Adoption curves

Adoption curves describe how transformative technologies and monetary goods spread through a population over time. The canonical references are Everett Rogers's Diffusion of Innovations (1962) — five adopter categories from innovators (2.5%) through laggards (16%) — and Geoffrey Moore's Crossing the Chasm (1991), which identifies the early-adopter to early-majority gap as the decisive transition. Mathematical operationalizations are S-shaped: logistic, Gompertz, Bass, and the Weibull CDF (Stephen Perrenod's preferred form for Bitcoin given its flexibility for early-phase growth). Bitcoin's position as of 2026 is contested but most analyses place it mid-chasm-crossing, with institutional adoption accelerating since the 2024 spot-ETF approvals. The framework is load-bearing for the The Power Law model: Santostasi and Perrenod treat price as compounding on a user base that itself grows as a power-law function of time.

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Why this note matters

The adoption-curve framework is the demand-side dynamic underlying every serious long-term Bitcoin price model. Three reasons it’s load-bearing:

1. It is the mechanism beneath the Power Law. The Power Law model presents price as $P(t)=A\cdot t^n$; the underlying why is that Bitcoin’s user base grows as a power-law function of time and value scales with users (Metcalfe-style). Without an adoption-curve frame, the Power Law is descriptive; with one, it has a causal mechanism.
2. It locates Bitcoin within a broader empirical pattern. Transformative technologies — electricity, automobiles, telephones, the internet, smartphones — all follow S-curve adoption; anchoring Bitcoin’s trajectory in 150 years of cross-technology evidence strengthens the case.
3. It addresses “what stage is Bitcoin in?” rigorously. Cycle positioning, allocation decisions, and time-horizon expectations all depend implicitly on where Bitcoin sits on its adoption curve — connecting also to Monetization S-curve — and the note engages the counter-argument that monetary goods may not follow standard tech-adoption curves.

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Rogers’ diffusion of innovations framework

Everett Rogers’s Diffusion of Innovations (Free Press, 1962; multiple subsequent editions) is the canonical study of how new technologies, ideas, and practices spread through populations. The framework is built on extensive empirical study of adoption patterns — from hybrid corn in 1950s Iowa to medical-practice changes among physicians to consumer-technology rollouts.

The five adopter categories, defined by where they fall on a normal-distribution model of adoption time:

Category	Share	Description
Innovators	2.5%	Risk-tolerant; willing to adopt before evidence is in; often technically engaged; pay the highest costs in money, time, and uncertainty
Early adopters	13.5%	Opinion leaders; visible community members; adopt once early evidence is available; bridge from innovators to mainstream
Early majority	34%	Deliberate adopters; wait for substantial evidence; adopt when peer pressure and clear utility align
Late majority	34%	Skeptical adopters; adopt only when most others have; respond to economic pressure (cost of not adopting)
Laggards	16%	Tradition-anchored; resist adoption until the alternatives are gone; adopt only when forced

The categories produce a cumulative S-curve — slow early growth (innovators only), then acceleration through the early-adopter and early-majority phases, then plateau as the late majority and laggards complete adoption.

Five characteristics that determine adoption speed, in Rogers’s framework:

1. Relative advantage — does the new technology measurably improve on the alternative?
2. Compatibility — does it fit with existing values, experiences, and practices?
3. Complexity — how hard is it to understand and use?
4. Trialability — can it be experimented with at low cost before commitment?
5. Observability — can adopters see the benefits in others before adopting themselves?

Technologies that score well on all five adopt faster; technologies that score poorly on one or more adopt slower or fail to adopt at all.

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Moore’s chasm

Geoffrey Moore’s Crossing the Chasm (HarperBusiness, 1991) refines Rogers’s framework for high-technology products with a specific structural observation: there is a discontinuity between early adopters and the early majority that many technologies fail to cross.

The chasm is the gap between:

• Early adopters who are tolerant of imperfection, willing to construct their own use cases, and motivated by the technology’s potential
• Early majority who require complete, well-supported, conventional-looking products and adopt only when their peers have already validated the technology

Many technologies that achieve strong early-adopter penetration fail at the chasm because the early-majority requirements (reliability, ease of use, peer validation, mainstream applicability) are substantially different from what the early adopters needed. Crossing the chasm requires either re-engineering the product, repositioning the marketing, or both.

