Allee Effects and Small World Networks
It sometimes takes reading about a subject several times from different sources to pick up on something that should have been evident the very first time but somehow wasn’t. Or at least that’s how it is with me. For instance, I recently read Clay Shirky’s book “Here Comes Everybody” and was struck by his description of small world networks as being characterized by a balance of small, densely connected groups and large, sparsely connected groups. Now, I’d previously read several other books about aspects of network theory and had ooh’ed and aah’ed at the examples and explanations of the 6 degree of separation phenomenon, synchronicity and other network effects, but this unexpected component of the small world network structure, first described in 1998 by Duncan Watts (Six Degrees) and Steve Strogatz (Sync), had never really taken hold in my brain until now.
And I think maybe the reason for my sudden but belated awareness is another phenomenon that I stumbled upon a few years back as I was researching success and failure in “founder”, or start-up, groups.
Warder Clyde Allee, an ecologist and zoologist working in the first half of the twentieth century, was particularly interested in how animal groups form and survive. In the classical model of population dynamics, over-crowding typically has a negative effect on group growth. What Allee discovered was that under-crowding can also be a problem. Below some critical density, small populations – as founder groups typically are - can be at increased risk of group decline and even extinction due to effects such as:
- Inability to find mates during the mating season;
- Reduction in foraging efficiency (and other activities) in the absence of cooperating hunters;
- Increased vulnerability to predators when group defensive behaviours (e.g. herding, flocking, schooling) become ineffective at low prey numbers
- Failure to attract pollinators when there are few conspecifics nearby to create an appealing place to forage
Achieving density is thus a prerequisite to population growth. But only up to a point. At some threshold the problem of under-crowding, also known as positive density dependence, (or Allee effects, after their discoverer) suddenly gives way to the problem of over-crowding, or negative density dependence (the classical model).
In other words, it’s important for small populations to get to critical density, but as they grow they need to spread out. And maybe something like this is what’s happening in small world networks. Small groups thrive at high densities (positive density dependence) and perhaps even need that density to retain group coherence or identity and thus survival. But large groups - where individuals begin to compete with one another - thrive at lower densities (negative density dependence). And I’m guessing that this is a recurring, dynamic pattern. New groups will form over time and only those that develop dense connections will persist. But as they grow they will tend to dissipate, leaving the leaders and fast followers with new groups to found and more Allee effects to overcome.