text stringlengths 30 4k | source stringlengths 60 201 |
|---|---|
t
i
n
U
y
r
a
r
t
i
b
r
A
(
L
R
0
.5
1
1.5
2
nhc/eB
2.5
3
3.5
Figure by MIT OpenCourseWare, adapted from
R. E. Prange and S. M. Girvin, The Quantum Hall Effect, Springer-Verlag, Berlin, 1987.
Figure 4: Longitudinal and transverse resistance for the quantum Hall effect.
(From Prange and Girvin.)
sample. It depends on the... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
B0/hc,
gives us the number of Landau levels that must be occupied as a function of
B0 (at fixed n),
N(B0) = nhc/eB0,
(90)
which is exactly the independent variable in Fig. 4.
We know from the analysis of the previous section that the Hall conduc-
tance is quantized when a Landau level is exactly full. Fig. 4 together wi... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
2/h. As B0 decreases, the Landau level fills. The conductance is
exactly e2/h. As B0 decreases further electrons are forced into higher localized
levels between the first and second Landau levels. These electrons do not
conduct, so the Hall conductance stays fixed at e2/h. When B0 decreases
still further, the second Landa... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
R. E. Prange and S. M. Girvin,
The Quantum Hall Effect, Springer-Verlag, Berlin, 1987.
Figure 5: Localized and extended states in a “realistic” two-dimensional
electron gas. (From Prange and Girvin.)
MIT OpenCourseWare
http://ocw.mit.edu
8.06 Quantum Physics III
Spring 2016
For information about citing these materials... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/3e3e804176b9c697b16abd811b5dfa89_MIT8_06S16_Supplementry.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.438 Algorithms for Inference
Fall 2014
4 Factor graphs and Comparing Graphical Model
Types
We now introduce a third type of graphical model. Beforehand, let us summarize
some key perspectives on our first two.
Fir... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/3e3e9934d12e3537b4e9b46b53cd5bf1_MIT6_438F14_Lec4.pdf |
efficiently represent conditional independencies in the distribution of interest, which
are expressed by the removal of edges, and which similarly reduces the complexity of
inference.
4.1 Factor graphs
Factor graphs are capable of capturing structure that the traditional directed and
undirected graphical models abov... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/3e3e9934d12e3537b4e9b46b53cd5bf1_MIT6_438F14_Lec4.pdf |
), medicare (x3), and foreign aid (x4) with constraints
3
0.5
0.25
0.01
x1 ≤
x2 ≤
x3 ≤
x4 ≤
and finally we need to decrease spending by 1, so x1 + x2 + x3 + x4 ≥
1. If we were
interested in picking uniformly among the assignments that satisfy the constraints,
we could encode this distribution conveniently wi... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/3e3e9934d12e3537b4e9b46b53cd5bf1_MIT6_438F14_Lec4.pdf |
hence be disconnected. Thus,
there can’t be any 3 node cliques, so all of the edges are maximal cliques, and there
are O(n2) edges. In general there can be exponentially many maximal cliques in an
undirected graph (See Problem Set 2).
4.2.2 Converting Factor Graphs to Directed Models
Take a topological ordering of... | https://ocw.mit.edu/courses/6-438-algorithms-for-inference-fall-2014/3e3e9934d12e3537b4e9b46b53cd5bf1_MIT6_438F14_Lec4.pdf |
Circuits Maximus v1.6
Documentation
MIT 6.01 Introduction to EECS I
Fall 2011
Contents
1 Getting Started
1.1 What Is CMax? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Running CMax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 File Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Circuit Simulator
3.1 Common Simulation Error Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
try
hitting ctrl+a, which will show a window with information about CMax. You should see that the CMax you are
running is version 1.6; if you do not see a window, or you see a different version number, please make sure you have
opened CMax using the instructions above.
CMax consists of two main components, the circu... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
to be placed in locations on the board where there are no holes; be careful
not to leave any elements disconnected. CMax does not, however, allow duplicate elements to be placed on top of
one another.
