Hegselmann-Krause Model with Delay

• The paper presents a model where agents adjust opinions based on historical (delayed) averaging, yielding rigorous consensus proofs.
• It distinguishes between transmission-type and reaction-type delays, highlighting the role of instantaneous damping in system stability.
• The study extends to state-dependent and distributed delays, providing analytical frameworks for consensus in complex opinion dynamics.

A Hegselmann–Krause-type model with delay is an opinion-dynamics system in which agents align not with contemporaneous opinions alone, but with retarded or history-averaged information. In this literature, delay appears in several mathematically distinct forms: transmission-type delay, reaction-type delay, time-variable pointwise delay, distributed delay, and state-dependent delay induced by finite speed of information propagation. The common macroscopic question is whether the opinion diameter converges to zero, typically written as $d_x(t)\to 0$ or $d(t)\to 0$, which is the standard notion of asymptotic consensus. Recent work shows that the answer depends strongly on the delay mechanism and on the structure of the communication weights: some delayed HK systems exhibit unconditional consensus for arbitrary finite delay, whereas others require explicit small-delay assumptions (Haskovec, 2020, Haskovec, 21 Jul 2025).

1. Classical background and delayed formulations

The classical Hegselmann–Krause model is a discrete-time bounded-confidence dynamics. For $n$ agents with opinions $x_t(i)\in\mathbb{R}^d$, the update is

$$x_{t+1}(i)=\frac{1}{|N_t(i)|}\sum_{j\in N_t(i)} x_t(j), \qquad N_t(i)=\{j\in[n]: \|x_t(i)-x_t(j)\|_2\le 1\}.$$

Equivalently, if $P_t$ is the simple random-walk matrix of the communication graph $G_t$, then $X_{t+1}=P_tX_t$. In that setting, the central convergence notion is freezing time, namely the first time $T$ for which the configuration becomes stationary. A refined energy argument gives a dimension-independent bound $O(n^4)$ for the maximal freezing time (Martinsson, 2015).

Delayed HK-type models replace the instantaneous neighbor state $d(t)\to 0$0 by delayed information. A generic transmission-type continuous-time form is

$d(t)\to 0$1

while a reaction-type variant delays the whole interaction,

$d(t)\to 0$2

The literature also studies variable delays $d(t)\to 0$3, distributed delay integrals over $d(t)\to 0$4, and state-dependent delays $d(t)\to 0$5. Communication weights are treated either in “classical” form, with $d(t)\to 0$6, or in normalized form, with $d(t)\to 0$7, which makes the interaction term a convex average. Consensus is typically measured by

$d(t)\to 0$8

These definitions recur across particle and mean-field models (Haskovec, 21 Jul 2025).

2. Normalized delayed HK dynamics and unconditional consensus

A central delayed HK system is the communication-type model with normalized weights

$d(t)\to 0$9

with

$n$0

variable delay $n$1, initial history on $n$2, and influence function satisfying only

$n$3

No monotonicity is required. The exact renormalization identity

$n$4

is the structural reason the delayed interaction is a convex combination of delayed positions (Haskovec, 2020).

For $n$5, every solution reaches global asymptotic consensus: $n$6 Moreover, the decay is exponential, with an explicit rate determined by a Gronwall–Halanay-type relation. The result is unconditional with respect to the maximal delay $n$7, the initial data, and the decay rate of $n$8 at infinity. The paper explicitly emphasizes that, unlike earlier delayed HK results, there is no small-delay hypothesis; from that point of view, the result is described as optimal (Haskovec, 2020).

The proof rests on two preparatory estimates. First, the group remains uniformly bounded in radius: $n$9 Hence all delayed distances are bounded by $x_t(i)\in\mathbb{R}^d$0, and one may define

$x_t(i)\in\mathbb{R}^d$1

which yields the uniform lower bound

$x_t(i)\in\mathbb{R}^d$2

Second, a convexity lemma for renormalized weights gives

$x_t(i)\in\mathbb{R}^d$3

whenever the convex coefficients have a common lower bound $x_t(i)\in\mathbb{R}^d$4. This converts normalized delayed averaging into a contraction estimate for the diameter. The delay is then absorbed through the Halanay inequality

$x_t(i)\in\mathbb{R}^d$5

whose exponential decay holds for any finite $x_t(i)\in\mathbb{R}^d$6 (Haskovec, 2020).

