Scaled Signed Averaging (SSA)

• Scaled Signed Averaging (SSA) is a distributed averaging framework for directed signed networks that replaces the traditional weight-balance requirement with an improved Laplacian potential.
• It introduces a node-specific scaling matrix derived from principal minors of the induced unsigned Laplacian to ensure exact signed-average consensus in balanced networks and stability in unbalanced ones.
• The framework offers two continuous-time protocols and a fixed-time extension, providing practical insights into convergence acceleration and precise consensus metrics.

Scaled Signed Averaging (SSA) is a framework for distributed averaging over directed signed networks in which graph-theoretic scaling is built into the consensus dynamics through a diagonal matrix $W$ constructed from principal minors of an induced unsigned Laplacian. In the formulation developed for signed networks, SSA replaces the weight-balance requirement of classical signed-consensus analyses with an “improved Laplacian potential” and yields two nearest-neighbor protocols that achieve signed-average consensus exactly when the underlying signed digraph is structurally balanced, while driving the state to zero when it is structurally unbalanced (Du et al., 2019).

1. Problem formulation and consensus objective

The SSA framework considers a network of $n$ agents,

$V=\{v_1,\ldots,v_n\},$

with scalar agent states $x_i\in\mathbb{R}$. Information flow is modeled by a signed directed graph

$\mathcal{G}=(V,E,A),$

where $A=[a_{ij}]\in\mathbb{R}^{n\times n}$ satisfies

$$a_{ij}\neq 0 \iff (v_j\to v_i)\in E,\qquad a_{ii}=0,$$

and the digon-sign-symmetry condition $a_{ij}a_{ji}\ge 0$ is assumed. Positive $a_{ij}$ represents a cooperative interaction and negative $a_{ij}$ an antagonistic interaction. The in-degree matrix is

$n$0

and the signed Laplacian is

$n$1

(Du et al., 2019).

Within this setting, SSA distinguishes two asymptotic behaviors. The first is signed-average consensus, defined by the existence of signs $n$2 such that

$n$3

This is the bipartite-consensus case in which agents split into two sign classes with equal-magnitude limiting values. The second is state stability, meaning $n$4. The convergence theorems are stated under the standing assumption that $n$5 is strongly connected (Du et al., 2019).

A central motivation for SSA is that the standard quadratic potential used in signed-consensus analysis coincides with $n$6 only under weight balance. SSA removes that restriction by introducing a graph-dependent scaling that compensates for weight-unbalanced directed topologies (Du et al., 2019).

2. Graph constructions and the scaling matrix

SSA is built from two auxiliary graph constructions. The first is the induced unsigned digraph

$n$7

with Laplacian

$n$8

For each $n$9, $V=\{v_1,\ldots,v_n\},$0 denotes the $V=\{v_1,\ldots,v_n\},$1 principal minor obtained by deleting row $V=\{v_1,\ldots,v_n\},$2 and column $V=\{v_1,\ldots,v_n\},$3 (Du et al., 2019).

The scaling matrix is then defined as

$V=\{v_1,\ldots,v_n\},$4

These $V=\{v_1,\ldots,v_n\},$5 are the graph-theoretic quantities that supply the “scaled” part of SSA. They depend only on the induced unsigned digraph and vary across nodes when the topology is weight-unbalanced (Du et al., 2019).

The second construction is the mirror undirected signed graph, whose adjacency is

$V=\{v_1,\ldots,v_n\},$6

with Laplacian

$V=\{v_1,\ldots,v_n\},$7

A key identity is

$V=\{v_1,\ldots,v_n\},$8

In addition, if $V=\{v_1,\ldots,v_n\},$9 is strongly connected, then the mirror graph $x_i\in\mathbb{R}$0 is connected. Structural balance is also preserved under the construction: $x_i\in\mathbb{R}$1 and similarly for structural unbalance (Du et al., 2019).

These constructions convert a directed signed-network problem into a form in which symmetric energy methods apply without assuming ordinary weight balance.

