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I have an undirected network and I am currently analyzing both three and 4 node motifs in the network. However, I can't find any articles which describe the biological significance of 3 of 4 node motifs in my network. Comparing with randomly generated networks, I am able to deduce that the number of three node motifs and 4 node motifs are significantly higher. However, I am unable to infer any biological significance for this finding.

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    $\begingroup$ Can you clarify what is your network? Is it a network of genes, protein-protein interactions or something else? Also do I understand right that you randomly generate networks without considering the fact that most biological networks are actually scale-free networks? $\endgroup$ – Maxim Kuleshov Oct 4 '17 at 21:43
  • $\begingroup$ @MaximKuleshov Its a transcriptional regulatory network price.systemsbiology.org/research/… $\endgroup$ – The Last Word Oct 4 '17 at 22:49
  • $\begingroup$ @MaximKuleshov I understand that TRNs are directed and bipartite comprising of transcription factors and the genes they regulate. I was just curious about motifs in co-expression networks. $\endgroup$ – The Last Word Oct 4 '17 at 23:00
  • $\begingroup$ @MaximKuleshov Also, I understand that Biological networks are scale free networks, I actually split my network into 8 clusters and the whole comparison to random networks is to prove that the clusters in themselves are real. $\endgroup$ – The Last Word Oct 4 '17 at 23:01
  • $\begingroup$ What is the structure of the 3 of 4 node motifs of your network for which there is no description of biological significance? - Which type of random network did you compare to? $\endgroup$ – tsttst Nov 4 '17 at 23:36
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I am not 100% sure that I understand the question, but I am going to try to answer, based on the following assumptions:

  1. The "number of 3 and 4 node motifs" is not very clear. If I understand correctly, it should be a quantity determined in large part by the degree distribution. You could rewire your network to lose all information about the TRN other than degree distribution and this number is probably very similar (or it is a quite weird network, which would make me suspicious of the data).
  2. Therefore, the number of such motifs of size $k$ is not particularly interesting biologically, specifically due to technical issues such as incomplete ascertainment of edges, etc. The true network may have many more edges, and this will change this number, so the degree distribution is not itself interesting- it is in fact the thing you want to control for.
  3. Therefore, when you compare with randomly generated networks and find different numbers of motifs of $k$ nodes, I suspect the random networks were generated with a different degree distribution. (It is pretty easy to generate networks of the same degree distribution, using e.g. the rewire() function in igraph)
  4. Therefore, I interpret the question to be more specifically: "Among all 3 and 4 node motifs, a subset are overrepresented in my TRN compared to randomized graphs of the same degree distribution. What are some ways to interpret these motifs?" This is the kind of question that is traditionally asked with TRNs in my experience.

As an example of how other groups have analyzed TRN motifs, I suggest looking at Figures 5 and 6 of this paper. For instance, the "feed-forward" motif is overrepresented for links involving some TFs. I believe that it is standard to compare motif distributions to other biological networks, such as the C. elegans neural network. There are references in the Cell paper that could probably help you out further. The wikipedia article on network motifs also looks to have some information, and there are other resources on the internet if you google.

While there is not necessarily a strict biological interpretation of such motifs, they nonetheless can be informative in identifying specific patterns peculiar to different TFs or master regulators.

Caveat It is possible that I have misunderstood the question or made bad assumptions- my graph theory is superficial. But if I am wrong and you are actually interested in the simple number of motifs of size $k$, that observation is of interest as a "I have a weird graph" theoretical problem, not as a "this is biologically interesting" problem. As a biologist I would care much more about which specific motifs of size $k$ you have, rather than the summed count of all motifs of size $k$.

On the other hand, forgetting about motifs for a moment, a "weird" network could be quite topologically interesting biologically. For instance, do your different clustered components associate with different pieces of biology, like sugar metabolism vs. morphogenesis? That would be expected, but it has very little to do with motifs- they may just be a side effect of that functional topology. In that case, it would be not only degree distribution but also that topology that you would have to control for to make interesting statements about motifs.

good luck!

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