Social Network Analysis
KSE 652 Social Computing Systems:
Design and Analysis
Uichin Lee
Oct. 16, 2013
Network
• Networks are a representation of interaction structure among units.
– In the case of social and economic networks, these units (nodes) are
individuals or firms.
• At some broad level, the study of networks can encompass the
study of all kinds of interactions.
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Information transmission.
Web links.
Information exchange.
Trade.
Credit and financial flows.
Friendship.
Trust.
Spread of epidemics.
Diffusion of ideas and innovation.
Themes
• Graph theory (e.g., strong/weak ties, 6 degrees of
separation, structural balance)
• Game theory (e.g., group interaction modeling,
auction)
• Markets and strategic interaction on networks
• Information network (e.g., web, blogging, search
engine)
• Network dynamics
– Population effects (social practice/network effects)
– Structural effects (social contagion/cascading effects)
Visual Example – 1
The social network of friendships within a 34-person karate club
provides clues to the fault lines that eventually split the club apart
(Zachary, 1977)
Zachary, Wayne W. "An Information Flow Model for Conflict and Fission in Small Groups."
Journal of Anthropological Research 33, no. 4 (1977): 452-473.
Visual Example – 2
The network of loans among financial institutions can be used to
analyze the roles that different participants play in the financial
system, and how the interactions among these roles affect the
health of individual participants and the system as a whole (Bech
and Atalay 2008)
The Topology of the Federal Funds Market." European Central Bank Working Paper Series No. 986, December 2008
Visual Example – 3
The network structure of political blogs prior to the 2004 U.S.
Presidential election reveals two natural and well-separated
clusters (Adamic and Glance, 2005)
"The Political Blogosphere and the 2004 U.S. Election: Divided They Blog.“ KDD 2005
Visual Example – 4
In a network representing international
trade, one can look for countries that
occupy powerful positions and derive
economic benefits from these
positions
In some settings, such as this map of
Medieval trade routes, physical
networks constrain the patterns of
interaction, giving certain participants
an intrinsic economic advantage based
on their network position.
Visual Example – 5
The spread of an epidemic disease (such as the tuberculosis
outbreak shown here) is a form of cascading behavior in a network.
The similarities and contrasts between biological and social
contagion lead to interesting research questions. (Andre et al. 2007)
Transmission Network Analysis to Complement Routine Tuberculosis Contact Investigations." American J. of Public Health 97
Visual Example – 6
Social networks based on communication and interaction can be
constructed from the traces left by on-line data. In this case, the
pattern of email communication among 436 employees of HP
Research Lab is superimposed on the official organizational hierarchy
Visual Example – 7
• When people are influenced by the behaviors of their neighbors in
the network, the adoption of a new product or innovation can
cascade through the network structure.
• Here, e-mail recommendations for a Japanese graphic novel spread
in a kind of informational or social contagion. (Leskovec 2007)
"The Dynamics of Viral Marketing." ACM Transactions on the Web 2007
Graphs
• Graph: a way of specifying relationships among a
collection of items
• A graph consists of a set of objects (called nodes); and
these objects are connected by links (called edges)
Two graph types: (a) an undirected graph, and (b) a directed graph.
Graphs
• Path: a sequence of nodes with the property that each
consecutive pair in the sequence is connected by an
edge
• Cycle: a path with at least three edges in which the first
and last nodes are the same, but otherwise all the
nodes are distinct
13-node Internet graph from December 1970
Graphs
• Connectivity: a graph is connected if for every
pair of nodes, there is a path between them
• A directed graph is called weakly connected if
replacing all of its directed edges with undirected
edges produces a connected (undirected) graph
Graphs
• Connected component of a graph is a subset of the
nodes such that (1) every node in the subset has a path
to every other; and (2) the subset is not part of some
larger set with the property that every node can reach
every other
• Giant component contains a large fraction of nodes
Collaboration graph of
biological research center SGPP
Connectivity of directed graphs
Giant weakly connected component (GWCC)
Disconnected components (DC)
Giant in-component (GIN),
contains all vertices from
which the GSCC is
approachable
GSCC: Giant strongly connected
component: the set of vertices reachable
from its every vertex by a directed path
Giant strongly connected component of directed networks,
http://arxiv.org/pdf/cond-mat/0103629.pdf
Giant out-component
(GOUT), the set of
vertices approachable
from the GSCC by a
directed path
Tendrils: the rest of the
GWCC, i.e., the vertices which
have no access to the GSCC
and are not reachable from it
Graphs
• The Small-world phenomenon (aka. six degrees of separation);
world looks “small” when you think of how short a path of friends it
takes to get from you to almost anyone else
• Erdos number (distance to Paul Erdos); Bacon number (distance to
Kevin Bacon)
A histogram from Travers and
Milgram’s paper on their small-world
experiment. For each possible length
(labeled “number of intermediaries”
on the x-axis), the plot shows the
number of successfully completed
chains of that length. In total, 64
chains reached the target person,
with a median length of six.
