Mining multiplex interaction relationships from usage records in social networks

Abstract

Social networks have become increasingly popular and are commonly used in everyday life. They also become the most convenient places to send information or receive advertisements. The multiplex network is an important study topic in social networks, in which many features could be appropriately represented in different layers. In this paper, we propose an approach to find the multiplex interaction relationships based on the action records of users on social networks. The multiplex user interactions are found and divided into three levels: high, normal and low. They are then used to check the friend and the follower relations such that users can find which friends or followers are active or not. In the experiments, the parameters are chosen based on Dunbar’s number, which is the number of social relationships that humans can have with high confidence. The results show the proposed approach is effective in helping users know the truly close friend relationships on a social network.

Keywords

Social network adjacency matrix multiplex interaction relationship multiplex network Dunbar’s number

1. Introduction

Social networks are platforms through which users can contact others easily. They are usually represented by a graph with each node revealing a user and each link showing the connections between two users. Recently, Facebook has attracted many users who communicate and post data on it. Based on a report from Facebook for the first quarter in 2017, there are 1.94 billion users, which have grown 17%, compared with the previous year [5].

The vigorous growth of users on Facebook also affects the number of their friends on it. Another report from Pew Internet & American Life Project and Berkman Center revealed that Facebook is the most commonly used social network for teenagers in America, and each user, on average, has over 400 friends on Facebook [13]. Therefore, there may be some friends on Facebook that users do not know in the real world, even some annoyingly malicious accounts. However, it is difficult and time-consuming for users to recognize those accounts that they seldom interact with but do exist on their friend lists. Thus, there is a need for checking users’ friend lists to help them recognize the closer relationships on a social network.

On a social network, a connection may exist between two users with some action relationships. A user may do some actions to their reachable users even if he/she does not directly connect to them. The four most commonly conducted actions on a social network are like, comment, share, and tag. In the paper, we thus propose an approach for checking users’ friend relationships through the action records on a social network. The proposed approach is divided into two phases. In the first phase, a multiplex graph on a social network is formed to classify the connections of a user into three levels of interaction based on the actions that other users took on him/her. In the second phase, the multiplex graph is compared with the user’s friends or followers so that the difference can be known to the user for annually or automatically revising the relationship of friends.

The remainder of this paper is organized as follows. Some related works are reviewed in Section 2. The proposed algorithm for forming the multiplex graph is stated in Section 3. An example is given to show the execution of the algorithm in Section 4. The designed approach for comparing the multiplex graph with the adjacent graph which represents the friend and follower relationship of a user is described in Section 5, with an example given in Section 6. Some experiments are given in Section 7. Finally, the conclusions and future works are given in Section 8.

2. Review of related works

In this section, some related studies are briefly reviewed.

2.1 User data

The usage data of users, which are collected under the consideration of the features of a user, can be used to obtain useful information. Ruan et al. crawled user usage data over several sessions, then profiled online social behaviors into several features to make each user a personal profile [15]. Lin et al. proposed a merchant recommendation system that first captured the location information and the comments that users had left and then rated the merchant by separating the comments into multiple aspects [11]. Heartherly et al. classified the levels of edges based on Bayes Classification scheme [8]. Ji et al. investigated adding connectivity links in an interdependent network to improve robustness and to compare the proposed method to four common-used link addition strategies [9]. Berlingerio et al. defined multidimensional networks where more than one connection might reside between two nodes [3].

2.2 Multiplex networks

The network theory has become an important tool for analyzing complex systems [12, 14]. Kivelä et al. made a good survey on multilayer networks [10]. From the concept of a multilayer network, a social network can be separated into several layers, with each layer representing different features of the social network. Battiston et al. presented a general framework and described multiplex networks with either unweighted or weighted links [2]. They proposed a comprehensive formalism to deal with systems composed of several layers with binary or weighted links and made a clear distinction about the different levels of description of a multiplex network.