For Bitcoin specifically, the chasm question is whether the asset has transitioned (or is transitioning) from early-adopter to early-majority phase. The substantive evidence for “chasm crossed” includes:

• Spot ETF approval (January 2024) — the canonical institutional-validation event
• Sovereign treasury adoption (El Salvador 2021; further national-level positioning since)
• Corporate treasury adoption (Strategy/MicroStrategy, Tesla brief, growing corporate count)
• Mainstream financial-press treatment as an asset class rather than a curiosity
• Banking and brokerage integration — major institutions offering Bitcoin custody and trading

The substantive evidence for “chasm not yet crossed” includes:

• Global holding rates remain below 5% — substantially below the ~16% threshold that marks early-majority transition
• Most institutional investors still treat Bitcoin as speculative rather than as a strategic monetary asset
• Regulatory friction in many jurisdictions limits straightforward early-majority access
• Consumer technology for Bitcoin (wallets, transaction UX, recovery) remains substantially harder than mainstream financial UX

The honest reading is that Bitcoin is at or in the chasm transition, with credible arguments on both sides. The 2024-2028 cycle is likely to resolve the question — either Bitcoin completes the institutional transition and enters early-majority adoption substantively, or it stalls in the chasm and the mainstream transition is deferred.

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Mathematical forms of S-curves

The qualitative Rogers framework operationalizes through several mathematical forms. Each is a specific functional form for the cumulative-adoption curve $P(t)$ — the fraction of the eventual saturation population that has adopted by time $t$.

Logistic function

$P(t)=\frac{L}{1+e^{-k(t-t_0)}}$

Where $L$ is the saturation level, $k$ is the growth rate, and $t_0$ is the midpoint (inflection point of the S-curve). The logistic is the most-cited functional form in the adoption literature and the easiest to fit.

Properties:

• Symmetric around the midpoint
• Asymptotic to $L$ as $t\to \infty$
• Maximum growth rate at $t=t_0$ (the 50% adoption point)

Limitations: The symmetry assumption is often violated empirically — many technologies show asymmetric adoption (slow start, faster middle, slow finish; or the reverse).

Gompertz function

$P(t)=L\cdot e^{-e^{-k(t-t_0)}}$

The Gompertz is an asymmetric S-curve with earlier inflection than the logistic — adoption accelerates faster in the early phase and decelerates slower in the late phase. It often fits better than the logistic for technology adoption.

Properties:

• Asymmetric (inflection at ~37% of saturation, not 50%)
• Better fit for many empirical adoption curves
• Common in actuarial science and mortality modeling, where similar asymmetry appears

Weibull cumulative distribution function

$P(t)=L\cdot \left(1-e^{-(t/\lambda {)}^k}\right)$

The Weibull CDF is the most flexible of the standard S-curve forms, with two parameters ($\lambda$ scale, $k$ shape) that produce a family of curves ranging from rapid-early-saturation to slow-early-rapid-late patterns.

Stephen Perrenod has argued the Weibull CDF is the appropriate form for Bitcoin — see his Substack writing — because the flexibility allows for the specific early-phase dynamics Bitcoin has shown (slow initial uptake during the cypherpunk era, acceleration after each halving, asymmetric institutional acceleration post-2024).

The Weibull form is also mathematically compatible with the Power Law: in the early-to-middle phase of a Weibull S-curve, the cumulative-adoption curve approximates a power-law-in-time relationship. This compatibility is part of why Power Law and adoption-curve frameworks integrate cleanly.

Bass diffusion model

$\frac{dP}{dt}=(p+q\cdot P/L)(L-P)$

Frank Bass’s 1969 model splits adoption into innovation (parameter $p$, independent-of-others adoption) and imitation (parameter $q$, network-effects adoption proportional to current adopter base). The Bass model is widely used in marketing and product-launch forecasting and produces an asymmetric S-curve similar in shape to the Gompertz.