2.1.1 Resistors
When a new resistor is added to the protoboard, its value is determined by the value of the prototy... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
Clicking on the power supply button will place power and ground on the lower rails. Clicking on the
power supply button with Shift held down will place power and ground on the upper rails.
2.1.4 Voltage Probes
You can use voltage probes to measure voltages on the virtual protoboard, just like using a multimeter to m... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
more new windows containing the simulator’s output will appear. Some circuit elements (such as po
tentiometers) require inputs; these inputs are specified via a simulation file, which is a Python script that creates
these inputs. You can load a simulation file by selecting Sim
Load Sim File from the menu (keyboard shor... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
for example. Examine your wiring. Maybe you inadvertently
used the same column of holes for two purposes. At worst, you can systematically remove wires until the problem
goes away, and that will tell you what the problem was.
• Element [‘Wire’, ‘b47’, ‘b41’] not connected to anything at node b41
The name ‘b41’ stan... | https://ocw.mit.edu/courses/6-01sc-introduction-to-electrical-engineering-and-computer-science-i-spring-2011/3e54268da5b5d7d46ef03047097d55e4_MIT6_01SCS11_cmax.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
18.014 Calculus with Theory
Fall 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-014-calculus-with-theory-fall-2010/3e9a6928277e7349ac5a8029711dc870_MIT18_014F10_ChFnotes.pdf |
6.776
High Speed Communication Circuits and Systems
Lecture 9
Enhancement Techniques for Broadband Amplifiers,
Narrowband Amplifiers
Massachusetts Institute of Technology
March 3, 2005
Copyright © 2005 by Hae-Seung Lee and Michael H.
Perrott
Shunt-Series Peaking
(cid:131) Series inductors isolate load capacitance fro... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
b/s Limiting Amplifier and Laser/Modulator
Driver in 0.18u CMOS”, ISSCC 2003, pp 188-189 and “Broadband ESD
Protection …”, pp. 182-183
- Also see "Circuit Techniques for a 40 Gb/s Transmitter in 0.13um CMOS",
J. Kim, et. al. ISSCC 2005, Paper 8.1
H.-S. Lee & M.H. Perrott
MIT OCW
Bandwidth Enhancement With ft Doubl... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
Io
M1
vin
Vbias
M2
vin
Ibias
Ibias
Vbias
Io
M1
M3
M2
Ibias
(cid:131) Use current mirror for bias (Battjes ft doubler)
- Inspired by bipolar circuits (see Tom Lee’s book, pp288-
290 (197-199))
(cid:131) Need to set Vbias such that current through M1 has the
desired current of Ibias
- The current through M2 will ideally... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
- Bandwidth decreases much slower than gain increases
(cid:131) Overall gain bandwidth product of amp can be increased
H.-S. Lee & M.H. Perrott
MIT OCW
Transfer Function for Cascaded Sections
Normalized Transfer Function for Cascaded Sections
0
-3
-10
-20
-30
-40
-50
-60
-70
)
B
d
(
i
n
a
G
d
e
z
i
l
a
m
r
o
N
n=1
n=2... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
ω1
ωu
(cid:131) Optimum gain per
stage is about 1.65
- Note than gain
per stage derived
from plot as
- Maximum is fairly
soft, though
(cid:131) Can dramatically
lower power (and
improve noise) by
using larger gain
per stage
0
0
10
H.-S. Lee & M.H. Perrott
5
15
n
20
25
30
MIT OCW
Motivation for Distributed Ampl... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
0
Zo
vout
Rs=Z0
delay
Zo
vin
M1
Zo
M2
Zo
M3
Zo
RL=Z0
(cid:131) Delay the outputs same amount as the inputs
- Now the signals match up
- We have also distributed the output capacitance
(cid:131) Benefit – high bandwidth
(cid:131) Negatives – high power, poorer noise performance,
expensive in terms of chip area
- Each t... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
) Note that conductances
add in parallel
(cid:131) Evaluate at s = jω
(cid:131) Look at frequencies about resonance:
H.-S. Lee & M.H. Perrott
MIT OCW
Tuned Amp Transfer Function About Resonance (Cont.)
