Mobile social group sizes and scaling ratio

Abstract

Social data mining has become an emerging area of research in information and communication technology fields. The scope of social data mining has expanded significantly in the recent years with the advance of telecommunication technologies and the rapidly increasing accessibility of computing resources and mobile devices. People increasingly engage in and rely on phone communications for both personal and business purposes. Hence, mobile phones become an indispensable part of life for many people. In this article, we perform social data mining on mobile social networking by presenting a simple but efficient method to define social closeness and social grouping, which are then used to identify social sizes and scaling ratio of close to “8”. We conclude that social mobile network is a subset of the face-to-face social network, and both groupings are not necessary the same, hence the scaling ratios are distinct. Mobile social data mining.

Appendices

Appendix

Terminology

• Associated User is a person who has made or received call(s) to or from center user.

• Call frequency is a number of calls.

• Callee is a person/entity receiving a call.

• Caller is a person/entity generating a call.

• Center user is a phone user for whom social closeness and group of all Associated Users are computed.

• High active user is a center user who has more than ten outgoing calls per day.

• Low active user is a center user who has less than six outgoing calls per day.

• Medium active user is a center user who has between six to ten outgoing calls per day.

• Talk time is call duration.

Mathematical equations and formulations

Normalized call frequency and normalized call duration

F (i, j) is the normalized call frequency (normalized to the maximum call frequency among all users with whom user i communicate) between user i and user j which is given by Eq. (3), and T(i, j) is the normalized call duration or talk time (normalized to the maximum talk time among all users with whom user i communicate) between user i and user j, which is given by Eq. (4).

$$ F(i,j) = {\frac{f(i,j)}{{\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ f(i,k)\} }}}, $$

(3)

$$ T(i,j) = {\frac{t(i,j)}{{\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ t(i,k)\} }}}, $$

(4)

where f(i, j) is the total number of calls or call frequency between user i and user j, t(i, j) is the total call duration or talk time between user i and user j, and U _i  = {1, 2, …, N} is the set of all users associated with user i (i.e., all users who have made/received calls to/from user i with total of N users).

Mathematical proof of property 1

From Eqs. (1), (3), and (4), S(i, j) can be defined as

$$ S(i,j) = \sqrt {\left( {1 - {\frac{f(i,j)}{{\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ f(i,k)\} }}}} \right)^{2} + \left( {1 - {\frac{t(i,j)}{{\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ t(i,k)\} }}}} \right)^{2} } , $$

(5)

and S(j, i) can be defined as

$$ S(j,i) = \sqrt {\left( {1 - {\frac{f(j,i)}{{\mathop {\text{max}}\limits_{{m \in U_{j} }} \{ f(j,m)\} }}}} \right)^{2} + \left( {1 - {\frac{t(j,i)}{{\mathop {\text{max}}\limits_{{m \in U_{j} }} \{ t(j,m)\} }}}} \right)^{2} } $$

(6)

Since f(i, j) = f(j, i) and t(i, j) = t(j, i), Eq. (6) can be rewritten as

$$ S(j,i) = \sqrt {\left( {1 - {\frac{f(i,j)}{{\mathop {\text{max}}\limits_{{m \in U_{j} }} \{ f(j,m)\} }}}} \right)^{2} + \left( {1 - {\frac{t(i,j)}{{\mathop {\text{max}}\limits_{{m \in U_{j} }} \{ t(j,m)\} }}}} \right)^{2} } $$

(7)

If social closeness is symmetric, i.e., S(i, j) = S(j, i), then

$$ \sqrt {\left( {1 - {\frac{f(i,j)}{{\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ f(i,k)\} }}}} \right)^{2} + \left( {1 - {\frac{t(i,j)}{{\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ t(i,k)\} }}}} \right)^{2} } = \sqrt {\left( {1 - {\frac{f(i,j)}{{\mathop {\text{max}}\limits_{{m \in U_{j} }} \{ f(j,m)\} }}}} \right)^{2} + \left( {1 - {\frac{t(i,j)}{{\mathop {\text{max}}\limits_{{m \in U_{j} }} \{ t(j,m)\} }}}} \right)^{2} } , $$

(8)

where equality holds if and only if $\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ f(i,k)\} = \mathop {\text{max}}\limits_{{m \in U_{j} }} \{ f(j,m)\}$ and $\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ t(i,k)\} = \mathop {\text{max}}\limits_{{m \in U_{j} }} \{ t(j,m)\}$.

