Multiplex Synchrony Alignment Function

• MSAF is a spectral functional that generalizes the single-layer SAF to optimize phase synchronization in multiplex networks by integrating structural and dynamical information.
• It provides explicit constructions for undirected duplex and directed networks using effective Laplacians, pseudoinverse methods, and singular value decomposition.
• MSAF serves as an optimization objective through tuning natural frequencies, rewiring network structures, and permuting node assignments to enhance synchrony.

Searching arXiv for the supplied topic and related synchronization papers to ground the article in current literature. The Multiplex Synchrony Alignment Function (MSAF) is a spectral functional for synchronization optimization in multiplex networks of phase oscillators. In its explicit form, it was derived for duplex multiplex networks with frustrated and non-frustrated Kuramoto dynamics, where it integrates structural properties of both layers and dynamical information encoded in effective frequency vectors, and its minimization yields optimal or perfect synchronization in the strong-coupling, small-phase-difference regime (Kundu et al., 2019). A directed extension replaces orthogonal Laplacian eigenmodes by singular modes of modified directed multiplex Laplacians and yields analogous optimization rules for directed duplexes with phase-lag (Das et al., 2 Apr 2026). In the broader synchronization literature on multiplex networks, the same object can be understood as a natural alignment functional built from supra-Laplacian operators and Master Stability Function (MSF) logic, although a general multiplex review explicitly notes that it does not define or name a “Multiplex Synchrony Alignment Function” (Battiston et al., 5 May 2026).

1. Conceptual placement within multiplex synchronization

MSAF generalizes the single-layer Synchrony Alignment Function (SAF) to multiplex settings. In the monolayer construction, near synchrony one writes the phase dynamics in Laplacian form, solves for the steady phase-locked state through the Laplacian pseudoinverse, and obtains a quadratic functional that weights projections of the centered frequency vector onto nontrivial Laplacian modes by the inverse squared eigenvalues. The central idea is that synchronization is improved when intrinsic heterogeneity projects weakly onto soft structural modes and more strongly onto stiff ones (Battiston et al., 5 May 2026).

In multiplex networks, the same logic is no longer layer-local. The relevant structural object is not a single Laplacian but an effective operator that already mixes the layers. In the undirected duplex setting, the MSAF is defined separately for each layer, but each layer’s effective Laplacian-like operator and effective frequency vector contain contributions from both layers (Kundu et al., 2019). In the directed duplex setting, the same principle holds, except that nonsymmetry requires a singular-value decomposition rather than an orthogonal eigen-decomposition (Das et al., 2 Apr 2026).

A recurrent misconception is to treat MSAF as a standard object of the general multiplex synchronization review literature. The 2026 review on dynamical processes in multiplex networks develops the supra-Laplacian and MSF machinery for synchronization, but explicitly states that it does not define or name an MSAF and does not explicitly treat heterogeneous natural frequencies in its synchrony analysis (Battiston et al., 5 May 2026). This suggests that MSAF occupies an intermediate position between classical Kuramoto alignment theory and general multiplex synchronizability theory: it is not merely an eigenratio criterion, but neither is it independent of MSF-style spectral stability analysis.

2. Spectral and supra-Laplacian foundations

The general multiplex synchronization framework is formulated on a multiplex network $\mathcal M$ with $N$ nodes and $M$ layers, intralayer adjacency matrices $A^{[\alpha]}$, and interlayer coupling matrices $C_i$. The relevant structural operators are the intralayer Laplacians

$L^{[\alpha]} = K^{[\alpha]} - A^{[\alpha]},$

and, for synchronization under linear and layer-independent coupling $H=V$, the supra-Laplacian

$$\mathcal L^s = \mathcal L^L + \frac{w}{g}\,\mathcal L^I, \qquad \mathcal L^L = \bigoplus_{\alpha=1}^M L^{[\alpha]}, \qquad \mathcal L^I = L^I \otimes I_N.$$

The synchronized-state stability is then governed by the eigenvalues $\{\lambda_h\}_{h=1}^{MN}$ and eigenvectors $\{\mathbf u_h\}$ of $N$0, with MSF stability condition

$N$1

For class II systems, $N$2 is the key quantity; for class III systems, the eigenratio $N$3 is also relevant (Battiston et al., 5 May 2026).

