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Diffusion of knowledge is expected to be huge when agents are open minded.The report concerns a more difficult diffusion case when communities are madeof stubborn agents. Communities having markedly different opinions are forexample the Neocreationist and Intelligent Design Proponents (IDP), on onehand, and the Darwinian Evolution Defenders (DED), on the other hand. The caseof knowledge diffusion within such communities is studied here on a networkbased on an adjacency matrix built from time ordered selected quotations ofagents, whence for inter- and intra-communities. The network is intrinsicallydirected and not necessarily reciprocal. Thus, the adjacency matrices havecomplex eigenvalues, the eigenvectors present complex components. Aquantification of the slow-down or speed-up effects of information diffusion insuch temporal networks, with non-Markovian contact sequences, can be made bycomparing the real time dependent (directed) network to its counterpart, thetime aggregated (undirected) network, - which has real eigenvalues. In order todo so, small world networks which both contain an $odd$ number of nodes arestudied and compared to similar networks with an $even$ number of nodes.  It is found that (i) the diffusion of knowledge is more difficult on thelargest networks, (ii) the network size influences the slowing-down orspeeding-up diffusion process. Interestingly, it is observed that (iii) thediffusion of knowledge is slower in IDP and faster in DED communities. It issuggested that the finding can be "rationalized", if some "scientific quality"and "publication habit" is attributed to the agents, as common sense wouldguess. This finding offers some opening discussion toward tying scientificknowledge to belief.

Slow-down or speed-up of inter- and intra-cluster diffusion of controversial knowledge in stubborn communities based on a small world network