Note: Symmetry of the social closeness here means that the social closeness between the user i and j perceived by the user i(S(i, j)) is the same as the social closeness perceived between the user i and j perceived by user j(S(j, i)). For example, suppose there are Subj. #1 and Subj. #2 who have been communicating with each other via mobile phones, so that they have established a mobile social relationship. In other words, Subj. #1 is an Associated User of Subj. #2, and Subj. #2 is also an Associated User of Subj. #2. Suppose we ask each subject (independently) to quantitatively identify the social closeness between them with a value from 0 to $\sqrt 2$ (where 0 implies the closest and $\sqrt 2$ implies the furthest). If Subj. #1 thinks that the social closeness between him and Subj. #2 is S(1, 2), and Subj. #2 thinks that the social closeness between him and Subj. #1 is S(2, 1). We say that social closeness between Subj. #1 and Subj. #2 is “symmetric” if S(1, 2) = S(2, 1). According to Eq. (5), the social closeness (S(i, j)) depends on four parameters: call frequency between user i and user j(f(i, j)), call duration between user i and user j(t(i, j)), the maximum call frequency among all Associated Users with whom user i communicate ($\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ f(i,k)\}$), and the maximum call duration among all Associated Users with whom user i communicate ($\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ t(i,k)\}$). Likewise, the social closeness (S(j, i)) from the user j’s perception depends on the same aforementioned parameters but from user j’s perspective: f(j, i ), t(j, i), $\mathop {\text{max}}\limits_{{m \in U_{j} }} \{ f(j,m)\}$, and $\mathop { \max }\limits_{{m \in U_{j} }} \{ t(j,m)\}$—as given by Eq. (6). Since the call frequency between user i and user j(f(i, j)) is always equal to the call frequency between user j and user i (f(j, i)) i.e., $f(i,j) = f(j,i)$, and the call duration between user i and user j(t(i, j)) is always equal to the call duration between user j and user i (t(j, i)) i.e., $t(i,j) = t(j,i)$—as shown in Eq. (7), therefore, this symmetry can only occur when $\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ f(i,k)\} = \mathop {\text{max}}\limits_{{m \in U_{j} }} \{ f(j,m)\}$ and $\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ t(i,k)\} = \mathop {\text{max}}\limits_{{m \in U_{j} }} \{ t(j,m)\}$—as shown in Eq. (8). A symmetric social closeness is rare because it can only happen under the mentioned condition ($\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ f(i,k)\} = \mathop {\text{max}}\limits_{{m \in U_{j} }} \{ f(j,m)\}$ and $\mathop {\text{max}}\limits_{{k \in U_{i} }} \{ t(i,k)\} = \mathop {\text{max}}\limits_{{m \in U_{j} }} \{ t(j,m)\}$).

Dirac’s delta functions and probability density function f(s)

Probability density function f(s) is given by Eq. (9) where $\delta$ is Dirac’s delta function and N is the number of grouping clusters. Probability density function represents the density of probability at each point s in the sense that the probability that S is in a small interval in the vicinity of s is approximately f(s)h where the probability Pr[s < S ≤ s + h] ≈ f(s)h. The Dirac’s delta is a generalized function representing an infinitely sharp peak bounding unit area—the function that has value of zero everywhere except at zero (s _i  = 0) where its value is infinitely large by which its total integral is equal to one

$$ f(s) = \sum\limits_{i = 1}^{N} \delta \left( {s - s_{i} } \right). $$

(9)

(H, q)-derivative

(H, q)-derivative is given by Eq. (10), which is a q-analog of the ordinary derivative where parameter q recovers the usual definition of a derivative as q → 1. For q ≠ 1, it is not just a derivative. It compares the relative variations of f(s) and of s when s is magnified by the finite factor q. Intuitively, the (H, q)-derivative tests the scale invariance property of the function f(s). It is a natural tool for describing discrete scale invariance as a fixed finite q compares f(s) with f(sq) at s magnified by a fixed factor, and thus it also compares f(sq) with f(sq ²), f(sq ²) with f(sq ³) and so on. $D_{q}^{H = 1} f(s)$ recovers the standard q derivative $D_{q}^{{}} f(s)$. There are two control parameters: the discrete scale factor q devised to characterize the log-periodic structure, and the exponent H formulated to account for a possible power-law dependence, i.e., to correct for the existence of trends in semi-log/log–log plots (Zhou et al. 2002)

$$ D_{q}^{H} f(s)\,\underline{{\underline{\Updelta } }}\, {\frac{f(s) - f(qs)}{{[(1 - q)s]^{H} }}}.$$

(10)

The Lomb periodogram

The Lomb periodogram or Lomb power $P(\omega )$ is given by Eq. (11). The Lomb periodogram is a method of estimating a frequency spectrum based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. It is also known as Least-squares spectral analysis (LSSA).