Within this formalism, a natural multiplex generalization of SAF is obtained by replacing the monolayer Laplacian $N$4 by $N$5 and the monolayer frequency vector by a multiplex frequency vector $N$6. In the near-synchronous, phase-locked regime,

$N$7

with $N$8, so that

$N$9

If $M$0 is symmetric, spectral expansion gives

$M$1

and therefore

$M$2

This leads to the multiplex alignment functional

$M$3

which is directly proportional to the squared phase dispersion (Battiston et al., 5 May 2026).

This reconstruction is not presented as a named MSAF in the review itself. However, it is fully consistent with the review’s statement that all the mathematical ingredients needed to reconstruct a multiplex generalization of the SAF are present: phase/oscillator dynamics on multiplexes, construction of Laplacians and supra-Laplacians, spectral analysis, and MSF-based stability conditions (Battiston et al., 5 May 2026).

3. Explicit MSAF for undirected duplex phase oscillators

The first explicit MSAF construction in the supplied material concerns a duplex multiplex of phase oscillators with possible phase frustration. For node $M$4 in layer $M$5, the dynamics are

$M$6

$M$7

Here $M$8 is both the intra-layer coupling strength and the inter-layer coupling strength, while $M$9 are phase-lag parameters for layers 1 and 2 (Kundu et al., 2019).

Linearization around a coherent state with small phase differences yields

$A^{[\alpha]}$0

$A^{[\alpha]}$1

with frustration-modified frequencies

$A^{[\alpha]}$2

At steady state in a co-rotating frame,

$A^{[\alpha]}$3

where the effective multiplex Laplacians are

$A^{[\alpha]}$4

$A^{[\alpha]}$5

and the effective multiplex frequencies are

$A^{[\alpha]}$6

$A^{[\alpha]}$7

These definitions make explicit that each layer’s phase-locked state depends on both layers’ structure and both layers’ modified frequencies (Kundu et al., 2019).

The order parameter in layer $A^{[\alpha]}$8 is approximated by

$A^{[\alpha]}$9

which defines the Multiplex Synchrony Alignment Function

$C_i$0

If $C_i$1, then

$C_i$2

The weighting by $C_i$3 gives low-eigenvalue modes the dominant contribution; consequently, good synchronization corresponds to effective frequencies that are orthogonal, or nearly orthogonal, to low-eigenvalue modes and aligned with large-$C_i$4 modes (Kundu et al., 2019).

The same paper gives an exact condition for perfect synchronization in layer $C_i$5,

$C_i$6

and obtains the degree-frequency prescriptions

$C_i$7

for a chosen target coupling $C_i$8. Thus, for prescribed coupling and phase-lag, perfect synchronization is achieved by choosing node frequencies linear in the degree sequence of the corresponding layer (Kundu et al., 2019).

4. Directed multiplex generalization

The directed extension treats duplex networks of Sakaguchi–Kuramoto oscillators with directed adjacency matrices $C_i$9, out-degrees $L^{[\alpha]} = K^{[\alpha]} - A^{[\alpha]},$0, and directed Laplacians

$L^{[\alpha]} = K^{[\alpha]} - A^{[\alpha]},$1

The phase dynamics are

$L^{[\alpha]} = K^{[\alpha]} - A^{[\alpha]},$2

$L^{[\alpha]} = K^{[\alpha]} - A^{[\alpha]},$3

After linearization, one obtains effective frequencies

$L^{[\alpha]} = K^{[\alpha]} - A^{[\alpha]},$4

modified Laplacians

$L^{[\alpha]} = K^{[\alpha]} - A^{[\alpha]},$5

$L^{[\alpha]} = K^{[\alpha]} - A^{[\alpha]},$6

and modified effective frequencies

$L^{[\alpha]} = K^{[\alpha]} - A^{[\alpha]},$7

$L^{[\alpha]} = K^{[\alpha]} - A^{[\alpha]},$8

As in the undirected case, each layer’s effective quantities already contain both layers (Das et al., 2 Apr 2026).