For Bitcoin, the Bass framing is particularly natural because the innovation/imitation split maps cleanly to technical interest (early innovators) versus network-effects-driven adoption (the dominant mechanism in the early-majority phase).

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Bitcoin’s current position in its adoption curve

Locating Bitcoin within the framework requires definitions of “adoption” and “saturation”:

Defining adoption. Adoption can be measured at multiple levels:

1. Awareness — knows what Bitcoin is. Global rates: ~70-85% in OECD countries; lower in developing economies.
2. Holdings — owns any non-zero amount. Global rates: ~5-10% varying by region; ~10-15% in the US; ~5-7% in Europe.
3. Strategic allocation — holds Bitcoin as a deliberate portfolio position rather than incidental exposure. Substantially lower: ~1-3% globally.
4. Operational use — sends or receives Bitcoin meaningfully (not just holds). Substantially lower again: ~0.5-1% globally.
5. Sovereign use — held as a treasury or reserve asset. Currently only a handful of nations directly; many indirectly via miner-state relationships.

Defining saturation. Saturation depends on Bitcoin’s eventual role:

• If Bitcoin becomes a store-of-value asset class alongside gold and equities, saturation likely caps at ~30-50% of households globally (similar to the gold-holder rate historically).
• If Bitcoin becomes a reserve asset held principally by institutions and sovereigns, saturation in household-holding terms remains modest (perhaps 10-20%) but in institutional terms reaches very high levels.
• If Bitcoin becomes a medium-of-exchange / unit-of-account monetary system, saturation approaches 100% in functional terms.

Where Bitcoin sits in 2026, in the holdings sense:

• ~5-10% global holdings places Bitcoin between innovators (2.5%) and early adopters (16% cumulative).
• The trajectory through 2024-2026 has been substantial acceleration — ETF approvals, institutional flows, sovereign positioning, corporate treasury growth.
• The chasm-crossing question (early adopters to early majority, ~16% threshold) is genuinely live: most analyses suggest 2026-2030 will be the decisive window.

The Bitcoin-specific adoption curve, with Weibull-style asymmetry:

• 2009-2013: Innovator phase (cypherpunk and early-technical communities)
• 2013-2017: Early-adopter phase (Bitcoin-native communities; first wave of retail)
• 2017-2021: Early-majority entry (institutional curiosity; first retail mainstream cycle)
• 2021-2024: Chasm-transition period (institutional infrastructure building; regulatory clarity)
• 2024+: Early-majority adoption (post-ETF; institutional and sovereign integration)
• 2028+: Hypothesized acceleration into late-majority adoption if current trajectory holds

This curve is consistent with both the Power Law trajectory (The Power Law model) and Boyapati’s four-phase monetization framework (Vijay Boyapati, Store of value vs medium of exchange vs unit of account).

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The connection to the Power Law

The Power Law model and the adoption-curve framework are not competing — they are layered. The adoption curve is the demand-side dynamics; the Power Law is the price-trajectory dynamics. The connection runs through Metcalfe-style network value:

The compound argument:

1. Bitcoin’s user base $U(t)$ grows as a power-law function of time: $U(t)\propto t^u$ where $u\approx 3$ in Santostasi’s framework.
2. Network value scales with some power of user base (Metcalfe: $V\propto U^2$; Reed: $V\propto 2^U$; in practice, generalized to $V\propto U^{\gamma }$ with $\gamma \approx 2$).
3. Composing the two: $V(t)\propto (t^u{)}^{\gamma }=t^{u\gamma }$.
4. With $u=3$ and $\gamma =2$, this yields $V(t)\propto t^6$ — close to the empirically-fitted Power Law exponent of ~5.7.

This composition is the mechanistic derivation that the Santostasi-Perrenod 2026 Scientific Bitcoin Institute paper formalizes. The adoption curve is the load-bearing first step. See The Power Law model for the price-side discussion and Metcalfe’s Law applied to Bitcoin for the network-value side.

