13} for calculation of maximum Lyapunov exponent transverse to synchronous manifold. The necessary condition requires the maximum Lyapunov exponent to be negative. In Sec. \ref{sbd_numerical}, we have elaborated the whole process with a suitable example.

	\section{Numerical Illustrations}\label{numerical}

	In order to illustrate our theoretical findings, here we consider three cases of different isolate node dynamics and nonlinear coupling functions. Specifically, we address two ideal chaotic systems, the Lorenz \cite{strogatz2018nonlinear} and R\"{o}ssler \cite{rossler1976equation} systems, and as a real-world instance of neuronal evolution, the Sherman model \cite{sherman1994anti,jalil2012spikes}. For numerical simulations, we consider that each layer of the multilayer network consists of $N$ number of nodes interacting through two structurally different tiers at every instance of time. To keep things simple, we take the interlayer connections as one to one between the nodes of adjacent layers and dynamics of isolated nodes in both layers are identical. In order to investigate the intralayer synchronous state, we define the synchronization error as,
	\begin{equation} \label{eq.14}
		E_{intra}=\lim\limits_{T\to\infty}\dfrac{1}{T}\int_{t_{trans}}^{t_{trans}+T}E_{intra}(t)~dt,
	\end{equation}
     where,~~$E_{intra}(t)=\sum\limits_{k=2}^{N}\dfrac{\|{\bf x}_k(t)-{\bf x}_1(t)\|+\|{\bf y}_k(t)-{\bf y}_1(t)\|}{2(N-1)}$.

\par Here $\|\cdot\|$ symbolizes the Euclidean norm, $t_{trans}$ determines the transient of the numerical simulation and $T$ is a sufficiently large positive number. Asymptotic stability of $E_{intra}$ will imply each layer is synchronized. For intralayer synchrony to emerge, we have used the threshold value of $E_{intra}$ as $10^{-5}$. Numerical simulations of the multilayer network are executed using  the $4^{th}$ order Runge-Kutta algorithm, with step of integration $\delta t$= $0.01$ (for chaotic oscillators) and $0.001$ (for neuron model), over a span of time $T=1000$ after an initial transient $t_{trans}=2000$. The initial condition of each dynamical node is chosen randomly from the phase-space of isolate node dynamics.  At each instant, we rewire all tiers of both layers separately with probability $f \delta t$, where $f$ is the frequency of rewiring process. All the numerical results are obtained by taking average over 10 network realizations and initial conditions. {At first, we will present the numerically obtained results based on the fast-switching stability criterion.}

\subsection{Numerical illustration of the fast switching approach}
\par In the next three sub-sections, we investigate the synchronization state numerically by means of synchronization error $E_{intra}$ and concurrently validate our analytical findings corresponding to fast switching approach for the three structurally different networked systems with $N=200$ nodes and different nonlinear coupling functions in each occasion.

\subsubsection{Coupled Lorenz systems with sine coupling functions} \label{lorenz_numerical}
\par We first consider multilayer hypernetwork consisting of coupled Lorenz oscillators interacting with sine coupling functions. Then the corresponding  equation of motion reads as:
\begin{equation} \label{eq.15}
\begin{array}{lll}

	\mbox{Layer-1 :} \\
\dot{x}_{1i}=\sigma(y_{1i}-x_{1i})+\epsilon_1\sum\limits_{j=1}^{N}\mathscr{A}^{[1,1]}_{ij}(t)\sin(x_{1j}-x_{1i}) \\~~~~~~~~~~+\lambda \sin(x_{2i}-x_{1i}),\\
\dot{y}_{1i}=x_{1i}(\rho-z_{1i})-y_{1i}+\epsilon_2\sum\limits_{j=1}^{N}\mathscr{A}^{[1,2]}_{ij}(t)\sin(y_{1j}-y_{1i}),\\
\dot{z}_{1i}=x_{1i}y_{1i}-\alpha z_{1i},\\\\

 \mbox{Layer-2 :} \\
\dot{x}_{2i}=\sigma(y_{2i}-x_{2i})+\epsilon_1\sum\limits_{j=1}^{N}\mathscr{A}^{[2,1]}_{ij}(t)\sin(x_{2j}-x_{2i})\\~~~~~~~~~~+\lambda \sin(x_{1i}-x_{2i}),\\
\dot{y}_{2i}=x_{2i}(\rho-z_{2i})-y_{2i}+\epsilon_2\sum\limits_{j=1}^{N}\mathscr{A}^{[2,2]}_{ij}(t)\sin(y_{2j}-y_{2i}),\\
\dot{z}_{2i}=x_{2i}y_{2i}-\alpha z_{2i},\hspace{50pt}i=1,2,\dots,N.
\end{array}
\end{equation}
where each isolate unit is in a chaotic state with system parameters $\sigma=10$, $\alpha=8/3$, and $\rho=28$. The intralayer connection topology for tier-1 is represented by Erd{\"o}s-R{\'e}nyi random network \cite{erdos2011evolution} with probability $p_{rand}$ and for tier-2 the intralayer connection topology is small-world network \cite{watts1998collective} with average node degree $2k_{sw}$ and link rewiring probability $p_{sw}$.

\begin{figure}[ht]
\centerline{\includegraphics[scale=0.43]{fig2.png}}
\caption{{\bf Coupled Lorenz oscillators:} Intralayer synchronization error in ($\epsilon_{1}$,$\epsilon_{2}$) parameter plane. For (a) $f=10^{-4}$, (b) $f=10^{-2}$, (c) $f=10^{0}$ and (d) $f=10^{2}$. Other parameter values are $d_{sw}=4$, $p_{sw}=0.1$, $p_{rand}=0.015$, $\lambda=0.1$. The solid {magenta} line in (d) is the theoretical prediction of intralayer synchronization threshold corresponding to MLE $=0$. White region represents stable intralayer synchronization state.}\label{fig2}
\end{figure}
For various rewiring frequencies, the intralayer synchronization error as a function of tier-1 coupling strength $\epsilon_{1}$ and tier-2 coupling strength $\epsilon_{2}$ is depicted in Fig. \ref{fig2}. The coherence and incoherence regions, portrayed in white and black, are plotted in Figs. $\ref{fig2}(a)$- $\ref{fig2}(d)$ for slow rewiring frequency $(f=10^{-4})$ to very fast rewiring frequency $(f=10^{2})$, respectively. For lower rewiring frequency (Fig. \ref{fig2}(a)), i.e, when the intralayer links are nearly steady with time, higher tier-2 coupling strength is needed to achieve synchrony and critical transition point of $\epsilon_2$ decrease as the value of $\epsilon_1$ increases. In Fig. \ref{fig2}(b) when rewiring frequency f is increased to $f=10^{-2}$, a small enhancement of synchronization region is observed in $(\epsilon_1,\epsilon_2)$ plane. For further increment of $f=10^0$ and $f=10^2$, a notable enhancement in coherent region is perceived in Figs. \ref{fig2}(c) and \ref{fig2}(d).

