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Distributed Systems & Control
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Advanced Topics in Control 2015
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Lecture 5: Lefties & Laplacians
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Brief announcements
1
summary of exercise
session is posted &
prints available here
2
printed copies of
homework solutions
available here
recap: discrete averaging
3
second homework is
posted with due date
Friday March 27
(last lecture, exercise session, & chapters 5)
|
Our main result for discrete-time averaging algorithms
Now we have a full picture of . . .
lecture & exercise by Tyler
(implicitly assume self-loops everywhere)
Main result for discrete-time averaging algorithms
4
3
4
3
Let A be row-stochastic. The following statements are equivalent:
1
2
3
the eigenvalue 1 is simple and all other eigenvalues have magnitude
strictly smaller than 1.
1
2
1
2
limk→∞ Ak = 1w T for some w ≥ 0 satisfying 1T w = 1.
the digraph associated to A contains a globally reachable node and
the subgraph of globally reachable nodes is aperiodic.
In either of the three equivalent cases the solution to x + = Ax satisfies




A=


? ? ? ?
? ? ? ?
? ? ?
?











A=


? ? ? ?
? ? ? ?
? ? ?
?
?







lim`→∞ x(`) = 1w T x0 ,
where wi > 0 if and only if node i is globally reachable.
lim x(`) = 1 · 0 0 0 1 · x0
`→∞
lim x(`) = 1 · ? ? ? ? · x0
`→∞
Organization of today’s lecture
Left Eigenvectors
The Laplacian
Matrix
(weight design &
centrality metrics)
(Chapter 4)
we start off with the weight design
(Chapter 6)
(Sections 5.4 & 5.5 & Exercises 4.6 & 5.1)
Design of weights: the equal-neighbor model
Undirected graph & averaging of neighbors’ values:
xi+ = average xi , {xj , for all neighbor nodes j} .
1/3
1/3
3
4
3
1/3
1
2

1/2
1/4
x(k + 1) = 
 0
0
|
discussion on board
1/3 1/3
1/2
1/4
1/4
1
4
1/3
1/4
1/2
1/4
2

1/2 0
0
1/4 1/4 1/4
 x(k)
1/3 1/3 1/3
1/3 1/3 1/3
{z
}
A˜
Design of weights: the Metropolis–Hastings model

1

,



 1 + max{d(i), d(j)}
X
A˜ij =

1−
A˜ih ,


{i,h}∈E


0,
if {i, j} ∈ E & i 6= j,
if i = j,
otherwise.
5/12
5/12
3
4
3/4
3
1/3
1/4
1/4
1/4
1
1
2

