We have motivated our notion of individuality by the uniqueness of user milieus. Clearly, this motivation strongly depends on the particular interpre-tation of the word “milieu”, and it is the purpose of this section to provide a formal definition for it. Indeed, modeling a social network by a formal context suggests an immediate definition that is both simple and, as we find, convincing.
LetK= (U , A, I) be a formal context representing a social network. Then for each useru∈U we define themilieuofusimply as the set{u}′of attributes common tou. Moreover, ifV ⊆U is a set of users, then themilieu ofV is the set of attributes common to all users inV, i.e., V′. Using this definition of user milieus, we want to measure the individuality of a social networkKby the amount of milieus that occur inK. Indeed, we shall be a bit more careful here, and propose a notion ofk-group individualityas a measure to quantify the number of milieus that occur inKas the milieu of groups of sizek, in the sense of how many of the milieus occurring in our social networkKdefine groups of size exactlyk, compared to the number of all groups of sizek. Then, the more individuality a social network contains, the more individual groups of a certain size can be defined through their milieu. Conversely, if a social network is quite homogeneous, then defining certain subgroups of individuals by their milieu is improbable.
This approach can naturally be rephrased in terms of formal concept analysis: measuring individuality inKfor user groups of sizekis the question of how many subsetsV ⊆U with|V|=kcan be expressed in terms ofV =B′ for someB⊆A. In other words, we ask for the number ofextents of sizekinK
9.3. MODELS OF INDIVIDUALITY IN SOCIAL NETWORKS 151 and use this number to measure thek-group individuality inK. The following
definition captures this idea.
Definition 9.1
Let K= (U , A, I) be a formal context. Define the set Extk(K) as the set of extents ofKof sizek, i.e.,
Extk(K)≔{V ⊆U |V =V′′,|V|=k}. Then thek-group individualitygik(K) ofKis
gik(K)≔
|Extk(K)| min{(︁|U|
k
)︁,2|A|}. (9.1) Note that we also normalize by the factor min{(︁|U|
k
)︁,2|A|}, because this is the maximal number ofk-groups definable by their milieu, and thus allows compa-rability between individuality of different networks. The used normalization is not optimal, as fork larger than 1 the value of gik(K) rapidly decreases.
However, so far the authors are not aware of other normalization approaches.
On a side note, one may also consider the dual measure taking the intents of sizek, which would help to measure and describe the individuality of a social network from the attribute point of view.
In terms of measuring the individuality in a social network, the value gi1(K) is of particular interest, as this is the percentage of users in this network uniquely determinable by their milieu. In this case, we shall also talk about theuser individualityui(K)≔gi1(K) of a social networkK.
Recalling our example from Figure 3.5 on Page 44, we first compute the extent sets. As we see in Figure 3.6, the concept lattice consists of four elements (apart from the top and bottom ones), and consequently there are four different extents. Indeed we obtain
Ext1={{userB}},
Ext2={{userB,userD},{userA,userC}}, Ext3={{userA,userB,userC}}.
Therefore, gi1(Kmisn) = 14, since only one user has a unique interest that is not covered by another user. We also obtain gi2(Kmisn) = 13, demonstrating that in
this network the individuality of “pairs” of users is higher than for individual users. Finally, gi3(Kmisn) = 14, showing that there is only one group of size three.
The network would be changed considerably if userC would have liked ballet instead of cabaret. In this context, which we want to call misn’, there would be three extents of size one and therefore gi1(Kmisn’) =34. Additionally, the number of extents of size two would be four, resulting in gi2(Kmisn’) =23. In short, by not being a copy of the interest of userA, userC can shift the individuality of the network massively by one interest change.
A remark on computingk-group individuality is in order. Using methods from formal concept analysis, the overall computational effort can be reduced to compute only extents of sizeat mostk. More precisely, the algorithm of Next-Closure [43] is able to enumerate closed sets of arbitrary closure operators in a particular order. Exploiting the fact that·′′ is a closure operator allows us to compute all extents ofKwith only polynomial overhead. Furthermore, Next-Closure can be extended to compute only extents of size at mostk, further reducing the overall computation costs. A drawback is that Next-Closure cannot be extended to only compute extents of sizek, a disadvantage that is not of profound severity, sincek-group individuality is usually computed for valuesk= 1,2, . . . , ℓup to some limitℓ∈N.
Note that group individuality also allows detecting the presence of large homogeneous groups, i.e., groups of users with the same milieu. Clearly, such a group of sizekexists if and only if gik(K)>0. In other words, the set
gid(K)≔{k∈N|gik(K)>0}
can be seen as a quantity for theindividuality distributionin the social network represented byK.
Finally, another aspect of group individuality that we want to consider in this work is the question of how muchinformationis necessary to define the milieu of a group of sizek. In terms of our modeling of social networks as formal contexts, we reformulate the question to ask how many attributes are necessaryon averageto define a unique group of sizekthat is itself identifiable through its unique milieu. This gives rise to the following definition.
Definition 9.2
LetKbe a formal context and letk∈gid(K). Define thek-group average milieu
9.4. EXPERIMENTAL RESULTS 153