2.3 Influence diagram

An influence diagram is a graphical representation for modeling the relationship of factors [16]. It was first developed in the mid-1970s by decision analysts with an intuitive semantic display to be understood easily. It can be thought of as a generalization of a Bayesian network and can be effectively used to solve decision-making problems. It uses the nodes of various colors and shapes to depict the key elements, including decisions, uncertainties, and objects. It is now adopted widely in many fields, including Engineering.

3. Forming a multiplex graph of a user from action records

To differentiate the level of each direct connection on a social network, the main idea is about the rate of interaction from a user to their connected users. In this paper, all connections will be classified into three levels: high interaction, normal interaction, and low interaction. The rate of interaction is calculated from each user’s usage records of the actions (like, comment, share, and tag) on a social network. The rate $l_{ij}^{w}$ of the action “like” from a user $i$ to $j$ are defined as follows:

$\displaystyle l_{ij}^{w}=\frac{L_{ij}}{TA({\textit{like},i})},$

where $L_{ij}$ denotes how many likes are clicked by a user $i$ to his reachable user $j$ and $TA(\textit{like},i)$ is the action clicks of “like” from user $i$ to all the other users. The other three rates $c_{ij}^{w}$ , $s_{ij}^{w}$ and $t_{ij}^{w}$ for the actions comment, share, and tag can be defined similarly.

Normally, users may have different usage habits and their own preferences on the actions to a social network. For example, some users prefer to like every post on Facebook rather than like the post they have interests in. Considering this phenomenon, we choose the second-largest rate from the four action rates and compare them with two thresholds to classify the level of interaction between two users. The process is shown in Fig. 1.

Figure 1.

The flow chart to form a multiplex graph from action records.

The algorithm is described below.

INPUT: Four users’ usage matrices

L

C

T

, and

S

, in respect to the four actions, like, comment, tag, and share, in a social network.

OUTPUT: An interaction matrix

I

of the social network.

1. Set

i=

1 and

j=

2 //

i

and

j

are used to index the current element to be processed

2. for each element

(i,j)\in

the interaction matrix

I

3. Calculate

l_{ij}^{w}

c_{ij}^{w}

t_{ij}^{w}

and

s_{ij}^{w}

4. Sort the four weights of

l_{ij}^{w}

c_{ij}^{w}

t_{ij}^{w}

and

s_{ij}^{w}

in descending order.

5. Denote the second largest weight in the sorted list as

r_{2}

6. Save the largest weight at the element (

i, j

) of the interaction matrix

I

7. if

r_{2}\geqslant\alpha

then

8. Assign the label of the user pair (

i, j

) as high interaction.

9. else if

r_{2}\geqslant\beta

then

10. Assign the label of the user pair (

i, j

) as normal interaction.

11. else

12. Assign the label of the user pair (

i, j

) as low interaction.

13. Put the label at the element (

i, j

) of the interaction matrix

I

14. Output the interaction matrix

I

as the result

4. An example

In this section, an example is given to illustrate the proposed algorithm. Assume that there are ten users in a social network, denoted as { $A, B, C, D, E, F, G, H, I, J$ }. Also, assume the details of the four individual actions (like, comment, tag, share) for the users are recorded separately in the system. For example, Table 1 shows an example of the ten users’ clicking numbers of like in this case.

Table 1
The usage record of like for the users in this example

Like	A	B	C	D	E	F	G	H	I	J
A	0	71	32	35	3	7	64	89	2	15
B	90	0	19	35	78	0	0	0	0	0
C	0	38	0	0	67	0	0	0	0	0
D	36	25	0	0	85	80	41	84	0	0
E	12	47	44	86	0	63	0	93	100	0
F	50	23	13	77	19	0	52	31	2	79
G	29	17	91	58	21	73	0	9	64	59
H	13	41	38	30	69	53	0	0	0	0
I	15	68	47	0	0	0	0	0	0	0
J	33	58	28	22	25	84	59	21	92	0

