Signum Consensus Protocol Analysis

• Signum consensus protocol is a type of consensus dynamics where agents update their state using only the sign of relative differences, reducing data transmission compared to magnitude-based methods.
• It is analyzed in continuous-time systems over arbitrary weighted directed graphs, with nonsmooth analysis and Filippov solutions handling discontinuities due to bounded disturbances.
• The protocol’s performance is quantified using the Polarization Index, which serves as an exact measure of dissensus growth and a criterion for achieving strong consensus.

The signum consensus protocol denotes a class of consensus dynamics in which each agent updates its state using only the sign of relative state differences rather than their magnitude. In the most developed formulation currently available, the protocol is studied for continuous-time systems over arbitrary weighted directed graphs with bounded disturbances, where the discontinuity set has codimension greater than one and the relevant Filippov sliding vector is not uniquely determined (Li et al., 7 Sep 2025). In adjacent literature, the same terminology also refers to single-bit consensus laws based on the sign of innovations (Doostmohammadian, 2020), while related work on “sign consensus” in balanced social networks uses a linear signed protocol and explicitly does not employ a signum nonlinearity (Pasquale et al., 2021). The modern control-theoretic treatment therefore combines nonsmooth analysis, graph-theoretic structure, extremal-set dynamics, and opinion-dynamics interpretations.

1. Canonical continuous-time formulation

In the formulation on arbitrary weighted directed graphs, the communication network is a weighted directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{W})$, with agent set $\mathcal{V}=\{1,\dots,n\}$, edge set $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$, and weight matrix $$\mathcal{W}=[w_{ij}]\in\mathbb{R}_{\ge 0}^{n\times n}$$ (Li et al., 7 Sep 2025). The edge $(j,i)\in\mathcal{E}$ means agent $i$ can compare its state with agent $j$’s; $w_{ji}>0$ iff $(j,i)\in\mathcal{E}$. Each agent $i$ has scalar state $\mathcal{V}=\{1,\dots,n\}$0, and disturbances or actuator errors are modeled by $\mathcal{V}=\{1,\dots,n\}$1, componentwise bounded by

$\mathcal{V}=\{1,\dots,n\}$2

The continuous-time signum consensus protocol with disturbances is

$\mathcal{V}=\{1,\dots,n\}$3

In stacked form,

$\mathcal{V}=\{1,\dots,n\}$4

with

$\mathcal{V}=\{1,\dots,n\}$5

This model does not impose symmetry, balance, or undirectedness on $\mathcal{V}=\{1,\dots,n\}$6 (Li et al., 7 Sep 2025). No a priori connectivity assumption is required for existence, and strong connectivity alone does not guarantee consensus. That feature distinguishes this formulation from results for single-bit consensus with a spanning-tree condition (Doostmohammadian, 2020) and from linear signed models on undirected, connected, clustering-balanced graphs (Pasquale et al., 2021).

A second, simpler signum protocol appears in the single-bit consensus literature: $\mathcal{V}=\{1,\dots,n\}$7 where each edge communicates only $\mathcal{V}=\{1,\dots,n\}$8 (Doostmohammadian, 2020). For vector states, the corresponding update is

$\mathcal{V}=\{1,\dots,n\}$9

In that setting, the protocol is motivated by reduced communication and computation load, rather than by bounded disturbances or higher-codimension sliding analysis.

2. Discontinuity manifolds and Filippov solutions

The principal analytical difficulty in the arbitrary directed-graph setting is that the right-hand side is discontinuous whenever a sign argument is zero. The set of discontinuity points is

$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$0

Intersections of several such hyperplanes produce manifolds of higher codimension, up to the consensus line $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$1 of codimension $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$2 (Li et al., 7 Sep 2025). On such manifolds, the sliding vector is not uniquely determined.

To capture mutually independent evaluations of each sign function and disturbances at discontinuity points, the analysis adopts Filippov’s parametric-combination definition. The dynamics are rewritten as

$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$3

with set-valued maps

$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$4

$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$5

and

$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$6

A Filippov solution is any absolutely continuous trajectory $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$7 with $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$8 almost everywhere (Li et al., 7 Sep 2025).

