Article Preview
TopIntroduction
Structural analysis of complex networks enables to explore social networks, sector economics, lexical taxonomies, etc. Networks are generally represented by graphs, and the aim is to study relevant interactions between individuals (nodes). Those interactions can be of various types such as dependency, influence, etc.
However, the graph theory framework does not allow to easily model relations at different levels (e.g. individual to group, group to group), thus limiting both modelling possibilities and levels at which structural analysis can be performed (Dalud-Vincent et al., 2001). To overcome such limitations, (Dalud-Vincent et al., 2001) proposed to use the pretopological framework, presented in (Belmandt, 1993; Brissaud et al., 2011) as an extension of graph theory.
The concept of pretopology was first introduced in the 1970’s (Brissaud, 1971, 1975). It results from a weakening of the axiomatic of topology, the latter being seen as a mathematical way to formalize human perception based on the concept of similarity. The notion of similarity can be declined mainly in two ways: proximity and approximation. Proximity can be a relation between two objects, an object and a set of objects, or between two sets of objects. On the other hand, given a discrete set 
, the notion of closure 
applied to a point 
(resp. a set of points 
) enables a simple formalization of the concept of approximation: 
(resp. 
) is what one sees when looking at 
(resp. 
). Doing so, it provides a means to implement how humans are perceiving patterns. However, in both cases, constraints brought by the axiomatic of topology often fail to process real data efficiently. In particular, the idempotence of the operator ad allows to generate only one possible approximation of an individual or a set. Unlike topology, a pretopology is defined by a function called pseudo-closure which is not (necessarily) idempotent. It thus offers the opportunity to follow an approximation process step by step as well as to model the notion of perception threshold.
Further developed from the 1980’s (Auray et al., 1979; Duru, 1980; Hubert Emptoz, 1983), the pretopological framework permits to study weak topological structures, in particular discrete and finite structures based on models generated step by step (propagation phenomenon), and describing for example information spreading in complex networks.
Pretopology found its first applications in social sciences and econometrics (Auray et al., 1979; Duru, 1980), social networks (Basileu et al., 2012; Bui, 2018; Dalud-Vincent et al., 2001; Levorato, 2011), pattern recognition (Hubert Emptoz, 1983), and image analysis (Arnaud et al., 1986; Lamure, 1987; Piegay, 1997; Piegay et al., 1995; Selmaoui et al., 1993). More recently, researchers have brought the pretopological framework in domains such as machine learning (Le et al., 2007) or text exploration (Cleuziou et al., 2011).