The document discusses assortativity and dissortativity in complex networks. It defines assortativity as "nodes with similar degree connect preferably" and dissortativity as "nodes with low degree try to connect with highly connected nodes." Several studies are summarized that examine assortativity in social networks like Twitter. Happiness was found to be assortative in Twitter reciprocal reply networks. Distance from one's typical location on Twitter was also found to correlate with expressed happiness levels. Assortative networks are described as being more resilient to attacks while disassortative networks are more vulnerable.

1. Assortativity and
Dissortativity
Complex Networks
Jaqueline Passos do Nascimento

2. Definitions
“nodes with similar degree connect preferably”
(assortative mixing)
“nodes with low degree try to connect with
highly connected nodes”
(dissortativity)
Xulvi-Brunet, R., & Sokolov, I. (2005). Changing Correlations In Networks: Assortativity And Dissortativity.

3. Definitions
ASSORTATIVITY OR ASSORTATIVE MIXING
Social networks show the property that nodes having many
connections tend to be connected with other highly
connected nodes.
DISSORTATIVITY
Technological and biological networks show the property
that nodes having high degrees are preferably connected
with nodes having low degrees.
Xulvi-Brunet, R., & Sokolov, I. (2005). Changing Correlations In Networks: Assortativity And Dissortativity.

4. Definitions

6. Definitions
“A friendship network may be highly assortative if it
connects individuals who are at similar locations or have
similar musical tastes.
A heterosexual network on the other hand will be highly
disassortative since partners will tend to be of the opposite
sex.
However, few networks are entirely assortative or
disassortative: most will exhibit both properties to some
degree depending on the particular characteristic.”

7. How to calculate metrics?
Spearman x Pearson
The Spearman correlation coefficient is the Pearson correlation coefficient applied to the ranks of the
degrees at each end of links in the network, is a non-parametric test that does not rely on normally
distributed data and is much less sensitive to outliers.

8. How to calculate metrics?
Newman
where ji,ki are the degrees of the vertices at the ends of the ith edge, with i = 1...M
M edges

9. Spearman Example

10. Spearman Example

11. Spearman Example

12. Assortativity coefficient
A structural metric of great interest in the research of social networks, which characterizes the degree similarity of
adjacent nodes, is the degree-degree correlation, that is “who is connected to who?”
The correlation is characterized by the assortativity r and defined as the Pearson correlation coefficient:
where i and j are the remaining degrees at the two ends of an edge and the ⟨·⟩ notation represents the average over all
links.
If a network’s assortativity coefficient is negative, a hub tends to be connected to non-hubs, and vice versa.
When r > 0, we call the network to have an assortative mixing pattern
when r < 0, disassortative mixing.
An uncorrelated network exhibits the neutral degree-mixing pattern whose r = 0.
Hu, H., & Wang, X. (2009). Disassortative mixing in online social networks. EPL (Europhysics Letters), 18003-18003. Retrieved September 2,
2014, from http://cs.fit.edu/~rmenezes/Teaching/Entries/2014/8/17_CSE5656__Complex_Networks.html

13. Relation to Homophily in Social
Sciences
similarity breeds connection

14. Relation to Homophily in Social
Sciences

15. Relation to Homophily in Social
Sciences
From the perspective of sociology and psychology, in real life everyone would
like to have intercourse with elites in a society; however the elites would rather
communicate with the people with the same social status as theirs, which may
lead to the assortative mixing pattern in the real-world social networks.

17. Relation to Homophily in Social
Sciences
Patterns of friendship between individuals for example are strongly affected by
the language, race, and age of the individuals in question, among other things.
Friendship is usually found to be assortative by most characteristics.
Assortative mixing can have a profound effect on the structural properties of a
network. For example, assortative mixing of a network by a discrete
characteristic will tend to break the network up into separate communities. If
people prefer to be friends with others who speak their own language, for
example, then one might expect countries with more than one language to
separate into communities by language.

18. Applications
A viral marketer attempting to advertise a new product could benefit from
considering specific sets of users on a social space who are homophilous with
respect to their interest in similar products or features.
Understanding the impact of homophily on diffusion is likely to have potential in
addressing the propagation of medical and technological innovations, cultural
bias, in understanding social roles and in distributed social search.

19. Study #1
Twitter reciprocal reply networks exhibit assortativity
with respect to happiness.
Bliss, C. A., Kloumann, I. M., Harris, K. D., Danforth, C. M. & Dodds, P. S. Twitter reciprocal reply networks exhibit assortativity with respect to happiness. Journal of Computational Science 3(5), 388–397 (2012).