On $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$9, for each pair $$\mathcal{W}=[w_{ij}]\in\mathbb{R}_{\ge 0}^{n\times n}$$0 with $$\mathcal{W}=[w_{ij}]\in\mathbb{R}_{\ge 0}^{n\times n}$$1, the term $$\mathcal{W}=[w_{ij}]\in\mathbb{R}_{\ge 0}^{n\times n}$$2 ranges over $$\mathcal{W}=[w_{ij}]\in\mathbb{R}_{\ge 0}^{n\times n}$$3. Because different sign functions and disturbances vary independently, $$\mathcal{W}=[w_{ij}]\in\mathbb{R}_{\ge 0}^{n\times n}$$4 becomes a convex polytope. Let

$$\mathcal{W}=[w_{ij}]\in\mathbb{R}_{\ge 0}^{n\times n}$$5

$$\mathcal{W}=[w_{ij}]\in\mathbb{R}_{\ge 0}^{n\times n}$$6

Then the extremal derivatives satisfy

$$\mathcal{W}=[w_{ij}]\in\mathbb{R}_{\ge 0}^{n\times n}$$7

$$\mathcal{W}=[w_{ij}]\in\mathbb{R}_{\ge 0}^{n\times n}$$8

The common velocity at which tied maximal or minimal agents move together is therefore selected from a set rather than from a singleton. This intrinsic non-uniqueness produces multiple Filippov solutions, including consensus and dissensus trajectories for the same initial condition (Li et al., 7 Sep 2025).

A common misconception is to treat all sign-based protocols as analytically equivalent. The available literature shows otherwise. The single-bit consensus paper notes that one may set $$\mathcal{W}=[w_{ij}]\in\mathbb{R}_{\ge 0}^{n\times n}$$9 or adopt a set-valued nonsmooth interpretation if needed, but its main convergence proofs are framed through Lyapunov arguments under spanning-tree assumptions (Doostmohammadian, 2020). By contrast, the arbitrary directed-graph treatment is built around non-uniqueness on codimension-$(j,i)\in\mathcal{E}$0 discontinuity manifolds (Li et al., 7 Sep 2025). The social-network paper on “sign consensus” is different again: it states that no signum nonlinearity is used (Pasquale et al., 2021).

3. Polarization Index and spread growth

The central invariant used to quantify disagreement in the arbitrary directed-graph model is the range seminorm

$(j,i)\in\mathcal{E}$1

For any nonempty $(j,i)\in\mathcal{E}$2, define

$(j,i)\in\mathcal{E}$3

and the hyperrectangle

$(j,i)\in\mathcal{E}$4

Then

$(j,i)\in\mathcal{E}$5

(Li et al., 7 Sep 2025).

The Autonomy Index is defined geometrically as

$(j,i)\in\mathcal{E}$6

where $(j,i)\in\mathcal{E}$7. Intuitively, $(j,i)\in\mathcal{E}$8 is the maximal common velocity the agents in $(j,i)\in\mathcal{E}$9 can sustain as an extremal group when tied. The Polarization Index is then

$i$0

It is the sharp upper bound on $i$1 over all Filippov selections, outside consensus (Li et al., 7 Sep 2025).

A closed-form arithmetic expression is obtained for $i$2 when $i$3: $i$4 Hence

$i$5

The spread of any Filippov solution satisfies

$i$6

Moreover, for every $i$7, there exists at least one Filippov solution $i$8, with some disturbance consistent with the bounds, such that $i$9 and

$j$0

(Li et al., 7 Sep 2025). The supremum is therefore attained at all forward times before consensus. This suggests that $j$1 is not merely a comparison quantity but an exact extremal growth rate for dissensus.