Table 1 shows the action numbers of each user in the example. Each row reveals the “like” record of a user to others, while each column represents the receiving “like” record of a user who was liked due to his/her posts in a social network. Each element $L_{ij}$ in the above matrix denotes how many likes are clicked by a user $i$ to his reachable user $j$ . For instance, $L_{AB}$ is 71 in Table 1, representing $A$ has clicked 71 likes to $B$ ’s posts, and $L_{BA}$ is 90, denoting $B$ has clicked 90 likes to $A$ ’s posts. Since in a social network, users seldom click like to their own posts, we thus set $L_{ii}$ at 0 for any user $i$ . The details of the other three actions, comment (C), share (S), and tag (T), are recorded in the same way as shown in Tables 2–4. All of the four matrices are, in general, asymmetric.

Table 2

The usage record of comment for the users in the example

Com	A	B	C	D	E	F	G	H	I	J
A	0	15	45	31	31	29	5	29	3	30
B	32	0	2	46	4	9	27	9	1	5
C	4	12	0	20	34	40	49	12	41	27
D	9	24	26	0	36	30	0	0	0	0
E	5	32	50	43	0	0	0	0	0	0
F	28	39	37	30	21	0	6	6	36	21
G	9	38	6	5	10	2	0	0	0	0
H	26	50	7	27	4	18	1	0	15	4
I	5	24	50	37	43	30	43	49	0	26
J	9	16	37	38	40	25	41	50	12	0

Table 3

The usage record of tag for the users in the example

Tag	A	B	C	D	E	F	G	H	I	J
A	0	2	10	9	8	0	0	0	0	0
B	5	0	0	2	6	0	0	0	0	0
C	0	1	0	0	3	4	8	0	0	0
D	10	1	0	0	8	0	0	3	0	0
E	0	0	0	0	0	0	0	0	0	0
F	0	5	1	0	2	0	7	0	6	10
G	0	0	0	0	0	2	0	0	0	0
H	9	0	0	0	0	0	0	0	1	0
I	0	0	0	0	8	9	5	7	0	0
J	2	0	1	0	0	9	0	0	8	0

Table 4

The usage record of share for the users in the example

Share	A	B	C	D	E	F	G	H	I	J
A	0	27	0	0	0	0	0	0	0	0
B	8	0	0	15	14	0	0	0	0	0
C	0	20	0	0	26	17	0	0	0	0
D	16	29	0	0	0	0	0	5	0	0
E	25	10	28	18	0	23	20	6	30	17
F	0	11	21	0	16	0	10	0	2	4
G	0	0	0	0	0	30	0	0	0	0
H	8	17	4	28	2	1	27	0	17	17
I	0	0	0	0	26	7	30	21	0	0
J	23	1	25	5	22	25	0	0	20	0

From the four detailed usage matrices for the four actions, the table of actions ( $T A$ ) can be calculated and shown in Table 5. Each row in the table reveals the number of occurrences of an action for different users, while each column represents a user’s clicking numbers of the four types of actions. Thus, each element $TA(a,i)$ in the table represents the number of an action $a$ done by a user $i$ . For example, in Table 5, $TA(\textit{like},A)$ is 399, representing that the user $A$ has clicked 399 likes on the social network.

Table 5

The table of actions in this example

Action	A	B	C	D	E	F	G	H	I	J
Like	399	222	105	351	445	346	421	244	130	422
Com.	218	171	239	125	130	224	70	152	307	268
Tag	29	13	16	22	0	31	2	10	29	20
Share	27	37	63	50	177	64	30	121	84	121

With these matrices, the proposed method can work step by step as follows. The index variable $i$ is set at 1, and the other $j$ is set at 2 in the beginning. The four corresponding usage weights are calculated. Thus for user 1 to user 2, there are four action rates found out for the four actions. For example, from the cell (1, 2) of the usage matrix of the action like in Table 1, $L_{12}=$ 50, and from the table of actions in Table 5, $TA(\textit{Like},1)$ is 200. Thus, the rate of the interaction like from user 1 ( $A$ ) to user 2 ( $B$ ) is calculated as:

