Victor Lee

What are Social Networks?

Role and Position Analysis

Equivalence Models for Roles

Block Modelling

Not just Facebook and MySpace… A social network is a collection of actors

who are joined by pairwise ties. Actors may be individual persons or organizations

Ties are any type of relationship between two actors, such as friendship, kinship, financial exchange, influence, or prestige.

As a Graph Actor vertex or node Tie edge or link Could be directed/undirected,

weighted/unweighted

As an Adjacency Matrix N x N matrix, N = number of vertices aij = weight of edge from i j

11

22

44

33

0100

0001

0100

0110

Social Position = collection of actors similarly embedded in a network Similar sets of ties to other actors Not based on adjacency, proximity, or reachability

Example: Nurses at different hospitals occupy the position nurse, even though they don't work with each other, or even the same doctors or patients.

Social Role = pattern of relationships that an actor has with other actors May include both direct and compound relations

Example of Kinship Relations: combinations of relations marriage and descent. Direct: sister, husband, son Combination: sister-in-law, uncle, grandson

Position is a grouping;Role is what characterizes the group

Positional analysis Separate actors into subsets of positions

Partitioning Each position is a mathematical equivalence class

Role Analysis Discover and describe patterns of relationships

Pattern Mining or Motif Discovery Global: look at the full set of edges Local: look at the neighborhood of an individual

Positional, then Rolea. Group actors into equivalence classes

(positions)b. Describe each position with an aggregate role

description.

Role, then Positionala. Describe the relationships of each individualb. Group actors that have equivalent or similar

patterns or relations

An equivalence class C is a set of ordered pairs in which the following properties hold: Reflexivity: (a,a) ∈ C Symmetry: if (a,b) ∈ C, then (b,a) ∈ C Transitivity: if (a,b) and (b,c) ∈ C, then (a,c) ∈

C

An equivalence relation for set A partitions A into equivalence classes {C1, …, Ck}

Goal: define a rule-based equivalence relation that will partition a set of actors into positions and roles

Three common definitions: Structural Equivalence

Automorphic Equivalence

Regular Equivalence

Two actors are structurally equivalent if they have identical ties to and from the other actors (Lorrain and White 1971).

Example: C1 = {1, 4} C2 = {2,5} C3 = {3}

1 4

2 5

3

00100

10010

00000

00100

10010

Two actors are automorphically equivalent if they have identical ties to equivalent actors. Must have same number of ties Recursive definition

If we color the vertices by position, Vertices are equivalent if their neighborhoods

consist of the same number of the same colors Example: Two families with exactly the

same number of children, parents, etc.

A graph isomorphism of graphs G and H is a bijective mapping of vertices f(V(G)) V(H) such that all the edges remain the same That is, e(a,b) ∈ G if and only if e(f(a),f(b)) ∈ H

A graph automorphism is when the isomorphism is to G itself That is, we rearrange the vertex labels of G

Example: Every possible labeling of a clique is automorphically equivalent What about a ring? A binary tree?

Two actors are regularly equivalent if they have ties to equivalent positions. Need not have the same number of ties Recursive definition

If we color the vertices by position, Vertices are equivalent if their neighborhoods

consist of the same set of colors But the quantity of each color does not matter

Example: Two families with different numbers of children, parents, etc.

Structural: connect to exactly the same neighbors {5,6}, {8,9}, singletons

Automorphic: connect to the same distribution of colors {5,6,8,9}, {2,4}, singletons

Regular: connect to the same colors {5,6,7,8,9}, {2,3,4}, {1}

1

2 3 4

985 6 7

Consider the matrix representations Assume we arrange rows and columns by

equivalence class, creating blocks What do you notice about each block? What rule would each type of equivalence follow?

If we can make a simple statement about each block: “Every relation within the block is (almost) always

1.” “Every relation within the block is (almost) always

0.” We can form an image matrix/reduced

graph

In automorphic equivalence, we expect to see the same number of 1’s in each row and each column within a block “Every relation within block Bij has a probability pij

of being present.” In-relations can be different from out-relations, so

Bij may be different from Bji

How would you construct a block model based on regular equivalence?

What if we have weighted edges?

Exact equivalence is often too strict

Also want to know how similar are two actors

We can then cluster together similar actors

Euclidean distance Each vertex or group is a dimension in space Distance between row vectors & column vectors of

two actors:

Correlation (Pearson product-moment) If two actors are equivalent, the correlation

between their rows and columns will be +1

Hierarchical Clustering

CONCOR Iteratively: compute

row & column correlations,then split

PCA might be superior

Relational Algebra Given multiple types of edges (say, Friend and Trust) Form an image matrix for each edge type (F and T) Image multiplication? Example: R = F x T

rxy means “x has a friend z who trusts y”

Optimal Partitioning? Exact equivalence clear idea of “best” partition Using similarity trade-off between tight variance

with a block and having a small number of blocks Other tools and methods to compute

similarity and to partition?

These slides are based onWasserman and Faust (2004), Social Network Analysis: Methods and Applications, Ch. 9 – 12.

Additional Reading Concise online book on social network analysis:

www.faculty.ucr.edu/~hanneman/nettext/index.html

Review paper from the sociologist’s perspective:Borgatti, S. and Everett, M., Notions of Position in Social Network Analysis, Sociological Methodology (22), 1-35, 1992.