4. Consensus criteria and time bounds

The strongest consensus statement in the arbitrary directed-graph setting is a necessary and sufficient criterion for Strong Consensus: all Filippov solutions reach consensus for every initial condition if and only if

$j$2

(Li et al., 7 Sep 2025). If $j$3 and $j$4, then

$j$5

is a least upper bound on consensus time. Consensus occurs no later than $j$6, and the bound is tight in the sense that some solution attains equality.

These results depart from more classical signum-based consensus statements. In the single-bit consensus framework, the protocol

$j$7

achieves consensus if and only if the communication network has a spanning tree, and convergence is finite-time (Doostmohammadian, 2020). With

$j$8

the paper derives

$j$9

and hence

$w_{ji}>0$0

For undirected symmetric weights, the final consensus value is the average,

$w_{ji}>0$1

because the sum of states is preserved (Doostmohammadian, 2020).

The contrast is instructive. In the single-bit setting, a spanning tree is both necessary and sufficient for consensus. In the arbitrary directed-graph setting with bounded disturbances and Filippov parametric combinations, no analogous purely topological condition is sufficient; instead, consensus is characterized by the sign of $w_{ji}>0$2 (Li et al., 7 Sep 2025). Strong connectivity alone does not guarantee consensus. This suggests that weighted balance between internal and external influence, rather than connectivity by itself, controls whether disagreement can persist.

5. Computational evaluation of the Polarization Index

Direct evaluation of the Polarization Index is combinatorial because it maximizes over disjoint nonempty subsets $w_{ji}>0$3 (Li et al., 7 Sep 2025). A reduced-domain quantity is introduced: $w_{ji}>0$4 with

$w_{ji}>0$5

An equivalent integer max–min formulation for $w_{ji}>0$6 is

$w_{ji}>0$7

Here $w_{ji}>0$8 encodes the partition $w_{ji}>0$9 versus $(j,i)\in\mathcal{E}$0, while $(j,i)\in\mathcal{E}$1 selects the weakest nodes on each side (Li et al., 7 Sep 2025).

By relaxing the inner problem to continuous $(j,i)\in\mathcal{E}$2 and taking its LP dual, the paper obtains a mixed-integer linear program: $(j,i)\in\mathcal{E}$3 Let $(j,i)\in\mathcal{E}$4 be the optimal value. Then

$(j,i)\in\mathcal{E}$5

The high-level procedure is: compute $(j,i)\in\mathcal{E}$6 by solving the MILP; if $(j,i)\in\mathcal{E}$7, conclude $(j,i)\in\mathcal{E}$8; if $(j,i)\in\mathcal{E}$9, optionally refine by searching disjoint pairs $i$0 that improve $i$1 beyond the complement case, or by exact enumeration or pruning; then use the sign of $i$2 or of $i$3 when $i$4 to decide Strong Consensus and compute $i$5 (Li et al., 7 Sep 2025). Although worst-case complexity remains exponential, numerical examples show problems up to $i$6 solvable in minutes. The paper characterizes this as a low-average-complexity algorithm.

6. Opinion dynamics, community structure, and related usages

The opinion-dynamics interpretation uses the same discontinuous dynamics as the arbitrary directed-graph protocol: $i$7 Opinions are real numbers, social influence is a directed weighted graph, and individuals do not see exact opinions; they only judge others as “more radical” or “more moderate” relative to themselves (Li et al., 7 Sep 2025). Disturbances encode stubbornness or exogenous media influence, for example

$i$8

or

$i$9

Within this interpretation, the quantity

$\mathcal{V}=\{1,\dots,n\}$00

is the internal weighted in-degree of node $\mathcal{V}=\{1,\dots,n\}$01 within $\mathcal{V}=\{1,\dots,n\}$02, while

$\mathcal{V}=\{1,\dots,n\}$03

is the external weighted in-degree. A subset $\mathcal{V}=\{1,\dots,n\}$04 is a Strong Community, or LS-set, if

$\mathcal{V}=\{1,\dots,n\}$05

The paper proves that $\mathcal{V}=\{1,\dots,n\}$06 is a Strong Community if and only if $\mathcal{V}=\{1,\dots,n\}$07 (Li et al., 7 Sep 2025). Thus $\mathcal{V}=\{1,\dots,n\}$08 acts as a continuous measure of community strength through the worst-node margin.