\begin{figure}[ht]
\centerline{\includegraphics[scale=0.42]{fig3.png}~~~~~}
\caption{{\bf Coupled Lorenz oscillators:} Intralayer synchronization error in (a) ($\epsilon_{1}$,f) and (b)  ($\epsilon_{2}$,f)  plane. For (a) $\epsilon_{2}=1.0$ and for (b)  $\epsilon_{1}=1.0$. Other parameter values are $d_{sw}=4$, $p_{sw}=0.1$, $p_{rand}=0.015$, $\lambda=0.1$. { The solid magenta circles in both the subfigures indicates analytically derived critical value of coupling strengths for sufficiently fast switching of intralayer network topologies. White region represents stable intralayer synchronization state. }}\label{fig3}
\end{figure}
\par Figure \ref{fig3} portrays variation of intralayer synchronization in the parameter plane $(\epsilon_1,f)$ and $(\epsilon_2,f)$. For $\epsilon_2=1.0$, the synchronized and desynchronized regions are plotted by varying $\epsilon_1$ and $f$ in Fig. \ref{fig3}(a). At small values of $\epsilon_1$ up to $2.5$, no synchronization occurs by varying rewiring frequency from slow to fast switching. Beyond $\epsilon_1=2.5$, higher rewiring frequency required to achieve synchrony and the critical point of $f$ reduces with increasing value of $\epsilon_1$. Similarly, Fig. \ref{fig3}(b) demonstrates the coherent and incoherent domains in the $(\epsilon_2,f)$ plane for fixed value of $\epsilon_1=1.0$.

	\par Next we proceed through the stability of synchronized state based on the fast switching approach. If $\bar{\mathscr{L}}^{[1]}$ and $\bar{\mathscr{L}}^{[2]}$ are the time-average Laplacian matrices corresponding to tier-1 (random network) and tier-2 (small world network) for both the layers, then
	\begin{equation*}
		\bar{\mathscr{L}}_{ij}^{[1]}= \begin{cases}
			-p_{rand}, & \mbox{for} \;\;i\neq j \\
			(N-1)p_{rand}, & \mbox{for} \;\;i = j
		\end{cases}
	\end{equation*}
	and

	\begin{equation*}
		\bar{\mathscr{L}}_{ij}^{[2]}= \begin{cases}
			-(1-p_{sw}), & \mbox{for} \;\; i-k_{sw}\leq j \leq i+k_{sw} \;\; \mbox{and} \;\; i\neq j \\
			2k_{sw}, & \mbox{for} \;\; i = j \\
			-\frac{2k_{sw}p_{sw}}{N-2k_{sw}-1}, & \mbox{otherwise}.
		\end{cases}
	\end{equation*}

Clearly $\bar{\mathscr{L}}^{[1]}$ commutes with the other Laplacian $\bar{\mathscr{L}}^{[2]}$ and the interlayer adjacency matrix (in this case identity matrix). So, we can recast the transverse error equation in the reduced form as Eq.\eqref{eq.10}, which in this case becomes,
\begin{equation} \label{eq.16}
\begin{array}{l}
\delta\dot{x}_{1i}= \sigma(\delta y_{1i}-\delta x_{1i}) -\epsilon_{1}\bar{\gamma}_{i}^{[1]}\delta x_{1i}     \\~~~~~~~~~~+ \lambda \cos(x_2-x_1)(\delta x_{2i}-\delta x_{1i}),     \\
\delta\dot{y}_{1i}=(\rho-z_1)\delta x_{1i}-\delta y_{1i} -x_{1}\delta z_{1i}-\epsilon_2 \bar{\gamma}_{i}^{[2]} \delta y_{1i}, \\
\delta\dot{z}_{1i}=y_{1}\delta x_{1i} +x_{1}\delta y_{1i}-\beta \delta z_{1i}, \\\\
\delta\dot{x}_{2i}= \sigma(\delta y_{2i}-\delta x_{2i}) -\epsilon_{1} \bar{\gamma}_{i}^{[1]}\delta x_{2i}   \\~~~~~~~~~~+\lambda \cos(x_1-x_2)(\delta x_{1i}-\delta x_{2i}),     \\
\delta\dot{y}_{2i}=(\rho-z_2)\delta x_{2i}-\delta y_{2i} -x_{2}\delta z_{2i}-\epsilon_2 \bar{\gamma}_{i}^{[2]} \delta y_{2i}, \\
\delta\dot{z}_{2i}=y_{2}\delta x_{2i} +x_{2}\delta y_{2i}-\beta \delta z_{2i},   \hspace{10 pt} i=2,...,N,
\end{array}
\end{equation}
where $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are the state variables of the synchronization manifold.

\par The necessary condition for synchronization requires maximum Lyapunov exponent obtained by simulating Eq.\eqref{eq.16}, to be negative. We verify the numerically obtained intralayer synchronization region in the $(\epsilon_1,\epsilon_2)$ plane by means of maximum transverse Lyapunov exponent for adequately fast switching. The critical curve for synchronization, characterized by MLE = 0 is drawn in solid {magenta} line in Fig. \ref{fig2}(d). In this parameter space, the regions above and below the threshold curve respectively depict coherent and incoherent state, which exactly agrees with the numerically obtained result on the basis of $E_{intra}$. Hence, for sufficiently fast switching the stability analysis of coherent state through the time-averaged network formation exactly signifies the coherence in time-varying network.