Graphs
• Small-world network: high clustering and low diameter?
ring network
“Collective Dynamics of ‘Small-World’ Networks,” Watts and Strogatz, Nature (1998)
Strong and Weak Ties
• Granovetter’s work on “strength of weak ties”
• Issues:
– How does information flow through a social
network?
– How can different nodes play structurally distinct
roles in this process?
– How these structural considerations shape the
evolution of the network itself over time?
Triadic Closure
• If two people in a social network have a friend in common,
then there is an increased likelihood that they will become
friends themselves at some point in the future
• Why? Opportunity of meetings, trust of relationship,
incentive (of removing latent stress of managing different
circles)
The formation of the edge between B and C illustrates the effects of
triadic closure, since they have a common neighbor A.
Clustering Coefficient
• The clustering coefficient of a node A is defined
as the probability that two randomly selected
friends of A are friends with each other.
A: 1/6
A: 1/2
Bridges and Local Bridges
• A bridging edge is an edge whose removal causes its
endpoints to lie in two different components
• Local bridge: if endpoints have no friends in common
– i.e., if removal will increase the distance between two endpoints
– this distance (after removal of a local bridge) is called the span
of a local bridge
bridging edge
local bridge
Strong Triadic Closure
• Each edge of the social network is labeled here as either a strong tie
(S) or a weak tie (W), to indicate the strength of the relationship.
• The labeling in the figure satisfies the Strong Triadic Closure
Property at each node: if the node has strong ties to two neighbors,
then these neighbors must have at least a weak tie between them
Tie Strength and Network Structure
• How to measure tie-strength?
• Who-talks-whom network (cellphone; Onnela 2007):
total minutes spent on phone calls between the two
ends of the edge
• Neighborhood overlap of an edge connecting A & B
– i.e., (# nodes who are neighbors of both A and B) /
(# nodes who are neighbors of at least one of A or B)
– The strength of weak ties predicts that neighborhood
overlap should grow as tie strength grows
Structure and tie strengths in mobile communication networks. J.-P. Onnela, J. Saramaki, J. Hyvonen, G.
Szabo, D. Lazer, K. Kaski, J. Kertesz, and A.-L. Barabasi. Proc. Natl. Acad. Sci. USA, 104:7332–7336, 2007.
Tie Strength and Network Structure
• Neighborhood overlap ratio of A-F
Tie Strength on Facebook
• Marlow et al.’s analysis on friendship links
• Link categories: mutual, one-way communications
(ever communicated?), maintained relationships
(viewing profile or newsfeeds?)
Four different views of a Facebook user’s network neighborhood
Cameron Marlow, Lee Byron, Tom Lento, and Itamar Rosenn. Maintained relationships on Facebook, 2009.
Tie Strength on Facebook
• The number of links corresponding to maintained relationships,
one-way communication, and reciprocal communication as a
function of the total neighborhood size for users on Facebook
Cameron Marlow, Lee Byron, Tom Lento, and Itamar Rosenn. Maintained relationships on Facebook, 2009.
Tie Strength on Facebook
Self-report tie-strength
(of each friend)
Usage data
Eric Gilbert and Karrie Karahalios , Predicting Tie Strength With Social Media CHI 2009
number of friends
Tie Strength on Twitter
number of followees
The total number of a user’s strong ties (# friends) as a function of the number of
followees he or she has on Twitter
Bernardo A. Huberman, Daniel M. Romero, and Fang Wu.
Social networks that matter: Twitter under the microscope. First Monday, 14(1), January 2009.
Embeddedness
• Embeddedness of an edge in a network: the number of
common neighbors shared by the two endpoints
• Sociology research: if two individuals are connected by an
embedded edge (i.e., embeddedness > 0), it is easier for
them to trust one another
Structural Holes
• Contrast between densely-knit groups and boundaryspanning links is reflected in the different positions of
nodes A and B in the underlying social network (B: local
bridge)
B’s position offers advantages:
• Info access
• Amplifier for creativity
• Social gatekeeping
Social Capital
• Somewhat difficult to define, but “Consensus is growing in
the literature that social capital stands for the ability of
actors to secure benefits by virtue of membership in social
networks/social structure” (Alejandro Porte 1998)
• Just as a screwdriver (physical capital) or a university
education (cultural capital or human capital) can increase
productivity (both individual and collective), so do social
contacts affect the productivity of individuals and groups
(Putnam 2000)
• Social capital as a tension between closure (groups) and
brokerage (interactions) [Burt 2000], relating to “bonding”
and “bridging” social capital, respectively [Putnam 2000]
Alejandro Portes. Social capital: Its origins and applications in modern sociology. Annual Review of Sociology, 24:1–24, 1998.
