Social Ties and Link Prediction
Kristina Lerman
University of Southern California
CS 599: Social Media Analysis
University of Southern California
1
Link Prediction
Will nodes 33 and
28 become friends
in the future?
What about
nodes 27 and 4?
Does network structure
contain enough
information to predict
what new links will
form in the future?
Who to follow
Strength of social ties (review)
• Strong ties
– surrounded by many mutual friends
– characterized by lots of shared time together
• Weak ties
– have few mutual friends
– Serve as bridges to diverse parts of the network
– Provide access to novel information
The Link-Prediction Problem for Social
Networks (Liben-Nowell & Kleinberg)
To what extent can the evolution of a social network be
modeled using features intrinsic to the network itself?
• Formalize the link prediction problem
– Given a snapshot of a network, infer which new
interactions between nodes are likely to occur in the
future
• Propose link prediction heuristics based on measures for
analyzing the “proximity” of nodes in a network.
• Evaluate link prediction heuristics on large coauthorship
networks. Future coauthorships can be extracted from
network topology.
The intuition
• In many networks, people who are “close” belong to the same
social circles and will inevitably encounter one another and
become linked themselves.
• Link prediction heuristics measure how “close” people are
x
x
y
y
Red nodes are close to each other
Red nodes are more distant
Link prediction heuristics
• Local
x
•
•
•
•
Common neighbors (CN)
Jaccard (JC)
Adamic-Adar (AA)
Preferential attachment (PA) …
• Global
y
• Katz score
• Hitting time
• PageRank …
Local link prediction heuristics
•Link prediction heuristics
– Common neighbors (CN)
x
y
CN  3
• Neighborhood overlap
– Jaccard (JC)
– Adamic-Adar (AA)
– Preferential attachment (PA)

Local link prediction heuristics
• Link prediction heuristics
x
– Common neighbors (CN)
– Jaccard (JC)
• Fraction of common neighbors
y
CN
JC 
dx  dy  CN
– Adamic-Adar (AA)
– Preferential attachment (PA)
Link prediction heuristics
•Link prediction heuristics
– Common neighbors (CN)
– Jaccard (JC)
– Adamic-Adar (AA)
x
• Nmbr common neighbors, with
each neighbor z attenuated by
log of its degree
y
– Preferential attachment (PA)
AA 

zCN
1
log d z
Local link prediction heuristics
•Link prediction heuristics
x
y
PA  d x d y
–
–
–
–
Common neighbors (CN)
Jaccard (JC)
Adamic-Adar (AA)
Preferential attachment (PA)
• Better connected nodes are
more likely to form more links
Global link prediction heuristics
•Link prediction heuristics
– Katz score
x
• Measures number of paths
between two nodes,
attenuated by their length
– Hitting time
y
• Expected time for a random
walk from x to reach y
– …
Data
• Collaboration networks of physicists
– Core nodes: authors who published at least 3 papers
during the training period and at least 3 papers during test
period
• Training data: graph G(t0, t0’) of collaborations during time
period [t0, t0‘] with V core nodes and Eold edges
• Test data: graph G(t1, t1’) of collaborations during a later time
period [t1, t1’] with V core nodes and Enew edges
Evaluation metric
•Link prediction algorithm
• Score node pairs using a heuristic p
• New links more likely among high scoring pairs
• Each link prediction heuristic p outputs a ranked list L of new
collaborations: pairs in VxV-Eold.
• Focus evaluation on new links Enew* between core nodes
• Performance metric: How many of the top n pairs in ranked
list L are the actual new nodes in Enew*?
Results
Heuristics vs
random predictor
Results
Heuristics vs graph
distance predictor
Summary
• Graph-based link prediction heuristics outperform random
guess by a factor of 40
• However, they still predict only 16% of new collaborations at
best, leaving much room for improvement.