$\displaystyle l_{12}^{w}=\frac{71}{399}.$

Similarly, the rates of the other three actions for (1, 2) are found as follows:

$\displaystyle c_{12}^{w}=\frac{C_{12}}{TA(\textit{Comment},1)}=\frac{15}{218},$ $\displaystyle t_{12}^{w}=\frac{T_{12}}{TA(\textit{Tag},1)}=\frac{2}{29},\text{and}$ $\displaystyle s_{12}^{w}=\frac{S_{12}}{TA(\textit{Share},1)}=\frac{27}{27}.$

In this example, $I(1,2)$ is $s_{12}^{w}(=1)$ , and the second-largest rate for user 1 to user 2 is $l_{12}^{w}=$ 0.177. Assume $\alpha=$ 0.15 and $\beta=$ 0.03. In this example, $r_{2}$ for (1, 2) is 0.177, which is larger than $\alpha=$ 0.15. Thus, the interaction level for user 1 to user 2 is high interaction. The label is thus put into the element (1, 2) of the interaction matrix $I$ . Note that the interaction value is put by the largest value $s_{12}^{w}(=1)$ .

Table 6 shows the result in this example, in which each element $I(i,j)$ consists of a value that represents the largest interaction rate and a label of its interaction level. For example, the element $I(1,2)$ is $(1,H)$ , representing that the largest interaction rate from user 1 ( $A$ ) to user 2 ( $B$ ) is 1, and the interaction level is high.

Table 6

The interaction matrix of a social network

Inter.	A	B	C	D	E	F	G	H	I	J
A	0	1, H	0.35, H	0.31, N	0.28, N	0.16, N	0.16, L	0.22, N	0.05, L	0.13, L
B	0.41, H	0	0.09, L	0.41, H	0.46, H	0.05, L	0.16, L	0.05, L	0.01, L	0.03, L
C	0.02, L	0.36, H	0	0.08, L	0.64, H	0.27, H	0.5, H	0.05, L	0.17, L	0.11, L
D	0.46, H	0.58, H	0.21, L	0	0.36, H	0.24, H	0.12, L	0.24, N	0, L	0, L
E	0.14, N	0.25, N	0.39, H	0.33, H	0	0.14, N	0.11, L	0.21, N	0.23, H	0.1, L
F	0.15, N	0.17, H	0.33, H	0.22, N	0.25, N	0	0.23, H	0.09, L	0.19, H	0.32, H
G	0.13, N	0.54, N	0.22, N	0.14, N	0.14, N	1, N	0	0.02, L	0.15, L	0.14, L
H	0.9, H	0.33, H	0.16, N	0.23, H	0.28, L	0.22, N	0.22, L	0	0.14, N	0.14, L
I	0.12, L	0.52, N	0.36, H	0.12, L	0.31, H	0.31, N	0.36, H	0.25, H	0	0.09, L
J	0.19, N	0.14, N	0.21, N	0.14, N	0.18, N	0.45, H	0.15, N	0.19, N	0.4, H	0

Figure 2.

The flow chart of finding a merged multiplex network for adjusting the original relationships.

Table 7

A symmetric matrix representing the friend relationship

	A	B	C	D	E	F	G	H	I	J
A	0	0	0	1	0	0	0	1	0	1
B	0	0	0	1	1	0	0	0	0	0
C	0	0	0	0	1	1	0	0	0	0
D	1	1	0	0	1	0	0	0	0	0
E	0	1	1	1	0	1	0	0	0	0
F	0	0	1	0	1	0	1	0	1	1
G	0	0	0	0	0	1	0	0	0	0
H	1	0	0	0	0	0	0	0	0	0
I	0	0	0	0	1	1	0	0	0	0
J	1	0	0	0	0	1	0	0	0	0

5. Adjusting the relationships of friends and followers

The follower relationship is usually asymmetric because it could be either unidirectional or bidirectional on Facebook or Twitter. The friend relationship is, however, usually symmetric in a social network. In the second phase, we merge the interaction matrix, which is output from the first phase 1, and the adjacency matrix coming from the friend and follower relationships of users on the social network. The output is a merged multiplex network, which will point out the relationships to be added on, kept in, or deleted from the ones of friends and followers. The flow diagram is shown in Fig. 2.

