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Network Theory and the Formation of Groups Without Evolutionary Forces

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Abstract

This paper presents a modified random network model to illustrate how groups can form in the absence of evolutionary forces, assuming groups are collections of entities at any level of organization. This model is inspired by the Zero Force Evolutionary Law, which states that there is always a tendency for diversity and complexity to increase in any evolutionary system containing variation and heredity. That is, in the absence of evolutionary forces, the expectation is a continual increase in diversity and complexity at any level of biological hierarchy. I show that, when modeled, this expectation of increasing variation results not only in the formation of groups, but also in a higher probability of group formation than is found in a model that is purely random.

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Notes

  1. This is often referred to in the graph theory and network literature as a clustering coefficient, and random networks are known for their extremely low clustering coefficients.

  2. This example is also useful because it illustrates the point that there can be degrees of connectivity between entities. This is not apparent in the ZFEL model since it is designed such that entities are either connected or they are not. Weighted networks incorporate weighted edges, or different edge strengths, and I hope to address this issue in future research.

  3. This denominator was also used when measuring 3-cycles and 4-cliques in the random model, except for changing the number of edges being chosen (see “Appendix 1” for details).

  4. This denominator was also used for measuring 3-cycles and 4-cliques in the ZFEL model, except for changing the number of edges being chosen (see “Appendix 1” for details).

  5. It is not necessary for each new node state to be unique; this method was just for ease of calculation and evaluation. A different model, where at time T 1 differentiated nodes change from state 0 to 1, and then at T 2 differentiated nodes change from either state 0 to 1 or 1 to 2, could also be used; however, the results in this paper would be the same under both methods because the only real constraint is that edges do not form between two nodes in state 0.

  6. If this is still confusing, consider the following example: At time T 8 there are four nodes, A, B, C, and D in the states 0, 5, 3, and 7, respectively. There are also the following edges: {A, D} {A, C} and {C, B}. At time T 9 the four nodes, A, B, C, and D are in states 0, 5, 8, and 7, respectively. Because node C changed states, only the edge {A, D} is carried over to T 9 and every other possible pair of nodes has the same probability of forming a new edge.

  7. A great amount of integration and interconnectedness among nodes is also known as high modularity in network theory. Although the definition of modularity varies in biology, the use of the term in network theory is consistent with an operational definition of modularity sometimes used in biology (see Dassow and Munro 1999, p. 312).

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Acknowledgments

I am eternally grateful to Eric Bair for helping me program, and for running large numbers of simulations for me. I am also incredibly thankful to Tim Schwuchow for helping me create equations to represent my model as well as for giving me moral support. Thanks also goes to Robert Brandon, Daniel McShea, Alex Rosenberg, Carl Simpson and David McCandlish, and to the Philosophy of Biology reading group at Duke University for giving me comments on my ideas when they were in their first stages.

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  1. Department of Philosophy, Duke University, 201 West Duke Building, Durham, NC, 27708, USA

    Leonore Fleming

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Correspondence to Leonore Fleming.

Appendices

Appendix 1

In the following equations, n is the total number of nodes. In the ZFEL model, because two undifferentiated nodes cannot form an edge, x represents the number of differentiated nodes. Similar to cluster calculation (see “Methods”), to calculate the probability of a cycle of size 3 (3-cycle) forming in the random model, Eq. 3 divides the number of possible graphs producing a 3-cycle by the total number of possible graphs that can be formed with n nodes and three edges. To calculate the same probability in the ZFEL model, Eq. 4 does the same thing while taking into account the number of nodes that have differentiated, represented by x. To form a cycle in the ZFEL model either all three nodes are differentiated, represented by \( \left( {\begin{array}{*{20}c} x \\ 3 \\ \end{array} } \right) \), or only two of the three nodes must have differentiated. There are three ways to create a cycle with two differentiated nodes. For example, if we look at nodes A, B, and C again, with edges {A, B}, {B, C} and {A, C}, a cycle can have either A, B or C undifferentiated as long as the other two are differentiated. This is represented by \( (n - x) \cdot \left( {\begin{array}{*{20}c} x \\ 2 \\ \end{array} } \right) \), which assumes one undifferentiated node and chooses two differentiated nodes to make a cycle. The denominators of both cycle equations were explained in “Methods”.

$$ \frac{{\left( {\begin{array}{*{20}c} n \\ 3 \\ \end{array} } \right)}}{{\left( {\begin{array}{*{20}c} {\frac{n \cdot (n - 1)}{2}} \\ 3 \\ \end{array} } \right)}} $$
(3)
$$ \frac{{\left( {\begin{array}{*{20}c} x \\ 3 \\ \end{array} } \right) + (n - x) \cdot \left( {\begin{array}{*{20}c} x \\ 2 \\ \end{array} } \right)}}{{\left( {\begin{array}{*{20}c} {\frac{n \cdot (n - 1) - (n - x) \cdot (n - x - 1)}{2}} \\ 3 \\ \end{array} } \right)}} $$
(4)

Lastly, to calculate the probability of a clique of size 4 (4-clique) forming in the random model, Eq. 5 divides the number of possible graphs producing a 4-clique by the total number of possible graphs that can be formed with n nodes and six edges, and for the ZFEL model, Eq. 6 does the same thing while taking into account the number of nodes that have differentiated, represented by x. Because all nodes must be differentiated and able to form edges in a clique, for the ZFEL model, the four nodes must be chosen from the number of differentiated nodes, represented by \( \left( {\begin{array}{*{20}c} x \\ 4 \\ \end{array} } \right) \). The denominators of both clique equations were explained in “Methods”.

$$ \frac{{\left( {\begin{array}{*{20}c} n \\ 4 \\ \end{array} } \right)}}{{\left( {\begin{array}{*{20}c} {\frac{n \cdot (n - 1)}{2}} \\ 6 \\ \end{array} } \right)}} $$
(5)
$$ \frac{{\left( {\begin{array}{*{20}c} x \\ 4 \\ \end{array} } \right)}}{{\left( {\begin{array}{*{20}c} {\frac{n \cdot (n - 1) - (n - x) \cdot (n - x - 1)}{2}} \\ 6 \\ \end{array} } \right)}} $$
(6)

For all of the computations in this paper, the value of n was 100, and for the ZFEL model, the following values were input for x: 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100. When x is 100 in the ZFEL model, it is equal to the random model, since in both cases all nodes are able to form edges.

Appendix 2

Cycles are more difficult to measure over time than clusters because of the quick shift from few or no cycles emerging (when edge numbers are low) to a somewhat complicated mess of many interconnected cycles (when edge numbers increase). Part of the problem is, unlike clusters, in graph theory, a node can be in more than one cycle at a time, meaning that when edge numbers increase only by a little, the number of cycles gets extremely difficult for the program to count. This was also a problem with cliques because a node can be in more than one clique at the same time as well. The ZFEL model had a much larger “window” in which cycles could be counted, however, most of those results could not be compared to the random model. Therefore, the most useful data comparing the two models was the longest cycle size over five different edge formation probabilities after ten generations (see Fig. 12). There are only two data points to display with the random model (the arrow is the second data point), however, the arrow represents the steep increase expected in the random model based on individual runs even though the 100-run simulation data could not be calculated. The trend in longest cycle length in the ZFEL model also increases rapidly, but at a much slower rate.

The minimum clique size measured was two nodes (an edge). In Fig. 13, which shows the average clique size in relation to the number of cliques, it is clear the difficulty of a 3-clique or larger forming in both models, even when the number of cliques is high. This means a 3-clique would likely form in an already very highly connected network. This can be seen in Fig. 14, where the largest clique size is compared to the number of cliques, and gives similar results to Fig. 13. Even at very large numbers of cliques, both models still barely produce cliques of size four or five. These numbers are the averages over 100 simulations. Unlike the other data, because clique size is so variable, it is quite possible that within a set of parameters, one simulation run could produce a clique of say, size 6, while the majority of runs only produce cliques of size 2. It is good to keep this in mind when interpreting the results. Figures 13 and 14 have logarithmic trend lines added.

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Fleming, L. Network Theory and the Formation of Groups Without Evolutionary Forces. Evol Biol 39, 94–105 (2012). https://doi.org/10.1007/s11692-011-9140-9

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