\subsubsection{Coupled R\"{o}ssler systems with nonlinear coupling functions}
\par To further elucidate that the time-varying interactions significantly contributes for the emergence of intralayer synchronization, we consider the multilayer hypernetwork \eqref{eq.1} with isolated node dynamics as the R\"{o}ssler oscillators, interacting with nonlinear coupling functions. Then the equation of motion of the multilayer network can be described as follows
	\begin{equation} \label{eq.17}
		\begin{array}{lll}

			\mbox{Layer-1 :} \\
			\dot{x}_{1i}=-y_{1i}-z_{1i}+\epsilon_1\sum\limits_{j=1}^{N}\mathscr{A}^{[1,1]}_{ij}(t)(\alpha-x_{1i})(x_{1j}-\beta)^2,\\
			\dot{y}_{1i}=x_{1i}+ay_{1i}+\epsilon_2\sum\limits_{j=1}^{N}\mathscr{A}^{[1,2]}_{ij}(t)(\alpha-y_{1i})(y_{1j}-\beta)^2 \\~~~~~~~~~~+\lambda(\alpha-y_{1i})(y_{2i}-\beta)^2,\\
			\dot{z}_{1i}=b+(x_{1i}-c)z_{1i},\\\\

			\mbox{Layer-2 :} \\
			\dot{x}_{2i}=-y_{2i}-z_{2i}+\epsilon_1\sum\limits_{j=1}^{N}\mathscr{A}^{[2,1]}_{ij}(t)(\alpha-x_{2i})(x_{2j}-\beta)^2,\\
			\dot{y}_{2i}=x_{2i}+ay_{2i}+\epsilon_2\sum\limits_{j=1}^{N}\mathscr{A}^{[2,2]}_{ij}(t)(\alpha-y_{2i})(y_{2j}-\beta)^2\\~~~~~~~~~~+\lambda(\alpha-y_{2i})(y_{1i}-\beta)^2,\\
			\dot{z}_{2i}=b+(x_{2i}-c)z_{2i},\hspace{20pt}i=1,2,\dots,N.
		\end{array}
	\end{equation}
	The system parameter values are chosen as $a=0.2$, $b=0.2$, $c=5.7$, and the two coupling parameter values chosen as $\alpha=0.37$, $\beta=-0.37$. The intralayer connection mechanisms for tier-1 and tier-2 are considered as random network with constant node degree $k_1=2$ and $k_2=3$ to guarantee the invariance condition of intralayer synchronization. The intralayer coupling functions are acting through $x$ and $y$ variable corresponding to tier-1 and tier-2, respectively, and interlayer coupling is through $y$ variable.

\begin{figure}[ht]
\centerline{\includegraphics[scale=0.425]{fig4.png}~~~~~~~~}
\caption{{\bf Coupled R\"{o}ssler oscillators:} Intralayer synchronization error in ($\epsilon_{1}$,$\epsilon_{2}$) parameter plane. For (a) $f=10^{-4}$, (b) $f=10^{2}$. Other parameter values are $k_1=2$, $k_2=3$, $\lambda=0.1$. The solid {magenta} curve in (b) represents the critical curve for which MLE $=0$, the regions below and above the critical curve denotes the unstable and stable synchronization state respectively. White region represents stable intralayer synchronization state.}\label{fig4}
\end{figure}
\par We have plotted $E_{intra} (\epsilon_1, \epsilon_2)$ in Fig. \ref{fig4} for slow and fast rewiring frequencies, respectively. A significant enhancement in synchronization region is noticed in Fig. \ref{fig4}(b) for sufficiently large rewiring frequency $(f=10^2)$ compared to the synchronization region for sufficiently slow rewiring frequency $(f=10^{-4})$, depicted in Fig. \ref{fig4}(a). For fast switching, we validate the numerical results through time-average network construction with theoretical predictions obtained from Eq.\eqref{eq.10}, similarly as previous example. The analytical threshold values for synchronization are portrayed in solid {magenta} line imposed to Fig. \ref{fig4}(b), which shows that the numerical simulation for fast switching case is in very good agreement with the analytical derivation for intralayer synchronization threshold.

\begin{figure}[ht]
\centerline{\includegraphics[scale=0.4]{fig5.png}}
\caption{{\bf Coupled R\"{o}ssler oscillators:} Intralayer synchronization error in  ($\epsilon_{1}$,f) plane. For (a) $\epsilon_{2}=0.05$, (b)  $\epsilon_{2}=0.1$, (c)  $\epsilon_{2}=0.15$, and (d)  $\epsilon_{2}=0.2$. Other parameter values are $k_1=2=4$, $k_2=3$, and $\lambda=0.1$. {The solid magenta circles in each subfigure indicates analytically derived critical value of coupling strengths for sufficiently fast switching of intralayer network topologies. }}\label{fig5}
\end{figure}
\par Further, Fig. \ref{fig5} represents the region of coherence and incoherence in $(\epsilon_1,f)$ parameter plane for various values of tier-2 coupling strength $\epsilon_2$. For lower value of $\epsilon_2=0.05$, the synchronization region portrayed in Fig. \ref{fig5}(a). By increasing $\epsilon_2$ to $\epsilon_2=0.1$ the enhancement in synchrony region is shown in Fig. \ref{fig5}(b). Further increment of $\epsilon_2$ to $\epsilon_2=0.15$ (Fig. \ref{fig5}(c)) and $\epsilon_2=0.2$ (Fig. \ref{fig5}(d)) shows significant enhancement in synchronization region. In all this figure, the critical transition point for $\epsilon_1$ initially decreases as the rewiring frequency increases up to $f \approx 10^{-2}$. Beyond, $f=10^{-2}$ the critical point against $\epsilon_1$ is almost vertical. We found somewhat similar results as above when we plot the variation of synchronization error in $(\epsilon_2,f)$ plane for different values of $\epsilon_1$ (corresponding figures are not shown in the manuscript).

	\subsubsection{Coupled Sherman model of pancreatic beta cells interacting through  electrical and chemical synapses}

	We now scrutinize our framework to study the neuronal synchronization. Synchronization in neuronal networks is of enormous significance. Studies in neuroscience have identified existence of simultaneous interconnection between neurons through various synaptic transmissions, which can be perfectly schematized by multilayer networks \cite{sporns2010networks,bera2019intralayer}.  Here we consider ensemble of pancreatic beta cells, represented by paradigmatic Sherman model, interconnected concurrently through electrical and chemical synapses in a multilayer network framework. The dynamics of entire network is given by,

	\begin{widetext}
	\begin{equation} \label{eq.18}
		\begin{array}{l}
			\mbox{Layer-1 :} \\
			\tau\dot{V}_{1i}=-I_{Ca}(V_{1i})-I_{K}(V_{1i},n_{1i})-I_{S}(V_{1i},s_{1i})+\epsilon_{1} \sum\limits_{j=1}^{N}{\mathscr{A}_{ij}^{[1,1]}}(t)(V_{1j}-V_{1i}) +\epsilon_{2}(E_{S}-V_{1i})\sum\limits_{j=1}^{N}{\mathscr{A}_{ij}^{[1,2]}}(t)\Gamma(V_{1j}) \\~~~~~~~~~~~~~ +\lambda(E_{S}-V_{1i})\Gamma(V_{2i}), \\
			\tau\dot{n}_{1i}=\mu [n^{\infty}(V_{1i})-n_{1i}], \\
			\tau_{s}\dot{s}_{1i}=s^{\infty}(V_{1i})-s_{1i}, \\\\