The detailed algorithm is described below.

INPUT: The friend graph

F(V,E_{1})

, the follower graph

L(V,E_{2})

, and the interaction matrix

I

derived from Phase 1 in a social network.

INPUT: A merged multiplex graph

M(V,E_{H},E_{N},E_{L})

, where

E_{H}

E_{N},

and

E_{L}

represent the edge sets of the three labels (high, normal, low), respectively.

1. Merge the friend graph

F(V,E_{1})

and the follower graph

L(V,E_{2})

into the adjacency graph

A(V,E_{3})

by representing

E_{1}

as bidirectional edges and setting

E_{3}=E_{1}\cup E_{2}

2. Initially set

E_{H}

E_{N}

and

E_{L}

as empty sets

3. Set

i=

1 and

j=

2 //

i

and

j

are used to index the current element to be processed

4. for each element

(i,j)\in

the interaction matrix

I

and adjacency matrix

A

5. if

A(i,j)=

1 then // Case 1

6. if

I(i,j)

is labeled as high interaction then

7. Add the edge

e(i,j)

to the high- effect layer by

E_{H}=E_{H}\cup e(i,j)

8. else if

I(i,j)

is labeled as normal interaction then

9. Add the edge

e(i,j)

to the normal-effect layer by

E_{N}=E_{N}\cup e(i,j)

10. else if

I(i,j)

is labeled as low interaction then

11. classify the edge

e(i,j)

to the low-effect layer by

E_{L}=E_{L}\cup e(i,j)

12. if

A(i,j)=

0 then

13. if

I(i,j)

is labeled as high interaction then

14. Add a new edge

ne(i,j)

to the high-effect layer by

E_{H}=E_{H}\cup ne(i,j)

15. else if

I(i,j)

is labeled as normal interaction then

16. Add a new edge

ne(i,j)

to the normal-effect layer by

E_{N}=E_{N}\cup ne(i,j)

17. Output the multiplex graph

M(V,E_{H},E_{N},E_{L})

as the result

After outputting the multiplex graph $M(V,E_{H},E_{N},E_{L})$ , the multiplex graph of a social network is found and outputted for pointing out the possible adjustment of the friend and follower relationships of users. Below, an example is given to explain the proposed approach.

6. An example for adjusting the relationships

Following the example above, assume that the friend relationship for the social network is shown in Table 7, in which each element reveals a friend relationship between two users. For the example in Table 7, the element for user $A$ to user $D$ is 1 and vice versa, representing that $A$ and $D$ are mutual friends. On the contrary, the element for $A$ to $B$ is 0 and vice versa, representing that $A$ and $B$ are not friends. Note that the matrix is symmetric.

Table 8 reveals an example of the follower relationship, in which each element represents a follower relationship for a user $i$ following another user $j$ . For instance, the element for $C$ to $B$ is 1, representing $C$ is a follower of $B$ . On the contrary, the element for $B$ to $C$ is 0, representing $B$ is not a follower of $C$ .

Table 8
An example of follower relationship represented as a matrix

	A	B	C	E	F	G	H	I
A	0	1	1	1	1	0	0	0
B	1	0	0	0	0	0	0	0
C	0	1	0	0	0	1	0	0
D	0	0	0	0	0	0	1	0
E	0	0	0	0	0	0	1	0
F	0	1	0	0	0	0	0	0
G	0	0	0	0	0	0	0	0
H	0	0	0	0	0	0	0	1
I	0	0	0	0	0	1	1	0
J	0	0	1	0	0	0	0	1

In this example, the friend and the follower relationships are first merged into the adjacency graph shown in Fig. 3.

Figure 3.

The merged adjacency graph in the example.