The implication: As Bitcoin’s adoption curve transitions phases (chasm crossing, early majority, late majority), the Power Law’s underlying mechanism shifts. The current Power Law fit reflects early-to-middle-phase adoption dynamics; later phases will produce different exponents or framework breakdowns. The Power Law’s specific exponent ($n\approx 5.7$) is an emergent property of Bitcoin’s specific current adoption phase, not a permanent constant.

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Predictions and implications

The adoption-curve framework generates several specific quantitative implications:

Trajectory implications:

• If Bitcoin tracks a Weibull-CDF adoption curve with parameters consistent with the record since 2009, full early-majority adoption (~50%) is approximately a 2035-2045 horizon for most adoption definitions.
• The post-2024 acceleration (institutional, sovereign, corporate) is consistent with the early-majority transition the framework predicts.
• Cycle-by-cycle diminishing returns — see Diminishing returns thesis — are a natural consequence of S-curve dynamics: each cycle moves Bitcoin further up the curve, reducing the relative magnitude of further phase transitions.

Allocation implications:

• Long-horizon Bitcoin allocation (5-10+ years) captures the full trajectory through chasm crossing and early-majority adoption.
• Cycle-aware allocation can attempt to position around specific adoption-curve inflections (chasm crossing, early-majority entry, late-majority entry).
• DCA across the trajectory is well-supported by the framework’s smooth-trajectory assumption.

Time-horizon implications:

• Monetization is not imminent in the sense of complete saturation — even aggressive scenarios place full Bitcoin monetization 10-20+ years out.
• But Bitcoin is also not early in the sense of pre-chasm — most of the early-adopter phase has been traversed.
• The current decade (2026-2036) is the structural transition period for Bitcoin’s adoption — chasm crossing and early-majority entry.

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Comparison to historical technology adoption

Bitcoin’s adoption pattern can be compared to other major technology rollouts. Approximate timelines from initial commercial introduction to ~50% household adoption in the US:

Technology	Introduction	~50% adoption	Years
Electricity	1880s	1930s	~50
Automobile	1900s	1950s	~50
Telephone	1880s	1950s	~70
Radio	1920s	1940s	~20
Television	1940s	1960s	~20
Personal computer	1980s	~2010	~30
Internet	1990s	~2010	~15-20
Smartphone	2007	~2015	~8

Two patterns are visible:

1. Adoption is accelerating across technology generations — newer technologies hit 50% adoption faster than older ones, primarily because of compound network effects (each technology builds on prior infrastructure).
2. Communication and information technologies adopt fastest (radio, TV, internet, smartphone) because their network-effects are immediate and the personal cost of adoption is low.

For Bitcoin specifically, the question is which reference class is most appropriate:

• As monetary technology (closer to electricity or automobile): 50-70 year timeline → full adoption mid-21st century
• As financial infrastructure (closer to PC or internet): 30-year timeline → full adoption around 2040
• As information-technology (closer to smartphone): 15-20 year timeline → full adoption around 2030
• As none of these (sui generis monetary good): timeline genuinely uncertain

Most analyses place Bitcoin between “financial infrastructure” and “monetary technology” — implying full adoption is a 20-40 year horizon from 2026.

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Counter-arguments and tensions

Adoption curves are descriptive, not predictive

The argument: Adoption curves are useful for describing past technology rollouts but have limited predictive power. The historical pattern of S-curve adoption is real, but the specific timing, saturation level, and curve shape can only be known after the fact. Using adoption-curve frameworks to make specific predictions about Bitcoin’s future is curve-fitting in disguise.

Response: Partly right. The framework is much stronger as a backward-looking explanatory tool than as a forward-looking prediction tool. The honest reading is that adoption curves provide a trajectory shape with substantial uncertainty around timing and saturation. The framework’s value is in providing baseline expectations, not specific point predictions. Used alongside the Power Law and Boyapati’s monetization phases, it gives directional guidance with appropriate epistemic humility.

Monetary goods may not follow technology-adoption patterns

The argument: Bitcoin is a monetary good, not just a technology. Monetary goods historically follow different adoption dynamics — slow accretion of monetary status through Mengerian salability convergence, with the eventual transition to dominant money happening through displacement of prior monies rather than gradual S-curve adoption. The technology-adoption framework may be the wrong reference class.