Because $L^{[\alpha]} = K^{[\alpha]} - A^{[\alpha]},$9 is generally nonsymmetric, the spectral analysis uses a singular value decomposition

$H=V$0

with pseudoinverse

$H=V$1

Near synchrony,

$H=V$2

where the directed-layer MSAF is

$H=V$3

The classical eigenvector/eigenvalue expression is thus replaced by an SVD formula involving left singular vectors and singular values. The functional role remains the same: large projections onto nontrivial modes with small singular values degrade synchrony, whereas minimizing $H=V$4 drives $H=V$5 (Das et al., 2 Apr 2026).

The directed theory yields two frequency constructions. For perfect synchronization at a prescribed coupling $H=V$6,

$H=V$7

For broad-range optimal synchronization under fixed variance and zero mean, the optimal multiplex effective frequencies are concentrated on the dominant left singular mode: $H=V$8 with the corresponding natural frequencies obtained by back-solving the linear multiplex relations (Das et al., 2 Apr 2026).

5. Optimization objectives and emergent structural–dynamical correlations

MSAF is an optimization objective in three distinct senses. First, for fixed multiplex structure, one can design natural frequencies that minimize the alignment functional and hence maximize the layer-wise and global order parameters. Second, for fixed frequencies, one can rewire the network so that the modified multiplex operators better accommodate the imposed dynamical heterogeneity. Third, for fixed topology and a fixed set of frequencies, one can permute the assignments of frequencies to nodes so as to reduce the alignment cost (Kundu et al., 2019).

In the undirected duplex construction, minimizing

$H=V$9

maximizes both $$\mathcal L^s = \mathcal L^L + \frac{w}{g}\,\mathcal L^I, \qquad \mathcal L^L = \bigoplus_{\alpha=1}^M L^{[\alpha]}, \qquad \mathcal L^I = L^I \otimes I_N.$$0 and the global order parameter

$$\mathcal L^s = \mathcal L^L + \frac{w}{g}\,\mathcal L^I, \qquad \mathcal L^L = \bigoplus_{\alpha=1}^M L^{[\alpha]}, \qquad \mathcal L^I = L^I \otimes I_N.$$1

The paper reports that optimal frequencies lead to a significantly earlier onset of synchronization than Lorentzian, uniform, or normal frequency assignments, and that explicit degree-frequency prescriptions can yield perfect synchronization in one or both layers for prescribed coupling and frustration (Kundu et al., 2019).

In the directed duplex framework, MSAF is also minimized numerically by a directed link rewiring strategy aimed at minimizing the MSAF and by a swapping algorithm for optimally assigning fixed frequencies on both layers of a given directed duplex network. The rewiring procedure preserves the number of edges in each layer and accepts a proposed change only if both layers’ MSAFs decrease; the swapping procedure accepts a permutation only if both layers’ MSAFs decrease (Das et al., 2 Apr 2026).

Examination of synchrony-optimized directed networks uncovers three notable correlations: a positive relationship between frequency and out-degree, a negative correlation between neighboring frequencies, and an anti-correlation between mirror node frequencies across directed layers (Das et al., 2 Apr 2026). In the reported ER–ER and SF–SF cases, the correlation between $$\mathcal L^s = \mathcal L^L + \frac{w}{g}\,\mathcal L^I, \qquad \mathcal L^L = \bigoplus_{\alpha=1}^M L^{[\alpha]}, \qquad \mathcal L^I = L^I \otimes I_N.$$2 and out-degree is much stronger than the corresponding correlation with in-degree, with Pearson coefficients $$\mathcal L^s = \mathcal L^L + \frac{w}{g}\,\mathcal L^I, \qquad \mathcal L^L = \bigoplus_{\alpha=1}^M L^{[\alpha]}, \qquad \mathcal L^I = L^I \otimes I_N.$$3 and $$\mathcal L^s = \mathcal L^L + \frac{w}{g}\,\mathcal L^I, \qquad \mathcal L^L = \bigoplus_{\alpha=1}^M L^{[\alpha]}, \qquad \mathcal L^I = L^I \otimes I_N.$$4 (Das et al., 2 Apr 2026). This suggests that, in directed multiplex synchronization optimization, the dominant structural control variable is not degree in the undirected sense but out-degree as the source of effective driving.

