Starant Graphs

In this piece, the paper Deterministic Small-world Communication Networks, a paper from the year of 2000 by Comellas, Ozon, and Peters, is discussed. From the introduction, where some mismatch between sigmatoids, and intended senses is identified, to the conclusion, where the same mismatch appears in the shape of wonder, the findings are of surprising nature: major misunderstandings in the interpretation of the research or teaching or invention of others may lead to great new theories, which may lead to wonderful sets of new, and meaningful mathematical paradigms.

Counter-examples to the 6-degree theory, starant graphs and cycles, comellas and circulants, and so on.

This poster describes our work in Graph Theory so far. This work is based on the work of Comellas et al., which is based, on its turn, on the work of Watts and Strogatz, which is based, on its turn, on the work of Milgram. The so famous Six Degree Theory gets mentioned in it. The own Milgram had recognised it contained a fallacy.

Refining the results from the work of Comellas et al. regarding deterministic small-world networks, and intending to apply results deduced from theirs into traffic networks, we introduce new constraints, extend their work to networks of circulants, criticize the choice of the name ‘small worlds’ for large circulants, with a number greater than 64 for their vertices and, as a side result, we introduce a new form of graph: starants as a replacement of the circulants for the case of disease spread and social networks. In order to reach our goals we make use of standard combinatorial tools, graph analysis, and general algebraic procedures.

In this paper, we put two concepts together: Shortest paths and starant graphs. We calculate the costs involved in putting two randomly selected individuals in contact in a controlled network. That would be the costs in terms of public health. Disease spread became our main concern in what comes to the starant graphs in the year of 2002 because that is one of the directions the work of Comellas et al. and Watts et al. pointed at, and our work is inspired in theirs. Other factors, such as random, and unexpected, contact between individuals, are disregarded, so that if the individual visits the clinic that belongs to Mister X, his mate, but his usual doctor, Mister Y, is not there, and he is then served by Miss R, we will need new calculations, what means that we go from predictive power to disgrace power, and that frontally opposes our initial intentions with this work.

In this paper, we will explain the relevance of the starant graphs, graphs created by us in the year of 2002. They were basically circulant graphs with a star graph that connects to all the vertices of the circulant graphs from inside of them, but they did not exist as a separate object of study in the year of 2002, as for all we knew. We now know that they can be used to model even social networking interactions, and they do that job better than any other graph we could be trying to use there. With the development of our mathematical tools, lots of conclusions will be made much more believable and therefore will become much more likely to get support from the relevant industries when attached to new queries.