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In this paper we investigate phenomena of spontaneous emergence or purposefulformation of highly organized structures in networks of related agents. We showthat the formation of large organized structures requires exponentially large,in the size of the structures, networks. Our approach is based on Kolmogorov,or descriptional, complexity of networks viewed as finite size strings. Weapply this approach to the study of the emergence or formation of simpleorganized, hierarchical, structures based on Sierpinski Graphs and we prove aRamsey type theorem that bounds the number of vertices in Kolmogorov randomgraphs that contain Sierpinski Graphs as subgraphs. Moreover, we show thatSierpinski Graphs encompass close-knit relationships among their vertices thatfacilitate fast spread and learning of information when agents in theirvertices are engaged in pairwise interactions modelled as two person games.Finally, we generalize our findings for any organized structure with succinctrepresentations. Our work can be deployed, in particular, to study problemsrelated to the security of networks by identifying conditions which enable orforbid the formation of sufficiently large insider subnetworks with maliciouscommon goal to overtake the network or cause disruption of its operation.

On the existence of highly organized communities in networks of locally interacting agents