Graphs and Matrices Spring 2012 Mills College Dan Ryan Lecture Slides by Dan Ryan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.Dan RyanCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License

Four Representations of a Graph G={V,E} NODE List A B D E B A C D C A B D D A B C E A EDGE List AB AD AE BA BC BD CA CB CD DA DB DC EA A C B D E MATRIX

Matrix ROWSCOLUMNS A matrix ELEMENT The MAIN DIAGONAL SQUARE MATRIX RECTANGULAR MATRIX

Nomenclature: nrows by mcols x2 matrix 2x4 matrix 3x1 matrix 3x3 matrix

Matrix

Sometimes we say “Vector” Each row (or column) in a matrix can be thought of as an ordered set of numbers describing an object. For a graph matrix, a row or a column is a “vector” of a vertex’s connections The fourth vertex is connected to the first, second, and third, but not the fifth

Transpose The transpose of a matrix is a swapping of its rows and columns

Transposing a Matrix transpose i j

Matrix Arithmetic

Example: Convert 2-Mode to 1-Mode(s) 1.Write the rectangular incidence matrix I A 123 BDC ABCD GROUPS PEOPLE A “groups by people” matrix

Example: Convert 2-Mode to 1-Mode(s) 2.Compute transpose of I, I T ABCD A101 B011 C111 D101 A “groups by people” matrix A “people by groups” matrix

When we multiply matrices… Columns of first factor = rows of second factor (People x Groups) X (Groups x People) = (People x People) (Groups x People) X (People x Groups) = (Groups x Groups) I T I A GxG I T I A PxP =x = x

Degree

Mean(average) Degree

Degree and Edges

Adjacency Matrix

Matrix Arithmetic See also:

Directed Unweighted Undirected Weighted Undirected Unweighted Directed Weighted dichotomize symmetrize E A B C D E A B C D E A B C D E A B C D

Paths If there is a path from vertex A to vertex B… …then there is a sequence of edges connecting them E.g., if there is a 2-path from A to C then there is some vertex B such that there is an edge AB and an edge BC AC B Path A to C?

Put another way An edge from vertex j to vertex i means A ij =1 If there is a 2-path from j to i then there is a k such that A ik =1 and A kj =1. …i…j…k…  i  j  k  j i k

A ik =1 and A kj =1 That means: if there is a path from vertex j to vertex k to vertex i THEN there is a k such that A ik A kj =1 …i…j…k…  i  j  k  j i k a b c There is an edge from k to i There is an edge from j to k

Another way to see it… If vertex j is connected to vertex k AND Vertex I is connected to k THEN Row I will have a 1 in position k AND Column j will have a 1 in position k

Elements of Product Matrix are Vector Products = X i j A 2 ij is how many times the row version and the column version have 1s in the same place

…i…j…k…  i  j  k  Are there other paths from j to i? j i k

Look for vertices between j and i …i…j………  i  j    j i k

Example: Convert 2-Mode to 1-Mode(s) 2.Compute transpose of I, I T A 123 BDC ABCD

Degree Distribution “Distribution of X” = pattern of different values X takes on More specifically: the frequency of each value || 2 |||| ||| 3 |||| 4 |||| | 5 |||

Diagonal Elements of A 2 are Vertex Degree A= A2=A2=

Why? = A2A A X A Degree equals number of “out and back” paths of length 2

How about A 3 ? = A3A X A A2A2

What do these represent? = A3A X A A2A2

PRACTICE:COMPUTE A 2 FROM WXYZ W -100 TO X 0-11 Y 10-0 Z 0010 A W Z X Y FROM WXYZ W 0011 TO X 1010 Y 0100 Z 1000 A2A2

PRACTICE:COMPUTE A 3 FROM WXYZ W -100 TO X 0-11 Y 10-0 Z 0010 A W Z X Y FROM WXYZ W 0011 TO X 1010 Y 0100 Z 1000 A2A2 FROM WXYZ W 1010 TO X 1100 Y 0011 Z 0100 A3A3

…i…j…k…  i  j  k  Are there other paths from j to i? j i k