CSCI 599 Social Media Analysis
Link prediction in complex networks: a
survey
Presenter: Yuan Shi
USC ID: 7678039433
L Lu and T Zhou, “Link prediction in complex networks: a
survey”, Physica A 390(6):11501170 (2011)
Link Prediction
• Estimate the likelihood of the existence
of a link between two nodes, based on
observed links and the attributes of
nodes
• Application
– Biological networks: costly to identify links
between nodes through field/laboratorial
experiments
– Online social networks: predicting
friendship and recommending new friends
(predicting future links in evolving
networks)
Problem Description and Evaluation Metrics
• Undirected network G = (V, E)
• Universal set U containing |V|(|V|-1)/2 possible links
• Task: Find out missing links in U – E.
• Evaluation: randomly split E into two sets: training set
ET , probe/validation set EP
• k-folder cross validation
– Randomly partition into k subsets
– Each time one subset is selected as probe set, the others as
training set
– Repeat k times, each with a different probe set
Evaluation Metrics
• A link prediction algorithm gives a ranking on each link
• AUC (area under the receiver operating characteristic curve)
– Focus the whole list of ranks
– The probability that a randomly chosen missing link is
given a higher score than a randomly chosen nonexistent
link
• Precision
– Focus on the top ranks
– Take top-L predicted links, among which Lr links are right,
the precision is Lr/L
Similarity-Based Algorithms
• Assign a score sxy to each pair of nodes x and y
• The attributes of nodes are generally hidden -> focus
on structural similarity: two nodes are linked if they
have similar network structure
• Similarity indices
– Local similarity Indices: only use local information
– Global similarity indices: use global information, more
accurate but costly
– Quasi-local indices: a tradeoff between local and global
Local similarity Indices
• 10 indices are discussed.
• Common neighbors (CN)
set of neighbors
• Resource Allocation Index (RA)
degree of note z
Intuition:
• Similarity(x, y) = the amount of resource y received from x
• x sends some resource to y, with their common neighbors as
transmitters
• Each transmitter has a unit of resource and will equally
distribute it to all its neighbors
• Adamic-Adar Index (AA)
Local similarity Indices - Evaluation
Metric: AUC.
Each number averaged by 10 implementations.
Real-world networks
PPI: protein-protein interaction
NS: co-authorship
Grid: electrical power-grid
PB: US political blogs
INT: router-level Internet
USAir: US air transportation
CN and AA have second
best performance
RA performs the best
Global similarity Indices
• 7 indices are discussed. Some examples are:
• Katz Index
• Average Commute Time
Laplacian matrix
• Random Walk with Restart (direct application of PageRank
algorithm)
• Global indices
– Pros: more accurate than local indices
– Cons: 1) time-consuming; 2) global topological information may not be
available
Quasi-local Indices
• 3 indices are discussed.
• Local Path Index (LP)
– Outperforms local indices like RA, AA and CN
– Performs competitively to global indices with much less
computational cost
• Local Random Walk (LRW): at time step t,
q is initial configuration function, e.g.
• Superposed Random Walk (SRW): at time step t,
Some experiments show LRW and SRW performs better than LP
Maximum Likelihood Methods
• Methodology
– Assume some organizing principles of the network structure
– Rules and parameters are obtained by maximizing the
likelihood of the observed structure
– Likelihood of any non-observed link can be calculated
according to those rules and parameters
• Pros: provide valuable insights into the network
organization
• Cons: Time consuming; Prediction accuracy is not very high
Example: Hierarchical Structure Model
Assumption:
1. Each internal node r associated with a probability pr
2. Probability of linking a pair of leaves equals to pr’ where r’ is their
lowest common ancestor
Some statistics of the graph
By maximizing the likelihood,
Prediction:
1. Sample a large number of dendrograms with probability
proportional to
their likelihood
2. Compute the link probability by averaging the corresponding
probability
over all sampled dendrograms
Application
• Reconstruction of Networks
– Not easy to reconstruct the “true” network since generally no one
knows how many links are missing
– Reliability of a network
Global optimization is difficult -> use greedy algorithms
• Classification of Partially Labeled Networks
– Predict the labels of these unlabeled nodes based on the known labels
and the network structure
– Approach: add artificial links between every pair of labeled and
unlabeled nodes
Application
• Evaluation of Network Evolving Mechanisms
– link prediction algorithm tells the factors resulting in the existence of
links
– Example: Similarity indices for the Chinese city airline network
CN: topological effects
DIS: geographical distance
POPU: population
GDP
TI: third sector of GDP, named the
tertiary industry
Outlook
• Link prediction in directed networks
• Multi-dimensional networks, where links could have different
meanings (e.g. positive/negative)
• Hybrid algorithms to combine different similarity indices
• Leveraging external information (e.g. attributes) to improve
accuracy
• Time-series link prediction approach considering the temporal
evolutions of link occurrences
Romantic partnerships and the dispersion of
social ties
Romantic Partnerships and the Dispersion of Social Ties
(Backstrom & Kleinberg)
• Questions
– Who are the most important individuals in a person’s
social neighborhood?
