complex networks - assortativity

Assortativity and Dissortativity

Complex NetworksJaqueline Passos do Nascimento

“nodes with similar degree connect preferably” (assortative mixing)

“nodes with low degree try to connect with highly connected nodes” (dissortativity)

Definitions

Xulvi-Brunet, R., & Sokolov, I. (2005). Changing Correlations In Networks: Assortativity And Dissortativity.

DefinitionsASSORTATIVITY OR ASSORTATIVE MIXING

Social networks show the property that nodes having many connections tend to be connected with other highly connected nodes.

DISSORTATIVITYTechnological 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.

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.”

Definitions

Spearman x Pearson

How to calculate metrics?

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.

Newman

How to calculate metrics?

M edges

where ji,ki are the degrees of the vertices at the ends of the ith edge, with i = 1...M

Spearman Example

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

Assortativity coefficient

similarity breeds connection

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.


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.


ApplicationsA 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.

Study #1Twitter 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).

Study #1Method

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.


Study #1

Love 8.42Special 7.20

Sad 2.38Die 1.74

Study #1

Study #1

Study #1In 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.


Study #2Happiness 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.

Study #2

Study #2Words 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’.

Study #3Happiness 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.”

Study #3

Resilience

Newman, M. (2002). Assortative Mixing in Networks. Physical Review Letters. Retrieved September 2, 2014, from http://arxiv.org/pdf/cond-mat/0205405.pdf

“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.”

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.

Resilience

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.

Spreading

Thank you

complex networks - assortativity

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