$$ P(\omega ) = {\frac{1}{{2\sigma^{2} }}}\left\{ {{\frac{{\left[ {\sum\limits_{s} f(s){ \cos }\omega (s - \tau (\omega ))} \right]^{2} }}{{\sum\limits_{s} \mathop { \cos }\nolimits^{ 2} \omega (s - \tau (\omega ))}}} + {\frac{{\left[ {\sum\limits_{s} f(s){ \sin }\omega (s - \tau (\omega ))} \right]^{2} }}{{\sum\limits_{s} \mathop { \sin }\nolimits^{2} \omega (s - \tau (\omega ))}}}} \right\}, $$

(11)

where $\sigma^{2}$ is the variance of f(s) and $\tau (\omega )$ is given by (12)

$$ \tau (\omega ) = {\frac{1}{2\omega }}{ \arctan }\left\{ {{\frac{{\sum\limits_{s} { \sin }2\omega s}}{{\sum\limits_{s} { \cos }2\omega s}}}} \right\}. $$

(12)

Survey of mobile social groups

The following document is the survey that we have used for our analysis of mobile social groups and its validation in Sect. 2.2:

- - - - - Survey begins here - - - - -

Behavior analysis of mobile phone users

This project is aimed to provide a better understanding of behavior, pattern, and social structure of mobile phone users, as well as facilitate research and development of mobile social applications as we take an early evolutionary steps toward a new era of mobile and pervasive computing, which is aimed to enhance quality of life with more sensitive and responsive mobile devices.

Survey process

The survey process is carried in two simple steps. First, you will download the call record details (records having details of each call you have dialed or received) from your service provider’s website. The required information for the survey from the downloaded call records are explained in the Data Collection process. You are requested to bring the downloaded information (soft copy) to a 15-min session. Then, you will review how to identify mobile social closeness for your associated callers/callees in the Mobile Social Closeness Identification process and provide your social closeness for each associated call ID in the Feedback process.

Data collection

You are requested to download the call detail records from your cellular service providers for the last 3 months (longer period preferred). You would be able to download these call records in an Excel sheet format (we can help you if have any problem in this regard). Next, you need to merge the call records of all those months into a single excel file (again we can help you if required). The call information for our survey is described in the following table. When necessary, you may have to remove some unnecessary data (fields) in the excel sheet.

 

Date	Start time	Type	Anonymous call ID	Talk time
1/5/2008	2:20 PM	Incoming	C1	2
1/6/2008	3:15 PM	Outgoing	C2	28
…	…	…	…	…

• 1. Date: Date when the call has taken place. This should be in the format MM/DD/YYYY.

• 2. Start Time: Start time of the call. This should be in the format HH:MM AM/PM.

• 3. Type: The call type i.e., whether the call is an “Incoming” or an “Outgoing” call. Some service providers record this field as “Incoming” for an incoming call and the destination location for an outgoing call.

• 4. Anonymous Call ID: You can choose an anonymous Call ID to each of the caller/callee. If you are unable to do that, we can run your data through our system and generate a set of anonymous IDs to each caller/callee.

• 5. Talk time: The amount of time spent during the call.

Mobile social closeness identification

You are requested to attend a 15-min session for the data analysis. You are requested to identify the Social Closeness for each Call ID as following:

• Enter “1” if Call ID indicates the person who is a Socially Closest Member:

  These are the people with whom you maintain the highest socially connectivity. Most of the calls you receive, come from individuals within this category. You receive more calls from them and you tend to talk with them for longer periods. Typically, the face-to-face social tie of these people is family member, friend, and colleagues.

• Enter “2” if Call ID indicates the person who is a Socially Near Member:

  People in this group are not as highly connected as family members and friends, but when you connect to them, you talk to them for considerably longer periods. Mostly, you observe intermittent frequency of calls from these people. These people are typically neighbors and distant relatives.

• Enter “3” if Call ID indicates the person who is a Socially Distant Member:

These individuals have less connection with your social life. These people call you with less frequency. You acknowledge them rarely. Among these would be, for example, a newsletter group or a private organization with whom you have previously subscribed. This group also includes individuals who have no previous interaction or communication with you. You have the least tolerance for calls from them e.g., strangers, telemarketers, fund raisers.

Feedback

Your call records will be processed using our system to extract all distinct Call IDs, and then you will be asked to identify Social Closeness for each Call ID as shown in an example below.

 

Anonymous call ID	Social closeness
C1	1
C2	3
C3	2
C4	1
.	.
.	.
.	.

- - - - - Survey ends here - - - - -