A STUDY OF FORMAL AND INFORMAL RELATIONS OF RUSSIAN-SPEAKING FACEBOOK USERS

Gubanov Dmitry

Introduction

Social network is interpreted, first, as a social structure comprising a set of nodes (individuals) and a set of relation links (friendship, communication etc.) defined on the first set and, second, as Internet implementation of this social structure.

Links among nodes can be interpreted differently, we can speak about of different social networks with the same set of nodes. In particular, links may be strong (e.g., regular correspondence) and weak (e.g., message exchange once a year), formal (one-time fixation) and informal (confirmed repeatedly).

We consider formal “friendship” links and partly informal “commenting” links – both in strong and weak variants. The following question is being searched for: whether informal links are conditioned by formal ones (or whether formal friendship link exist by themselves and user communication is executed through other channels).

The answer to this question seems to be important to solve various problems, for instance, to predict the existence of formal/informal links among users of a social network or to study the information propagation through social network links (our motivation).

About dataset sources. Anonymized dataset on Russian-speaking Facebook segment is provided for research by Digital Society Laboratory LLC. Commenting ties were considered from June 2012 till June 2013, friendship relations were regarded within September 2013.

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Friendships network

Basic characteristics of the friendships network

Number of Russian speaking Facebook segment users makes 3.3 million (3,279,156)

Number of friendship ties among them is equal to 77.6 million (77,639,757), a user has 47 friends on the average

Distribution of friends number is similar to power law (gamma=2.24): 20% of users have not more than 3 friends, 80% of users have not more than 45 friends.

One largest connected component (3.1 M users) prevails in the distribution of friendship network connected components; isolates are most widely spread (197 K users), the number of other components does not exceed 2 dozens of users and is found considerably less often.

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Friendship weak and strong relations

If users happen to have a common friend, friendship link between the users is called a strong one. Friendship link strength of a user

𝑤𝑠𝑓 𝑢 = 𝑣∈𝑓𝑟𝑖𝑒𝑛𝑑𝑠(𝑢) 𝐼

𝑠𝑓 𝑢, 𝑣𝑓𝑟𝑖𝑒𝑛𝑑𝑠(𝑢) ,

where 𝐼𝑠𝑓 𝑢, 𝑣 = 1, 𝑖𝑓 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑠𝑡𝑟𝑜𝑛𝑔 𝑓𝑟𝑖𝑒𝑛𝑑𝑠ℎ𝑖𝑝 𝑙𝑖𝑛𝑘 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑢 𝑎𝑛𝑑 𝑣

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

The more friends a user has, the stronger friendship links he possesses (they are proved by common friends).

Friends number

Friendsh

ip strength

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Links between friends of a user

Do friends of a certain user have friendly relations? We use two metrics: a) connectivity of the user’s friends; b) the number of connected components in the set of the user’s friends.

Connectivity of the user’s friends u

𝑐𝑑 𝑢 =2 ∙ 𝑒𝑓

𝑑(𝑑 − 1),

where ef is a number of links among

the user’s friends u,

d is the number of friends of the user u.

It is typical for a user to have 10% friendship links between his friends from their maximal possible number. In general, for the friendship network friends connectivity of users has the value of 0.2.

Сonnectivity of friendsNumber of use

rs

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Links between friends of a user.

The number of connected components in the set of a user’s friends.

As the number of friends in the user’s neighborhood is increasing, the isolates and the largest components are dominating.

As the number of the friends increases, at first the number of the largest components increases reaching its peak, and then it decreases up to zero, it is replaced by the next largest component. In the user’s friends network there is one largest connected component, which includes a greater part of his friends.

Number of friends

Percentage of

components

Percentage of use

rs

in c

omponents

Number of friends

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Friendship of similar users

What users’ characteristics explain the existence of links among them?

We consider the number of their friends as a characteristic of users’ similarity. To make a more precise estimate of the fact that users have friendly relations with similar users there may be used assortativity mixing:

𝑟 =1

𝜎𝑞2

𝑗𝑘

𝑗𝑘(𝑒𝑗𝑘 − 𝑞𝑗𝑞𝑘) ,

where 𝑒𝑗𝑘 is a probability that randomly selected graph edge is incident to j + 1 and k + 1 degree vertices, 𝑞

𝑗is a probability

that for a randomly selected edge the degree of vertices incident to

it is equal to j + 1, and 𝜎𝑞2 = 𝑘 𝑘

2 𝑞𝑘− ( 𝑘 𝑘𝑞𝑘)

2.