A related direct result covers fixed transmission-type delay for both classical and Motsch–Tadmor normalized weights. Under continuity and strict positivity of $x_t(i)\in\mathbb{R}^d$7, with no monotonicity assumption, no decay-rate condition, no small-delay assumption, and no smallness restriction on initial diameter, every solution reaches global asymptotic consensus. That proof proceeds by explicit shrinkage of the group diameter on finite time intervals and does not use Lyapunov functionals or nonnegative matrix theory (Haskovec, 2020).

3. Small-delay Lyapunov regimes and distributed delay

Another major branch of the theory treats time-variable or distributed delays by Lyapunov functionals. In one time-variable-delay model,

$x_t(i)\in\mathbb{R}^d$8

with either symmetric or normalized weights, non-increasing $x_t(i)\in\mathbb{R}^d$9, and delay satisfying $$x_{t+1}(i)=\frac{1}{|N_t(i)|}\sum_{j\in N_t(i)} x_t(j), \qquad N_t(i)=\{j\in[n]: \|x_t(i)-x_t(j)\|_2\le 1\}.$$0, $$x_{t+1}(i)=\frac{1}{|N_t(i)|}\sum_{j\in N_t(i)} x_t(j), \qquad N_t(i)=\{j\in[n]: \|x_t(i)-x_t(j)\|_2\le 1\}.$$1, exponential consensus is proved under the explicit smallness condition

$$x_{t+1}(i)=\frac{1}{|N_t(i)|}\sum_{j\in N_t(i)} x_t(j), \qquad N_t(i)=\{j\in[n]: \|x_t(i)-x_t(j)\|_2\le 1\}.$$2

The argument couples a Dini-derivative estimate for the diameter with a delay term

$$x_{t+1}(i)=\frac{1}{|N_t(i)|}\sum_{j\in N_t(i)} x_t(j), \qquad N_t(i)=\{j\in[n]: \|x_t(i)-x_t(j)\|_2\le 1\}.$$3

and closes the estimate by means of a delay-aware Lyapunov functional (Choi et al., 2019).

A distributed-delay analogue takes the form

$$x_{t+1}(i)=\frac{1}{|N_t(i)|}\sum_{j\in N_t(i)} x_t(j), \qquad N_t(i)=\{j\in[n]: \|x_t(i)-x_t(j)\|_2\le 1\}.$$4

with

$$x_{t+1}(i)=\frac{1}{|N_t(i)|}\sum_{j\in N_t(i)} x_t(j), \qquad N_t(i)=\{j\in[n]: \|x_t(i)-x_t(j)\|_2\le 1\}.$$5

and

$$x_{t+1}(i)=\frac{1}{|N_t(i)|}\sum_{j\in N_t(i)} x_t(j), \qquad N_t(i)=\{j\in[n]: \|x_t(i)-x_t(j)\|_2\le 1\}.$$6

Under $$x_{t+1}(i)=\frac{1}{|N_t(i)|}\sum_{j\in N_t(i)} x_t(j), \qquad N_t(i)=\{j\in[n]: \|x_t(i)-x_t(j)\|_2\le 1\}.$$7, nonincreasing $$x_{t+1}(i)=\frac{1}{|N_t(i)|}\sum_{j\in N_t(i)} x_t(j), \qquad N_t(i)=\{j\in[n]: \|x_t(i)-x_t(j)\|_2\le 1\}.$$8, $$x_{t+1}(i)=\frac{1}{|N_t(i)|}\sum_{j\in N_t(i)} x_t(j), \qquad N_t(i)=\{j\in[n]: \|x_t(i)-x_t(j)\|_2\le 1\}.$$9, and the smallness condition

$P_t$0

the particle system satisfies

$P_t$1

The proof uses a Lyapunov functional with an explicitly delayed integral term and a uniform lower bound $P_t$2 derived from boundedness of the trajectories (Paolucci, 2020).

