Directional Stubbornness

• Directional stubbornness is a form of selective resistance that asymmetrically restricts change along specific coordinates or evaluative polarities.
• It is modeled using techniques such as discourse sheaves, directed network influence, and state-dependent updating to capture one-sided dynamics.
• Engineering realizations translate these concepts into practical systems by configuring agent compliance, activation-space steering, and selective updating mechanisms.

to=arxiv_search.search 开号网址json {"^^^^^^^^3query3 stubbornness\"^^^^^^^^3 OR ti:\3^^^^^^^^"directional stubbornness\"","max_results":^^^^3all:\3query3^^^^,"sort_by":"submittedDate","sort_order":"descending"} to=arxiv_search.search 天天彩票与你同行json {"^^^^^^^^3query3 directional opinion dynamics sheaf refusal social power","max_results":^^^^3all:\3query3^^^^,"sort_by":"submittedDate","sort_order":"descending"} Directional stubbornness denotes a family of models in which resistance to change is asymmetric with respect to a direction, source, coordinate, or evaluative polarity. Taken together, the literature suggests that the phrase does not denote a single standardized formal object. In some work it is explicit: selected subspaces of an opinion stalk are clamped while complementary directions remain free, producing a forced sheaf equation rather than a homogeneous consensus problem (&&&^^^^3query3^^^^&&&). In other work it is indirect: fixed-opinion anchors pull network equilibria upward or downward, one opinion is protected only in one interaction direction, or a model is evaluated by whether it accepts helpful pressure more readily than harmful pressure (&&&^^^^3all:\3^^^^&&&, &&&^^^^3 OR ti:\3^^^^&&&, Kim et al., 12 Jun 2026). By contrast, several papers on stubbornness are important precisely because they do not formalize a directional notion, instead using scalar resistance, diagonal susceptibility, or topological rigidity (Pramanik, 29 Jan 2025, Tian et al., 2019).

^^^^3all:\3^^^^. Conceptual scope and definitional variants

A useful synthesis is that directional stubbornness concerns selective resistance rather than resistance simpliciter. The selectivity may be geometric, as in subspace-valued opinion coordinates; sign-pattern based, as in orthant-specific dependence; interaction-specific, as in one-sided population-protocol transitions; or normative, as in distinguishing beneficial from harmful nudges (&&&^^^^3query3^^^^&&&, Amo et al., 28 May 2025, &&&^^^^3 OR ti:\3^^^^&&&, Kim et al., 12 Jun 2026).

Formal setting	Directional primitive	Resulting asymmetry
Discourse sheaves	PRESERVED_PLACEHOLDER_^^^^3query3^^^^	Only selected opinion directions are clamped (&&&^^^^3query3^^^^&&&)
Undecided-state dynamics	One-sided persistence parameter PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^ in PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^ interactions	Opinion ^^^^3all:\3^^^^ resists elimination only in one conflict direction (&&&^^^^3 OR ti:\3^^^^&&&)
LLM compliance evaluation	$A=\mathrm{BCR}/\mathrm{HCR}$	Helpful and harmful pushback are separated (Kim et al., 12 Jun 2026)
Multivariate dependence	$\alpha\in\{-1,1\}^n$	Dependence is measured by orthant/sign pattern (Amo et al., 28 May 2025)

Two recurrent boundary cases clarify the concept. In the soccer control model, stubbornness is a scalar control $u(s)\in[0,1]$, optimized as a state-dependent intensity but not defined by directional asymmetry in state space (Pramanik, 29 Jan 2025). In the FJ-augmented DeGroot-Friedkin model, stubbornness is $1-\theta_i$, a scalar self-anchoring parameter; any asymmetry comes from the directed network $C$, not from stubbornness itself (Tian et al., 2019). These cases are adjacent to directional stubbornness, but they are not direct instances of it.

^^^^3 OR ti:\3^^^^. Network opinion dynamics: anchors, directed ties, and social power

In opinion dynamics on networks, direction often enters through fixed anchors and directed influence topology rather than through a separate directional stubbornness coefficient. In the stubborn-agent placement problem, agents in $\mathcal V_0$ keep fixed scalar opinions $\theta_i(t)=\theta_i(0)$, and non-stubborn equilibria satisfy

PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^

This makes the equilibrium of persuadable agents a linear image of the stubborn agents’ fixed opinions (&&&^^^^3all:\3^^^^&&&). Because PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^ is the sensitivity matrix from anchor values to downstream equilibria, increasing a stubborn opinion cannot decrease any equilibrium non-stubborn opinion in the model’s monotone regime. The paper’s harmonic influence centrality and optimization results sharpen that interpretation: for the mean objective, the targeting set function is monotone and submodular, yielding a greedy PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^-approximation guarantee (&&&^^^^3all:\3^^^^&&&). This is an indirect but rigorous model of directional stubbornness through fixed-opinion anchors.