Stephen P. Borgatti, Candace Jones, and Martin G. Everett. Network measures of social capital. Connections, 1998.
Ronald S. Burt. The network structure of social capital. Research in Organizational Studies, 22:345–423, 2000.
Robert D. Putnam. Bowling Alone: The Collapse and Revival of American Community. Simon & Schuster, 2000.
Betweenness Measure
and Graph Partitioning
• Betweenness of an edge: total amount of flow it carries,
counting flow between all pairs of nodes using this edge
• Flow between A and B divides itself evenly along all the
possible shortest paths from A to B (if there are k shortest
paths from A and B, then 1/k units of flow pass along each
one
Betweenness Measure
and Graph Partitioning
• Breaking a network down into a set of tightly-knit regions
with sparser interconnections between the regions
• General approaches to graph partitioning
– Divisive: identifying/removing “spanning links” between
densely-connected regions
– Agglomerative: gluing nodes together into regions (bottom up)
• Girvan and Newman’s divisive method (greedily removing
an edge with highest edge betweeness)
Betweenness Measure
and Graph Partitioning
Homophily
• Homophily
– Tendency of being similar to our friends; birds of a feather
flock together!
• Understanding possible link formation:
– Intrinsic (structural) factors: a friendship formed as two
people are introduced through a common friend
• Related to “triadic closure” properties; increased opportunities,
trust of relationship, incentives of facilitating relationship
– Contextual factors (or social context): a friendship formed
as two people attended the same school or work for the
same company
• How a network’s surrounding contexts can drive the formation of
its links?
Homophily
Homophily can produce a division of a social network into densely-connected,
homogeneous parts that are weakly connected to each other. In this social network from a
town’s middle school and high school, two such divisions in the network are apparent: one
based on race (with students of different races drawn as differently colored circles), and
the other based on friendships in the middle and high schools respectively
Race, school integration, and friendship segregation in America. American Journal of Sociology, 2001
Measuring Homophily
• Testing homophily (i.e., testing the change of random mixing)
– In a network with no homophily, friendships are being formed as if
there were random mixing across the given characteristics
• Example: a network with p% of males, q% of females (p+q=1)
– If the fraction of cross-gender edges is significantly less than 2pq, then
there is evidence for homophily
Using a numerical measure, one can
determine whether small networks such
as this one (with nodes divided into two
types) exhibit homophily:
p = 2/3, q = 1/3
2pq = 4/9 = 8/18
=> 8 cross gender edges vs.
5 cross gender edges (reality)
=> This shows some evidence of homophily
Mechanisms Underlying Homophily
• Selection: individual characteristics drive the
formation of links
• Social influence: existing links in the network
serve to shape people’s (mutable) characteristics
• Interplay of selection and social influence
– Ex) teenage friends: teenagers seek out social circles
composed of people like them + peer pressure causes
them to conform to behavioral patterns within their
social circles
Nicholas A. Christakis and James H. Fowler. The spread of obesity in a large social network over 32 years.
New England Journal of Medicine, 357(4):3700–379, July 2007.
Homophily and Health
• Longitudinal study (32 years) of ~12K people,
– Obesity status: BMI (body mass index) = weight /
height^2
– Social network structure: list of friends, siblings,
spouse, neighbors
• Results: non-obese people clustered in the
network in a fashion consistent with homophily
• Why? Selection (friending w/ similar obesity
status), social influence (i.e., behavioral influence
among friends), confounding effects (homophily
vs. other factors than obesity)
Homophily and Spatial Segregation
• Global patterns of spatial segregation can arise
from the effect of homophily operating at a local
level (Schelling model)
The tendency of people to live in
racially homogeneous
neighborhoods produces spatial
patterns of segregation that are
apparent both in everyday life
and when superimposed on a
map — as here, in these maps of
Chicago from 1940 and 1960. In
blocks colored yellow and orange
the percentage of AfricanAmericans is below 25, while in
blocks colored brown and black
the percentage is above 75.
Chicago, 1940
Chicago, 1960
Markus M. M¨obius and Tanya S. Rosenblat. The process of ghetto formation: Evidence from Chicago, 2001.