– What are the defining structural signatures of a person’s
social neighborhood?
• Contributions
– Dispersion: a new measure for estimating tie strength
– Characterize romantic relationships in terms of network
structure
– Empirical study of this characteristic across Facebook
population
Who are the most important people in one’s
social neighborhood?
• Following Granovetter, researchers use number of mutual
friends (embeddedness) to identify strong ties
– Close friends, who share much time together
– Emotionally intense interactions
A-B tie is highly embedded
in the network
A-B tie is not embedded
in the network
C
D
E
F
C
A
A
D
B
B
E
F
Romantic ties
• Embeddedness is not able to identify “significant others”
(romantic relationships, e.g., spouse, partner, boy/girlfriend)
• Ego network – social neighborhood of an individual, showing
all his/her friends and links between them
Ego network
of an
individual
Who is the
“significant other”?
Social foci
• People have large clusters of friends corresponding to well-defined foci of
interaction in their lives
– These links have high embeddedness but are not very strong ties
• In contrast, romantic partners may have lower embeddedness, but they
often involve mutual friends from different foci
Co-workers
Ego network
of an
individual
College friends
Embeddedness vs dispersion
Embeddedness:
u and v have many
mutual neighbors.
Links u-b, u-c, and
u-f have
embeddedness 5
Link u-h has
embeddedness 4
Dispersion:
mutual neighbors of u
and v are not wellconnected to one
another, and hence u
and v are the only
intermediaries
between these
different parts of the
network.
Link u-h has high
dispersion: u and h are
the only intermediaries
between c and f
Link dispersion
disp(u,b)=1
s(u,h)=4
Evaluation
• Egonetworks of 1.3 million Facebook users, selected
uniformly at random from among all users of age at least 20,
with between 50 and 2000 friends, who list a spouse or
relationship partner in their profile
• Rank all friends by importance. Attempt to identify romantic
partners
• Measure: Precision of the first position, Pr@1
Performance – Pr@1
• How well does dispersion predict the “significant other”? –
precision of the top-ranked person in the individual’s egonet
– Beats others measures of interaction between users
• viewing of profiles, sending of messages, and co-presence at
events (photos)
Performance as a function of neighborhood size
• Performance is best when the neighborhood size is around
100 nodes (56%), & drops moderately (to 33%) as the egonet
size increases by an order of magnitude to 1000
• Interaction features are better for larger neighborhoods, due
to users with larger neighborhoods being more active
Performance as a function of user’s time on site
Best performance when combining features
• Predict relationship status of users
– Ground truth: 60% of users are in a relationship
• Demographic features (age, gender, country, and time on
site) work better than network-based features (dispersion)
• Best performance combining demographic and network
features
How does performance vary based on age of the
relationship?
Marriage
• Performance of dispersion measures increases as people
approach time of their marriage
Persistence of relationships
• Transition probability from the status ‘in a relationship’ to the
status ‘single’ over a 60-day period. The transition
probabilities decrease monotonically, and by significant
factors, for users with high normalized or recursive dispersion
to their respective partners.
Summary
• Graph structure contains information predictive of individual
relationships
– New collaborations
– Romantic partnerships
• In many cases, graph-based algorithms outperform featurebased machine learning algorithms
• These suggest complex interactions between personal
relationships and global network structure