A bipartition $\mathcal{V}=\{1,\dots,n\}$09 is a Satisfactory Partition when both $\mathcal{V}=\{1,\dots,n\}$10 and $\mathcal{V}=\{1,\dots,n\}$11 are Strong Communities. If such a partition exists, then

$\mathcal{V}=\{1,\dots,n\}$12

which obstructs Strong Consensus (Li et al., 7 Sep 2025). Conversely, the absence of any Strong Community is a sufficient condition for Strong Consensus, and the absence of a Satisfactory Partition is a necessary condition. This links dissensus to internally cohesive and externally weakly connected groups.

The numerical and conceptual examples reinforce this interpretation. Even with strongly connected graphs and zero disturbance, non-unique Filippov solutions exist: some trajectories reach consensus, others exhibit persistent dissensus (Li et al., 7 Sep 2025). When $\mathcal{V}=\{1,\dots,n\}$13, the spread can grow at rate $\mathcal{V}=\{1,\dots,n\}$14; when $\mathcal{V}=\{1,\dots,n\}$15, all simulations converge before $\mathcal{V}=\{1,\dots,n\}$16. Under stubborn disturbances, internally cohesive communities preserve their opinion range more effectively, whereas weaker internal connectivity leads to dissensus and spread.

Related literature uses neighboring terminology differently. The paper on “Tripartite and Sign Consensus for Clustering Balanced Social Networks” studies undirected, signed, weighted, connected graphs with three clusters and dynamics

$\mathcal{V}=\{1,\dots,n\}$17

where cooperative and antagonistic influences are encoded directly in $\mathcal{V}=\{1,\dots,n\}$18 and “sign consensus” means that one cluster converges to a positive value, another to zero, and the third to a negative value (Pasquale et al., 2021). That work explicitly states that no signum nonlinearity is used. It is therefore related by outcome terminology rather than by protocol class.

7. Position within the research landscape

Three strands of work delimit the present meaning of signum consensus. The first treats signum-based consensus as a communication-efficient protocol using only the sign of innovations; it proves consensus iff a spanning tree exists, finite-time convergence, switching-topology extensions, and average preservation for undirected symmetric weights (Doostmohammadian, 2020). The second uses “sign consensus” to describe a limiting sign pattern in clustering-balanced signed networks, but employs a linear modified DeGroot law with tuned stubbornness and no discontinuous signum term (Pasquale et al., 2021). The third, and most technically general, analyzes the discontinuous signum consensus protocol on arbitrary weighted directed graphs with bounded disturbances and codimension-greater-than-one discontinuity manifolds (Li et al., 7 Sep 2025).

The main novelties of the arbitrary directed-graph framework are explicit in the source: treatment of higher-codimension discontinuity via Filippov’s parametric-combination map; a closed-form Polarization Index expressed through node-level weight balances; a necessary and sufficient condition for Strong Consensus given by $\mathcal{V}=\{1,\dots,n\}$19; a least upper bound on consensus time $\mathcal{V}=\{1,\dots,n\}$20; a practical MILP-based evaluation method; and an opinion-dynamics interpretation linking dissensus to Strong Communities and Satisfactory Partitions (Li et al., 7 Sep 2025).

The stated limitations are equally specific. The graph is static, weights are nonnegative, and states are scalar. Time-varying topology, signed weights, and higher-dimensional opinions would require extensions (Li et al., 7 Sep 2025). The adopted Filippov notion is suited to settings with implementation or perception errors; other Filippov variants could yield different solution sets. A plausible implication is that the signum consensus protocol should not be regarded as a single theorem but as a family of sign-based interaction laws whose qualitative behavior depends sharply on graph class, disturbance model, and solution concept.