			\mbox{Layer-2 :} \\
			\tau\dot{V}_{2i}=-I_{Ca}(V_{2i})-I_{K}(V_{2i},n_{2i})-I_{S}(V_{2i},s_{2i})+\epsilon_{1} \sum\limits_{j=1}^{N}{\mathscr{A}_{ij}^{[2,1]}}(t)(V_{2j}-V_{2i})+\epsilon_{2}(E_{S}-V_{2i})\sum\limits_{j=1}^{N}{\mathscr{A}_{ij}^{[2,2]}}(t)\Gamma(V_{2j})
			\\~~~~~~~~~~~~~~~+\lambda(E_{S}-V_{2i})\Gamma(V_{1i}), \\
			\tau\dot{n}_{2i}=\mu[n^{\infty}(V_{2i})-n_{2i}], \\
			\tau_{s}\dot{s}_{2i}=s^{\infty}(V_{2i})-s_{2i},
		\end{array}
	\end{equation}
     \end{widetext}
	where $I_{Ca}(V)=g_{Ca}m^{\infty}(V-E_{Ca})$, $I_{K}(V,n)=g_{K}n(V-E_{K})$, $I_{S}(V,s)=g_{S}s(V-E_{K})$.
	$V_{li}$ is the membrane potential corresponding to the reversal potential $E_{Ca}=0.025$V, $E_{K}=-0.075$V. The time constants and maximum conductance are $\tau=0.02$, $\tau_{s}=5$, $g_{Ca}=3.6$, $g_{K}=10$, and $g_{S}=4$.  $\mu=1$, an auxiliary scaling factor, manages the time
	scale of the persistent potassium channels. The values of the gating variables at steady state are
	\begin{equation*}
	\begin{array}{l}
		m^{\infty}(V)=\{1+\exp[-83.34(V+0.02)]\}^{-1}, \\n^{\infty}(V)=\{1+\exp[-178.57(V+0.016)]\}^{-1}, \;\mbox{and}\\ s^{\infty}(V)=\{1+\exp[-100(V+0.035345)]\}^{-1}.
	\end{array}
	\end{equation*}

	\par We consider the neurons in a layer are interconnected simultaneously subject to electrical gap junctional coupling via electrical synapses and  chemical ion transportation via chemical synapses. The adjacency matrices corresponding to electrical synapses are $\mathscr{A}^{[l,1]}$, $l=1,2$, obtained from small-world network with average degree $k_{sw}$ and edge rewiring probability $p_{sw}$. $\mathscr{A}^{[l,2]}$, $l=1,2$ describes the structure of connections via chemical synapses, deliberated by random network with constant node degree $k_c$. Further, the neurons of two different layers are connected through chemical synapses.  The sigmoidal input-output function
	\begin{eqnarray*}
		\Gamma(V)=\frac{1}{1+\exp(\lambda_{s}(V-\Theta_s))} ,
	\end{eqnarray*} describes the procedure for initiation of nonlinear chemical synapse. The synaptic reversal potential is $E_s=-0.02$. Slope of the sigmoidal function, and synaptic firing threshold are determined by the real-valued constants $\lambda_{s}$ and $\theta_{s}$, which are fixed at $\Theta_s=-0.045$ and $\lambda_{s}=-1000$.

\begin{figure}[ht]
\centerline{\includegraphics[scale=0.45]{fig6.png}}
\caption{{\bf Coupled Sherman model:} Intralayer synchronization error in $(\epsilon_1,f)$ plane. For (a) $\lambda=0.0$, (b) $\lambda=0.02$, (c) $\lambda=0.04$, and (d) $\lambda=0.05$. Other parameter values are $\epsilon_2=0.015$, $p_{sw}=0.15$, $k_{sw}=4$, and $k_{c}=4$ respectively. {The solid magenta circles in each subfigure indicates analytically derived critical value of coupling strengths for sufficiently fast switching of intralayer network topologies. }}\label{fig6}
\end{figure}
\par Figure \ref{fig6} portrays the result corresponding to intralayer synchronization in $(\epsilon_{1},f)$ parameter plane for various interlayer coupling strength $\lambda$. In the absence of interlayer coupling $(\lambda=0)$, the coherent and in-coherent regions are portrayed in Fig. \ref{fig6}(a) and by introducing $\lambda=0.02$, the enhancement in synchronization region is obtained in Fig. \ref{fig6}(b). Further increments of $\lambda$ to $\lambda=0.04$, and $0.05$, shows more enhancement, which are delineated in Figs. \ref{fig6}(c) and \ref{fig6}(d). Figures \ref{fig7}(a)-\ref{fig7}(d) represents similar results in $(\epsilon_{2},f)$ parameter plane for increasing value of interlayer coupling strength $\eta$. Apart from this, we notice that in neuronal network also the threshold to achieve synchronization lowers with the increase in rewiring frequency $f$. For sufficiently fast switching $(f=10^{2})$, $E_{intra} (\epsilon_1,\epsilon_2)$ is reported in Fig. \ref{fig8}, along with the analytical conjecture obtained from Eq.\eqref{eq.10} (shown in solid {magenta} line laid over the diagram), which confirms that the numerical result is in good agreement with our analytical derivation.

\begin{figure}[ht]
\centerline{\includegraphics[scale=0.44]{fig7.png}~~~~~~~~}
\caption{{\bf Coupled Sherman model:} Intralayer synchronization error in $(\epsilon_2,f)$ plane for (a) $\lambda=0.0$, (b) $\lambda=0.02$, (c) $\lambda=0.04$, and (d) $\lambda=0.05$. Other parameter values are $\epsilon_1=0.35$, $p_{sw}=0.15$, $k_{sw}=4$, and $k_{c}=4$ respectively.}\label{fig7}
\end{figure}

\begin{figure}[ht]
	\centerline{
		\includegraphics[scale=0.6]{fig8.png}}

	\caption{  {\bf Coupled Sherman model:} Intralayer synchronization error in  $(\epsilon_1,\epsilon_2)$ parameter plane. The other parameter values are $\lambda=0.02$, $p_{sw}=0.15$, $k_{sw}=4$, and $k_c=4$. The solid {magenta} continuous line represents the curve corresponding to synchronization threshold MLE $=0$ which separates the synchrony and desynchrony region.}
	\label{fig8}
\end{figure}

	\subsection{Numerical results of Simultaneous Block Diagonalization Method} \label{sbd_numerical}

