A line of research (e.g. [1]) has considered the question of self-similarity and self-dissimilarity in complex networks.
Very generally, a self-similar object is similar to a part of itself. A natural application of this definition to networks would be to say that a network is self-similar if its graph is composed of subgraphs that are structurally similar to the graph as a whole. Still, this is not yet a precise, quantifiable definition - for instance, what does it mean to say that the graph is "composed" of subgraphs?
Please comment and discuss.
[1]: http://arxiv.org/abs/cond-mat/0503078
Verrrrrrry interesting question!
Here is the full paper without the paywall... http://arxiv.org/abs/cond-mat/0503078
And then these by other authors... http://nlsc.ustc.edu.cn/phpcms/uploadfile/2012/0217/20120217063444214.pdf
http://arxiv.org/abs/0710.2092
I see repeating patterns in human networks... but do they meet the definition of self-similarity/fractalness??? Not obvious...
Graph composed of subgraphs...
Easy to compose a single graph component from "triangles" -- both open and closed.
When you allow for "overlap" of open and closed triangles, almost an unlimited number of graphs can be created. Whether they are self-similar is another, more difficult question. Core-periphery structure is often seen in human networks. The core-periphery structures often repeats at different levels of focus/aggregation.
Attached is a graph of an actual business organization showing only Simmelian Ties (Krackhardt, 1998) -- closed triangles of strong ties only. Notice core-periphery in each cluster and in the overall organization.
Interesting question!
I would say YES a complex network is self-similar, because it contains many nested substructures that possess far more small structures than large ones. In this paper, we illustrated that a complex network can be decomposed into many pieces, which tend to form the scaling hierarchy of far more small communities than large ones:
https://www.researchgate.net/publication/271771290_Defining_Least_Community_as_a_Homogeneous_Group_in_Complex_Networks
Article Defining Least Community as a Homogeneous Group in Complex Networks
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