1. A STUDY OF FORMAL AND
INFORMAL RELATIONS OF
RUSSIAN-SPEAKING
FACEBOOK USERS
Gubanov Dmitry

2. 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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3. 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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4. 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
𝑤 𝑠𝑓
𝑢 = 𝑣∈𝑓𝑟𝑖𝑒𝑛𝑑𝑠(𝑢) 𝐼 𝑠𝑓
𝑢, 𝑣
𝑓𝑟𝑖𝑒𝑛𝑑𝑠(𝑢) ,
where 𝐼 𝑠𝑓
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
Friendshipstrength
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5. 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
𝑐 𝑑 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 friendsNumberofusers
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6. 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.
Numberoffriends
Percentageof
components
Percentageofusers
incomponents
Numberoffriends
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7. 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
𝜎 𝑞
2
𝑗𝑘
𝑗𝑘(𝑒𝑗𝑘 − 𝑞𝑗
𝑞 𝑘
) ,
 where 𝑒𝑗𝑘 is a probability that randomly selected graph edge is
incident to j + 1 and k + 1 degree vertices, 𝑞𝑗
is a probability
that for a randomly selected edge the degree of vertices incident to
it is equal to j + 1, and 𝜎 𝑞
2
= 𝑘 𝑘2
𝑞 𝑘
− ( 𝑘 𝑘𝑞 𝑘
)
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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8. 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 =
𝐸 𝑐
𝑉 ∗( 𝑉 −1)/2
= 1.3 ∗ 10−6
and 𝑃 𝑆𝐶 =
𝐸 𝑠𝑐
𝑉 ∗( 𝑉 −1)/2
= 6.6 ∗ 10−9
.
 𝑃 𝐶|𝐹 =
𝑃 𝐶,𝐹
𝑃 𝐹
=
𝐸 𝑓∩𝐸с
𝐸 𝑓
= 0,041 and 𝑃 𝑆𝐶|𝐹 =
𝑃 𝑆𝐶,𝐹
𝑃 𝐹
=
𝐸 𝑓∩𝐸 𝑠с
𝐸 𝑓
= 0,0003.
 𝑃 𝐶|𝑆𝐹 =
𝑃 𝐶,𝑆𝐹
𝑃 𝑆𝐹
=
𝐸 𝑠𝑓∩𝐸с
𝐸 𝑠𝑓
= 0,165 and 𝑃 𝑆𝐶|𝑆𝐹 =
𝑃 𝑆𝐶,𝑆𝐹
𝑃 𝑆𝐹
=
𝐸 𝑠𝑓∩𝐸 𝑠с
𝐸 𝑠𝑓
= 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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9. Interdependence of formal and informal
relations
 To what extent formal links (friendship links) are conditioned by
informal ones?
 𝑃 𝐹 =
𝐸 𝑓
𝑉 ∗( 𝑉 −1)/2
= 1.4 ∗ 10−5
and 𝑃 𝑆𝐹 =
𝐸 𝑠𝑓
𝑉 ∗
𝑉 −1
2
= 3.4 ∗ 10−6
 𝑃 𝐹|𝐶 =
𝑃 𝐶,𝐹
𝑃 𝐶
=
𝐸 𝑓∩𝐸с
𝐸 𝑐
= 0.45 and 𝑃 𝑆𝐹|𝐶 =
𝑃 𝑆𝐹,𝐶
𝑃 𝐶
=
𝐸 𝑠𝑓∩𝐸с
𝐸 𝑐
= 0.42
 𝑃 𝐹|𝑆𝐶 =
𝑃 𝐹,𝑆𝐶
𝑃 𝑆𝐶
=
𝐸 𝑓∩𝐸 𝑠с
𝐸 𝑠𝑐
= 0.68 and 𝑃 𝑆𝐹|𝑆𝐶 =
𝑃 𝑆𝐶,𝑆𝐹
𝑃 𝑆𝐶
=
𝐸 𝑠𝑓∩𝐸 𝑠с
𝐸 𝑠𝑐
= 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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10. 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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11. 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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12. 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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