6. Other synchronization manifolds, scope, and limitations

MSAF is not tied exclusively to complete synchronization across all nodes and layers. The literature on inter-layer synchronization in multiplex networks defines a distinct invariant manifold

$$\mathcal L^s = \mathcal L^L + \frac{w}{g}\,\mathcal L^I, \qquad \mathcal L^L = \bigoplus_{\alpha=1}^M L^{[\alpha]}, \qquad \mathcal L^I = L^I \otimes I_N.$$5

on which each node is synchronous with its own replicas across layers even if nodes within the same layer are not synchronized with each other. For two identical layers with diffusive replica coupling, the transverse variational equation is

$$\mathcal L^s = \mathcal L^L + \frac{w}{g}\,\mathcal L^I, \qquad \mathcal L^L = \bigoplus_{\alpha=1}^M L^{[\alpha]}, \qquad \mathcal L^I = L^I \otimes I_N.$$6

and inter-layer synchronization is stable when the maximum transverse Lyapunov exponent is negative (Sevilla-Escoboza et al., 2015).

That paper does not explicitly define an MSAF. However, it states that the transverse variational equations, Lyapunov exponents, and synchronization errors provide the exact mathematical structure such a function would have to depend on. This suggests a manifold-specific alignment functional, for example an inter-layer synchrony alignment function identified with the maximum transverse Lyapunov exponent

$$\mathcal L^s = \mathcal L^L + \frac{w}{g}\,\mathcal L^I, \qquad \mathcal L^L = \bigoplus_{\alpha=1}^M L^{[\alpha]}, \qquad \mathcal L^I = L^I \otimes I_N.$$7

or with the inter-layer synchronization error

$$\mathcal L^s = \mathcal L^L + \frac{w}{g}\,\mathcal L^I, \qquad \mathcal L^L = \bigoplus_{\alpha=1}^M L^{[\alpha]}, \qquad \mathcal L^I = L^I \otimes I_N.$$8

as an empirical proxy (Sevilla-Escoboza et al., 2015). The same work distinguishes three regimes in the $$\mathcal L^s = \mathcal L^L + \frac{w}{g}\,\mathcal L^I, \qquad \mathcal L^L = \bigoplus_{\alpha=1}^M L^{[\alpha]}, \qquad \mathcal L^I = L^I \otimes I_N.$$9-plane: Region A (inter-layer synchronization without intra-layer synchronization), Region B (coexistence of inter-layer and intra-layer synchronization), and Region C (intra-layer synchronization without inter-layer synchronization), indicating that synchronization optimization in multiplexes can be manifold-dependent rather than reducible to a single scalar notion (Sevilla-Escoboza et al., 2015).

The main limitations of current MSAF formulations are explicit in the source material. The undirected duplex construction is built on linearization around a coherent state with small phase differences, assumes undirected, unweighted layers, and is presented for a duplex with identical intra- and inter-layer coupling strengths (Kundu et al., 2019). The directed extension removes the undirected restriction but remains a duplex theory of phase oscillators with a common coupling strength $\{\lambda_h\}_{h=1}^{MN}$0 and uses SVD of modified Laplacians under near-synchrony assumptions (Das et al., 2 Apr 2026). The general multiplex review, by contrast, provides a broader spectral and MSF framework, but focuses on identical oscillators in its synchrony analysis and therefore does not itself furnish an explicit heterogeneous-frequency MSAF (Battiston et al., 5 May 2026).

Taken together, these results place MSAF at the intersection of multiplex spectral theory, phase-oscillator reduction, and synchronization optimization. In its explicit forms, it is a quadratic spectral functional of effective multiplex frequencies and effective multiplex Laplacians; in its broader interpretation, it is the natural alignment criterion obtained whenever supra-Laplacian structure and oscillator heterogeneity jointly determine the phase-locked dispersion.