Affiliation
• Being part of a group (e.g., company, organization);
frequenting a particular place; pursuing a particular hobby
or interest
• These are all activities that (when shared between two
people) tend to increase the likelihood that they will
interact and hence form a link in the social network
• Such activities are called “foci”, i.e., focal points of social
interaction, constituting “social, psychological, legal, or
physical entities around which joint activities are organized
(e.g., workplaces, voluntary organizations, hangouts, etc.)”
Affiliation Network
• An affiliation network is a bipartite graph that
shows which individuals are affiliated with which
groups or activities.
• Here, Anna participates in both of the social foci
on the right, while Daniel participates in only one.
Social Affiliation Network
• A social-affiliation network shows both the friendships
between people and their affiliation with different social
foci.
• (co-evolution) structural pattern of memberships can reveal
subtleties in the interactions among both members and
groups
Social Affiliation Network
• Closure process: closing the third edge of a
triangle in the network
Triadic closure
(selection; forming
links to others who
Focus
Focal closure share characteristics
with you)
(social influence)
Membership closure
Tracking Closure Process Online
• We can address these questions empirically using
network data as follows.
– (i) We take two snapshots of the network at different
times.
– (ii) For each k, we identify all pairs of nodes who have
exactly k friends in common in the first snapshot, but who
are not directly connected by an edge.
– (iii) We define T(k) to be the fraction of these pairs that
have formed an edge by the time of the second snapshot.
This is our empirical estimate for the probability that a link
will form between two people with k friends in common.
– (iv) We plot T(k) as a function of k to illustrate the effect of
common friends on the formation of links
Tracking Closure Process Online
• T(0) is the rate at which link formation happens
when it does not close a triangle, while the values
of T(k) for larger k determine the rate at which
link formation happens when it does close a
triangle.
• Comparison between T(0) and these other values
addresses the most basic question about the
power of triadic closure.
Tracking Closure Process Online:
Triadic closure
Quantifying the effects
of triadic closure in an
e-mail dataset
T(k) = 1-(1-p)^k
T’(k) = 1-(1-p)^k-1
Number of common friends
Suppose that for some small probability p, each common friend that two people have
gives them an independent probability p of forming a link. So if two people have k
friends in common, the probability they fail to form a link on any given day is (1−p)^k
Empirical analysis of an evolving social network. Science, 311:88–90, 2006.
Tracking Closure Process Online:
Focal closure
Quantifying the effects of
focal closure in an e-mail
dataset
# common foci (=# of common classes)
Empirical analysis of an evolving social network. Science, 311:88–90, 2006.
Prob. of joining a community
Tracking Closure Process:
membership closure
# friends who are already member
Quantifying the effects of membership closure in a large online dataset: The plot shows
the probability of joining a LiveJournal community as a function of the number of friends
who are already members. Friendships are designated by users in their profiles, and foci
correspond to membership in user-defined LiveJournal communities
Group formation in large social networks: Membership, growth, and evolution, KDD 2006
Tracking Closure Process: Interplay
• Similarity in Wikipedia: if editor A has edited the Wikipedia articles
on Ithaca NY and Cornell University, and editor B has edited the
articles on Cornell University and Stanford University, then their
similarity under this measure is 1/3, since they have jointly edited
one article (Cornell) out of three that they have edited in total
(Cornell, Ithaca, and Stanford).
Similarity
The average similarity of two editors
on Wikipedia, relative to the time (0)
at which they first communicated.
Each unit corresponds to a single
Wikipedia action taken by either of
the two editors. The curve increases
both before and after the first contact
at time 0, indicating that both
selection and social influence play a
role; the increase in similarity is
steepest just before time 0.
# of edits after first communication
Feedback effects between similarity and social influence in online communities, KDD 2008
Structural Balance
• Balanced if a complete graph is divided into two
sets of mutual friends
• Applications: international relationships (politics)
mutual
antagonism
between
all sets
If a complete graph can be divided into two
sets of mutual friends, with complete
mutual antagonism between the two sets,
then it is balanced. Further, this is the only
way for a complete graph to be balanced.
A complete graph is weakly balanced precisely
when it can be divided into multiple sets of
mutual friends, with complete mutual
antagonism between each pair of sets.
Summary
• Graphs: connectivity, component, giant
component, small-world phenomenon
• Strong/weak ties: triadic closure, clustering
coefficients, bridges/local bridges, strong triadic
closure, social capital, graph partitioning
(betweenness)
• Homophily: testing, underlying mechanisms,
implications (health, spatial segregation)
• Affiliation: social affiliation network; closure
process
• Structural balance