A topic that I’ve researched a bunch over the years is the interaction between people in a network of relationships and an outsider who wants to tell or sell them something. You know something about the structure of the network that people belong to: who talks to who. Which people do you send messages or promotions to?

A pretty robust feature of this problem is that if you want to reach or persuade as many people as you can with as little effort as possible, you don’t necessarily just want to start off by targeting all the people with the most connections.

Take a simple example of a network with small world properties. These are “six degrees of separation” type networks, where you don’t necessarily have loads of connections, but you can reach pretty much anyone in a surprisingly small number of steps.

Let’s say that people share information with their network neighbors. You want to tell all of these people something, but you want to send as few messages as possible—you want communication across the network to do some of the work for you. So you figure out (or, maybe, buy from the network operator) information on who the best connected people in the network are, and you send three of them messages:

When these folks talk to their neighbors, here is who got the message:

Here’s a different way you could have sent three messages:

And here’s who got the message after these three talk to their neighbors:

For the same number of messages, this time, you reached everyone. It might be a little less intuitive to “waste” some messages on people with relatively few connections, but in a network like this, it gives you more bang for your buck. The reason is that this time you avoided redundant messaging, where the same little area of the network got the message many times, while some other areas never heard it at all.

The idea here is that if you want to blanket a network with information, it pays to think about the connectivity—the coverage—of sets of nodes together, rather than thinking only about the connectivity of individual nodes. Dispersed word-of-mouth is more lucrative than concentrated word-of-mouth.

Now, this is certainly a very stylized example. There are plenty of good reasons why you might want to disproportionately target the “dense” area of the network—maybe it pays for people to hear your message from more than one friend, maybe messages travel further than just immediate neighbors, and so on. But the spirit of the set connectivity approach can still teach us something in more complicated settings too.

For example, would targeting based on demographic characteristics look more like the set connectivity targeting example or the individual connectivity targeting example? If someone who runs a network offers to sell you ad targeting to a bunch of nodes, how would you like that bundle to be composed?

This example was pretty simple. You just want to send the fewest messages to inform everyone. What I’ve done in my research is to look at problems of this type that are a little trickier to think through.

First: what if you have to persuade people as well as inform them? That is, what if people are skeptical that you’re a good person or have a good product, so that on top of letting them know you exist, you have to convince them you’re worth their time or money? This is called a signaling game in economics, and what I did is to make it so that the signaler is trying to persuade a network of people who share information with their neighbors.

There’s a well-established concept in economics called money burning for signaling situations. You try to prove how confident you are that you have a great product that will attract loyal customers by setting fire to a bunch of money in public: “see? I will make all of this back from you and more. Trust me!”

What I found in the networked case is that the dispersed word-of-mouth set pops back up here. The amount of money you have to burn is the smallest when you combine it with targeting the dispersed set. The reason is that targeting that set maximizes the number of people who will learn how good you are from their neighbors, and, therefore, minimizes the number of people you have to convince indirectly by burning money.

The idea of expensive, dispersed launch promotions is not totally alien:

That’s Colin Kaepernick with his personalized Beats, Lady Gaga on the cover of “Poker Face” with her personalized Beats, and Lil’ Wayne with his diamond-encrusted(!) personalized Beats. No matter what kind of thing you’re a fan of, chances are you’ve seen someone you follow with a one-of-a-kind pair of headphones hanging around their neck.

The published paper for signaling to a network is here, or an ungated pre-print is here.

A second bit of research: what if you’re not the only person trying to get information out to the network? What if there’s a second person out there with the same goal? You would like to inform as many of people as possible with as little effort as possible, but you also don’t want to be beaten to the punch by your opponent.

The dispersed word-of-mouth set shows up again here, but not always. Life is a lot more complicated with two competing informers. It matters a lot how they compete: if someone finds out about both you and your opponent, how costly is your fight over them? It matters a lot how expensive it is to send messages: if messages are super cheap, do you just send messages to everyone to avoid the risk of being beaten to the punch?

It turns out that you and your opponent may end up economizing on messages—dispersing your word-of-mouth—only in very special cases. If you don’t compete too fiercely and if sending messages is quite expensive, you might reach an accord where you target the dispersed set from before. But if those conditions change, you might end up in either an fragile, patchwork slicing up of the network into monopolized turf, or in a messaging arms race that saturates the network in messages from both of you.

The weird thing here is that competition among informers can upend the incentives of the operator of the network. If they are setting a price to send messages to a given node, they might have a new incentive to cut the price of messages to the point where they induce that arms race. And you’re left wondering why you weren’t able to exploit word-of-mouth to save you money.

The published paper for competitive targeting is here, or an ungated pre-print is here.

I still do some work on questions like these, among other things, and I have a couple of things in the works about outsiders interacting with networks. There are loads of examples out there that fit that description, so there’s a lot to think about. Feel free to drop me a line if you have any thoughts or suggestions!