Response: This is a substantive point. The Mengerian framework (see Origins of money, Carl Menger, Saifedean Ammous) suggests monetary goods can have phase-transition dynamics — the displacement of silver by gold in the late 19th century happened relatively suddenly once the structural shift occurred. Bitcoin’s case may be hybrid: technology-adoption dynamics in the early phases (where Bitcoin functions as a speculative asset and store of value), monetary-displacement dynamics in the later phases (where Bitcoin functions as the dominant monetary good). The frameworks are complementary, not competing.

The chasm question is contested

The argument: It is genuinely unclear whether Bitcoin has crossed the chasm or remains in early-adopter phase. Different metrics give different answers. The framework provides a vocabulary but not a definitive empirical answer.

Response: True. The chasm-crossing question is one of the live empirical questions about Bitcoin’s current state. The framework’s value here is in providing the vocabulary and the categorization, not in producing a definitive answer. Different definitions of “adoption” give different answers, and the honest reading is that Bitcoin is in transition.

Bitcoin may not follow any standard adoption curve

The argument: Bitcoin’s specific characteristics — pseudonymous network, no central marketing, regulatory friction, technology-and-monetary hybrid nature, halving-cycle dynamics — may produce an adoption pattern that doesn’t fit any standard S-curve form. Trying to force Bitcoin into the framework may obscure rather than illuminate.

Response: Plausible at the margin but not at the core. Bitcoin’s empirical adoption pattern through 2026 has been substantially S-curve-shaped, with the specific deviations (halving-cycle acceleration, post-ETF acceleration) being understandable within the framework as superimposed cyclical structure rather than as framework breakdown. The framework is approximate; Bitcoin’s specifics produce deviations; the overall trajectory remains S-curve-shaped.

Saturation is uncertain

The argument: Adoption curves require a saturation level $L$. Bitcoin’s eventual saturation depends on what role it ends up playing (asset class, reserve asset, dominant money), which is genuinely uncertain. Different saturation assumptions produce wildly different curve fits.

Response: Fair. The framework’s predictions depend on the saturation assumption. The honest practice is to specify saturation scenarios explicitly (Bitcoin as ~5% of household financial assets vs. ~25% vs. ~50%) and explore the implications of each. Saturation uncertainty is a real limitation, but it doesn’t invalidate the framework — it just means predictions should be presented as conditional on specific saturation scenarios.

Adoption curves smooth over regime change

The argument: A smooth S-curve assumes continuous adoption dynamics, but real adoption is regime-driven — major catalysts (regulatory clarity, institutional integration, sovereign adoption cascades, fiat hyperinflation events) produce discontinuous acceleration that smooth-curve frameworks underestimate. The “gradually then suddenly” pattern (Parker Lewis) is in tension with the smooth-curve framing.

Response: Substantive point. The framework captures the “gradually” portion well; it under-captures the “suddenly” portion. Major regime-change events should be treated as deviations from the smooth curve rather than as features of the smooth curve. The Power Law model has the same limitation (smooth trend; under-captures regime change). Both frameworks should be paired with explicit regime-change scenario analysis rather than treated as complete forecasting tools.

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Open questions for further development

• What is Bitcoin’s actual eventual saturation level? The framework requires this and the answer depends on what role Bitcoin plays in the eventual global monetary system.
• Has Bitcoin crossed the chasm or remains pre-chasm? The empirical question is genuinely contested and may only be answerable retrospectively.
• Are Bitcoin’s halving cycles a feature of the adoption curve or a deviation from it? The log-periodic framework (see Log-periodic cycles and the Perrenod-Santostasi wave model) treats them as oscillations around a smooth trend; alternative framings treat them as adoption-curve milestones.
• How should the framework be updated as institutional adoption mechanics replace retail-driven adoption? The post-2024 dynamics suggest a different mechanism in the early-majority phase than was present in the early-adopter phase.
• What relationship does the framework have with the Monetization S-curve note in the Economics section? The two notes treat similar territory with different framings (technology adoption vs. Mengerian salability); reconciliation deserves explicit treatment.
• Does the framework imply specific cycle-top and cycle-bottom expectations for the 2024-2028 cycle? If Bitcoin is mid-chasm-crossing, cycle dynamics may differ from prior cycles where Bitcoin was in clearer early-adopter territory.
• How does the framework engage Lyn Alden’s fiscal-dominance framework? Fiscal-dominance dynamics could produce regime-change adoption that the smooth-curve framework would miss.

