Introduction to Matrices and Statistics in SNA

Laura L. HansenDepartment of Sociology

UMBSNA WorkshopJuly 31, 2008

(SOURCE: Introduction to Social Network Methods, Robert A. Hanneman and Mark Riddle, http://faculty.ucr.edu/~hanneman/nettext/)

When to Use Matrices

• Smaller networks can be visually represented in graphs. However, when there are a large number of nodes (actors, relationships, etc.) in the network, it is more useful to start with a matrix.

• Matrices are also useful tools in order to use math and statistics in SNA.

What exactly is a matrix?

• Rectangular arrangement of a set of elements.• An “l by j” matrix has l rows, j columns• The elements (cells) are identified by their addresses.• Example: 3 by 6 Matrix – element 1,1 is the entry in

the first row and first column.

1, 1 1, 2 1, 3 1, 4 1, 5 1, 6

2, 1 2, 2 2, 3 2, 4 2, 5 2, 6

3, 1 3, 2 3, 3 3, 4 3, 5 3, 6

Vectors

• A matrix consisting of a single row is called a row vector.

A = ( 1 2 3)

• A matrix consisting of a single column is called a column vector.

B = 1

2

3

Diagonals in Matrices

• There are two ways of representing the diagonal in a matrix (Actor A by Actor A)

  Pacey Theo Laura

Pacey 0 1 1

Theo 1 0 1

Laura 1 1 0

  Pacey Theo Laura

Pacey - 1 1

Theo 1 - 1

Laura 1 1 -

Types of Matrices

• Matrices are generally square (i.e. 4 by 4) but can also be rectangular as in the case of row and column vectors.

• The 3-dimensional matrix include rows, columns and levels. Each level has the same rows and columns as each other level.

The “adjacency” Matrix

• Most common form of matrix in SNA is a very simple square matrix with as many rows and columns as there are actors or things.

• An adjacency matrix can be symmetric or asymmetric.

• The relationships are binary: 0 if there is no relationship, 1 if there is a relationship.

Symmetric Adjacency Matrix

• Everyone likes everyone else.   Pacey Theo Laura

Pacey - 1 1

Theo 1 - 1

Laura 1 1 -

Pacey

Theo Laura

Asymmetric Adjacency Matrix

• Not everyone likes everyone else.

  Red Sox Yankees Padres Patriots Giants Chargers

Red Sox - 0 1 1 0 1

Yankees 0 - 0 0 1 0

Padres 1 1 - 1 1 1

Patriots 1 0 1 - 0 0

Giants 0 1 1 0 - 0

Chargers 1 1 1 1 1 -

Asymmetric Graph

RS

Y C

G P

PATS

Transposing a Matrix

• Rows and columns get exchanged; i becomes j and j becomes i.

• Example: Transpose of a directed adjacency matrix will show the degree of reciprocity of ties (directed graph).

The Math of Matrices

• Addition/Subtraction: add and subtract each element of two or more matrices. Used when you wish to reduce the complexity of multiple relationships in matrices.

• Multiplication: Unusual, but useful tool. Matrices have to be “conformable” – the number of rows in the first matrix has to be equal to the number of columns in the second. Note: X*Y is not the same as Y*X. The order of the matrices matter.

Statistical Tools in SNA

• Distinctions between “orthodox” statistics and statistics in SNA:

- The relationship is between actors, not attribute variables.

- Standard statistical computations, including estimated standard errors,

probability doesn’t work with network data.

- Observations are not on independent samplings of the population.

Descriptive Statistics• 1 2 3• WEALTH #PRIORS #TIES• --------- --------- ---------• 1 Mean 42.563 25.938 13.875 (proportion of possible

ties)• 2 Std Dev 35.704 25.071 13.527 • 3 Sum 681.000 415.000 222.000 (best with valued data)• 4 Variance 1274.746 628.559 182.984• 5 SSQ 49381.000 20821.000 6008.000• 6 MCSSQ 20395.938 10056.938 2927.750• 7 Euc Norm 222.218 144.295 77.511 (sq rt of the sums of

square)• 8 Minimum 3.000 0.000 1.000• 9 Maximum 146.000 74.000 54.000• 10N of Obs 16.000 16.000 16.000

(UCINET6, Padgett Data, Tools>Statistics>Univariate)NOTE: One statistic not given in UCINET6 – coefficient of variation (standard

deviation/means times 100)

Hypothesis Testing

EXAMPLE:

If we want to test ego density (ratio of the degree of an actor to the maximum number of ties possible), and we propose that all egos in a network will communicate with every other ego in the network, we would expect the density to be 1.0.

Bivariate Statistics: Correlation Between Two Networks with the Same Actors

• Do the patterns of ties for one relation among a set of actors align with the patterns of ties for another relation among the same actors?

• You can compare a number of network measures between matrices, including density and centrality. You can also test for a difference between tie strengths of two relations.

QAP: The Quadratic Assignment Procedure

Useful in helping to get around the issue of non-independent observations in SNA.

Examples:How does level of trade between countries vary as

a function of language similarity, etc.?Do companies with overlapping board of directors

tend to perform similarly on the stock market?Are people more likely to be friends if they share

similar characteristics, such as being about the same age?

(Source: William Simpson, Harvard Business School, http://fmwww.bc.edu/RePEc/nasug2001/simpson.pdf)