\par In this subsection, we investigate the validity of our obtained theoretical results associated with simultaneous block diagonalization framework. For this, we consider the example of coupled Lorenz oscillators with sine coupling function described earlier in Eq.\eqref{eq.15}. The parameter values are taken same as before (See Sec. \ref{lorenz_numerical}). Each layer of the multilayer hypernetwork consists of $N=8$ nodes interconnected through two different tiers. The network architecture of each tier are taken as complete graph of N nodes by randomly removing two edges. In this scenario, we consider more systematically temporal networks that alternate between two different configurations. Particularly, the networks corresponding to each tier changes between two networks alternatively in odd and even time spans, i.e., each adjacency matrices $\mathscr{A}^{[1,1]}(t)$, $\mathscr{A}^{[1,2]}(t)$, $\mathscr{A}^{[2,1]}(t)$, and $\mathscr{A}^{[2,2]}(t)$ have two network realizations corresponding to odd and even times. The interlayer connections are all to all, i.e., one node in a layer is linked with each node of the other layer.

\begin{figure}[ht]
\centerline{\includegraphics[scale=0.4]{fig9.png}}
\caption{ Variation of intralayer synchronization error in (a) $(\epsilon_1,\epsilon_2)$, (b) $(\epsilon_1,\lambda)$ plane. The other parameter values are $\lambda=0.1$ for (a), and for (b) $\epsilon_2=0.05$, respectively. The solid {magenta} lines corresponds to the intralayer synchronization threshold obtained from analytical prediction.}\label{fig9}
\end{figure}
We first scrutinize the region of synchronization and desynchronization by evaluating the synchronization error $E_{intra}$. The corresponding results are portrayed in Fig. \ref{fig9}. Figure \ref{fig9}(a) describes $E_{intra}(\epsilon_1,\epsilon_2)$ for a fixed interlayer coupling strength $\lambda=0.1$. As $\epsilon_2$ increases the decreasing critical transition point of $\epsilon_1$ to achieve synchrony, confirms enhancement of intralayer synchronization. Figure \ref{fig9}(b) illustrates $E_{intra}(\epsilon_1, \lambda)$ for fixed intralayer coupling strength $\epsilon_2=0.05$ for tier-2. The synchronization threshold against $\epsilon_1$ is almost same with increasing $\lambda$ up to $\lambda \approx 0.5$. For $\lambda$ just greater than $0.5$, the critical point for $\epsilon_1$ suddenly decreases to $\epsilon_1=0.47$. Beyond that the critical coupling to achieve synchronization decreases as $\lambda$ increases. Similar phenomenon is also observed in $(\epsilon_{2},\lambda)$ parameter plane for fixed $\epsilon_1$ (corresponding figure is not shown here).

\begin{figure}[ht]
\centerline{\includegraphics[scale=0.85]{fig10.png}}
\caption{The block representations of all the non-commuting matrices in the set $S_{1}$ after applying SBD algorithm are portrayed from (a) to (i). Here the non-zero elements are colored in light to deep black according to their increasing values and the zero elements are colored in white.}\label{fig10}
\end{figure}

\par Next, we study the analytical stability of the synchronization state using SBD approach. As the network changes its configuration between even and odd times, the Laplacians associated with each tier also varies. Therefore, as discussed in Sec.\ref{sbd}, the variational equation \eqref{eq.12} will have a set of non-commuting matrices        $ \\S_1=\{U^{[1,1]}(\mbox{odd}), U^{[1,1]}(\mbox{even}),U^{[1,2]}(\mbox{odd}), U^{[1,2]}(\mbox{even}),\\\dots,U^{[2,2]}(\mbox{odd}), U^{[2,2]}(\mbox{even}),\mathscr{B}\}$. Here $U^{[l,\beta]}(\mbox{odd})$ and $U^{[l,\beta]}(\mbox{even})$, $l,\beta=1,2$ are associated with the network configuration at even and odd times. Applying SBD approach, we transform the above set of non-commuting matrices in common block diagonal form (Fig. \ref{fig10}). We notice that the set of matrices transforms into block diagonal matrices with blocks of sizes $1\times 1$, $2 \times 2$, $3 \times 3$ and $4 \times 4$. Among them two $1\times 1$ blocks are associated with the parallel modes to synchronization manifold and all the other blocks are associated with the transverse modes to synchronization manifold. Since, the maximum size of block is $4 \times 4$, we can rewrite the higher-dimensional transverse system in terms of lower-dimensional subsystems as,
	\begin{equation} \label{eq.19}
		\begin{array}{l}
			\delta\dot{x}_{1i}= \sigma(\delta y_{1i}-\delta x_{1i}) -\epsilon_{1}\sum\limits_{j=1}^{4}{\mathscr{C}_{ij}^{[1,1]}(t)}\delta x_{1i} \\~~~~~~~~~~~~ -\lambda \mathscr{D}_{ii}^{[1]}\cos(x_2-x_1)\delta x_{1i},     \\

			\delta\dot{y}_{1i}=(\rho-z_1)\delta x_{1i}-\delta y_{1i} -x_{1}\delta z_{1i}-\epsilon_2 \sum\limits_{j=1}^{4}{\mathscr{C}_{ij}^{[1,2]}(t)} \delta y_{1i}, \\

			\delta\dot{z}_{1i}=y_{1}\delta x_{1i} +x_{1}\delta y_{1i}-\alpha \delta z_{1i}, \\\\

			\delta\dot{x}_{2i}= \sigma(\delta y_{2i}-\delta x_{2i}) -\epsilon_{1}\sum\limits_{j=1}^{4}{\mathscr{C}_{ij}^{[2,1]}(t)}\delta x_{2i}
			\\~~~~~~~~~~~~ -\lambda \mathscr{D}_{ii}^{[2]} \cos(x_1-x_2)\delta x_{2i},     \\

			\delta\dot{y}_{2i}=(\rho-z_2)\delta x_{2i}-\delta y_{2i} -x_{2}\delta z_{2i}-\epsilon_2 \sum\limits_{j=1}^{4}{\mathscr{C}_{ij}^{[2,2]}(t)}\delta y_{2i}, \\

			\delta\dot{z}_{2i}=y_{2}\delta x_{2i} +x_{2}\delta y_{2i}-\alpha \delta z_{2i},    \hspace{10 pt} i=1,2,3,4.

