When is a small world reached?

In “People of Peace in Small Worlds,” I explored the “small worlds” concept and how it is related to church planting movements.

Here is a representative picture of a “small world” that I have used in some of my presentations:

In this, you can see some of the less-connected participants (the smaller dots) and the more-connected participants (the larger dots, especially orange and red). The more-connected “hubs” are thus the principle keys to this swarm.

In Chapter 2 of Nexus, the author asks one of the most famous and valuable questions in small world or graph theory. Read the scenario below and see if you can figure out how it relates to evangelization and church planting movements.

Imagine you have been given the task of building roads to connect up the towns in an undeveloped country. At the moment, there are no roads at all—just 50 isolated towns scattered across the map. Linking them together is your task, but it is not quite that simple—you also face some constraints. To begin with, even if you request a road to be built in a precise location, the ever-incompetent Department of Roadways will simply ignore you and put it somewhere else, between a random pair of towns. Ask and it will be built—but there is no telling where.

To make matters worse, the country has very little money, and so you want to build as few roads as possible. The question then is this: how many will be enough? Given unlimited funds, you could command the Department of Roadways to keep building until every last pair of towns was linked together. To link each of the fifty towns to all forty-nine others would take 1,225 roads. That would certainly suffice.

But: what is the smallest number of roads you need to build to be reasonably sure that drivers can go between any two towns without ever leaving the pavement?

In world evangelization, we certainly don’t have unlimited funds. We’ve tried “saturation” approaches before—the equivalent of building a road between every town, or every house, or every whatever. But that’s not scalable in the long run. Nexus continues:

This problem is one of the most famous in graphic history. It need not involve towns and roads. It can be expressed in terms of houses and telephone links, people and links of acquaintances, a pack of dogs connected randomly by leads, or any of a thousand other things. The essential problem would be the same, and it is by no means an easy one, so you should not worry if you cannot see the answer.

Whew! In fact, this is the research that a famous mathematician named Paul Erdos solved in 1959. And, as Nexus tells us, it turns out that the completely random placement of 98 roads is adequate to ensure that the great majority of towns are linked. This represents just 8% of the total of 1,225 roads that could be built. “Putting links in places at random is not quite as inefficient as it may seem.”

“More surprisingly, the percentage required dwindles as the network gets bigger.” For a network of 300 points, out of 50,000 possible links, only 2% are needed. For 1,000 points, just 1% are needed. For 10 million points, just 0.0000016.

Within this model, there is perhaps the seed of an answer for the missiological question of when a people group is reached.

First, we can see that based on graph theory, very few connections are required for everyone to be connected. You work it out based on the formula: in a network with N vertices (nodes, dots, people), the links required are ln(N)/N, where ln(N) is the natural logarithm of N. Yes, complicated: but you can say the maximum is probably going to be 2% of connections.

But here’s the trick in this particular model: we are talking not about the “presence” of the Gospel within every people group (e.g. a church for every people) but rather a “path” for the Gospel within every people group, and the freeflow of the Gospel over that path. In order for the Gospel to transit the path, however, the “hub connections”—the people along the path—have to be willing to pass it on. It has to be “sticky” and “viral”—easily heard, easily understood, easily reproduced.

Now, as I mentioned in the last article, every person has a “decision threshold”—factors that determine whether or not the person is going to be involved in following Jesus and telling others. These factors typically have very little to do with the believability or understandability of a doctrine, and a lot more to do with the social network of the Gospel. It’s very telling that Jesus said “YOU are the light of the world”—not a particular book, but we His disciples. I am not suggesting the Bible isn’t important—for it certainly is, it is our defining authority—but the believability of the Word often rests upon the believability and credibility of the witnesses—we, his disciples.

This decision threshold can in some ways be the number of credible witnesses that a person sees. So in the attached video, I’m highlighting both the graphic problem above, as well as how a church planting movement can ripple through a small world—and where it can break down. After you see the video, think about this: how would you handle the few remaining “holdouts”? What do you think might be some appropriate approaches?