2.1 The Decentralized Physical Infrastructure Network (DePIN) Concept

The decentralized physical infrastructure concept was brought to the attention of the academic world a few years ago. Broadly, it refers to the decentralization of any physical infrastructure network, from electricity grids and city-wide heating systems to wireless communication and storage networks, etc. DePINs are developed for a variety of purposes as well: better reliability compared to centralized networks, lower capital investment, shorter response latency, etc. It is worth noting that DePIN refers to two distinct cases. In the first case, it refers to a decentralized network structure that, for example, uses locally generated power (solar panels) instead of one big power plant, or uses wireless mesh networks instead of one network configured in a star structure. In the other case, DePIN refers to building an infrastructure network in a decentralized way by motivating community or multiple participants to build together, instead of building the network by its operator only.

DePIN has multiple sectors: cloud servers, wireless networks, sensor networks, automobile networks, energy networks, smart grid networks etc [15]. Like with other decentralized systems, this network model is subject to the blockchain trilemma: a trade-off has to be made among decentralization, scalability and security.

The cost model of the DePIN is important as a lot of them rely on the token incentive to bootstrap the network. The cost of the network has to be well modeled to function as the base of the tokenomics. The cost model is developed with a focus on designing (or modifying) an infrastructure system to provide a particular infrastructure product or services (water, natural gas, electricity, internet etc.) for a set of n locations or subnetworks (towns, buildings, etc.). Each location has some known demand for the product and also has the ability to produce this product locally with a production cost that varies with geography. In order to satisfy the demand at each location, one can either produce materials locally, or build an interconnection to some nearby locations that can produce them less expensively. A formula has been proposed [16] which can be used for optimizing the cost-efficiency of a prospective DePIN . The formula has been simplified as follows:

where 𝑂 stands for the optimal investment cost coefficient. It is a normalized item and independent of the number of locations. This allows us to study the optimal scheme of a centralized tree / star structure vs. a system of decentralized subnetworks or a mesh connection structure. The same formulation can be applied to networks of different spatial scales: a large country with many nodes (for instance), or a small city with fewer nodes, can be modeled by optimizing the value for this parameter. In most cases, the actual cost shall be close to linearly proportional to the number of locations. 𝐶 is a vector representing the marginal cost of producing one unit of a good, and 𝐾 is a vector representing the levelized cost of the production capacity needed for a node to produce at a rate of one unit per unit time. For example, if a particular power plant technology costs $1000/𝒌𝑾 to build, it would have a levelized capacity cost of 𝒌 = $79.2 per 𝒌𝑾-year when amortized over a 20-year horizon at a 5% discount rate. If its production cost were $10 per MWh, it would have an annual production cost of 𝒄 = $87.66/𝒌𝑾-year(8766 hours per year). 𝐺 represents the quantity of goods produced, 𝑤√𝑁 is an interconnection cost parameter (cost per unit length-capacity), 𝑁 is the number of locations, 𝑤 is the cost, 𝐿 is the matrix of distance between each node, and 𝐹 is the flow capacity. The smallest value of w refers to the situation wherein the mean component size is at or near the size of the network, while for the larger values of w, the component size is far smaller than the size of the network. For the case 𝑤 = 1, it is hardly optimal to build any interconnection.

Reliability plays an enormous role in the design of infrastructure systems. 𝑷 measures the cost of not serving demand in response to a set of node or link outages (perturbations). These costs are assumed to follow a linear function of the amount of unserved load over all perturbations. 𝒓 is the reliability cost parameter, ranging from 0 to 1, and ∆𝐷 is the change (loss) of demand that results from perturbation, which is always a non-positive value. As a whole this formulation allows us to observe how network size and structure changes as we increase the relative importance of reliability. If 𝒓 = 0, demand losses are effectively deemed irrelevant. On the other hand, as r increases, we hypothesize that networks are likely to become more meshed and more likely to include surplus production capacity. On the one hand, small, local networks will be more robust to failures and thus may be more optimal when reliability is very important. On the other hand, large interconnected systems provide a high level of redundancy, which also brings tremendous value. A tradeoff has to be made.

The model under discussion is designed to find fundamental properties of the DePIN system that do not depend on the size of the network. Thus, the cost function is designed so that both the production and the construction terms grow linearly with n. It has been observed, at least for the case of uniformly distributed node locations, that the distances between randomly selected node pairs decreases according to l ~ 1 ⁄ √𝑁𝑁 , as a a result ensuring that linear growth of the interconnection cost term requires that we scale the relevant term by √𝑁. The similar consideration given to the reliability item - the reliability term, is divided by 𝑁 so that this term also roughly increases linearly with 𝑁, as ∆𝐷 is proportional to the number of perturbations, which is also proportional to the number of the nodes.

This formula explains the value created by production:

𝑉 = (𝑅 − 𝑂) ∙ 𝑁

where 𝑹 is the revenue per node, 𝑶 is the optimal cost per node, and 𝑵 is the node matrix.

As for the decentralized telecommunication network, it is also important to note that Metcalfe's law applies. Metcalfe's law states that the value of a telecommunications network is proportional to the square of the number of connected users of the system (𝒏²). It works with both telecom networks as well as networks such as the Internet, social networks, and the World Wide Web.

However, the number n refers to the number of end devices, like cell phones, while in the Formula , N refers to the number of locations or subnetworks, like a gateway. In this situation

𝑉 = 𝐴 ∙ 𝑛² + (𝑅 − 𝑂) ∙ 𝑁

where 𝑨 is the network value coefficient for the telecommunication network. Often, it refers to the value generated by the information flow, while R is the direct value generated by the connectivity.