The document discusses the concepts of homophily and influence within social networks, highlighting how individuals tend to connect with similar peers and how these connections can affect behaviors such as obesity. It explores methods for measuring social influence and correlation through various models, including the Independent Cascade and Linear Threshold models. Additionally, it addresses techniques for optimizing influence maximization in marketing using social network data.

1. Homophily and Influence in
Social Networks
Nicola Barbieri
nicolabarbieri1@gmail.com
References:
Maximizing the Spread of Influence through a Social Network, Kempe et Al 2003
Influence and Correlation in Social Networks,Anagnostopoulos et Al 2008
Feedback Effects between Similarity and Social Influence in Online Communities, Crandall et Al 2008
Community Detection and Mining in Social Media, Lei Tang and Huan Liu 2010
Learning Influence Probabilities In Social Networks, Goyal et al 2010
Sparsification of Influence Networks, Mathioudakis et Al 2011
Influence Propagation in Social Networks:A Data Mining Perspective, Bonchi 2011

2. The hidden influence of SNs
• We're embedded in complex and so ubiquitous social networks:
how do they affect our lives?
• Widower effect (dying of a broken heart):
• when I die, my wife's risk of death can double in the first year
• the widowhood effect it’s not restricted to husbands and wives
nor to pairs of people
• Obesity epidemic:
• Every dot is a person
• dot size proportional to people's
body size
• Yellow dots: clinically obese

4. Analysis of the Spread of Obesity
• Your friend is obese: your risk of obesity is 45 percent higher
• Your friend's friends are obese: your risk of obesity is 25 percent
higher
• Your friend's friend's friend is obese: your risk of obesity is 10 percent
higher.
• Only when you get to your friend's friend's friend's friends that there's
no longer a relationship between that person's body size and your
own body size.
• What might be causing this phenomenon?
• As I gain weight, it causes you to gain weight
• I form my tie to you because you and I share a similar body size
• We share a common exposure to something

5. :-)
:-(
:-/

6. Influence and Correlation in SNs
• The availability of rich data from popular Social Networks makes it
possible to analyze user actions at an individual level in order to
understand user behavior at large
• How user’s actions can be correlated to his/her social connections?
• What is the source of the correlation?
• We are concerned with individuals performing a certain action for the
first time, e.g., purchasing a product, visiting a web-page, or tagging a
photo with a particular tag
• After an agent performs the action, we say that the agent has become
active
• Social correlation: for two nodes u and v that are adjacent in G, the
events that u becomes active is correlated with v becoming active.

7. Models of Social Correlation
• Homophily: is the tendency of individuals to choose friends with similar
characteristic.
Individuals often befriend others who are similar to them, and hence perform
similar actions
“Birds of a feather flock together”
• Confounding: the correlation between actions of adjacent agents in a
social network can be explained by assuming an external influence
both the choices of individuals to become friends and their choice to become
active are affected by the same unobserved variable
• Influence: the action of individuals can induce their friends to act in a
similar way
a user buys a product because one of his/her friends has recently bought the
same product

8. Models of Social Correlation

9. Homophily:Analyzing similarity
over time
• How does the similarity between two people vary in the time win-
dow around their first interaction with each other?
• An elevated level of similarity just before meeting indicates a type of
selection at work, while increasing similarity following this meeting
provides evidence for social influence.
• Average cosine similarity of user pairs
as a function of the number of edits
from time of first interaction, for
Wikipedia
• Baseline: average similarity for pairs of
users who have not interacted
• Separate plots are shown for pairs of
users with different activity levels (at
least k edits before and k edits after the
first interaction)
Selection
Influence?
Avg similarity pairs of user who have not interacted

10. Identifying Social Influence
• Identifying situations where social influence is the source of
correlation is important.
• In the presence of social influence, an idea, norm of behavior, or a
product diffuses through the social network like an epidemic.
• Activation Process: in each of the time steps [1,...,T] each non-
active agent decides whether to become active:
• The probability of becoming active for each agent u is a function
p(x) of the number x of other agents v that have an edge to u and
are already active

