It was the purpose of this work to introduce a new measure on social networks that incorporates the notion of individuality in social networks, an approach that has not been examined before. For this we made use of ideas from formal concept analysis to provide a notion of milieu definability. Based on this, we developed in a natural way the notions of group individuality, individuality distribution, andaverage milieu size. Conducting experiments on real-world data-sets, we were able to show that these new measures were both independent of previously known metrics like ASP and ALCC and allowed differentiating further otherwise similar networks. To sum up, we claim to have shown that the measures of individuality introduced in this work are both natural and meaningful.
This work has only started the study of our individuality measures, and it has not reached its end. For example, so far we have investigated individuality
9.6. CONCLUSIONS 169
Table 9.7: Concept probability (cp) and separation index (si) for Brunson Club Membership network for concepts of extent size one or two.
extent |B| cp si
{U1} 3 4.09E-02 1.07E-01 {U4} 3 5.84E-02 1.03E-01 {U6} 4 1.88E-02 1.03E-01 {U7} 3 5.31E-03 1.67E-01 {U8} 4 3.84E-04 1.90E-01 {U10} 3 4.22E-01 8.11E-02 {U12} 4 2.58E-02 1.00E-01 {U13} 7 2.67E-03 1.06E-01 {U14} 5 1.56E-02 1.00E-01 {U15} 5 2.65E-03 1.16E-01 {U16} 6 1.88E-03 1.13E-01 {U17} 5 9.18E-04 1.43E-01 {U18} 5 3.52E-03 1.14E-01 {U20} 3 3.21E-01 8.33E-02 {U21} 3 2.32E-01 8.82E-02 {U23} 5 8.15E-03 1.04E-01 {U8 U20} 2 1.50E-01 2.22E-01 {U12 U18} 3 4.22E-01 1.50E-01 {U14 U22} 4 9.78E-02 1.74E-01 {U14 U13} 4 1.34E-01 1.60E-01 {U16 U13} 5 1.56E-02 1.89E-01 {U17 U13} 4 5.12E-03 2.35E-01 {U19 U13} 5 5.19E-02 1.72E-01 {U15 U5} 3 2.18E-01 1.62E-01 {U18 U1} 2 6.81E-01 1.38E-01 {U23 U18} 3 2.18E-01 1.54E-01
only on real-world networks, where this notion has a natural interpretation.
However, we have not even started to look at individuality in networks that do not stem from real-world networks, and we do not know what values of individuality to expect there. In a similar vein, one could ask in how far group individuality is suitable to distinguish real-world networks from artificial ones.
Another aspect that requires further research is the scaling factor for k-group individuality. To improve comparability, we divide the number of extents of size k by (︁|G|
k
)︁, the theoretical maximal number of such extents.
Due to this scaling,k-group individuality is always between zero and one.
However, this maximum is never achieved in practice and results in almost-zero values ofk-group individuality for larger values ofk, making those values virtually useless. Finding a better approach to scalek-group individuality is subject to further investigations.
In our experiments, the running times of our algorithms never posed a problem. However, for larger networks, measuring group individuality can represent a serious challenge: our methods require in the worst case the computation of the whole concept lattice of the representing formal context, and this lattice can be exponentially large. This somehow limits the useful-ness of our approach, and further investigations are necessary to explore the possibilities of measuring group individuality of real-world networks.
The networks we have considered in this paper were bi-modal networks from the start, and the actual modeling of finding a suitable attribute set was not an issue. However, for uni-modal networks, finding a suitable set of attributes for a contextual representation may be difficult. To what extent group individuality can be adapted to this kind of networks remains an open problem and is subject to future research.
To establish the small world character of our used data sets, we employ the approach of using null models – something we have not yet done for our individuality measures. One of the main reasons for this is that generating null models for bi-modal networks has received attention from the research community only recently [106], and a proper evaluation is still missing.
Our preliminary results on user individuality embed well in the theory of network individuality seem promising. Provided some sociological validation of this approach one may want to expand on those ideas.
9.6. CONCLUSIONS 171 A particular kind of social network that is not covered with our contextual
representation are so-calledtripartitenetworks, sometimes also called folk-sonomies[57]. The corresponding structure in formal concept analysis is the one of atriadic formal context, and generalizing group individuality to those structures is also a promising line for future research.
Collaborative Conceptual Exploration 10
The ideas and constructions in this chapter are published in [53]. They are a result of an ongoing collaboration with Jens Zumbrägel and Sergei Obiedkov.
The author of [98] implies that the “knowledge explosion [. . .] has made anything like a comprehensive survey of the major fields of knowledge impos-sible”. Hence, in addition to the absence of polymaths in contemporary times there even are, or will be, no experts for a field or subfields of knowledge, like branches of mathematics, physics, biology, languages or history, etc.
Keeping this in mind we build up on the idea of attribute exploration, as introduced in Section 3.5, and try to overcome some of the limitations of the classical approach. For this we simulate an expert for a domain of knowledge via a social approach. In contrast to a single domain expert we try to engage a social collective of partial domain experts. This collective forms itself a bipartite social network consisting of the partial experts and the attribute subsets (domain parts) they are experts for. To accomplish this approach we embed the attribute exploration process in a general collaborative interactive learning scheme, which we will call consortium. We model how such a scheme could be implemented and investigate what the abilities and limitations of such a bipartite social network of partial domain experts are.
10.1 Introduction
As discussed in Section 3.5, the algorithm for exploration is a well-known FCA approach for learning. This algorithm is able to explore a domain by consulting some domain expert. The result is a formal concept lattice which contains all formal concepts discovered in the domain. For this it is crucial that the algorithm has access to a domain expert for the whole domain, to whom it uses a minimal number of queries (which may still be exponential in the size of input, i.e., the size of the relation between objects and attributes as discussed before).
However, the availability of a domain expert is often not given in practice.
Moreover, even if it exists, such an expert might not be able or willing to answer the possibly exponential number of queries. The purpose of this work package is to provide a solution in this case, at least for some of such tasks, given a certain collaborative scenario. More precisely, suppose that we have a coveringM=⋃︁
i∈INi of the attribute setM together with a set oflocal experts pi onNi, then we propose aconsortial expertfor the domain. As is easy to see, such an expert is in general less capable of handling queries than a domain expert. Nonetheless, depending on the form ofM={Ni|i∈I}our approach may still be able to answer a significant amount of non-trivial queries.
In this work we provide a first complete characterization of (weak) local experts in order to define what aconsortiumis, what can be explored and what next steps should be focused on.