3. Improved Laplacian potential and its spectral meaning

In classical signed-consensus analysis one often considers

$x_i\in\mathbb{R}$2

but this coincides with $x_i\in\mathbb{R}$3 only when the graph is weight-balanced. SSA modifies the potential by inserting the node-dependent scaling $x_i\in\mathbb{R}$4 (Du et al., 2019): $x_i\in\mathbb{R}$5

This improved Laplacian potential satisfies the exact identities

$x_i\in\mathbb{R}$6

The first equality links the potential to the original directed signed graph through $x_i\in\mathbb{R}$7, and the second links it to the mirror signed graph through a symmetric quadratic form (Du et al., 2019).

Its definiteness properties encode the network’s balance structure. For a strongly connected $x_i\in\mathbb{R}$8,

• if $x_i\in\mathbb{R}$9 is structurally balanced, then $\mathcal{G}=(V,E,A),$0 and $\mathcal{G}=(V,E,A),$1 are positive semidefinite, with

$\mathcal{G}=(V,E,A),$2

where $\mathcal{G}=(V,E,A),$3 is the gauge such that $\mathcal{G}=(V,E,A),$4;

• if $\mathcal{G}=(V,E,A),$5 is structurally unbalanced, then $\mathcal{G}=(V,E,A),$6 is positive definite, so

$\mathcal{G}=(V,E,A),$7

This is the analytic core of SSA. The scaling does not merely reweight the protocol; it furnishes a Lyapunov-type measure whose nullspace and positivity directly separate bipartite consensus from asymptotic stability (Du et al., 2019).

4. Distributed SSA protocols and convergence theorems

SSA yields two nearest-neighbor continuous-time protocols. The first uses the mirror signed graph: $\mathcal{G}=(V,E,A),$8 which in vector form is

$\mathcal{G}=(V,E,A),$9

The second applies the scaling directly to the original signed Laplacian: $A=[a_{ij}]\in\mathbb{R}^{n\times n}$0 with vector form

$A=[a_{ij}]\in\mathbb{R}^{n\times n}$1

Since $A=[a_{ij}]\in\mathbb{R}^{n\times n}$2, the spectrum of $A=[a_{ij}]\in\mathbb{R}^{n\times n}$3 is exactly the same as that of a weighted Laplacian (Du et al., 2019).

Protocol	Node-level update	Vector form
Protocol 1	Uses $A=[a_{ij}]\in\mathbb{R}^{n\times n}$4 and $A=[a_{ij}]\in\mathbb{R}^{n\times n}$5	$A=[a_{ij}]\in\mathbb{R}^{n\times n}$6
Protocol 2	Uses $A=[a_{ij}]\in\mathbb{R}^{n\times n}$7 and $A=[a_{ij}]\in\mathbb{R}^{n\times n}$8	$A=[a_{ij}]\in\mathbb{R}^{n\times n}$9

For both protocols, the convergence statements are identical under strong connectivity. Signed-average consensus holds if and only if $$a_{ij}\neq 0 \iff (v_j\to v_i)\in E,\qquad a_{ii}=0,$$0 is structurally balanced, and state stability holds if and only if $$a_{ij}\neq 0 \iff (v_j\to v_i)\in E,\qquad a_{ii}=0,$$1 is structurally unbalanced (Du et al., 2019).

For Protocol 1, the proof uses

$$a_{ij}\neq 0 \iff (v_j\to v_i)\in E,\qquad a_{ii}=0,$$2

which yields

$$a_{ij}\neq 0 \iff (v_j\to v_i)\in E,\qquad a_{ii}=0,$$3

If the graph is balanced, LaSalle’s invariance principle gives convergence to

$$a_{ij}\neq 0 \iff (v_j\to v_i)\in E,\qquad a_{ii}=0,$$4

hence bipartite consensus, and the left/right eigen-projector calculation recovers the exact signed average

$$a_{ij}\neq 0 \iff (v_j\to v_i)\in E,\qquad a_{ii}=0,$$5

If the graph is unbalanced, $$a_{ij}\neq 0 \iff (v_j\to v_i)\in E,\qquad a_{ii}=0,$$6 for all $$a_{ij}\neq 0 \iff (v_j\to v_i)\in E,\qquad a_{ii}=0,$$7, so the origin is asymptotically stable (Du et al., 2019).