Note that the real implementation of the above concept can be done by matrices. The matrix for the friend relationship and the one for the follower relationship can be directly merged by a union operation to become the result of the adjacency matrix.

Then the index variable $i$ is set at 1, and the other $j$ is set at 2 in the beginning. Then the element $A(1,2)$ in the adjacency matrix $A$ is compared with the element $I(1,2)$ in the interaction matrix $I$ in Step 4. Since the connection $A(1,2)=$ 1 in the adjacency matrix and $I(1,2)$ is labeled as high interaction, an edge $e(A,B)$ is added to the higheffect layer. As another example, when $i=$ 4 and $j=$ 6, the connection $A(4,6)=$ 0 in the adjacency matrix while $I(4,6)$ is labeled as high interaction. Thus, we add an edge $ne(A,I)$ to the higheffect layer by marking it as a new edge. After the execution of the algorithm, the different layers of the resultant multiplex graph $M(V,E_{H},E_{N},E_{L})$ are shown in Figs 4 and 6. In Figs 4 and 5, the dash edges are the new edges after the comparison. In Fig. 6, the solid edge is the original relationship, which can be removed after the comparison.

Figure 4.

The high-effect layer of the multiplex network in the example.

Figure 5.

The normal-effect layer of the multiplex network in the example.

Figure 6.

The low-effect layer of the multiplex network in the example.

7. Experimental results

In this section, the experiments conducted for the proposed approach are described and discussed. The experiments were implemented in MATLAB on a personal computer with an Intel Core i5 4570 with 3.20 GHz and 8 GB RAM. Simulated data were used in the experiments to show the effects of the thresholds that were applied in the proposed approach. The data in the usage records were randomly generated in a way similar to real data, e.g., clicking “like” is the most frequently used action for most users.

Experiments were done to show the influence of the user number on the interaction relationship. The two parameters were set as $\alpha=$ 0.15 and $\beta=$ 0.03. The dataset used here were produced randomly, with the numbers of users ranging from 1000 to 10000. Figure 7 shows the relation between the amount of high interaction and the number of users. The amount of high interactions decreased when the number of users increased. The decrease was especially sharp before 2000 users.

Figure 7.

The relation between the amount of high interactions and the number of users when $\alpha=$ 0.15 and $\beta=$ 0.03.

Figure 8 shows the relation between the amount of low interaction and the number of users. The amount of low interactions increased when the number of users increased.

Figure 8.

The amount of low interaction relationship with different numbers of users when $\beta=$ 0.03.

Figure 9.

The amounts of high interactions for different numbers of users when $\alpha=$ 0.0019.

Figure 10.

The amounts of high interactions for different numbers of users when $\alpha=$ 0.0017.

Figure 11.

The amounts of low interactions for different numbers of users when $\beta=$ 0.0011.

Next, the value of $\alpha$ was modified to a lower value to ensure the amounts of high interactions for different numbers of users could be easily recognized. Figures 9 and 10 show the relation between the amount of high interactions and the number of users for $\alpha=$ 0.0019 and $\alpha=$ 0.0017.

Figures 9 and 10 show the amount of connections was around 120000 to 160000 for 1000 users for the value of $\alpha$ being within 0.0017 and 0.0019. That was, each user had, on average, 120 to 160 high interactions. We may choose $\alpha$ based on Dunbar’s number, which is the number of social relationships that humans can have [1, 4, 6]. It is a confidence interval from 100–230 with 95% confidence. Thus, for a given social network, we could adjust and determine the value of $\alpha$ in terms of the amount of high interactions to fit Dunbar’s number.

Figure 11 shows that the amount of low interaction relationships would increase along with the increase of the user number. However, on a social network such as Facebook, the limitation of the number of friends for a user is 5000. That is, when the number of users gets larger, there will be more strangers on a social network, which makes a large amount of low interactions. Thus, our approach is quite consistent with real situations.