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Canonical sources for this note

Foundational adoption-theory literature

• Everett Rogers, Diffusion of Innovations (Free Press, 1962; multiple editions through 2003) — the foundational text on adoption categorization and dynamics
• Geoffrey Moore, Crossing the Chasm (HarperBusiness, 1991) — the high-tech-specific refinement
• Frank Bass, “A New Product Growth Model for Consumer Durables” (Management Science, 1969) — the canonical mathematical-marketing framework

Bitcoin-specific adoption-curve analyses

• Stephen Perrenod, various Substack articles on adoption and S-curves (stephenperrenod.substack.com) — particularly the Weibull-CDF analyses and the integrated power-law/adoption frameworks
• Giovanni Santostasi, “The Bitcoin Power Law Theory” and related work — adoption as the underlying mechanism of the Power Law trajectory
• Santostasi and Perrenod, “A Mechanistic Derivation of the Bitcoin Price Power Law” — the Scientific Bitcoin Institute paper deriving the Power Law from adoption + Metcalfe-style network value
• Vijay Boyapati, The Bullish Case for Bitcoin (essay 2018; book 2021) — the four-phase monetization framework, an adoption-curve adjacent treatment

Empirical-adoption studies

• Various Glassnode and Checkonchain reports on address-count growth and wallet-distribution evolution
• Various exchange-publish institutional-adoption reports (Coinbase, Fidelity, BlackRock institutional research)
• Statista and similar consumer-research surveys on Bitcoin awareness and holdings rates

Background mathematical literature

• Various probability-and-statistics textbooks treating the Weibull, logistic, and Gompertz distributions
• Eric von Hippel, Democratizing Innovation (2005) — adjacent framework on user-driven innovation
• Clayton Christensen, The Innovator’s Dilemma (1997) — adjacent framework on technology disruption

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Related notes

• The Power Law model — the price-trajectory framework that compounds on the adoption curve
• Monetization S-curve — the Mengerian-monetary adjacent framework in the Economics section
• Metcalfe’s Law applied to Bitcoin — the network-value side of the Power Law derivation
• Stock-to-flow model — alternative supply-side framework engaging the same trajectory question
• Diminishing returns thesis — natural consequence of S-curve dynamics
• Log-periodic cycles and the Perrenod-Santostasi wave model — cyclical structure around the smooth trend
• Four-year halving cycles — supply-schedule cyclicality interacting with adoption-curve dynamics
• Lindy effect and Bitcoin — adjacent survival-based framework
• Store of value vs medium of exchange vs unit of account — Boyapati’s four-phase monetization framework, adoption-curve adjacent
• Origins of money — Mengerian salability framework underlying monetary-good adoption
• Bitcoin as emergent money — emergence framework intersecting adoption dynamics
• Hard money vs fiat money — hardness as precondition for monetary-good adoption
• Portfolio approaches to Bitcoin — practical allocation implications of adoption-curve positioning
• Long-term price models and cycles — sub-MOC for the price-models area
• Network effects and Metcalfe’s Law — the network-economics framework underlying adoption-driven value compounding
• Giovanni Santostasi — Power Law adoption-mechanism originator
• Stephen Perrenod — Weibull-CDF adoption framing; mechanistic derivation co-author
• Vijay Boyapati — four-phase monetization framework
• Lyn Alden — empirical-macro framework intersecting adoption dynamics
• Saifedean Ammous — Mengerian-salability framework underlying monetary-good adoption
• Carl Menger — foundational salability theorist; monetary-good adoption dynamics
• Robert Metcalfe — Metcalfe’s Law; network-value side of the adoption-driven price derivation
• James Check — on-chain analyst engaging adoption-related metrics
• Ryan - On-Chain Mind — on-chain analyst engaging adoption-related metrics