		\end{array}
	\end{equation}

	\begin{equation} \label{eq.20}
		\begin{array}{l}
			\delta\dot{x}_{1i}= \sigma(\delta y_{1i}-\delta x_{1i}) -\epsilon_{1}{\mathscr{C}_{ii}^{[1,1]}(t)}\delta x_{1i} \\ ~~~~~~~~~~ -\lambda \mathscr{D}_{ii}^{[1]}\cos(x_2-x_1)\delta x_{1i},     \\

			\delta\dot{y}_{1i}=(\rho-z_1)\delta x_{1i}-\delta y_{1i} -x_{1}\delta z_{1i}-\epsilon_2 {\mathscr{C}_{ii}^{[1,2]}(t)} \delta y_{1i}, \\

			\delta\dot{z}_{1i}=y_{1}\delta x_{1i} +x_{1}\delta y_{1i}-\alpha \delta z_{1i}, \\\\

			\delta\dot{x}_{2i}= \sigma(\delta y_{2i}-\delta x_{2i}) -\epsilon_{1}{\mathscr{C}_{ii}^{[2,1]}(t)}\delta x_{2i} \\ ~~~~~~~~~ -\lambda \mathscr{D}_{ii}^{[2]} \cos(x_1-x_2)\delta x_{2i}),     \\

			\delta\dot{y}_{2i}=(\rho-z_2)\delta x_{2i}-\delta y_{2i} -x_{2}\delta z_{2i}-\epsilon_2 {\mathscr{C}_{ii}^{[2,2]}(t)}\delta y_{2i}, \\

			\delta\dot{z}_{2i}=y_{2}\delta x_{2i} +x_{2}\delta y_{2i}-\alpha \delta z_{2i},   \hspace{10 pt} i=5,6,7

		\end{array}
	\end{equation}
where $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are the synchronized solutions. The matrices $\mathscr{C}^{[l,\beta]}(t)$, and $\mathscr{D}^{[l]}$, $l,\beta=1,2$ are the block diagonalized form of the intralayer Laplacians $\mathscr{L}^{[l,\beta]}(t)$ and the interlayer adjacencies $\mathscr{B}^{[1]},\mathscr{B}^{[2]}$, after discarding the $1\times1$ blocks associated with parallel mode. The maximum Lyapunov exponent corresponding to this lower-dimensional subsystems \eqref{eq.19}, \eqref{eq.20} gives the condition for stable intralayer coherent state of the temporal multiplex hypernetwork. In Figs. \ref{fig9}(a) - \ref{fig9}(c), the continuous {magenta} lines represent the curves corresponding to the synchronization threshold MLE $=0$. The regions below and above these curves respectively depict MLE $>0$ and MLE $<0$. Value of negative MLE gives stable synchronization state and positive MLE signifies unstable synchrony. These maximum Lyapunov exponent plots are found to be suitably confirmed with our numerical experiments.

\section{Conclusion}\label{conclusion}
\par Here we have considered a universal model attributing for temporal multilayer hypernetwork framework with arbitrary sorts of intralayer and interlayer coupling functions, and explicitly carried out the stability analysis of intralayer synchronous state. We derived the invariance condition of the coherent state and based on that condition look into the derivation of necessary condition for stable synchronization solution. The process of stability analysis is conducted by means of two well-known concepts - the fast switching approach and simultaneous block diagonalization technique. Employing fast switching approach, we show that the temporal network achieves stable intralayer synchronization state whenever corresponding static time-average structure is in stable intralayer coherent state. The condition for stable synchronous motion is derived with respect to time-averaged network structure, and we have delineated that, in some instances, our method resembles classical MSF approach. On the other hand, using simultaneous block diagonalization procedure, we derive the condition for stability of intralayer synchronous solution for temporal networks, without restricting to time-average network structure. In this case, we have shown that the transverse modes associated with synchronization manifold can be decoupled into finer form without any assumption. Finally, the analytical inferences are accompanied by several numerical results, which confirms the sustainability and universality of our approach. We expect, our study will pave the way for a new understanding of collective properties emerge in temporal multilayer framework. The use of our methods irrespective of coupling functions provide the scope to apply it for exploration of large class of coupling schemes associated with complex systems that can be represented by multilayer networks.