11. Measuring social correlation
• In the influence model, each individual flips an independent coin in
every time step to decide whether or not to become active
• Simple case: we measure this probability as a function of only one
variable, the number of already-active friends
• We can estimate the probability p(a) of activation for an agent with a
already-active friends as follows:
• The coefficient α measures social correlation: a large value of α
indicates a large degree of correlation
• We estimate α, β using maximum likelihood logistic regression

12. Measuring Social correlation
• Ya,t: number of users who at the beginning of time t had a active
friends and started using the tag at time t
• Ya = Σt Ya,t
• Na,t : number of users who at time t were inactive, had a active
friends, but did not start using the tag
• Na = Σt Na,t
• We compute the values of α and β that maximize the expression

13. The Shuffle Test
• If influence does not play a role, the timing of activations should be
independent of the timing of other agents.
• Let G be the social network, and W = {w1,...,wl} be the set of users that
are activated during the period [0,T].We computeYa and Na, and use
the maximum likelihood method to estimate α.
• We create a second problem instance with the same graph G and the
same set W of active nodes, by picking a random permutation π of
{1,...,l}.We computeY’a and N’a and the social correlation coefficient α′
• The shuffle test declares that the model exhibits no social influence if
the values of α and α′ are close to each other.

14. The Edge-reversal Test
• We reverse the direction of all the edges and run logistic
regression on the data using the new graph
• If the correlation is based on the fact that two friends often share
common characteristics, we intuitively expect reversing the edges
not to change our estimate of the social correlation significantly.
• Social influence spreads in the direction specified by the edges of
the graph, and hence reversing the edges should intuitively change
the estimate of the correlation.

15. Influence on Flickr

16. Influence Propagation in SNs
• A social network plays a fundamental role as a medium for the
spread of information, ideas, and influence among its members
• The basic assumption is that when users see their social contacts
performing an action they may decide to perform the action
themselves

17. Diffusion Models
• At a given timestamp, each node is either active (an adopter of the
innovation, or a customer which already purchased the product) or
inactive
• Each node’s tendency to become active increases
monotonically as more of its neighbors become active
• An active node never becomes inactive again
• Time unfolds deterministically in discrete steps
• As time unfolds, more and more of neighbors of an inactive node u
become active, eventually making u become active, and u’s decision
may in turn trigger further decisions by nodes to which u is
connected.

18. Independent Cascade Model
• When a node v first becomes active, say at time t, it is
considered contagious.
• It has one chance of influencing each inactive neighbor u with
probability pv,u, independently of the history thus far.
• If the tentative succeeds, u becomes active at time t + 1.
• The probability pv,u, that can be considered as the strength of the
influence of v over u

19. Linear Threshold Model
• A node v is influenced by each neighbor w according to a weight
bv,w such that
• Each node v chooses a threshold θv uniformly at random from the
interval [0, 1];
• This represents the weighted fraction of v’s neighbors that must
become active in order for v to become active.
• In step t, all nodes that were active in step t − 1 remain active, and we
activate any node v for which the total weight of its active neighbors is
at least θv:

20. Linear Threshold Model
Assume bw,v = 1/kv and that the threshold for each node is 0.5.

21. ICM vs LTM
• LTM is receiver-centered
• ICM is sender-centered
• LTM’s activation depends on the whole neighborhood of one node
• LTM, once the thresholds are sampled, the diffusion process is
determined
• ICM is specified by a stochastic process

22. Influence Maximization
• Viral marketing: suppose that we have data on a social network,
with estimates for the extent to which individuals influence one
another, and we would like to market a new product that we hope
will be adopted by a large fraction of the network
• The aim is to detect few “influential” nodes to target in order to
maximize the spread on the network
• Suppose that we want to push a new product in the market and we
are given:
• a social network
• the estimates of reciprocal influence between individuals
connected in the network
• Influence Maximization: how should one select the set of initial
users so that they eventually influence the largest number of users in
the social network ?