(cid:131) From previous slide
=0
(cid:131) Simplifies to RC circuit for bandwidth calculation
vout
vin
gmRp
Cp
Lp
Rp
... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
gain-bandwidth product:
(cid:131) The above expression is just like the low-pass and
independent of center frequency!
- In practice, we need to operate at a frequency less than
the ft of the device
H.-S. Lee & M.H. Perrott
MIT OCW
The Issue of Q
Cp
Lp
Rp
vout
iin=gmvin
Ztank
(cid:131) By definition
(cid:131) For par... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
vin
Vbias
(cid:131) At frequencies above and below resonance
H.-S. Lee & M.H. Perrott
MIT OCW
Purely
Capacitive!
Neutralization in Tuned Amplifier
Recall the neutralization for broadband amplifier
CN
-1
Zin
Cgd
RL
Id
vout
Rs
vin
Vbias
Cgs
M1
CL
For narrowband amplifier, the inverting signal can
be generated by a tapp... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
•Amplitude of oscillation grows again due to positive feedback
H.-S. Lee & M.H. Perrott
MIT OCW
Active Real Impedance Generator
Zin
Vin
Cf
Av(s)
Vout
Av(s) = -Aoe-jΦ
(cid:131) Input admittance:
Resistive component!
H.-S. Lee & M.H. Perrott
MIT OCW
This Principle Can Be Applied To Impedance Matching
Zin
Rs
Iout
M1
Ls
... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
permittivity dielectric material (ceramic), the size can be reduced
to a reasonable dimensions. With εr=10, the length of waveguide is only
about inch.
- Different configurations of filters can be built by combining sections of
- More appropriate at frequencies over GHz
series and parallel LC equivalents
(cid:131) ... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
s
o
L
n
o
i
t
r
e
s
n
I
0
5
10
15
20
25
30
35
40
45
50
2200
2450
Frequency [MHz]
0
1
2
3
4
5
6
7
8
9
10
2700
Figure by MIT OCW.
JRC NSVS754 2.4 GHz RF SAW Filter Charactersitic
H.-S. Lee & M.H. Perrott
Adapted from Japan Radio Co.
MIT OCW | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3e9c40e70c1fc4964f16372c7572daf1_lec9.pdf |
6.776
High Speed Communication Circuits
Lecture 7
High Freqeuncy, Broadband Amplifiers
Massachusetts Institute of Technology
February 24, 2005
Copyright © 2005 by Hae-Seung Lee and Michael H.
Perrott
High Frequency, Broadband Amplifiers
(cid:131) The first thing that you typically do to the input signal
is amplify i... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
in a given technology, use
minimum gate length, bias the transistor at maximum
(cid:131) When velocity saturation is reached, higher does
not give higher gm
(cid:131) In case fixed wiring capacitance is large, power
consumption must be also considered
H.-S. Lee & M.H. Perrott
MIT OCW
Gain-bandwidth Observations
(c... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
(cid:131) Consider Cgd in the MOS device as Cf
- Assume gain is negative
Zin
Vin
(cid:131) Input capacitance:
Cf
Av
Amp
Zout
Vout
ZL
Looks like much larger capacitance by |Av|
H.-S. Lee & M.H. Perrott
MIT OCW
Example: Miller Capacitance
Zin
Vin
Cf
Av
Amp
Zout
Vout
ZL
(cid:131) Output impedance:
This makes sense becaus... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
(+Cov1+Cov2)
Miller multiplication factor
H.-S. Lee & M.H. Perrott
MIT OCW
Add Resistive Feedback?