redpajama_set_name: RedPajamaArXiv


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You are here: Home / Politics / Iowa lawmakers applaud reversal on farm child labor rules
Iowa lawmakers applaud reversal on farm child labor rules
April 27, 2012 by O.Kay Henderson
The Obama Administration has permanently shelved controversial rules that would have limited what kids under the age of 16 can do on the farm. Iowa's politicians are applauding, figuratively, via written statements. You may read them below. (No statement from Senator Tom Harkin, chairman of the Senate HELP Committee — the L stands for Labor.)
(DES MOINES) – Gov. Terry E. Branstad and Lt. Gov. Kim Reynolds released a statement regarding the U.S. Department of Labor's withdrawal of the proposed rules that would have placed burdensome restrictions on the activities of young people in agriculture.
Gov. Branstad and Iowa Secretary of Agriculture Bill Northey sent a letter to U.S. Secretary of Labor Hilda Solis in November, 2011, expressing concerns over these regulations. Governor Branstad also discussed his continued concerns with U.S. Department of Labor officials and the congressional delegation during his February trip to Washington, DC. In addition, both he and Lt. Governor Reynolds discussed the proposed rules on Monday in a roundtable discussion with FFA members and State leaders at the Iowa Capitol.
Gov. Terry Branstad stated the following:
"I applaud this announcement by the U.S. Department of Labor to withdraw their proposed rules for young people participating in agriculture. The proposed rules were a prime example of federal overreach and it is unfortunate that they were proposed in the first place. That said, I am glad that common sense has prevailed. The parents, grandparents, aunts, uncles, and neighbors who care the most about the young people starting a career in agriculture are best positioned to determine the capabilities and safety of the kids they love. Agriculture continues to be a bright spot in the U.S. economy and we should continue to oppose regulations that lack common sense.
"I look forward to continuing to work with agricultural stakeholders, including the FFA and 4-H, to build upon successful grassroots initiatives that truly help to continually improve agricultural working practices for young people."
Lt. Governor Reynolds stated the following:
"I am heartened to see this change in course from the U.S. Department of Labor. The governor and I have heard from numerous Iowans that these proposed rules would have prevented young people from learning career and life skills through active participation in livestock operations and many aspects of crop production. In addition, there were significant concerns that the federal government was seeking to narrow the definition of the parental exemption in a way that would have been totally disjointed from the realities and structures of current family farms. The Governor and I discussed these rules with Iowa FFA students and Secretary Northey earlier this week. We would also like to thank those Members of the Iowa congressional delegation who were actively engaged in this issue, including the efforts of Congressman Tom Latham to preempt these regulations with legislation if the rules had not been withdrawn."
WASHINGTON, D.C. – Congressman Leonard Boswell (IA-3) issued a statement on the Department of Labor's withdrawal of the child farm labor rule.
"I am pleased the Department of Labor stopped pursing the issue of limiting youth work on the family farm. Reasoned heads prevailed," Boswell said. "There are better, already existing educational resources through farm groups to educate and promote safety for young farm hands. I will work with those groups to keep farm workers of all ages safe."
Washington, D.C. – Rep. Bruce Braley (IA-01) released the following statement after the US Department of Labor announced it was withdrawing a proposed regulation that would have made it more difficult for farmers and ranchers to hire youth to work in agriculture:
"The demise of the Obama administration's proposed rule to require children be a minimum age to work on farms is welcome news. A regulation prohibiting youths from working on farms would strike at the very core of agriculture across Iowa and the Midwest. This is Iowa. Working on the family farm is part of growing up. I know — I remember many hot summer days I spent as a kid detassling corn in the fields. I'll keep working to ensure misguided regulations like this one don't see the light of day."
In December, Braley wrote to Labor Secretary Hilda Solis urging her to drop the proposed rule.
Washington, D.C. – Congressman Dave Loebsack today praised the U.S. Department of Labor (DOL) for withdrawing a proposed rule dealing with youth who work in agriculture. Last week, Loebsack met with the DOL to urge them to review Iowans' concerns that part of the proposal would limit activities children could perform on family farms and the effect the updates could have on student education programs in rural areas.
"Iowa farmers have a long and proud tradition of feeding the world," said Loebsack. "After hearing from concerned family farmers from across Iowa and the nation, I am pleased the Department of Labor has taken this step to help protect Iowa's way of life. When I met with the Department of Labor, I stressed the importance of better cooperation and outreach to the agriculture community and raised concerns about this rule. I applaud their commitment to working in a cooperative way to make our farms safer for kids."
In addition to meeting with the DOL, Loebsack wrote to the Secretary of Labor to urge the Department to ensure Iowa farmers and families had the opportunity to be heard regarding these proposals and urged the Department to reconsider the provisions relating to the parental exemption for children helping on the farm, which had already been withdrawn prior to today.
Washington, DC- Congressman Steve King (R-IA) released the following statement today after the Department of Labor announced that it will withdraw its proposed rule to prohibit young people from working on family farms and participating in agriculture education programs including Future Farmers of America and 4-H.
"Once again President Obama's overreaching policies have outraged the American people," said King. "I'm happy to see the backlash surrounding this rule has prompted the Department of Labor to withdraw the proposed rule. The Department of Agriculture collaborated with the Department of Labor on this direct assault of American family farms. Family farms are the core of American culture, and for centuries they have played a large role in instilling a strong work ethic in young people. I know because I raised my three sons in rural Iowa and taught them the value of hard work."
"I'm pleased to have had the support of many farm families as I've worked to oppose this rule in order to ensure that young people will still have the same opportunities. This policy was not only unnecessary, it threatened the very way of life that I and so many others hold dear. When will President Obama realize that his out of touch policies do not reflect the values of the American people? His actions continue to show that he is disconnected with the people in the heartland of this great country."
Senator Chuck Grassley released the following statement after learning that the Department of Labor would be withdrawing the entirety of its proposed regulations that would have put stringent restrictions on young people working on family farms. Grassley, one of the only working family farmers in the U.S. Senate, has vigorously defended the opportunity for kids to work on family farms. He has said that generations of Iowans have cut their teeth working on the farm, whether for their own family, or a neighbor's farm and that young people are a valuable part of farming operations.
"It's good the Labor Department rethought the ridiculous regulations it was going to stick on farmers and their families. It would have been devastating to farm families across the country. Much of rural America was built on families helping families, neighbors helping neighbors. To even propose such regulations defies common sense, and shows a real lack of understanding as to how the family farm works. I'm glad the Obama administration came to its senses."
Former IDP chair says 'plant our flag,' and hold Iowa Caucuses first January 26, 2023
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redpajama_set_name: RedPajamaCommonCrawl