The same paper extends the static equilibrium picture to noisy communication with time-varying susceptibility. Non-stubborn agents update with PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^, where smaller PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^4 means more stubbornness, and convergence to the DeGroot equilibrium still holds under conditions such as

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^5

This implies that increasing resistance can slow directional movement without changing its asymptotic anchor-determined direction, provided learning remains sufficiently persistent (&&&^^^^3all:\3^^^^&&&).

A related but distinct line studies social power in directed FJ networks with stubborn individuals. There, susceptibility is PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^6, stubbornness is PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^7, and actual social power is

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^8

Directionality enters through the transpose PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^9: power accumulates along incoming ties, and the effective contribution from PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3query3^^^^ to PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^ is reweighted by relative stubbornness through PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^ (&&&^^^^3all:\3^^^^9&&&). The distributed perception dynamics converge exponentially to actual social power without requiring global matrix inversion, and the model shows that stubbornness can increase baseline power while simultaneously changing how incoming acknowledgment is converted into perceived influence (&&&^^^^3all:\3^^^^9&&&). This is directional in effect, though the stubbornness parameter remains agent-specific rather than edge-specific.

3. One-sided, state-dependent, and issue-dependent resistance

The most explicit one-sided formalization appears in the biased Undecided State Dynamics with preferred Opinion ^^^^3all:\3^^^^. The state space is PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^, and only one transition is modified: when a PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4-initiator meets a PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5-responder, it keeps Opinion ^^^^3all:\3^^^^ with probability PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6 and becomes undecided with probability PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7. The key quantity is the weighted bias

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8

with phase-transition threshold

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9

If $A=\mathrm{BCR}/\mathrm{HCR}$^^^^3query3^^^^, consensus on Opinion ^^^^3all:\3^^^^ occurs after $A=\mathrm{BCR}/\mathrm{HCR}$^^^^3all:\3^^^^ interactions; if $A=\mathrm{BCR}/\mathrm{HCR}$^^^^3 OR ti:\3^^^^, consensus on Opinion ^^^^3 OR ti:\3^^^^ occurs in the same order; and in the critical window the process still reaches consensus, but after $A=\mathrm{BCR}/\mathrm{HCR}$3, with either opinion able to survive (&&&^^^^3 OR ti:\3^^^^&&&). This is directional stubbornness in a strict transition-level sense.

A different mechanism appears in the concatenated FJ model over issue sequences. Within each issue,

$A=\mathrm{BCR}/\mathrm{HCR}$4

and across issues,

$A=\mathrm{BCR}/\mathrm{HCR}$5

After each issue, the vote is approximated by the median final opinion, and disagreement with that vote feeds back into stubbornness. In the increasing-stubbornness case,

$A=\mathrm{BCR}/\mathrm{HCR}$6

where $A=\mathrm{BCR}/\mathrm{HCR}$7 is distance from the median-vote outcome (&&&^^^^3 OR ti:\3 OR ti:\3^^^^&&&). Consensus is no longer automatic; only sufficient conditions are proved in special complete-graph cases, such as $A=\mathrm{BCR}/\mathrm{HCR}$8. In the opposite regime, where disagreement makes agents less stubborn, consensus is guaranteed under strong connectivity for $A=\mathrm{BCR}/\mathrm{HCR}$9, and simulations show faster convergence than with constant stubbornness (&&&^^^^3 OR ti:\3 OR ti:\3^^^^&&&). This is directionality with respect to agreement versus disagreement with collective choice.