8. Conclusions and future works

In this paper, we have utilized the usage records to find the multiplex interaction relationships on a social network. We have proposed an approach to find the interaction matrix based on the records and divided the user interactions into three levels: high, normal, and low. A connection is labeled as high interaction if it has a relatively higher interaction rate than the threshold $\alpha$ . Thus, even for users who do not use social networks frequently, the algorithm still works. For example, a value 0.1 of $r_{2}$ may be the rate of either 1/10 or 10/100, which will consequently have the same label. The interaction matrix can then be compared with the friend and the follower lists of users for helping users easily know the truly close friend relationships on a social network. From the comparison, some new relationships will be added that initially do not exist on the lists, and some old relationships will be deleted that exist initially but seem to have too few interactions.

In the future, we will continuously verify and improve the proposed approach. We may also apply the multiplex interaction relationships to other applications like information diffusion [7, 17].

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10.

Kivelä

Arenas

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Like	A	B	C	D	E	F	G	H	I	J
A	0	71	32	35	3	7	64	89	2	15
B	90	0	19	35	78	0	0	0	0	0
C	0	38	0	0	67	0	0	0	0	0
D	36	25	0	0	85	80	41	84	0	0
E	12	47	44	86	0	63	0	93	100	0
F	50	23	13	77	19	0	52	31	2	79
G	29	17	91	58	21	73	0	9	64	59
H	13	41	38	30	69	53	0	0	0	0
I	15	68	47	0	0	0	0	0	0	0
J	33	58	28	22	25	84	59	21	92	0

Com	A	B	C	D	E	F	G	H	I	J
A	0	15	45	31	31	29	5	29	3	30
B	32	0	2	46	4	9	27	9	1	5
C	4	12	0	20	34	40	49	12	41	27
D	9	24	26	0	36	30	0	0	0	0
E	5	32	50	43	0	0	0	0	0	0
F	28	39	37	30	21	0	6	6	36	21
G	9	38	6	5	10	2	0	0	0	0
H	26	50	7	27	4	18	1	0	15	4
I	5	24	50	37	43	30	43	49	0	26
J	9	16	37	38	40	25	41	50	12	0

Tag	A	B	C	D	E	F	G	H	I	J
A	0	2	10	9	8	0	0	0	0	0
B	5	0	0	2	6	0	0	0	0	0
C	0	1	0	0	3	4	8	0	0	0
D	10	1	0	0	8	0	0	3	0	0
E	0	0	0	0	0	0	0	0	0	0
F	0	5	1	0	2	0	7	0	6	10
G	0	0	0	0	0	2	0	0	0	0
H	9	0	0	0	0	0	0	0	1	0
I	0	0	0	0	8	9	5	7	0	0
J	2	0	1	0	0	9	0	0	8	0

Share	A	B	C	D	E	F	G	H	I	J
A	0	27	0	0	0	0	0	0	0	0
B	8	0	0	15	14	0	0	0	0	0
C	0	20	0	0	26	17	0	0	0	0
D	16	29	0	0	0	0	0	5	0	0
E	25	10	28	18	0	23	20	6	30	17
F	0	11	21	0	16	0	10	0	2	4
G	0	0	0	0	0	30	0	0	0	0
H	8	17	4	28	2	1	27	0	17	17
I	0	0	0	0	26	7	30	21	0	0
J	23	1	25	5	22	25	0	0	20	0

	A	B	C	D	E	F	G	H	I	J
A	0	0	0	1	0	0	0	1	0	1
B	0	0	0	1	1	0	0	0	0	0
C	0	0	0	0	1	1	0	0	0	0
D	1	1	0	0	1	0	0	0	0	0
E	0	1	1	1	0	1	0	0	0	0
F	0	0	1	0	1	0	1	0	1	1
G	0	0	0	0	0	1	0	0	0	0
H	1	0	0	0	0	0	0	0	0	0
I	0	0	0	0	1	1	0	0	0	0
J	1	0	0	0	0	1	0	0	0	0

