on Netwo'k k fF
Notes on Networks: Perron-Flobenius
Definition A matrix ,4 is nonnegative, written A ) 0, if its entries are all nonnegative real
numbers. ,4 is positive, -4 > 0, of its entries are all positive.
Notes and Homework
Let .4 be an n x n matrix. Recall the follovring definitions:
1. A nonzero vector v is an eigenvector with corresponding eigenvalue
A is diagonalizable if there is an invertible matrix P and a di
Ifomework 2. Suppose v1 and
Show that cfw_rr, rr i" a i"qufly
v2 &ro eigenvectors of -4 with eigenvalues A1 and 42, where lr # Iz.
What is the general statement here (when there
are more ttran 2 eigenvectors)? Can you prove it?
where c is a nonze
Shonr all work to receive fuIl credit!
1 State whether each of the following is tlue or Fal,se. If the statement
reason why, and if it's falre, provide a counterom,mple.
b true, give a brief
1. If .lcfw_ is auy matrix, then
6ym*,etric = orfho?onal'ly d;^1urta,cfw_i wble.
\'e're skipping the details of the proof, which
the network and follow it to a vertex
Consider the following problem: Randomly
at one end. What is the probability that the vertex you reach has degree a? Call this probability
total edge-ends. There are
To compute q6, think
for i in
net I i]:[len(net
]), net Ii ]l
I.cfw_ow, the degree of vertex 2 is:
net  
and the list of neighbors is
and the first neighbor of vertex two is
net  t1l tOl
we added an edge to the graph, say by connecting a new ve
t+u/ + 7.
on FR gr*p*
Notes and Homework
Let n be the number of nodes in our Erdos-R6nyi random network and let p be the fixed probability
of an edge existing between any pair of nodes i, arrd j. The probability dist
SUo* all ruork to receive fuIl creditl
Problem 1 Let A be the adjaceney mstrix of an undirmted network aad let e be
the rrector of all
ls. Give expremions for the ftrllorving in terms of A and e.
(a) The vector rvhase components ar
that for large n, the number of triangles in G(n, p) is approximately
(b) Show that for large n. the number of groups of 3 vertices with at least two of them connected
by edges is $nc2. Such a group of vertices is called a connected triple.
Problem 3 Cousider
the directed netwtrrk:
(a) Give the adjacency matrix for the networlt'
tz 3t r
o t looo
fer the internet' GiYe the Google
Assume that the aeiumrk above is a nrodel
netwtlrk' You c
tn - -1
But then exponentiating and writing S : (1_: llgives:
This equation can be graphed for different values of c, the
lVotu 0n. l,etu,or(
Definition A network is a collection of vertices, or nodes, connected by edges.
Definition The degree of a vertex is the number of edges that connect to that vertex.
Definition A network is directed if edges
i' rL "tJo*3 '"u"['i x "cfw_
A; f t
wcto,r Pcfw_to v
rcfw_ ;S *L*o ,$ A
rt Li-=o "*l is
o*cfw_'*rwix cfw_ uoJd* &*J \ nrodc tod'i'"+"rl
I r t- -ltt
Notes and Homework
4e"oUt"m 1. Let A be a matrix. Show that
the eigenvalues of AT
A areall non-negative.
n x n), it's orthogonally diagonalizable. 'Write Ar A :
QLQT where Q is an orthogonai matrix whos
That nreans p, if it exists, is an eigenve ctar of AD-1 with eigenvalue 1. If we rezrrrange this equation
where 1 is the n x
n identity matrix. Manipulating further we can write:
- AD-\p: (DD-r - AD-')p: (D - A)D-'p: LD-tp: o.
Math 345: Quiz I
all work to receive &rll credit!
Let.4 be an fl. x t& matrix, B be an rn x n matrix, ffid C be an ?x x t?t matrix. Let
f be an n x 1 vector and fi be an nr x L vector. Which of the following expressions
Math 345: Quiz 4
Show all work to receive
Problem 1 Let G(n,p) be a Poisson random graph u'ith n vertices and probabitity
p that two vertices are connected.
(a) \Ahat is the average degree of G(100,0.05)?
p(n- 1):.05(gg) :
Problem 3 Suppose .d is a 3 x 7 matrix
hss a SVD
with shgular values 3,2, and 0, and that
e What is therank of d?
r l4hite doum the matrix E.
r What are the eigenvalues of CrC?
Lo o o o d o o J