vout
vin
(gm1+gm2)(ro1||ro2)
(gm1+gm2)Rf
1
Bandwidth extended
and less sensitivity
to bias offset
(does not improve
GB, though)
M2
M1
vout
Cfixed
M4
M3
slope = -20 dB/dec
1
2πCtot(ro1||ro2)
gm1+gm2
2πCtot
1
2πCtotRf
f
... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
Vbias
M1
CL
vin
Vbias
(cid:131) Advantages
- PMOS gate no longer loads the signal
- NMOS device can be biased at a higher voltage (higher gm up
to velocity sat. limit)
(cid:131) Issue
- PMOS is not an efficient current provider (Id/drain cap Cgd+Cdb)
- Signal path is loaded by cap of Rf and drain cap of PMOS
(cid:131)... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
)
Miller multiplication factor
slope =
-20 dB/dec
f
gm1
2πCtot
gm2
2πCtot
(cid:131) Gain set by the relative sizing of M1 and M2
H.-S. Lee & M.H. Perrott
MIT OCW
Design of NMOS Load Amplifier
Vdd
M2
1
gm2
Id
vout
M1
Cfixed
M3
vin
Vbias
Ctot = Cdb1+Csb2+Cgs2 + Cgs3+KCov3 + Cfixed
(+Cov1)
Miller multiplication factor
(... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
dec
f
gm1
2πCtot
1
2πRLCtot
current to capacitance ratio)
(cid:131) This is the fastest non-enhanced amplifier topology
- Unsilicided poly is a low parasitic load (i..e, has a good
- Output can go near Vdd
VRL
(cid:131) Allows following stage to achieve high ft , but at the cost of
gain (max gain
- Linear settling b... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
1) In typical broadband amplifiers, the OCT estimate is too
pessimistic due to multiple poles at around similar
frequencies
H.-S. Lee & M.H. Perrott
MIT OCW
Open Circuit Time Constant Method
Assumptions: No zero near or ωh
Zero well below wh is handled by treating
corresponding capacitor as sort circuit
Negative re... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
o
+
-
H.-S. Lee & M.H. Perrott
Cn
Rno
MIT OCW
OCT’s for CS Amplifier
RL
R2o
Cgd
RS
R1o
Cgs
vo
CL
R3o
By inspection:
It takes some work to figure
H.-S. Lee & M.H. Perrott
MIT OCW
OCT’s for CS Amplifier
RL
+
it
-
vt
vo
gmvgs
RS
+
vgs
-
H.-S. Lee & M.H. Perrott
MIT OCW
OCT’s for CS Amplifier
(cid:131) If RS=0, then τ1o... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
1
g
+
mb
2
R
3
o
≈
R
S
,
eff
1
g
m
,2
eff
≈
g
m
R
4
o
=
r
b
⎛
⎜
⎜
⎜
⎜
⎝
R
L
+
R
S
,
eff
1
+
1
g
effm
,
1
+
g
mb
2
+
R
L
≈
R
L
2
⎞
⎟
⎟
⎟
⎟
⎠
H.-S. Lee & M.H. Perrott
MIT OCW
Cascode
(cid:131) Improves bandwidth in a single-stage amplifier
(cid:131) Problem in cascading:
- Bias point: Reduces ωt of the next stage. PMOS... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/3ebeaaf8fbb23e6ed5db7ede599ecc0d_lec7.pdf |
18.336 spring 2009
lecture 18
04/14/09
Intermezzo:
Boundary Conditions for Advection
Linear advection
�
ut + ux = 0
x ∈ [0, 1]
Image by MIT OpenCourseWare.
Upwind (first order) treats boundary conditions naturally correctly.
LF, LW: Need artificial/numerical boundary conditions at x = 1,
that acts as close to “do ... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/3ed0b03979b2d5868bbf5f44fdc5f3a2_MIT18_336S09_lec18.pdf |
)
+1 + Uj
Δt
+
n
n
+1) − f (Uj
−1)
f (Uj
2Δx
= 0
Image by MIT OpenCourseWare.
