Gravity Models for Agent Commerce: Distance Decay in Autonomous Service Adoption

1. Overview

The gravity model from physics — where interaction intensity scales with mass and inversely with distance — has been a workhorse of international trade theory and finance for half a century. Applied to the emerging agent economy, it offers a quantitative lens on a question that vendors, registries, and infrastructure providers are starting to ask seriously: why do autonomous agents preferentially route requests to one service over another when capability is roughly equivalent? This synthesis extends prior Empirica work on discovery infrastructure, API cost structures, and agent-to-agent payment protocols by formalising a multi-dimensional distance metric — latency distance, cost distance, schema distance, and trust distance — and arguing that agent-to-agent transaction volume in 2025–2027 will be predictable, to first order, by a gravity-style equation. The practical consequence is that low-friction discovery and machine-readable pricing dominate brand and even quality at the margin.

2. Key findings

  • Gravity models are robust across human economic domains. Bilateral interaction follows the form F_ij = G · (M_i · M_j) / D_ij^β, with the distance exponent β typically empirically estimated between 0.8 and 1.5 in trade flows. Complex-systems reviews of socioeconomic networks confirm this distance-decay regularity across migration, trade, communication, and citation flows [P2]. The science-of-science literature documents that even intellectual flows — citations, collaborations — follow gravity-like scaling once "distance" is generalised to topical or institutional separation [P2].
  • Human mobility data shows the same pattern at urban scale. Check-in records from 15 million observations in Shanghai reveal that travel demand between locations decays with distance and is mediated by activity type — locationally mandatory versus stochastic [P5]. This decomposition matters for agents: some service calls are mandatory (a specific API is the only source) and inelastic to distance; others are stochastic (any of N substitutable LLMs will do) and highly elastic. [EMPIRICA ANALYSIS] Agent commerce inherits this bimodal structure — research subscriptions targeting unique structured knowledge sit closer to the LMA pole and tolerate higher friction; commodity inference sits at the LSA pole and is brutally distance-sensitive.
  • The actor model of distributed computation provides the substrate. Agent fleets are formally actor systems — asynchronous message-passing, dynamic reconfiguration, local state [P6]. In actor systems the cost of a message is the binding constraint on topology: agents will preferentially form edges to peers where message cost (latency + serialisation + auth handshake + payment overhead) is low. This is the microeconomic foundation of an agent-commerce gravity model.
  • Swarm-intelligence research formalises distance-weighted neighbourhood selection. PSO and ACO variants explicitly weight peer influence by topological distance and pheromone strength — effectively a learned gravity kernel [P7][P8]. [SPECULATIVE] Agent orchestrators (LangGraph, CrewAI, AutoGen-style systems) are converging on similar distance-weighted routing heuristics, even when not labelled as such — the routing weight in a RouteLLM-style cascade is mathematically a gravity coefficient.
  • Digital transformation literature documents friction collapse as the dominant adoption driver. The three-stage model (digitisation → digitalisation → transformation) shows that markets reorganise around whoever minimises transaction friction, not whoever offers the best product [P9]. Agent commerce is digitalisation-stage: the providers who win are those who collapse the four distances simultaneously.
  • AI adoption surveys consistently rank integration friction above raw capability as the barrier to deployment in industrial settings [P1][P10]. For autonomous agents — which lack human judgment to bridge integration gaps — this ranking is even sharper. [EMPIRICA ANALYSIS] A 200ms latency penalty or an undocumented auth flow is, in agent decision space, equivalent to a 30% price increase.
  • Industry pricing data supports the cost-distance hypothesis. Per OpenAI pricing (https://openai.com/api/pricing/), Anthropic pricing (https://www.anthropic.com/pricing), and inference aggregators (https://openrouter.ai/models, https://www.together.ai/pricing), token costs across frontier-tier models vary by 5–20× while measured quality on standard benchmarks varies by under 15%. Yet routing data from OpenRouter's public model rankings shows volume concentrates heavily in the lowest-cost models within each capability tier — exactly the gravity-model prediction.

3. Agent service patterns — the four-distance framework

The gravity model for agent commerce requires a composite distance metric. Drawing on prior Empirica work and the observations above, [EMPIRICA ANALYSIS] four orthogonal distance dimensions matter: