The Emergence of Informative Higher Scales in Complex Networks

Figure 5

The emergence of scale in preferential attachment networks. (a) By repeatedly simulating networks with different degrees of preferential attachment ( values) with new edge per each new node, and running them through a greedy algorithm (described in Materials & Methods), we observe a distinctive peak of causal emergence once the degree of preferential attachment is above , yielding networks that are no longer “scale-free.” (b) The log of the ratio of original network size, , to the size of the macroscale network, . Networks with higher values—more star-like networks—show drastic dimension reductions, and in fact, all eventually reach the same of 2. Comparatively, random trees show essentially no informative dimension reductions.

(a)

(b)