For Protocol 2, the same if-and-only-if structure follows from the spectral characterization of $$a_{ij}\neq 0 \iff (v_j\to v_i)\in E,\qquad a_{ii}=0,$$8: $$a_{ij}\neq 0 \iff (v_j\to v_i)\in E,\qquad a_{ii}=0,$$9 has exactly one zero eigenvalue if and only if the graph is balanced, and is Hurwitz if and only if the graph is unbalanced (Du et al., 2019).

A common misconception is that structural balance should imply ordinary consensus. In directed signed networks, the relevant limiting behavior is generally bipartite consensus, and SSA identifies the correct invariant quantity as the signed average rather than the unsigned arithmetic average.

5. Fixed-time extension and numerical behavior

SSA also admits a fixed-time extension based on a two-term homogeneous feedback law applied to the mirror-graph disagreement term: $a_{ij}a_{ji}\ge 0$0 where $a_{ij}a_{ji}\ge 0$1 and $a_{ij}a_{ji}\ge 0$2, $a_{ij}a_{ji}\ge 0$3 are odd integers. In vector form, $a_{ij}a_{ji}\ge 0$4 with a 2-homogeneous mapping (Du et al., 2019).

Under strong connectivity, the fixed-time results separate the balanced and unbalanced cases:

• if $a_{ij}a_{ji}\ge 0$5 is balanced, fixed-time bipartite consensus is achieved within

$a_{ij}a_{ji}\ge 0$6

• if $a_{ij}a_{ji}\ge 0$7 is unbalanced, fixed-time stability is achieved within

$a_{ij}a_{ji}\ge 0$8

The Lyapunov function is again the improved potential,

$a_{ij}a_{ji}\ge 0$9

and the derivative estimate takes the form

$a_{ij}$0

which is compared with the scalar system

$a_{ij}$1

to obtain the time bound (Du et al., 2019).

The numerical example in the source considers $a_{ij}$2 agents on two strongly connected, weight-unbalanced digraphs, one balanced and one unbalanced, with

$a_{ij}$3

Under the original unscaled Altafini protocol

$a_{ij}$4

the balanced graph converges to approximately $a_{ij}$5, not the correct signed average $a_{ij}$6, while the unbalanced graph converges to zero. Under SSA Protocol 1 or Protocol 2, the balanced graph converges exactly to $a_{ij}$7,

$a_{ij}$8

and the unbalanced graph again converges to zero. Because the values $a_{ij}$9 differ from row to row, convergence under Protocol 2 is reported as slightly faster than under an unweighted version, illustrating that the SSA weights can accelerate agreement without changing the final value (Du et al., 2019).

6. Terminological scope and unrelated uses of “SSA”

The acronym “SSA” is not unique to signed-network averaging, and this creates a persistent source of confusion across fields.

In power electronics and switching-system theory, “SSA” commonly denotes state-space averaging rather than scaled signed averaging. An operator-theoretic reconstruction of classical state-space averaging shows that, for a piecewise-linear switching system with two subintervals, the classical averaged matrix

$a_{ij}$0

arises as the leading-order truncation of an exact matrix-logarithm reconstruction of the Poincaré map. That work also explains why state-space averaging relies on low-frequency and small-ripple assumptions and why it becomes fragile for converters with more than two subintervals per cycle (Yang et al., 20 Dec 2025). This usage is terminologically unrelated to distributed averaging over signed networks.

A second unrelated usage appears in transformer research. In the study “Improving in-context learning with a better scoring function,” scaled signed averaging is proposed as an alternative to Softmax in the attention mechanism. The abstract states that LLMs exhibit limitations on tasks involving first-order quantifiers such as all and some, as well as on in-context learning with linear functions; it identifies Softmax as a contributing factor, proposes scaled signed averaging, and reports that SSA dramatically improves performance on the target tasks while matching or exceeding Softmax-based encoder-only and decoder-only transformers across a variety of linguistic probing tasks (Naim et al., 20 Aug 2025).

These cross-domain usages do not share a common formalism. In signed networks, SSA refers to graph-scaled distributed averaging built from $a_{ij}$1, $a_{ij}$2, and $a_{ij}$3. In power electronics, SSA denotes a continuous-time approximation regime for switched systems. In transformer attention, SSA denotes a scoring-function alternative to Softmax. The shared acronym therefore has field-specific meaning, and precision requires identifying the domain explicitly.