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Homophily and Influence in Social Networks
Nicola [email protected]
References:Maximizing the Spread of Influence through a Social Network, Kempe et Al 2003Influence and Correlation in Social Networks, Anagnostopoulos et Al 2008Feedback Effects between Similarity and Social Influence in Online Communities, Crandall et Al 2008Community Detection and Mining in Social Media, Lei Tang and Huan Liu 2010Learning Influence Probabilities In Social Networks, Goyal et al 2010Sparsification of Influence Networks, Mathioudakis et Al 2011Influence Propagation in Social Networks: A Data Mining Perspective, Bonchi 2011
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
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
:-) :-( :-/
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.
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
Models of Social Correlation
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)
SelectionInfluence?
Avg similarity pairs of user who have not interacted
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
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
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
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 compute Ya 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 compute Y’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.
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.
Influence on Flickr
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
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.
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
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:
Linear Threshold Model
Assume bw,v = 1/kv and that the threshold for each node is 0.5.
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
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 ?
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)
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)
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
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
IM Process
How to learn influence Probabilities?
Learning Influence Probabilities• We are given:
• a social graph in the form of an undirected graph G = (V, E) where the nodes V 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&vnumber 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
Av2unumber of actions propagated from v to u in the training
set.
• 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
Continuous Time (CT) Models
• Influence probability may not remain constant in time
• The probability of v influencing its neighbor u at time t is:
•p0v,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
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
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
