arXiv subjects

John L. Pfaltz

Publications and source records attributed to John L. Pfaltz.

3 recordsLinked to original sources

arXiv2015

Domination and Closure

An expansive, monotone operator is dominating; if it is also idempotent it is a closure operator. Although they have distinct properties, these two kinds of discrete operators are also intertwined. Every closure operator is dominating; every dominating operator embodies a closure. Both can be the basis of continuous set transformations. Dominating operators that exhibit categorical pull-back constitute a Galois connection and must be antimatroid closure operators. Applications involving social networks and learning spaces are suggested

math.CO↗

arXiv2013

The Irreducible Spine(s) of Undirected Networks

Using closure concepts, we show that within every undirected network, or graph, there is a unique irreducible subgraph which we call its "spine". The chordless cycles which comprise this irreducible core effectively characterize the connectivity structure of the network as a whole. In particular, it is shown that the center of the network, whether defined by distance or betweenness centrality, is effectively contained in this spine. By counting the number of cycles of length 3 <= k <= max_length, we can also create a kind of signature that can be used to identify the network. Performance is analyzed, and the concepts we develop are illurstrated by means of a relatively small running sample network of about 400 nodes.

cs.DM↗

arXiv2012

Entropy in Social Networks

We introduce the concepts of closed sets and closure operators as mathematical tools for the study of social networks. Dynamic networks are represented by transformations. It is shown that under continuous change/transformation, all networks tend to "break down" and become less complex. It is a kind of entropy. The product of this theoretical decomposition is an abundance of triadically closed clusters which sociologists have observed in practice. This gives credence to the relevance of this kind of mathematical analysis in the sociological context.

math.CO↗