(no straightforward LW, since based on linear Taylor expansion)
Numerical Flux Function
Uj
n+1 − Uj
n
Δt
Fj
+
n − Fj−1
n
Δx
= 0
Upwind: Fj
n =
�
f (Uj
n)
f (Uj+1
n)
if
n
f (U +1
j
n
U +1−U
j
)−f (U n
j )
n
j
�
≥ 0
< 0
Im... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/3ed0b03979b2d5868bbf5f44fdc5f3a2_MIT18_336S09_lec18.pdf |
Op#miza#on Problems,
John Gu7ag
MIT Department of Electrical Engineering and
Computer Science
6.0002 LECTURE 2
1
Relevant Reading for Today’s Lecture
§ Chapter 13
6.0002 LECTURE 2
2
The Pros and Cons of Greedy
§ Easy to implement
§ Computa<onally efficient
§ But does not always yield the best solu<on
◦... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
#es
LeS-first, depth-first
enumera<on
Take
Don’tTake
Val = 170
Cal = 766
Val = 120
Cal = 766
Val = 140
Cal = 508
Val = 90
Cal = 145
Val = 80
Cal = 612
6.0002 LECTURE 2
Val = 30
Cal = 258
Val = 50
Cal = 354
Val = 0
Cal = 0
6
Image © source unknown. All rights reserved. This content is excluded from... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
value of a
solution to 0/1 knapsack problem and
the items of that solution""”
toConsider. Those items that nodes higher up in the tree
(corresponding to earlier calls in the recursive call stack)
have not yet considered
avail. The amount of space still available
6.0002 LECTURE 2
9
Body of maxVal (without ... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
us a befer answer
§ Finished quickly
§ But 28 is not a large number
◦ We should look at what happens when we have a more
extensive menu to choose from
6.0002 LECTURE 2
12
Code to Try Larger Examples
import random
def buildLargeMenu(numItems, maxVal, maxCost):
items = []
for i in range(numItems):
items.ap... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
was something not even a
Congressman could object to. So I used it as an
umbrella for my ac<vi<es.
-- Richard Bellman
6.0002 LECTURE 2
15
Recursive Implementa#on of Fibonnaci
def fib(n):
if n == 0 or n == 1:
return 1
else:
return fib(n - 1) + fib(n - 2)
fib(120) = 8,670,007,398,507,948,658,051,921
6.... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
2
18
Using a Memo to Compute Fibonnaci
def fastFib(n, memo = {}):
"""Assumes n is an int >= 0, memo used only by
recursive calls
Returns Fibonacci of n"""
if n == 0 or n == 1:
return 1
try:
return memo[n]
except KeyError:
result = fastFib(n-1, memo) +\
fastFib(n-2, memo)
memo[n] = result
return resu... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
Cal = 258
Val = 50
Cal = 354
Val = 0
Cal = 0
22
A Different Menu
Take
Don’t Take
6.0002 LECTURE 2
23
Need Not Have Copies of Items
Item
Value
Calories
a
b
c
d
6
7
8
9
3
3
2
5
6.0002 LECTURE 2
24
Search Tree
§ Each node = <taken, leS, value, remaining calories>
6.0002 LECTURE 2
25
Wha... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
128
256
512
1024
4
16
256
65,536
7
25
427
5,191
4,294,967,296
22,701
18,446,744,073,709 42,569
,551,616
Big
Really Big
Ridiculously big
83,319
176,614
351,230
Absolutely huge
703,802
6.0002 LECTURE 2
29
How Can This Be?
