A <-matrix(c(0, 1, 1,1, 0, 1,1, 1, 0),nrow =3, ncol =3, byrow =TRUE)rownames(A) <-c("Bob","Ann","Steve")colnames(A) <-c("Bob","Ann","Steve")A#> Bob Ann Steve#> Bob 0 1 1#> Ann 1 0 1#> Steve 1 1 0
From Graph Theory to Network Concepts to Descriptives
Social network concepts and their theoretical underpinnings
Span several branches of the social sciences (e.g. sociology, psychology, criminology, economy, etc)
these are crucial for understanding
The presence and absence of ties
How networks emerge and function
Which social processes and phenomena are implied
How to connect posed research question to network research design
Homophily
Actors are more likely to connect to other actors with similar attribute
The homophily principle
The phenomena of attribute similarity producing ties among actors
people’s personal networks are homogeneous with regard to many sociodemographic, behavioral, and intrapersonal characteristics [McPherson et. al.,2001, Link to paper]
Homophily
Actors are more likely to connect to other actors with similar attribute
The homophily principle
The phenomena of attribute similarity producing ties among actors
Also know as assortativity
The opposite: heterophily
Homophily
the tendency for people to connect to others with similar
behavior/beliefs/attributes
the tendency for people to change their
behavior/beliefs/attributes
according to the
behavior/beliefs/attributes
of those that they are tied to
Social Selection vs. Social Influence
can we separate social influence and selection for cross-sectional data?
Social Selection vs. Social Influence
YES!
by logical reasoning (fixed covariates)
for example ethnicity, age, etc
by theory
we can make an a priori choice
but always be ware of confounders!
proximity/social context (foci)
contextual factors
anything that affects both attributes and ties
Assortativity in
data("greys")
Assortativity in
Measures the level of homophily based on some node labeling or values.
Transitivity/Triadic Closure
when looking at three actors A, B and C, if actors A and B are connected, and A and C are connected, then actors B and C are also connected
the friend of my friend is my friend
Triad Census
The triad census consists of a classification of all directed triads into one of 16 different categories. The distribution can be compared against null models to test for the presence of configural biases (e.g., transitivity bias)
In empirical networks, we often observe a tendency towards high transitivity (“the friend of a friend is a friend”)
What do you think is the (local/global) transitivity of greys?
Transitivity
Transitivity
The value itself is not always enough to judge the level of transitivity of a network. We should also check if it deviates significantly from what would be expected by randomness (more on this later in the course)
A special case of transitivity applied to signed networks
Fritz Heider: underpinnings of structural balance theory (1940s)
Dorwin Cartwright & Frank Harary: formalised with graph theory (1950s)
Structural Balance
A triangle is balanced if
all ties are positive (“the friend of a friend is a friend”)
only one tie is positive (“the enemy of my enemy is my friend”)
The remaining configurations are said to be unbalanced.
Structural Balance
Extension: A network is balanced if i.a., it can be partitioned into two vertex subsets, such that intra-group edges are all positive and inter-group edges are all negative.
Degree distribution
The degree of a node in a network is the number of connections it has to other nodes.
The degree distribution is the probability distribution of the degrees over the whole network.
Empirical degree distributions are generally right skewed:
(many nodes have a few connections and few have many)
“preferential attachment”
“Matthew effect”
“the rich get richer”
Scale Free Networks
A scale-free network is a network whose degree distribution follows a power law (asymptotically). The fraction \(P(k)\) of nodes in the network having degree \(k\) is given by \(P(k) \sim k^\boldsymbol{-\gamma}\)
Degree distribution
Degree distribution are informative about
homogeneity/heterogeneity
the extent to which all actors have similar or dissimilar degrees
centralization
the extent to which the network is dominated by a single actor
(the extent to which a network looks like a star network)
Degree distribution in
er <-sample_gnp(n =5000, p =0.01)pa <-sample_pa(n =5000, power =1.5, m =2, directed =FALSE)
plot(degree_distribution(er))
plot(degree_distribution(pa),log ="xy")
Density in
The density of a network is defined as the fraction of the potential edges in a network that are actually present.
One can find a short chain of acquaintances (i.e. a shortest path), often of no more than a handful of individuals, connecting almost any two people on the planet.
Milgram’s small-world experiment: 6 (“6 degrees of separation”)
Average distance between users on Facebook (2016): 4.57
Paul Erdős published ~1525 papers with ~500 collaborators
The Erdős number of an individual is their distance to Paul Erdős in the co-authorship network
Bacon number
The Bacon number of an actor is their distance to Kevin Bacon in the movie co-appearance network.
The Zlatan number is the distance of football players to Zlatan Ibrahimovic in the “squad network” (blog post)
Diameter
The length of the longest shortest path is called the diameter of the network.
diameter(greys)#> [1] 8
Shortest Paths in
A shortest path is a path that connects two nodes in a network with a minimal number of edges. The length of a shortest path is called the distance between two nodes.
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Social Selection vs. Social Influence
can we separate social influence and selection for cross-sectional data?