Six Degrees of the Alpha Male: breeding networks to understand population structure

This post is part of a continuing series on ways to think more deeply about human biological diversity. In last month's post (One Evolutionary Trajectory, Many Processes), I discussed how dual-process models (such as the DIT model) might be used to include a new dimension to more traditional studies of population genetics. This example did not spend too much time on the specifics of what such a model would look like. Nevertheless, a dual-process model provides a broader view of the evolutionary process, particularly for highly social (and cultural) species like humans.

In this post, I will lay out another idea briefly mentioned in the "Long Game of Human Biological Variation" post. This involves the use of complex network theory to model the nature of structure in populations. To review, the null hypothesis (e.g. no structure) is generally modeled using an assumption of panmixia [1]. In this conception, structure emerges from interactions between individuals and demes (semi-isolated breeding populations). Thus, a deviation from the null model involves the generation of structure via selective breeding, reproductive isolation, or some other mechanism.

One way to view these types of population dynamics is to use a population genetics model such as the one I just described. However, we can also use complex network theory to better understand how populations evolve, particularly when populations are suspected to deviate from the null expectation [2]. Complex networks provide us with a means to statistically characterize the interactions between individual organisms, in addition to rigorously characterizing sexual selection and the long-range effects of mating patterns.

An example of a small-world network with extensive weak ties. Importantly, this network topology is not random, but instead feature shortcuts and extensive structure. Data represents the human brain. COURTESY: Reference [3].

Since attending the Network Frontiers Workshop (Northwestern University) last December, I have been toying around with a new approach called "breeding networks". The breeding network concept [2] involves using multilayered, dynamical networks to characterize breeding events, the creation of offspring, the subsequent breeding events for those offspring, and macro-scale population patterns that result. This allows us to characterize a number of parameters in one model, such as the effects of animal social networks on population dynamics [4]. This includes traditional network statistics (e.g. connectivity and modularity parameters) that translate into theoretical measures of fecundity, the diffusion of genotypic markers within a population, and structural independence between demic populations. But these statistics are determined by a meta-process, one that is explicitly social and behavioral.

Before we continue, it is worth asking why complex networks are relevant. No doubt you have heard of the "small-world network" phenomenon, which postulates that given a certain type of network topology, networks with many nodes and connections can be traversed in a very small number of steps [5]. This is the famous "six degrees" phenomenon in action. But complex networks can range from random connectivity to various degrees of concentration. This approach, which comes with its own mathematical formalisms, allows us to neatly characterize the behavior, physiology, and other non-genetic factors that result in the population dynamics that produces structured genetic variation.

An example of regular, small-world, and random networks, ordered by to what extent their connectivity is determined by random processes [6]. In breeding networks, non-random connectivity is determined by sexual selection (e.g. selective breeding). As sexual selection increases or decreases, it can change the connectivity of a population.

As complex networks are made up of nodes and connections, the connections themselves are subject to
connection rules. In some networks, these rules can be observed as laws of preferential attachment [7].
But in general, each node or class of nodes can have simple rules for preferring (or ignoring) association with one node over another. If this sounds like an informal selection rule, this is no accident. While complex network theory does not approach connectivity rules in such a way, breeding networks are expected to be influenced by sexual selection at a very fundamental (e.g. dyadic interaction) level.

The complex network zoo, and the three parameters (heterogeneity, randomness, and modularity) that define the connectivity of a network topology. Examples of specific network types are given in the three-dimensional example above, but breeding networks could fall anywhere within this space. COURTESY: Reference [8].

Another feature of breeding networks involves connectivity trends over time. For example, a founder population with a small effective population size might indeed be panmictic (in this case represented by a random network topology). However, as the population size increases and connectivity rules change, this topology can evolve to one with scale-free or even small-world properties. This is not only due to the selective nature of producing offspring, but difference in the fecundity of individual nodes.

Once you start paying around with this basic model, a number of alternative network structures [9] can be used to represent the null model. Types of configuration such as star topologies, hyperbolic trees, and cactus graphs can approximate inherent geographic structure in a population's distribution. These alternative graph topologies are the product of factors such as geography or migration, and may have pre-existing structure. The key is to use these features as the null hypothesis as appropriate. This will provide us with a better accounting of the true complexity involved in shaping the structural features of an evolving population.

A map showing the seasonal migration of shark populations in the Pacific, including aggregation points. COURTESY: The Fisheries' Blog.

NOTES:
[1] One model organism for understanding local and global panmixia is the aquatic parasite Lecithochirium fusiforme. For more, please see: Criscione, C.D., Vilas, R., Paniagua, E., and Blouin, M.S.   More than meets the eye: detecting cryptic microgeographic population structure in a parasite with a complex life cycle.
Molecular Ecology, 20(12), 2510-2524 (2011).

[2] The idea of breeding networks is similar to the idea of sexual networks, except that breeding networks are more explicitly tied to population genetics. This paper give good insight into how sexual selection factor into the formation of structured, complex networks: McDonald, G.C., James, R., Krause, J., and Pizzari, T.   Sexual networks: measuring sexual selection in structured, polyandrous populations. Proceedings of the Royal Societiy B, 368, 20120356 (2013).

[3] Gallos, L.K., Makse, H.A., and Sigman, M.   A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks. PNAS, 109(8), 2825-2830 (2012).

[4] For more on animal social networks and their relationship to evolution, please see the following references:

a) Oh, K.P. and Badyaev, A.V.   Structure of social networks in a passerine bird: consequences for sexual selection and the evolution of mating strategies. American Naturalist, 176(3), E80-89 (2010).

b) Kurvers, R.H.J.M., Krause, J., Croft, D.P., Wilson, A.D.M., Wolf, M.   The evolutionary and ecological consequences of animal social networks: emerging issues. Trends in Ecology and Evolution, 29(6), 326–335 (2014).

[5] For a definition of network diameter in context, please see: Porter, M.A.   Small-world Network. Scholarpedia, 7(2), 1739 (2012).

[6] This classification of idealized graph models is based on the Watts-Strogatz model of complex networks. For more information, please see: Watts, D.J. and Strogatz, S.H.   Collective dynamics of 'small-world' networks. Nature, 393, 440-442 (1998).

[7] This property of idealized graph models is based on the Barabasi-Albert model of complex networks. For more information, please see: Barabasi, A-L. and Albert, R.   Emergence of scaling in random networks. Science, 286(5439), 509–512 (1999).

[8] Sole, R.V. and Valverde, S.   Information Theory of Complex Networks: On Evolution and Architectural Constraints, Lecture Notes in Physics, 650, 189–207 (2004).

[9] Oikonomou, P. and Cluzel, P.   Effects of topology on network evolution. Nature Physics 2, 532-536 (2006).