Temporal inertia can also produce partial directional behavior without an explicit directional parameter. In jury deliberation, stubbornness is the probability $\alpha\in\{-1,1\}^n$^^^^3query3^^^^ of not reconsidering one’s current opinion, and it increases with holding time $\alpha\in\{-1,1\}^n$^^^^3all:\3^^^^: $\alpha\in\{-1,1\}^n$^^^^3 OR ti:\3^^^^ The effective accumulation rate $\alpha\in\{-1,1\}^n$3 is reduced in hung configurations, so resistance is state-dependent rather than side-specific (&&&^^^^3 OR ti:\3^^^^4&&&). In the Galam debate model, directional asymmetry is created by coupling collective-belief bias $\alpha\in\{-1,1\}^n$4 with side-specific inflexible fractions $\alpha\in\{-1,1\}^n$5 and $\alpha\in\{-1,1\}^n$6. The critical thresholds

$\alpha\in\{-1,1\}^n$7

state exactly how much side-specific stubbornness is required to neutralize or overturn an exogenous directional bias (&&&^^^^3 OR ti:\3^^^^5&&&).

A further adjacent case is nonlinear, extremity-based stubbornness. In the friction-inspired opinion model, the stubbornness term is

$\alpha\in\{-1,1\}^n$8

which is directional relative to the agent’s prejudice $\alpha\in\{-1,1\}^n$9 but not sign-asymmetric on a signed ideological axis (&&&^^^^3 OR ti:\3^^^^6&&&). The paper’s point is state-dependent persistence: extreme opinions are harder to move than moderate ones.

4. Subspace and orthant formalisms

The most explicit and technically complete definition of directional stubbornness is developed on discourse sheaves. Each vertex $u(s)\in[0,1]$^^^^3query3^^^^ has a private-opinion stalk $u(s)\in[0,1]$^^^^3all:\3^^^^, and stubbornness is specified by a subspace

$u(s)\in[0,1]$^^^^3 OR ti:\3^^^^

of clamped directions; the orthogonal complement

$u(s)\in[0,1]$3

contains free directions (&&&^^^^3query3^^^^&&&). The free components form a sheaf $u(s)\in[0,1]$4, and the vertex-cochain space decomposes as

$u(s)\in[0,1]$5

If $u(s)\in[0,1]$6 is the clamped component and $u(s)\in[0,1]$7 is the free component, then the free dynamics are

$u(s)\in[0,1]$8

and the equilibrium solves the sheaf Poisson equation

$u(s)\in[0,1]$9

The forcing term $1-\theta_i$^^^^3query3^^^^ is the projected disagreement induced by the clamped directions, so directional stubbornness converts harmonic extension into an affine forced problem (&&&^^^^3query3^^^^&&&).

The same paper develops a dual theory of selective learning of communication maps. If only a subset of incidences may adapt, the resulting gradient flow is sheaf diffusion on an auxiliary structure sheaf, and for the joint evolution of beliefs and expressions the Lyapunov energy

$1-\theta_i$^^^^3all:\3^^^^

is nonincreasing (&&&^^^^3query3^^^^&&&). The rate ratio between opinion updating and structural adaptation yields stagnation bounds distinguishing “belief resolution” from “rhetorical accommodation.” In this framework, directional stubbornness is literally subspace-valued.

A mathematically adjacent but conceptually different formalism is the theory of directional $1-\theta_i$^^^^3 OR ti:\3^^^^-coefficients. Here direction means a sign pattern

$1-\theta_i$3

and the generalized coefficient is

$1-\theta_i$4

The coefficients measure orthant-specific dependence rather than temporal persistence, and they satisfy identities such as

$1-\theta_i$5

The paper also gives rank-based estimators and asymptotic normality (Amo et al., 28 May 2025). This is not stubbornness in the behavioral sense, but it provides a precise language for directional alignment and directional avoidance in multivariate systems.

5. Engineering realizations: selective updating, refusal directions, and haptic cooperation

In LLM evaluation, a directly comparable notion is directionally calibrated updating. Compliance Asymmetry is defined as

$1-\theta_i$6

where BCR is the Beneficial Compliance Rate and HCR is the Harmful Compliance Rate (Kim et al., 12 Jun 2026). Across 9 models and 97^^^^3 OR ti:\3^^^^,^^^^3query3query3query3^^^^ nudge-condition responses, factual questions showed $1-\theta_i$7, while moral questions showed $1-\theta_i$8 (Kim et al., 12 Jun 2026). The paper interprets $1-\theta_i$9 as directional selectivity and $C$^^^^3query3^^^^ as direction-blind compliance. Chain-of-thought prompting increased both helpful and harmful compliance together, and Contextual Identity Prompting decreased both by nearly identical margins, so neither restored moral directional selectivity (Kim et al., 12 Jun 2026). In this setting, directional stubbornness is not refusal of all influence but selective resistance to harmful influence.

