Group Types in Social Media

Abstract

Dynamics of social systems are the result of the complex superposition of interactions taking place at different scales, ranging from the pairwise communications between individuals to the macroscopic evolutionary patterns of the full interaction graph. Social communities, namely groups of people originated by any spontaneous aggregation process, constitute the mid-ground between such two extremes. Groups are important constituents of social environments as they form the basis for people’s participation and engagement beyond their minute dyadic interactions. Communities in online social media have been studied widely in their static and evolutionary aspects, but only recently some attention has been devoted to the exploration of their nature. Besides the characterization of online communities along their spatio-temporal and activity features, the recent advancements in the emerging field of computational sociology have provided a new lens to study social aggregations along their social and topical dimensions. Using the online photo sharing community Flickr as a main running example, we survey some techniques that have been used to get a multi-faceted description of group types and we show that different types of groups impact on orthogonal interaction processes on the social graph, such as the diffusion of information along social ties. Our overview supports the intuition that a more nuanced description of groups could not only improve the understanding of the activity of the user base but can also foster a better interpretation of other phenomena occurring on social graphs.

Appendix

5.1.1 Correction Parameter for Standard Deviation

Standard formulation of standard deviation is:

$$\begin{aligned} \sigma ^2 = \sqrt{\frac{1}{N-1} \sum (t-\mu )^2} \end{aligned}$$

(5.15)

Given a list N values t that can assume in [0, 1], with a given mean $\mu$ the greater possible standard deviation would be achieved under a Bernoulli distribution with $t=1$ with probability p and $t=0$ with probability q. Under these circumstances we can write:

$$\begin{aligned} \sum (t-\mu )^2 = N \cdot p \cdot (0 - \mu )^2 + N \cdot q \cdot (1 - \mu )^2 \end{aligned}$$

(5.16)

which, under a Bernoulli distribution, can be rewritten as:

$$\begin{aligned} \sum (t-\mu )^2= & {} N \cdot (1 - \mu ) \cdot (0 - \mu )^2 + N \cdot \mu \cdot (1 - \mu )^2 \end{aligned}$$

(5.17)

$$\begin{aligned}= & {} N \cdot (1 - \mu )\mu ^2 + N \cdot \mu (1 + \mu ^2 -2\mu ) \end{aligned}$$

(5.18)

$$\begin{aligned}= & {} N \mu (1 -\mu ) \end{aligned}$$

(5.19)

Therefore, being $N \mu (1 -\mu )$ the maximum value for $\sum (t-\mu )^2$, we use it as normalization factor in Formula 5.2.

5.1.2 Correction Parameter for Skewness

Under a Bernoulli distribution with that assumes value 0 with probability p(0) and 1 with p(1), the mean $\mu$ is equal to p(1), while the median is given by:

$$\begin{aligned} median ={\left\{ \begin{array}{ll} 0 &{} if \quad p(0) > p(1) \\ 0.5 &{} if \quad p(0) = p(1) \\ 1 &{} if \quad p(0) < p(1) \end{array}\right. } \end{aligned}$$

(5.20)

In case $p(0) = p(1) = 0.5$ the normalization factor is not relevant so mean and median are equal and the difference would remain the same. In other cases, one can define the maximum difference ($max_{diff}$) given the mean $\mu$ as follows:

$$\begin{aligned} maxdiff ={\left\{ \begin{array}{ll} 1-\mu &{} if \quad p(0) < p(1) \\ \mu &{} if \quad p(0) > p(1) \end{array}\right. } \end{aligned}$$

(5.21)

Under a Bernoulli distribution taking values 0 and 1, the mean is equal to p(1). Also, p(0) is equal to the remaining $1 - \mu$. Given that, we can rewrite the equation as:

$$\begin{aligned} max_{diff} ={\left\{ \begin{array}{ll} 1-\mu &{} if \quad 1-\mu < \mu \\ \mu &{} if \quad 1-\mu > \mu \end{array}\right. } \end{aligned}$$

(5.22)

that can be finally rewritten as:

$$\begin{aligned} max_{diff} = \min (1-\mu ,\mu ) \end{aligned}$$

(5.23)

which we use it as normalization factor in Formula 5.3.