Sprint’s Marketing Claim and Non-Linearity

“In fact, Sprint’s reliability now… performs within 1 percent of AT&T and Verizon. Claim based on Sprint’s analysis of latest Nielsen drive test data for average network reliability in top 106 markets”

Sprint is essentially claiming that the alleged 1% increase in Verizon’s network reliability means that the extra money which Verizon charges, due to it’s more robust network, is not worth the cost. Presumably, this cost is also greater than 1.01 times Sprint’s. I’ll explain why I think this is a dubious claim.

Firstly, let’s establish a few things. A network is more reliable if it is more connective, something more connective must be so primarily due to more connections, and a network that has more connections must tend to be more reliable and connective (1). Therefore, Sprint is implying that the incremental connectivity benefit associated with Verizon’s marginal increase in connections (aka coverage) is not justified by the incremental cost — in other words, that the marginal ROI (2) for the increase in connections is negative.

The degree of a network’s reliability/connectivity must increase non linearly with coverage since more coverage generates more interactions among the nodes and every single interaction depends upon all of the others. It’s also a fact that since these networks are real-world “non-random” networks their connectivity is power law distributed within the system (a property exhibited by Pareto’s Law), meaning a minority of nodes will have a majority of the connections.

Owing to the importance of the interactions between connections and their unequal distribution, the properties of this network as a whole are fundamentally different from that of the individual components. Since the interactions of the average component is less than the average interactions of the whole, you cannot arrive at an understanding of the whole by aggregating the individual components, of which Jensen’s Inequality is proof (3). With aggregation being invalid, averages are also uninformative which alone dismisses Sprint’s average-based claim.

As a consequence of non-linearities, you can’t say that a small increase in coverage would result in a correspondingly small increase in connectivity, hence benefit. For instance, going from 300 to 301 neurons, an increase of about 0.3%, may more than double the complexity and so increasing a vastly complex network by 1% could increase the complexity by a huge amount — there are even instances where going from 1,000 to 1,001 increases complexity by billions of times.

To summarize, the presence of non-linearities means that Verizon’s small increase in coverage, which Sprint concedes, could easily generate outsize benefits likely outweighing the incremental cost of having Verizon.

Footnote 1: There are obviously other things that go into reliability beyond quantity of connections, such as quality of infrastructure, technology, operations, etc. but given that these are two market leaders and we know that Sprint has been investing heavily in their technology it would make sense that the main differentiator would be coverage driven by the size of its network.

Footnote 2: mROI = (incremental gain / incremental cost — 1) * 100

Footnote 3: The degree of error you’ll get if you do increases with the degree of non-linearity. Jenson’s Inequality proves that, under non-linearities, a function of an average is not an average of a function, and the difference increases with disorder.

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