These small-delay results are not universal across delayed HK-type models. A different time-variable-delay and distributed-delay framework assumes only that $P_t$3 is positive, bounded, and continuous, with no monotonicity requirement, and proves exponential consensus without any smallness assumption on $P_t$4. The key quantities are the window diameters

$P_t$5

which satisfy $P_t$6, together with a discrete contraction

$P_t$7

leading to

$P_t$8

This suggests that delay smallness is model-dependent rather than intrinsic to the entire delayed HK class (Continelli et al., 2022, Haskovec, 2020).

4. Transmission delay, reaction delay, and the role of instantaneous damping

A systematic overview distinguishes two delay mechanisms. In transmission-type delay,

$P_t$9

the self-term remains instantaneous. In reaction-type delay,

$G_t$0

the self-damping is delayed as well. The overview stresses that this distinction is decisive: transmission delay is compatible with asymptotic consensus for any delay length, whereas reaction delay generally requires small $G_t$1 (Haskovec, 21 Jul 2025).

The difference is already visible in the two-agent toy model. For $G_t$2, $G_t$3, and normalized weights, the transmission-delay system yields

$G_t$4

where $G_t$5. The instantaneous term $G_t$6 stabilizes the dynamics, and $G_t$7 for any $G_t$8. In contrast, the reaction-delay toy model becomes

$G_t$9

Its stability depends on the delay: if $X_{t+1}=P_tX_t$0, solutions decay without oscillation; if $X_{t+1}=P_tX_t$1, solutions oscillate but decay; if $X_{t+1}=P_tX_t$2, the equilibrium is unstable (Haskovec, 21 Jul 2025).

The same overview formulates four theorem scenarios. Transmission delay with classical weights gives global asymptotic consensus under a subconvexity bound $X_{t+1}=P_tX_t$3 and a positive lower bound on the weights; the proof uses direct $X_{t+1}=P_tX_t$4-type diameter estimates. Transmission delay with normalized weights gives exponential consensus through convexity and a Halanay-type inequality. Reaction delay with symmetric weights yields consensus for $X_{t+1}=P_tX_t$5 by an $X_{t+1}=P_tX_t$6-based Lyapunov functional exploiting conservation of the mean opinion. Reaction delay with non-symmetric weights yields exponential consensus under a stronger small-delay condition $X_{t+1}=P_tX_t$7 and a generalized Gronwall–Halanay argument (Haskovec, 21 Jul 2025).

A common misconception is that “delay” in itself forces a smallness hypothesis. The available results do not support that blanket statement. For transmission-type delayed HK models, several papers establish consensus for arbitrary finite delay under positivity assumptions; for reaction-type systems, the absence of instantaneous damping changes the stability mechanism and small-delay restrictions reappear (Haskovec, 21 Jul 2025, Haskovec, 2020).

5. State-dependent delay and finite speed of information propagation

A distinct HK-type line of work models finite speed of information propagation. Here the delay is endogenous and pair-dependent: $X_{t+1}=P_tX_t$8 The corresponding dynamics are

$X_{t+1}=P_tX_t$9

This is not a prescribed-delay system but an ODE with state-dependent, heterogeneous delays (Haskovec, 2020, Haskovec et al., 2023).

Well-posedness requires a subsonic constraint

$T$0

Under positivity and global Lipschitz continuity of $T$1, and Lipschitz initial histories with speed strictly less than $T$2, one obtains unique global solutions. In one spatial dimension, asymptotic consensus is unconditional. In higher dimensions, one 2020 result gives a sufficient exponential-consensus condition

$T$3

where $T$4 and $T$5 are extrema of $T$6 on the invariant ball containing the trajectories (Haskovec, 2020).

A later result sharpens the consensus theory. Assuming $T$7 is globally positive, uniformly Lipschitz, nonincreasing, normalized by $T$8, and still satisfies $T$9, every solution reaches global asymptotic consensus. The crucial estimate is a uniform delay bound

$O(n^4)$0

hence $O(n^4)$1 for some finite $O(n^4)$2. This bounded-delay reduction permits a slabwise contraction argument. The paper explicitly characterizes the assumptions as minimal and necessary for well-posedness and unconditional consensus, so the result is described as optimal (Haskovec et al., 2023).