3/4
1/4
A˜ = 
 0
0
1/4

1/4
0
0
1/4 1/4 1/4 

1/4 5/12 1/3 
1/4 1/3 5/12
2
4
discussion on board
Figure 1.
Russian trade routes in the 12th - 13th cenntries.
Russian river trade network
The Medieval River Trade Network of Russia
Revisited
Forrest
R. Pitts
Ur~ivers~ty
286
Q2: was that obvious from the
underlying network graph?
RUSSIAN
Hawa~~~
IN
THE
TRADE
t2-13TH
ROUTES
CENTURIES
Forrest R. Pitts
Medieval trade and communication
centrality & left eigenvectors
Q1: which turned out to be the
most central city in this network?
28.5
Social Networks,1 (1978179) 285-292
0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
along the rivers of Russia are consid-
Figure
routes Two
in the measures
12th - 13thare
cenntries.
ered 1.as aRussian
social trade
network.
presented. An intermediate nodeFigure 2.
Graph ofRussian
trade routes in the 12th - 13th centuries.
occurrence rate (Shimbel’s stress index) provides a measure of centrality.
The short-batch dista~~cesto all other places are su~~~~ledto provide a systemeffort measure of accessibility. Both measures show Moscow to have been
most central and accessible with aggregate least effort.
.~
Thirteen years ago I published (Pitts 1965) a short paper on the medieval
Russian river network (Figure I), in an effort to assess the centrality of the
urban places (Figure 2) on the network. Interest was focused on the light a
network approach would shed on the perennial controversy
concerning the
growth of Moscow.
Many historians have written about Moscow’s gradual rise to a position of
dominance over other towns. Most claim that the growth was a response to
favorable geographical conditions. The Russian historian Kluchevsky ( 19 1 I)
comments:
(Section 5.6)
“The political fortunes of Moscow were closely connected with its geographical position .” (p. 273)
RUSSIAN
TRADE
ROUTES
“Thus Moscow arose at the point of intersection of three great land roads - a geoTHE
t2-13TH
CENTURIES
graphical position which conferred important IN economic
advantages
upon the city and
its neighborhood.
. ..Boyars always followed the currents of popufar migration, so that
their genealogical records are evidence that at that period [time of Daniel, youngest
son of Alexander Nevsky, circa 1270 A.D.] the general trend was toward Moscow.
into the
city,
as into
a central
reservoir,
of all the Russian popular
Graphinflux
ofRussian
trade
routes
in the
12th - 13th
centuries.
FigureThis
2. steady
~__.position of
.~
forces threatened by external foes was primarily due to the geographical
Moscow.” (p. 276)
‘~Irnmunity from attack, so rare in those days [circa 1240 - 13601, caused the eastward
movement of Russian colonization to become reversed. That is to say, settlers began to
flow back from the old-established colonies of Rostov to the unoccupied lands of the
Principality of Moscow. This constituted the first condition which, arising out of the
geographical position of Moscow, contributed to the successful settlement of the Moscovite region. ...Another condition which... contributed to the growth of the Principaiity was the fact that Moscow stood upon a river which had always - even from the
most ancient times - been possessed of great commercial importance . .. a waterway
Scientific citation network
The most central city of Switzerland is . . .
*Department of Geography,
University
of Hawaii,
Honolulu,
HI 96822.
Q1: who is the most central scientist?
GRAPH
Q2: why?
RUSSIAN
IN
THE
TRADE
12-13TH
According to sbb.ch, it is
edge i → j: scientist i cites j
Many metrics beyond # citations:
h-index: h papers with at least h citations
i10-index: # papers with at least 10 citations
Q: are these metrics
any meaningful?
OF
ROUTES
CENTURIES
.
GRAPH
RUSSIAN
IN
THE
TRADE
12-13TH
OF
ROUTES
CENTURIES
~__.
Degree centrality
your turn: discuss with your neighbors
about a network of your interest and
think about the following questions:
1) which is the most central node?
2) why especially this one?
Eigenvector centrality
eigenvector centrality:
“node i is important if its
in-neighbors are important”
X
cev (i) = α
aji cev (j)
degree centrality is the
in-degree:
X
cdegree (i) = din (i) =
aji
j
example: Google Scholar
citation counter
computation: cdegree = AT 1
A derivative: Katz centrality
+ =
computation: cev
1
T
ρ(A) A cev
maxi cdegree (i) but cev (i) = 0
j
where α = 1/ρ(A)
example: Google PageRank
X
cpr (i) = ε ·
aji cpr (j)
j
|
{z
}
Katz centrality: “combination of degree and eigenvector centrality”
X
X
aji + α
aji cK (j)
cK (i) = α
j
j
| {z }
|
{z
}
cdegree (i)
=cev (i)
+ (1 − ε) ·
X
cpr (j)/n
j
|
{z
}
=random surfer
where ε ∈ ]0, 1[ (see Exercise 3)
=cev (i)
where α < 1/ρ(A)
computation: cK+ = αAT (cK + 1)
(see homework)
Closeness centrality
Betweenness centrality
prelims: the geodesic path
between two nodes k, j is the
shortest path from k to j.
The number of geodesic paths
from k to j passing through
node i ∈ {1, . . . , n} is gkij .
prelims: the geodesic distance
