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Describing the New Orleans Facebook Network

For these analyses I’ll start by taking the entire network of friendships. This network is inherently symmetric (except perhaps considering who initiated the friendship request). Later we’ll consider restrictions on the number of ties, or using an interaction network based on wall posts.

The full friendship network consists of 63,731 nodes, and 817,090 edges. Here’s the degree distribution for this network:

The following graph demonstrates that this network is scale-free. A network is scale-free when its the degree distribution will be linear when plotted on a log-log plot. Scale-free graphs have been shown to result from preferential attachment, among other mechanisms.

Egonet bias for friendships

Again considering the network of all Facebook friends, we can immediately note the extent of egonet bias overall.

Because those with few friends are likely to be friends with others who have few friends, this effect may be disproportionately rooted in those with many friends. The plot below shows that indeed the median egonet mean increases with the degree of the individual.

Bias for wall-posting in- and out- degree

In-degree will be the main focus of this section, as in-degree is the most directly visible quality of a friend. Posts on a friend’s wall will be available for the ego, and may show up on their feed. Thus, unlike out-degree, egos are sure to have some exposure to information about in-degree from their friends.

The true mean for in- and out- degree is 9.5176. The true median for in- degree in wall posts is 6. The true median for out- degree in wall posts is 6.

In-degree

  • The mean egonet mean wall-post in-degree is 9.031.

  • The median egonet mean wall-post in-degree 7.5.

  • The mean egonet median wall-post in-degree is 6.3491.

  • The median egonet median wall-post in-degree 5.

In the population, the average person sees their most popular friend as receiving 37.0921 posts.

  • 95.3053% of people have at least one friend who has received any wall posts from others in the network.
  • 81.855% of people have at least two friends who has received any wall posts from others in the network.

Out-degree

  • The mean egonet mean wall-post out-degree is 8.5058.

  • The median egonet mean wall-post out-degree 7.1304.

  • The mean egonet median wall-post out-degree is 6.4654.

  • The median egonet median wall-post out-degree 5.

Paper stats

Summary

  • The maximum number of friends is 1,098.

  • 84.2463% of people have LESS than 50 friends.

  • Only 106 people have more than 300 friends.

  • Those with more than 500 friends account for 1.0214% of all friends.

  • The minimum number of friends is 1, and 30,501 people have fewer than 10 friends.

  • The mean degree in the network is 25.6418 and the median is 10

  • The mean egonet mean degree is 58.3634.

  • The percentage of the 63,731 egonets whose mean degree is greater than the true mean is 73.0571%.

  • The percentage of egonets whose median degree is greater than the true median is 0%, with 0% of egonets having a median degree 10, the true population median.

  • The correlation between degree and mean egonet degree is 0.3974.

  • 90.8067% of egonets have mean degree greater than the ego’s.

Applications to particular types of ties

For the 31,082 people who have any posting partners, people have a mean of 5.0138 posting partners. We can determine the mean number of posting partners for each of their egonets of posting partners; we find that their posting partner egonets have a mean of 10.0356 posting partners.

Applications to particular types of people

63.0713% of people are posters.

62.4971% of people are posters with at least one posting friend.

( The average person who posts has 28.629 friendships with other people who post. )

We find that the average person who posts (and has posting friends) has 28.8921 friendships with other people who post. We can examine their egonets of people who post, and we find that the mean egonet has a mean of 56.362 friendships with people who post, which is considerably more than the average for people who post.

Egonet Biases Regarding Other Types of Ties

We find that for all people, the correlation between the number of posting partners and the number of other friends is 0.5709. In the data, people have an average of 2.4453 posting partners. We find that friends have an average of 7.6493 posting partners, and egonets have an average of 5.0074 posting partners.

Egonet Biases Regarding Other Types of People

new

The correlations between people’s numbers ofposter and non-poster friends is 0.8933.
Consequently, even though posters have an average of
5.3742 non-poster friends, their poster friends have a much larger average of non-perter friends (13.448), and their egonets of posters have a larger number of non-poster friends an average (10.4178).

Thus, if posters use their poster friends and egonets of posters to estimate how many non-poster friends posters generally have, they would greatly overestimate the numbers of non-poster friends of posters.

old

The correlations between number of friends who post and those who do not is 0.8844. Thus, it turns out that even though non-posters have an average of 2.302 non-poster friends, their egonets of posters have an average of 10.8431 non-poster friends. Their egonets have an average of 10.097 non-poster friends.

Similarly, even though non-posters have an average of 10.4495 poster friends, their friends have an average of 61.8008 friends who post, and their egonets have an average of 46.7158.

Some tests of Feld’s correlation hypothesis

1,151.535 819.4698

1,948.913 1,370.528

0.3974

0.9453 0.9388

28,324.66

Older version of statistics

Summary Statistics

  • 63.0713% of people have ever posted.

  • 28.3386% of people have who have ever posted have only posted once.

And the actual number of posts sent by each person

Posting partners in posting partners network

  • 31,082 people have posting partners.
  • People have an average of 5.0138 posting partners.
  • The average egonet of posting partners has 10.0356 posting partners.

Non-posters

  • The correlation between number of friends who are posting people and those who are just friends is 0.9976.

  • Non-posters have an average of 2.1824 non-posting friends

  • Egonets of non-posters have an average 9.3937 non-posting friends.

Posters

  • Posters who have a posting friend have an average of 5.4095 non-posting friends

  • Posting friends of posters have an average of 10.4178 non-posting friends.

Limit to those with posting partners

  • Those with any posting partners have a mean of 5.0138 posting partners.

  • Those egonets where the ego has any posting partners have a mean of 6.1399 posting partners.

The network of those who post

  • The average person who posts has 28.629 friendships with other people who post.

  • Those egonets where the ego posts has a mean of 53.9998 friends with others who post.

Posters using posting friends to estimate # of non-posting friends of posters

  • The average person who posts has 5.4095 friendships with people who don’t post.

  • The average person who posts has 28.629 friendships with other people who post.

Other types of ties: Posting partners in the whole network

  • For all people, the correlation between the number of posting partners and the number of other friends is 0.5709

  • For all people, the correlation between the number of non-posting partners and the number of other friends is 0.5709

  • For all people, the correlation between the number of posting partners and the number of friends is 0.6538

  • People have an average of 2.4453 posting partners

  • Friends have an average of 7.6493 posting partners.

  • The mean of egonets’ posting partners per person is 5.0074.

Other types of people: posters vs non-posters

The following table shows the correlation matrix for the number of friends, the number of friends who never post, the number of friends who post, and the number of friends who post a regularly (>10 in this dataset).

deg EgonetN_PostNone EgonetN_PostAny EgonetN_PostMuch
deg 1.0000000 0.9225019 0.9975598 0.9453494
EgonetN_PostNone 0.9225019 1.0000000 0.8933019 0.8229131
EgonetN_PostAny 0.9975598 0.8933019 1.0000000 0.9519369
EgonetN_PostMuch 0.9453494 0.8229131 0.9519369 1.0000000

  • The posting ego-nets of non-posters have a mean number of 10.8431 non posting friends.

  • Egonets of non-posters who have at least one poster have a mean number of 10.097 non-posters per person.

  • Non-posters have an average of 9.1788 posting friends

  • Non-posters’ friends have an average of 66.8852 posting friends

  • Non-poster egonets have an average of 39.4981 posting friends

Notes from conversation

Type of people… Posting partners and non-posting-partners Worried about the actual numbers of friends. Doesn’t want people that weren’t there for the whole (or most) time period

## [1] TRUE
## [1] 4.659204
## [1] 2.460758
## [1] 7.075299