6. Mean-field limits and delayed transport equations

The delayed HK literature contains a parallel mean-field theory in which particle systems converge to transport equations for probability measures. For the normalized delayed HK system, the particle model leads to

$O(n^4)$3

with delayed nonlocal velocity

$O(n^4)$4

For compactly supported initial data $O(n^4)$5, the support diameter satisfies

$O(n^4)$6

with exponential decay. The proof combines the particle consensus theorem with stability in the $O(n^4)$7 Wasserstein distance (Haskovec, 2020).

For distributed delay, the mean-field equation becomes

$O(n^4)$8

with force field

$O(n^4)$9

or its normalized version. Under compactly supported initial histories, one has existence and uniqueness of a weak solution $d(t)\to 0$00, represented by the characteristic flow

$d(t)\to 0$01

Consensus of the support diameter follows under the same small-delay condition as in the particle model, and the proof uses a stability estimate

$d(t)\to 0$02

together with approximation by empirical measures (Paolucci, 2020).

A related pointwise-delay continuum theory works in $d(t)\to 0$03-Wasserstein distance and proves existence, uniqueness, and stability for measure-valued solutions with compactly supported initial history. Because the particle consensus constants are uniform in $d(t)\to 0$04, the exponential particle estimate transfers to the limiting PDE (Choi et al., 2019). Another line removes small-delay assumptions entirely in pointwise and distributed delay PDEs under positive, bounded, continuous interaction kernels, obtaining exponential decay of the support diameter in both cases (Continelli et al., 2022).

7. Structured variants: intermittent attraction, leaders, non-universal graphs, and temporal confidence

The delayed HK framework has been generalized well beyond all-to-all homogeneous interaction. One extension allows “attractive-lacking interaction,” meaning that a scalar factor $d(t)\to 0$05 may weaken or suspend interaction on some intervals. The delayed system

$d(t)\to 0$06

still reaches exponential consensus if the cumulative interaction condition

$d(t)\to 0$07

holds on successive windows and $d(t)\to 0$08 is positive, bounded, and continuous. No small-delay assumption is required (Continelli et al., 2023).

Another extension treats non-universal interaction through a directed influence graph $d(t)\to 0$09. Under the common influencer assumption

$d(t)\to 0$10

the delayed HK-type system converges exponentially to consensus. When this condition fails, leaders can substitute for graph-theoretic connectivity: a fixed leader anchors the group, a controlled leader can steer the group to a prescribed target, and a two-leader configuration also admits asymptotic consensus estimates (Cicolani et al., 2024). A related controlled model with one leader and either pointwise time-varying delay or distributed delay shows that, under suitable smallness conditions on the delays, the leader can drive the group to any prefixed state (Paolucci et al., 2021).

Hierarchical and multipopulation models exhibit the same pattern. A delayed leader–follower system with leaders interacting only among themselves and followers interacting with both followers and leaders converges exponentially to consensus without smallness assumptions on the delay parameters; the same mechanism persists in two mean-field regimes (Choi et al., 11 Aug 2025). A two-population delayed HK-type model with leader-mediated cross coupling also reaches asymptotic consensus for any fixed $d(t)\to 0$11; the proof uses careful intervalwise estimates and a discrete contraction over windows of length $d(t)\to 0$12 (Cicolani et al., 2024).

A more radical modification replaces confidence in opinion distance by confidence in information recency. In the max-plus model with temporal confidence, communication timing is governed by max-plus algebra,

$d(t)\to 0$13

lag-vectors measure information age,

$d(t)\to 0$14

and the HK update becomes

$d(t)\to 0$15

Here the “confidence interval” is temporal rather than geometric. Under strong connectivity of the max-plus adjacency matrix, lag-vectors become periodic after a transient, and simulations show multiple consensus clusters and periodic oscillations in opinion values (Feinstein et al., 2021).

Across these variants, one broad pattern recurs. Strict positivity of interaction on the attained bounded region, together with either convexity of normalized weights, an invariant-range estimate, or a suitable windowed contraction, is the recurrent mechanism behind delayed consensus. The principal dividing line is not merely whether delay is present, but whether the delayed model retains enough instantaneous damping or accumulated attraction to dominate the memory term (Haskovec, 2020, Haskovec, 21 Jul 2025).