	A	B	C	E	F	G	H	I
A	0	1	1	1	1	0	0	0
B	1	0	0	0	0	0	0	0
C	0	1	0	0	0	1	0	0
D	0	0	0	0	0	0	1	0
E	0	0	0	0	0	0	1	0
F	0	1	0	0	0	0	0	0
G	0	0	0	0	0	0	0	0
H	0	0	0	0	0	0	0	1
I	0	0	0	0	0	1	1	0
J	0	0	1	0	0	0	0	1

Like	A	B	C	D	E	F	G	H	I	J
A	0	71	32	35	3	7	64	89	2	15
B	90	0	19	35	78	0	0	0	0	0
C	0	38	0	0	67	0	0	0	0	0
D	36	25	0	0	85	80	41	84	0	0
E	12	47	44	86	0	63	0	93	100	0
F	50	23	13	77	19	0	52	31	2	79
G	29	17	91	58	21	73	0	9	64	59
H	13	41	38	30	69	53	0	0	0	0
I	15	68	47	0	0	0	0	0	0	0
J	33	58	28	22	25	84	59	21	92	0

Com	A	B	C	D	E	F	G	H	I	J
A	0	15	45	31	31	29	5	29	3	30
B	32	0	2	46	4	9	27	9	1	5
C	4	12	0	20	34	40	49	12	41	27
D	9	24	26	0	36	30	0	0	0	0
E	5	32	50	43	0	0	0	0	0	0
F	28	39	37	30	21	0	6	6	36	21
G	9	38	6	5	10	2	0	0	0	0
H	26	50	7	27	4	18	1	0	15	4
I	5	24	50	37	43	30	43	49	0	26
J	9	16	37	38	40	25	41	50	12	0

Tag	A	B	C	D	E	F	G	H	I	J
A	0	2	10	9	8	0	0	0	0	0
B	5	0	0	2	6	0	0	0	0	0
C	0	1	0	0	3	4	8	0	0	0
D	10	1	0	0	8	0	0	3	0	0
E	0	0	0	0	0	0	0	0	0	0
F	0	5	1	0	2	0	7	0	6	10
G	0	0	0	0	0	2	0	0	0	0
H	9	0	0	0	0	0	0	0	1	0
I	0	0	0	0	8	9	5	7	0	0
J	2	0	1	0	0	9	0	0	8	0

Share	A	B	C	D	E	F	G	H	I	J
A	0	27	0	0	0	0	0	0	0	0
B	8	0	0	15	14	0	0	0	0	0
C	0	20	0	0	26	17	0	0	0	0
D	16	29	0	0	0	0	0	5	0	0
E	25	10	28	18	0	23	20	6	30	17
F	0	11	21	0	16	0	10	0	2	4
G	0	0	0	0	0	30	0	0	0	0
H	8	17	4	28	2	1	27	0	17	17
I	0	0	0	0	26	7	30	21	0	0
J	23	1	25	5	22	25	0	0	20	0

	A	B	C	D	E	F	G	H	I	J
A	0	0	0	1	0	0	0	1	0	1
B	0	0	0	1	1	0	0	0	0	0
C	0	0	0	0	1	1	0	0	0	0
D	1	1	0	0	1	0	0	0	0	0
E	0	1	1	1	0	1	0	0	0	0
F	0	0	1	0	1	0	1	0	1	1
G	0	0	0	0	0	1	0	0	0	0
H	1	0	0	0	0	0	0	0	0	0
I	0	0	0	0	1	1	0	0	0	0
J	1	0	0	0	0	1	0	0	0	0

	A	B	C	E	F	G	H	I
A	0	1	1	1	1	0	0	0
B	1	0	0	0	0	0	0	0
C	0	1	0	0	0	1	0	0
D	0	0	0	0	0	0	1	0
E	0	0	0	0	0	0	1	0
F	0	1	0	0	0	0	0	0
G	0	0	0	0	0	0	0	0
H	0	0	0	0	0	0	0	1
I	0	0	0	0	0	1	1	0
J	0	0	1	0	0	0	0	1