A complementary activation-space perspective treats stubborn behavior as a latent direction in representation space. COSMIC constructs candidate directions

$C$^^^^3all:\3^^^^

then scores them by concept inversion using cosine similarity, selecting a direction that makes harmless activations resemble naturally refusing harmful activations under addition, and harmful activations resemble naturally compliant harmless activations under ablation (Siu et al., 30 May 2025). The steering mechanism is output-independent, and the paper reports robustness in adversarial complete-refusal settings and on weakly aligned models (Siu et al., 30 May 2025). This suggests that at least some persistent behavioral tendencies can be operationalized as activation-space directions, although the paper also shows model-specific layer variation and non-monotonic steering effects.

A more literal control-theoretic interpretation appears in cooperative human-machine trajectory planning. Human stubbornness is estimated as a two-component vector,

$C$^^^^3 OR ti:\3^^^^

derived from force magnitude and mismatch between estimated human request and executed automation motion (Schneider et al., 2024). This is channel-specific rather than fully geometric: longitudinal and angular stubbornness are separated, but the estimate is based on absolute values and is therefore unsigned. The automation maps the estimate into utility loss and target utility,

$C$3

so that higher estimated human stubbornness yields greater automation compliance (Schneider et al., 2024). The “Stubborn” MARL environment similarly operationalizes insistence through the conditional probability

$C$4

the probability that an agent continues choosing its currently preferred side after $C$5 disagreement turns and private estimate gap $C$6 (Rachum et al., 2023). These engineering models are not identical, but both treat stubbornness as a measurable persistence policy rather than a static personality label.

6. Non-examples, boundary cases, and unresolved distinctions

Several influential papers are best read as clarifying what directional stubbornness is not. In the soccer stochastic-control model, stubbornness is a scalar control $C$7 in a one-dimensional goal-dynamics process,

$C$8

and the paper’s main result is a state-dependent optimal intensity derived from the stationarity condition

$C$9

The author explicitly treats the model as single-player, and the paper does not introduce vector-valued stubbornness, sign-sensitive persistence, or anisotropic resistance (Pramanik, 29 Jan 2025). It is therefore a model of optimal scalar stubbornness intensity, not directional stubbornness.

The FJ-augmented DeGroot-Friedkin model with stubborn individuals provides another boundary case. There, stubbornness is $\mathcal V_0$^^^^3query3^^^^, encoded by a diagonal matrix $\mathcal V_0$^^^^3all:\3^^^^, and the paper explicitly notes that it is not source-specific, sign-dependent, issue-dependent, or coordinate-wise (Tian et al., 2019). Any directional asymmetry comes from the directed influence matrix $\mathcal V_0$^^^^3 OR ti:\3^^^^, not from the stubbornness parameterization itself. The model proves, among other things, that autocracy cannot be achieved and democracy can occur under much broader topologies than in the original DF model (Tian et al., 2019). This is a theory of scalar self-anchoring in a directed network, not of directional stubbornness.

Topological rigidity is a third boundary case. In signed social-network dynamics with stubborn links, stubbornness means a subset of negative links is fixed at $\mathcal V_0$3 and cannot flip under structural-balance updates (&&&4^^^^3 OR ti:\3^^^^&&&). The key result is that final imbalance depends more on the location of stubborn links than on their fraction, and clustered stubborn communities can trap the system in high-energy states; the most stressful configuration is associated with roughly five stubborn communities in the large-$\mathcal V_0$4 analysis (&&&4^^^^3 OR ti:\3^^^^&&&). This is link-based and topological rather than directional in the sense of source-specific or sign-sensitive resistance.

Taken together, these distinctions suggest a useful taxonomy. Some models implement directional stubbornness directly, by clamping selected coordinates or protecting one opinion only in one transition. Others implement it indirectly, through anchor values, asymmetric networks, or evaluative asymmetry between helpful and harmful updates. A final group models stubbornness without directionality at all. The central unresolved issue is therefore not whether stubbornness matters, but which formal asymmetry is meant by “directional”: coordinate, sign pattern, edge orientation, side-specific inflexibility, temporal feedback, or content-sensitive updating (&&&^^^^3query3^^^^&&&, &&&^^^^3 OR ti:\3^^^^&&&, Kim et al., 12 Jun 2026).