23. Influence Maximization
• Both the Linear Threshold and Independent Cascade Models involve
an initial set of active nodes A0 that start the diffusion process
• σ(A) is the expected number of active nodes at the end of the
process, given that A is this initial active set.
• Given a parameter k find a k-node set of maximum influence.
• Both for IC and LT it is NP-hard to determine the optimum for
influence maximization
but ...
(continue)

24. Approximated Algorithm
• Given a propagation model m, if σm(S) is monotone and
submodular
then
• the optimal solution for influence maximization can be efficiently
approximated to within a factor of (1 − 1/e − ε) (slightly better than
63%)
• Monotonicity says as the set of activated nodes grows, the likelihood
of a node getting activated should not decrease
• Sub-modularity: the probability for an active node to activate some
inactive node u does not increase if more nodes have already
attempted to activate u (diminishing returns property)

25. Greedy Algorithm for IM
• The step 3 is #P-hard
• We can employ Monte Carlo simulation
• Heuristics to improve the efficiency of the Greedy
algorithm

26. Speeding up the Greedy
algorithm
• We aim to find a node with the maximal marginal gain
σ(S ∪ {v}) − σ (S)
• Exploit the submodularity!!!
σ(St ∪ {v}) − σ (St) ≥ σ(St+1 ∪ {v}) − σ (St+1)
• The marginal gain of adding a node v to a selected set S can only
decrease after we expand S
• Suppose we evaluate the marginal gain of a node v in one iteration and
find out the gain is ∆
• Those nodes whose marginal gain is less than ∆ in the previous iteration
should not be considered for evaluation because their marginal gains can
only decrease

27. IM Process
How to learn influence Probabilities?

28. Learning Influence Probabilities
• We are given:
• a social graph in the form of an undirected graph G = (V, E) where the
nodesV are users and (u,v) ∈ E represents a social tie between the
users
• a relation Actions(User,Action,Time), which contains tuples (u, a, tu)
indicating that user u performed action a at time tu
• We want to learn a function p : E → [0, 1] × [0, 1] assigning to both
Au number of actions performed by user u in the training set
Au&v
number of actions performed by both u and v in the
training set
Au|v
number of actions either u or v performs in the training
set
Av2u
number of actions propagated from v to u in the training
set.

29. • Jaccard Index
Static Models
• Bernoulli distribution: any time a contagious user v tries to
influence its inactive neighbor u, it has a fixed probability of making u
activate
• Partial Credit: each of the neighbors who have performed the action
before share the “credit” for influencing u to perform that action
Suppose user u performs an action a at time tu(a) and S its set of activated
neighbors Flickr social network and we consider
“joining a group” as the action

30. Continuous Time (CT) Models
• Influence probability may not remain constant in time
• The probability of v influencing its neighbor u at time t is:
•p0
v,u is the maximum strength of v influencing u (static models)
•τv,u can be estimated as the average time delay in propagating an
action from v to its neighbor u in the training set.
• The probability of u being influenced at time t by the combination of its
active neighbors is
If max {ptu(.)}≥θu , the activation threshold
of u, we conclude that u activates

31. Learning the parameters of the
IC Model
• The independent cascade model generates independent
propagation traces
• The set F+
α(v) of nodes that possibly influenced v are the nodes that
performed action α before v and within ∆t time
• The set F-
α(v) of nodes that definitely failed to influence v

32. where we have two contributes:
1. likelihood that at least one of the nodes in F+
α(v) succeed to influence v
EM Algorithm
• The likelihood Lα(G) of the trace can be written as
2. likelihood that the nodes in F-
α(v) fail
The probability values p(u, v) that maximize the total log-likelihood can be
computed using the following iterative formula