dij between two nodes i, j is the
length of the shortest path
connecting i and j.
closeness centrality:
“the nearest to everybody”
ccloseness (i) = Pn
betweenness centrality:
“fraction of shortest paths
passing through node i”
Pn
1
j=1 dij
j,k=1 gkij
Pn
i=1
j,k=1 gkij
cbetweenness (i) = Pn
example: Olten in Swiss railway
network
example: Moscow in Russian
river trade network
Conclusions on centrality
a different node may be the most central one in each metric
Internet Mathematics Vol. 10: 222–262
Axioms for Centrality
The Laplacian Matrix
(a) degree
(b) eigenvector
Downloaded by [ETH Zurich] at 04:48 16 March 2015
Paolo Boldi and Sebastiano Vigna
Abstract.
Given a social network, which of its nodes are more central? This question
has been asked many times in sociology, psychology, and computer science, and a whole
plethora of centrality measures (a.k.a. centrality indices, or rankings) were proposed to
account for the importance of the nodes of a network. In this study, we try to provide a
mathematically sound survey of the most important classic centrality measures known
from the literature and propose an axiomatic approach to establish whether they are
actually doing what they have been designed to do. Our axioms suggest some simple,
basic properties that a centrality measure should exhibit.
Surprisingly, only a new simple measure based on distances, harmonic centrality,
turns out to satisfy all axioms; essentially, harmonic centrality is a correction to Bavelas’s classic closeness centrality [Bavelas 50] designed to take unreachable nodes into
account in a natural way.
As a sanity check, we examine in turn each measure under the lens of information retrieval, leveraging state-of-the-art knowledge in the discipline to measure the
effectiveness of the various indices in locating webpages that are relevant to a query.
Although there are some examples of such comparisons in the literature, here, for the
first time, we also take into consideration centrality measures based on distances, such
as closeness, in an information-retrieval setting. The results closely match the data we
gathered using our axiomatic approach.
Our results suggest that centrality measures based on distances, which in recent years
have been neglected in information retrieval in favor of spectral centrality measures,
do provide high-quality signals; moreover, harmonic centrality pops up as an excellent
general-purpose centrality index for arbitrary directed graphs.
The proper centrality metrics
are still up for debate!
For every individual network,
it depends on the context . . .
222
(c) closeness
(d) betweenness
⃝
C
Taylor & Francis Group, LLC
ISSN: 1542-7951 print
(Section 6)
The Laplacian in mechanical networks of springs
The Laplacian in electrical networks of resistors
+
aij
Ci
xi
i
1/
ai
Ohm’s law gives the current
flowing from i to j as
j
j
xj
ci→j = aij (vi − vj )
where aij = 1/(resistance)ij is the
conductance & vi is the potential
each node i is subject to an elastic force
X
Felastic,i =
aij (xj − xi ) = −(Lx)i
j6=i
the total elastic energy is
1X
1
aij (xi − xj )2 = x T Lx
Eelastic =
{i,j}∈E
2
2
the Newtonian dynamics are
Mi x¨i = −
X
Di x˙i
−
aij (xj − xi )
j6=i
|{z}
|
{z
}
viscous friction
spring force
Kirchhoff’s current law says that at each node i:
X
X
cinj,i =
ci→j =
aij (vi − vj ) or
j6=i
j6=i
dissipation on resistor {i, j} is ci→j (vi − vj ) & total dissipated power is
X
Edissipated =
aij (vi − vj )2 = vT Lv.
{i,j}∈E
Faraday’s law at capacitor i is Ci v˙ i = cinj,i & network dynamics are
X
X
Ci v˙ i =
ci→j =
aij (vi − vj )
j6=i
properties of Laplacian matrices
cinj = L v
j6=i
discussion on board
A crash course in spectral partitioning
A quick example
given: an undirected, connected, & weighted graph
partition: V = V1 ∪ V2 , V1 ∩ V2 = ∅ , and V1 , V2 6= ∅
P
cut is the size of a partition: J = i∈V1 , j∈V2 aij
⇒ if xi = 1 for i ∈ V1 and xj = −1 for j ∈ V2 , then
J=
X
i∈V1 , j∈V2
n
1 X
1
aij =
aij (xi − xj )2 = x T Lx
2
2
i,j=1
combinatorial min-cut problem: minimizex∈{−1,1}n \{−1n ,1n } 12 x T Lx
relaxed problem: minimizey ∈Rn , y ⊥1n , ky |2 =1 12 y T Ly
6. THE LAPLACIAN MATRIX
⇒CHAPTER
minimum
is algebraic connectivity λ2 and minimizer is Fiedler vector v652
heuristic: xi = sign(yi ) ⇒ “spectral partition”
A quick example – cont’d
Reading assignment (lecture notes):
Chapter 5: Discrete Averaging Algorithms
Chapter 6: The Laplacian Matrix
Exercise session (Friday):
matrix
Fiedler vector
v2 adjacencyre-arranged
adj. with
matrix
Figure adjacency
6.1: The first panel
shows a randomly-generated
sparse
matrix A for a graph
1000
nodes. The second panel displays the eigenvector v˜2 which is identical to the normalized eigenvector v2 after
sorting the entries according to their magnitude, and the third panel displays the correspondingly sorted
˜
adjacency matrix A.
review of take-home messages
examples & additional facts
exercises & illustrations
questions regarding homework