For friendships network of Russian-speaking Facebook segment the assortativity index r=0.267 (for company managers network it is 0.276, while for the Internet it is -0.189)

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Interdependence of formal and informal relations

Event notation: F means that two randomly selected users are friends,

C means that there is a link of commenting between two randomly selected users,

SF means that two randomly selected users are “strong” friends,

SC means that there is a “strong” link of commenting between two randomly selected users.

To what extent informal links are conditioned by formal ones?

𝑃 𝐶 =𝐸𝑐

𝑉 ∗( 𝑉 −1)/2= 1.3 ∗ 10−6 and 𝑃 𝑆𝐶 =

𝐸𝑠𝑐

𝑉 ∗( 𝑉 −1)/2= 6.6 ∗ 10−9.

𝑃 𝐶|𝐹 =𝑃 𝐶,𝐹

𝑃 𝐹=

𝐸𝑓∩𝐸с

𝐸𝑓= 0,041 and 𝑃 𝑆𝐶|𝐹 =

𝑃 𝑆𝐶,𝐹

𝑃 𝐹=

𝐸𝑓∩𝐸𝑠с

𝐸𝑓= 0,0003.

𝑃 𝐶|𝑆𝐹 =𝑃 𝐶,𝑆𝐹

𝑃 𝑆𝐹=

𝐸𝑠𝑓∩𝐸с

𝐸𝑠𝑓= 0,165 and 𝑃 𝑆𝐶|𝑆𝐹 =

𝑃 𝑆𝐶,𝑆𝐹

𝑃 𝑆𝐹=

𝐸𝑠𝑓∩𝐸𝑠с

𝐸𝑠𝑓= 0,0013.

A strong friendship link increases commenting probability by more than 4 times as compared to an “ordinary” friendship link

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Interdependence of formal and informal relations

To what extent formal links (friendship links) are conditioned by informal ones?

𝑃 𝐹 =𝐸𝑓

𝑉 ∗( 𝑉 −1)/2= 1.4 ∗ 10−5 and 𝑃 𝑆𝐹 =

𝐸𝑠𝑓

𝑉 ∗𝑉 −1

2

= 3.4 ∗ 10−6

𝑃 𝐹|𝐶 =𝑃 𝐶,𝐹

𝑃 𝐶=

𝐸𝑓∩𝐸с

𝐸𝑐= 0.45 and 𝑃 𝑆𝐹|𝐶 =

𝑃 𝑆𝐹,𝐶

𝑃 𝐶=

𝐸𝑠𝑓∩𝐸с

𝐸𝑐= 0.42

𝑃 𝐹|𝑆𝐶 =𝑃 𝐹,𝑆𝐶

𝑃 𝑆𝐶=

𝐸𝑓∩𝐸𝑠с

𝐸𝑠𝑐= 0.68 and 𝑃 𝑆𝐹|𝑆𝐶 =

𝑃 𝑆𝐶,𝑆𝐹

𝑃 𝑆𝐶=

𝐸𝑠𝑓∩𝐸𝑠с

𝐸𝑠𝑐= 0.67

The probabilities increase, consequently, existence of commenting links indicate friendship (increases its probability). A strong commenting link increases friendship probability by more than 1.5 times as compared to an “ordinary” commenting.

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Interdependence of formal and informal relations. Conclusions

Commenting links are found considerably more seldom than friendship links

If one user comments another one (“ordinary” commenting link), then in half cases they are friends

If one user comments another one (strong commenting link), then in two of three cases they are friends

If users are friends, the only in one case of 25 there is a commenting link between them

If users are friends and they have a common friend, then in one case of six there is a commenting link between them

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Interdependence of formal and informal relations. Conclusions

Commenting links are found considerably more seldom than friendship links

If one user comments another one (“ordinary” commenting link), then in half cases they are friends

If one user comments another one (strong commenting link), then in two of three cases they are friends

If users are friends, the only in one case of 25 there is a commenting link between them

If users are friends and they have a common friend, then in one case of six there is a commenting link between them

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Conclusions

We define general characteristics of the network, give a definition to strong links.

We analyze strength of Facebook friendship links, study connectivity between a user’s friends, and give an answer to the question how users who have something in common are connected with each other.

We consider interrelation between friendship and commenting links

The obtained results are planned to be applied in future to modeling of behavior of social network users.

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