influence and correlation in social networks aris anagnostopoulos ravi kumar mohammad mahdian
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Influence and Correlation in Social Networks
Aris AnagnostopoulosRavi Kumar
Mohammad Mahdian
Preliminaries
- Correlations exist in users' behaviors
Preliminaries
- Correlations exist in users' behaviors - Representation: individuals are nodes of a social graph, G every node is "active" or "inactive" - Formally, correlation = if u and v are adjacent in G: the event that u becomes active is correlated with v becoming active
Preliminaries
- Correlations exist in users' behaviors - Representation: individuals are nodes of a social graph, G every node is "active" or "inactive" - Formally, correlation = if u and v are adjacent in G: the event that u becomes active is correlated with v becoming active
- Want to distinguish between different sources of social correlation
Models of Social Correlation
- Homophily = tendency for individuals to choose friends with similar characteristics / preferences
Models of Social Correlation
- Homophily = tendency for individuals to choose friends with similar characteristics / preferences
- Confounding = external influence from elements in the environment (confounding factors)
Models of Social Correlation
- Homophily = tendency for individuals to choose friends with similar characteristics / preferences
- Confounding = external influence from elements in the environment (confounding factors)
- Influence = the action of one individual induces another individual to act in a similar way.
Motivation
- Useful to know when social influence is the source of correlation
Motivation
- Useful to know when social influence is the source of correlation
- Viral marketing -> want to target select individuals
Motivation
- Useful to know when social influence is the source of correlation
- Viral marketing -> want to target select individuals
- Influence behavior -> create "role models" (e.g. in fashion)
Motivation
- Useful to know when social influence is the source of correlation
- Viral marketing -> want to target select individuals
- Influence behavior -> create "role models" (e.g. in fashion)
- We want to identify situations when such techniques can be applied.
Motivation
- Useful to know when social influence is the source of correlation
- Viral marketing -> want to target select individuals
- Influence behavior -> create "role models" (e.g. in fashion)
- We want to identify situations when such techniques can be applied.
- Also useful for analysis (predicting future state of network)
Modeling Influence
1. Graph G drawn according to some distribution
Modeling Influence
1. Graph G drawn according to some distribution 2. In each of the time steps 1, ..., T, each non-active agent decides whether to become active.
Modeling Influence
1. Graph G drawn according to some distribution 2. In each of the time steps 1, ..., T, each non-active agent decides whether to become active. 3. An agent becomes active with probability p(a), a function of the number of neighboring and active nodes.
or, alternatively,
Some remarks...
- The coefficient α measures social correlation.
Some remarks...
- The coefficient α measures social correlation.
- Since actions are stored, a represents the number of users active at any earlier time step
Some remarks...
- The coefficient α measures social correlation.
- Since actions are stored, a represents the number of users active at any earlier time step
- This model is relatively simplistic: - the probability does not vary between nodes - or as time passes
Some remarks...
- The coefficient α measures social correlation.
- Since actions are stored, a represents the number of users active at any earlier time step
- This model is relatively simplistic: - the probability does not vary between nodes - or as time passes
- However, these simplifying assumption are practical
Estimating α, β
- Can estimate using maximum likelihood logistic regression
- Maximize expression
whereis the number of users who at the beginning of time had a active friends and became active at time t
The Shuffle Test
- Idea: if influence does not play a role, then the timing of activations amongst users should be independent of each other:
Pr(a active before b) = Pr(b active before a)
The Shuffle Test
1. Estimate α for initial graph2. Randomly permute the order in which active nodes have been activated:
set the time of
3. Estimate α' for this configuration4. If the values for α and α' are close to each other, the model exhibits little or no social influence.
The Edge-reversal Test
1. reverse direction of all the edges 2. run the same logistic regression on the data using the new graph
If correlation is not due to influence, then α should not change
Generative Models
- No Correlation
- Influence
- Correlation, no influence
Generative Models - No Correlation
- network grows just as the real data - at every step, randomly pick n nodes, and make them active
Influence Model- network grows just as the real data - at every step, every inactive node flips a coin, with
Correlation, No Influence Model
- network grows just as the real data - Pick a subset S of G: - randomly pick centers, add a ball of radius 2 from each to S - do this until |S| reaches parameter L- Pick nodes to become active uniformly at random, from S
Distinguishing Influence: Shuffle Test
Influence:
Correlation:
Distinguishing Influence: Edge Reversal
Correlation:
Influence:
Real Data: the Flickr Dataset
- analyzed 800K users over 16 months - about 340K exhibited tagging behavior
- size of giant component: 160K
- 2.8M directed edges, 28.5% not mutual
- analyzed 1,700 tags independently - various types (event, color, object, etc) - various numbers of users - various growth patterns (bursty, smooth, periodic)
Distinguishing Influence in Flickr
Shuffle test
Distinguishing Influence in Flickr
Edge reversal test
Some Influence
- can discover traces of influence by looking at similar tags
Some Influence
- can discover traces of influence by looking at similar tags - for the tag "graffiti", the difference between αs was 0
- however, for the misspelling "grafitti", difference was slightly larger
- with even less common misspelling "graffitti", difference increased even more
Conclusions
- distinguishing between correlation and causation is difficult
Conclusions
- distinguishing between correlation and causation is difficult
- timing information can help answer the question (shuffle)
Conclusions
- distinguishing between correlation and causation is difficult
- timing information can help answer the question (shuffle)
- knowing of asymmetric social ties is also useful (edge-reversal)
Further research directions
- formal verification of results? (controlled experiments) - quantification of the strength of influence? - identify which nodes influence others - what if social ties are symmetric? - distinguishing between other forms of correlation
- distinguishing between different forms of social influence
Questions?