Generally, we measure decentralization by measuring its opposite, centralization. There are actually formal measures of centralization in graph theory, which are based on what proportion of links are in by what proportion of nodes. A highly centralized network will have many of its links accounted for by a small number of nodes.
Since you asked this as a "social sciences" question, I assume you're most interested in examples from political and economic systems.
A more centralized government is one in which more decisions are made by the top decision-makers, rather than delegated to intermediate levels of leadership. For example, federal law is more centralized than state law, which is more centralized than city ordinances. Depending on what proportion of important laws are made at each level, a government could overall become more centralized or decentralized. There is reason to think, in fact, that the US government has become more centralized over time, with the federal government taking on more and more roles previously reserved for state governments. (Is this good or bad? Hard to say. But it's definitely more centralized.)
A government could also be considered more decentralized if decisions are made by a large legislative body instead of a single executive official. In this sense the US government has definitely gotten more centralized, as the role of the President has greatly expanded over time.
Businesses, or even whole industries, can also be consider more or less centralized. An industry with a handful of major corporations that control it (such as aircraft manufacturing) is more centralized than an industry with a large number of competing producers (such as tomato farming). Even within a company, decision-making could be more decentralized if the management structure is set up to delegate more authority to middle managers or employees instead of top executives.
It sounds like you might also be interested in circumstances that might lead to greater or lesser decentralization. This is a much harder question; there isn't a lot of consensus withing political science on why some governments become more centralized than others. It may be related to complex cultural and historical factors. We do have some reason to think that countries with a lot of easily-extracted natural resources (such as gold or oil) tend to have more centralized governments, because the government often forms around that particular extractive industry. But even then there are clear exceptions, such as Norway, which produces a lot of oil but has a very decentralized government.
In economics at least, we do have the concept of a natural monopoly, or more generally a natural oligopoly. In either case, this is an industry where, given the amount of demand, and operating at optimal efficiency, there is a fixed number of companies that would sell in that market. Either more or fewer companies entering the market would end up producing less efficiently. A natural monopoly is the special case when the optimal number of companies is exactly 1. You could also have a natural duopoly (2), a natural triopoly (3), and so on.
A good example of a natural monopoly is municipal water; the setup cost for water lines is very large, but the marginal cost of providing water is very small, so it makes the most sense in terms of efficiency for one company to set up to provide water and then provide it to everyone in the city.
In the presence of a natural oligopoly, centralization becomes greater. The smaller the oligopoly, the stronger this centralization becomes.