20. Study #1
Method
From September 2008 to February 2009, they retrieved over 100 million tweets from the Twitter
streaming API service;
If the tweet was made using Twitter’s built-in reply function,3 the identification number of the message
being replied to (original message id) and the identification of the user being replied to (original user
id) were also reported.
Bliss, C. A., Kloumann, I. M., Harris, K. D., Danforth, C. M. & Dodds, P. S. Twitter reciprocal reply networks exhibit assortativity with respect to happiness. Journal of Computational Science 3(5), 388–397 (2012).

21. Study #1

22. Study #1
Love 8.42
Special 7.20
Sad 2.38
Die 1.74

23. Study #1

24. Study #1

25. Study #1
Bliss, C. A., Kloumann, I. M., Harris, K. D., Danforth, C. M. & Dodds, P. S. Twitter reciprocal reply networks exhibit assortativity with respect to happiness. Journal of Computational Science 3(5), 388–397 (2012).

26. Study #1

27. Study #1
In a study of over 6 million users, Cha et al. [10] found that users with the
highest follower counts were not the users whose messages were most
frequently retweeted. This suggests that such popular users (as measured by
follower count) may not be the most influential in terms of spreading
information, and this calls into question the extent to which users are influenced
by those that they follow.
Large degree nodes use words such as “you,” “thanks,” and “lol” more
frequently than small degree nodes, while the latter group uses words such as
“damn,” “hate,” and “tired” more frequently.
Bliss, C. A., Kloumann, I. M., Harris, K. D., Danforth, C. M. & Dodds, P. S. Twitter reciprocal reply networks exhibit assortativity with respect to happiness. Journal of Computational Science 3(5), 388–397 (2012).

28. Study #2
Happiness and the Patterns of Life: A Study of
Geolocated Tweets
37 million geolocated tweets were used to characterize the movement patterns
of 180,000 individuals, taking advantage of several orders of magnitude of
increased spatial accuracy relative to previous work. Employing the recently
developed sentiment analysis instrument known as the hedonometer, we
characterize changes in word usage as a function of movement, and find that
expressed happiness increases logarithmically with distance from an individual’
s average location.

29. Study #2

30. Study #2

31. Study #2
Words appearing on the right increase the happiness of the 2500 km distance
relative 1 km distance.
For example, tweets authored far from an individual’s expected location are
more likely to contain the positive words ‘beach’, ‘new’, ‘great’, ‘park’,
‘restaurant’, ‘dinner’, ‘resort’, ‘coffee’, ‘lunch’, ‘cafe’, and ‘food’, and less likely to
contain the negative words ‘no’, ‘don’t’, ‘not’, ‘hate’, ‘can’t’, ‘damn’, and ‘never’
than tweets posted close to home.
Words going against the trend appear on the left, decreasing the happiness of
the 2500 km distance group relative to the 1 km group. Tweets close to home
are more likely to contain the positive words ‘me’, ‘lol’, ‘love’, ‘like’, ‘haha’, ‘my’,
‘you’, and ‘good’.

32. Study #3
Happiness is assortative in online social networks
“General happiness or Subjective Well-Being (SWB) of Twitter users, as
measured from a 6 month record of their individual tweets, is indeed assortative
across the Twitter social network. To our knowledge this is the first result that
shows assortative mixing in online networks at the level of SWB.”

33. Study #3

34. Study #3

35. Resilience
“social networks that spread disease, appear to be assortative, and therefore
are resilient, at least against simple targeted attacks such as attacks on the
highest degree vertices. And yet at the same time the networks that we would
wish to protect, including technological networks such as the Internet, appear to
be disassortative, and are hence particularly vulnerable.”
“assortative networks percolate more easily and that they are also more robust
to removal of their highest degree vertices, while disassortative networks
percolate less easily and are more vulnerable. This suggests that social
networks may be robust to intervention and attack while technological networks
are not.”
Newman, M. (2002). Assortative Mixing in Networks. Physical Review Letters. Retrieved September 2, 2014, from http://arxiv.org/pdf/cond-mat/0205405.pdf

36. Resilience
Assuming that the goal of a vaccination program is to destroy network
connectivity so that the disease in question cannot spread, our findings suggest
that even targeted vaccination strategies would be less effective in assortative
networks than in disassortative or neutral ones because of the resilience of the
network to this type of attack.

37. Spreading
For the spreading phenomena in online communities, such as diffusion of
opinions, technical innovations or gossip, one can expect the things to be
spread to a larger segment of the population in disassortative networks than in
assortative ones.

38. Thank you