The prince continued to regard Nastasia with a sorrowful, but intent and piercing, gaze.
He had left things quiet and peaceful; the invalid was fast asleep, and the doctor, who had been called in, had stated that there was no special danger. Lebedeff, Colia, and Burdovsky were lying down in the sick-room, ready to take it in turns to watch. There was nothing to fear, therefore, at home.
"Let's go," said Rogojin, touching his shoulder. They left the alcove and sat down in the two chairs they had occupied before, opposite to one another. The prince trembled more and more violently, and never took his questioning eyes off Rogojin's face.
"As for yesterday's episode," continued Gania, "of course it was pre-arranged." Here he paused, as though expecting to be asked how he knew that. But the prince did not inquire. Concerning Evgenie Pavlovitch, Gania stated, without being asked, that he believed the former had not known Nastasia Philipovna in past years, but that he had probably been introduced to her by somebody in the park during these four days. As to the question of the IOU's she had spoken of, there might easily be something in that; for though Evgenie was undoubtedly a man of wealth, yet certain of his affairs were equally undoubtedly in disorder. Arrived at this interesting point, Gania suddenly broke off, and said no more about Nastasia's prank of the previous evening.
But by this time they had reached Gania's house.
The amiable and undoubtedly witty Prince N. could not but feel that he was as a sun, risen for one night only to shine upon the Epanchin drawing-room. He accounted them immeasurably his inferiors, and it was this feeling which caused his special amiability and delightful ease and grace towards them. He knew very well that he must tell some story this evening for the edification of the company, and led up to it with the inspiration of anticipatory triumph.
"What! surely not?" said Aglaya.
The prince did not die before his wedding--either by day or night, as he had foretold that he might. Very probably he passed disturbed nights, and was afflicted with bad dreams; but, during the daytime, among his fellow-men, he seemed as kind as ever, and even contented; only a little thoughtful when alone.
"I do so want to hear about it," repeated Adelaida.
Then the sky cleared in a moment. The prince seemed to arise from the dead; he asked Colia all about it, made him repeat the story over and over again, and laughed and shook hands with the boys in his delight.
"Besides," said Burdovsky, "the prince would not like it, would he?" So they gave up the pursuit.
"You don't seem to want to tell us," said Aglaya, with a mocking air.
"I want to explain all to you--everything--everything! I know you think me Utopian, don't you--an idealist? Oh, no! I'm not, indeed--my ideas are all so simple. You don't believe me? You are smiling. Do you know, I am sometimes very wicked--for I lose my faith? This evening as I came here, I thought to myself, 'What shall I talk about? How am I to begin, so that they may be able to understand partially, at all events?' How afraid I was--dreadfully afraid! And yet, how _could_ I be afraid--was it not shameful of me? Was I afraid of finding a bottomless abyss of empty selfishness? Ah! that's why I am so happy at this moment, because I find there is no bottomless abyss at all--but good, healthy material, full of life.
Gavrila Ardalionovitch nodded to the prince and entered the room hastily.
"One moment, my dear prince, just one. I must absolutely speak to you about something which is most grave," said Lebedeff, mysteriously and solemnly, entering the room with a bow and looking extremely important. He had but just returned, and carried his hat in his hand. He looked preoccupied and most unusually dignified.
"H'm!" grunted the astonished servant.
"She's mad--she's mad!" was the cry.
"Forgive a silly, horrid, spoilt girl"--(she took his hand here)--"and be quite assured that we all of us esteem you beyond all words. And if I dared to turn your beautiful, admirable simplicity to ridicule, forgive me as you would a little child its mischief. Forgive me all my absurdity of just now, which, of course, meant nothing, and could not have the slightest consequence." She spoke these words with great emphasis.
"Oh, don't think that I have no sense of my own humiliation! I have suffered already in reading so far. Which of you all does not think me a fool at this moment--a young fool who knows nothing of life--forgetting that to live as I have lived these last six months is to live longer than grey-haired old men. Well, let them laugh, and say it is all nonsense, if they please. They may say it is all fairy-tales, if they like; and I have spent whole nights telling myself fairy-tales. I remember them all. But how can I tell fairy-tales now? The time for them is over. They amused me when I found that there was not even time for me to learn the Greek grammar, as I wanted to do. 'I shall die before I get to the syntax,' I thought at the first page--and threw the book under the table. It is there still, for I forbade anyone to pick it up.
"Of course it is all, my friend. I don't doubt you for a moment," said Lizabetha Prokofievna with dignity.
He felt in a very curious condition today, a condition similar to that which had preceded his fits in bygone years.
"To this keen question I replied as keenly, 'The Russian heart can recognize a great man even in the bitter enemy of his country.' At least, I don't remember the exact words, you know, but the idea was as I say. Napoleon was struck; he thought a minute and then said to his suite: 'I like that boy's pride; if all Russians think like this child, then--' he didn't finish, but went on and entered the palace. I instantly mixed with his suite, and followed him. I was already in high favour. I remember when he came into the first hall, the emperor stopped before a portrait of the Empress Katherine, and after a thoughtful glance remarked, 'That was a great woman,' and passed on.
"I've--I've had a reward for my meanness--I've had a slap in the face," he concluded, tragically.
The young officer, forgetting himself, sprang towards her. Nastasia's followers were not by her at the moment (the elderly gentleman having disappeared altogether, and the younger man simply standing aside and roaring with laughter).
Hippolyte looked around at the laughing guests. The prince observed that his teeth were chattering as though in a violent attack of ague.
"It was Gogol, in Dead Souls, father," cried Colia, glancing at him in some alarm.
"I am base--base!" muttered Lebedeff, beating his breast, and hanging his head.
On the morning following the bacchanalian songs and quarrels recorded above, as the prince stepped out of the house at about eleven o'clock, the general suddenly appeared before him, much agitated.
"Yes, he would!" said Rogojin, quietly, but with an air of absolute conviction.
"I must say it's very nice of you to laugh. I see you really are a kind-hearted fellow," said Mrs. Epanchin.
"But if I beg you to make it up?" said Varia.
"Just a couple of words!" whispered another voice in the prince's other ear, and another hand took his other arm. Muishkin turned, and to his great surprise observed a red, flushed face and a droll-looking figure which he recognized at once as that of Ferdishenko. Goodness knows where he had turned up from!
"What you say is quite true," observed General Epanchin; then, clasping his hands behind his back, he returned to his place on the terrace steps, where he yawned with an air of boredom.
He did not dare look at her, but he was conscious, to the very tips of his fingers, that she was gazing at him, perhaps angrily; and that she had probably flushed up with a look of fiery indignation in her black eyes.
The prince sat silent for a long while. His mind was filled with dread and horror.
"What have you got there?" asked the prince, with some anxiety.
Of such people there are countless numbers in this world--far more even than appear. They can be divided into two classes as all men can--that is, those of limited intellect, and those who are much cleverer. The former of these classes is the happier.
So saying, Rogojin crossed the road.
"Nastasia Philipovna, is this really you? You, once so refined and delicate of speech. Oh, what a tongue! What dreadful things you are saying," cried the general, wringing his hands in real grief.
No sooner had the carriage driven off than the door opened once more; and Rogojin, who had apparently been awaiting them, let them in and closed it after them.
But the prince was not satisfied with what he had said to Rogojin. Only at this moment, when she suddenly made her appearance before him, did he realize to the full the exact emotion which she called up in him, and which he had not described correctly to Rogojin.
"There were a couple of old bullets in the bag which contained the pistol, and powder enough in an old flask for two or three charges.
"Who has been annoying her? Who has been tormenting the child? Who could have said such a thing to her? Is she raving?" cried Lizabetha Prokofievna, trembling with rage, to the company in general.
Little by little a sort of inspiration, however, began to stir within him, ready to spring into life at the right moment. When he did begin to speak, it was accidentally, in response to a question, and apparently without any special object.
He saw, for instance, that one important dignitary, old enough to be his grandfather, broke off his own conversation in order to listen to _him_--a young and inexperienced man; and not only listened, but seemed to attach value to his opinion, and was kind and amiable, and yet they were strangers and had never seen each other before. Perhaps what most appealed to the prince's impressionability was the refinement of the old man's courtesy towards him. Perhaps the soil of his susceptible nature was really predisposed to receive a pleasant impression.
"One more second and I should have stopped him," said Keller, afterwards. In fact, he and Burdovsky jumped into another carriage and set off in pursuit; but it struck them as they drove along that it was not much use trying to bring Nastasia back by force.
Meanwhile all these people--though friends of the family and of each other to a certain extent--were very far from being such intimate friends of the family and of each other as the prince concluded. There were some present who never would think of considering the Epanchins their equals. There were even some who hated one another cordially. For instance, old Princess Bielokonski had all her life despised the wife of the "dignitary," while the latter was very far from loving Lizabetha Prokofievna. The dignitary himself had been General Epanchin's protector from his youth up; and the general considered him so majestic a personage that he would have felt a hearty contempt for himself if he had even for one moment allowed himself to pose as the great man's equal, or to think of him--in his fear and reverence--as anything less than an Olympic God! There were others present who had not met for years, and who had no feeling whatever for each other, unless it were dislike; and yet they met tonight as though they had seen each other but yesterday in some friendly and intimate assembly of kindred spirits.
Rogojin suffered from brain fever for two months. When he recovered from the attack he was at once brought up on trial for murder.
"Oh, I forgive him with all my heart; you may tell him so if you like," laughed Evgenie.


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