  In this paper we investigate phenomena of spontaneous emergence or purposeful
formation of highly organized structures in networks of related agents. We show
that 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. We
apply this approach to the study of the emergence or formation of simple
organized, hierarchical, structures based on Sierpinski Graphs and we prove a
Ramsey type theorem that bounds the number of vertices in Kolmogorov random
graphs that contain Sierpinski Graphs as subgraphs. Moreover, we show that
Sierpinski Graphs encompass close-knit relationships among their vertices that
facilitate fast spread and learning of information when agents in their
vertices are engaged in pairwise interactions modelled as two person games.
Finally, we generalize our findings for any organized structure with succinct
representations. Our work can be deployed, in particular, to study problems
related to the security of networks by identifying conditions which enable or
forbid the formation of sufficiently large insider subnetworks with malicious
common goal to overtake the network or cause disruption of its operation.