§ Problem is exponen<al
§ Have we overturned the laws of the universe? ... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
of op<miza<on problems—
those with op<mal substructure and overlapping
subproblems
◦ Solu<on always correct
◦ Fast under the right circumstances
6.0002 LECTURE 2
31
The “Roll-over” Op#miza#on Problem
Score = ((60 – (a+b+c+d+e))*F + a*ps1 + b*ps2 + c*ps3 + d*ps4 + e*ps5
Objec<ve:
Given values for F, ps1, ps2,... | https://ocw.mit.edu/courses/6-0002-introduction-to-computational-thinking-and-data-science-fall-2016/3edc8db04a770f3da51086320c8fe4da_MIT6_0002F16_lec2.pdf |
Tracking Indoors
1
Pervasive Computing MIT 6.883 SMA 5508 Spring 2006 Larry
Rudolph
Location of what?
• Objects
• Static, Moveable, or Mobile
• Frequency of movement: door, desk,
laptop
• Dumb or Networked
• People
• Waldo asks “Where am i?”
• System asks “where’s Waldo?”
• Services
• applications, resourc... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/3ef14f365936b016b8aedc689b1e8843_l7_batscrickets.pdf |
)
•
• button (just one)
• rf transceiver
• Receivers in ceiling
• Base station
• periodically queries, then bats respond
• query time, recv time, room temp
• 330 m/s + .6*temp; >2 receivers ==>
location
More on BATs
• Deployment
• 50 staff members, 200 BATS, 750
Receivers, 3 Radio cells, 10,000 sq ft
offic... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/3ef14f365936b016b8aedc689b1e8843_l7_batscrickets.pdf |
to copyright restrictions.
How well does it
work?
Image removed due to copyright restrictions.
Image removed due to copyright restrictions.
Mobile
Application
Bat
Sensor
Resource
Monitors
Ouija
Proxy
Server
A
B
R
O
C
h
t
a
P
t
s
a
F
Spatial
Indexing
Proxy
Ouija
Proxy
Server
)
I
C
O
(
e
c
a
f
r
e
t
n
I
C
e
l
c
a
r... | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/3ef14f365936b016b8aedc689b1e8843_l7_batscrickets.pdf |
or virtual (instantiation of
program on some machine)
• Need scalable solution to connect
them
• RFIDs demand scalability
Pervasive Computing MIT 6.883 SMA 5508 Spring 2006 Larry
Rudolph | https://ocw.mit.edu/courses/6-883-pervasive-human-centric-computing-sma-5508-spring-2006/3ef14f365936b016b8aedc689b1e8843_l7_batscrickets.pdf |
6.034 Artificial Intelligence. Copyright © 2004 by Massachusetts Institute of Technology.
6.034 Notes: Section 10.1
Slide 10.1.1
So far, we've only talked about binary features. But real problems are typically characterized by
much more complex features.
Slide 10.1.2
Some features can take on values in a discrete... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
advantage of the notion of distance between values that the reals
affords us in order to build in a very deep bias that inputs whose features have "nearby" values
ought, in general, to have "nearby" outputs.
Slide 10.1.6
We'll use the example of predicting whether someone is going to go bankrupt. It only has two
f... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
10.1.10
The naive Euclidean distance isn't always appropriate, though.
Consider the case where we have two features describing a car. One is its weight in pounds and the
other is the number of cylinders. The first will tend to have values in the thousands, whereas the
second will have values between 4 and 8.
Slide... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
others. In such
cases, you might want to multiply them by a weight that will increase their influence in the distance
calculation.
Slide 10.1.15
Another popular, but somewhat advanced, technique is to use cross validation and gradient descent
to choose weightings of the features that generate the best performance ... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
of the nearest neighbor algorithm? It's sort of different from our other
algorithms, in that it isn't explicitly constructing a description of a hypothesis based on the data it
sees.
Given a set of points and a distance metric, you can divide the space up into regions, one for each
point, which represent the set of... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
of Technology.
Slide 10.1.28
Another issue is memory. If you gather data over time, you might worry about your memory filling
up, since you have to remember it all.
Slide 10.1.29
There are a number of variations on nearest neighbor that allow you to forget some of the data
points; typically the ones that are most... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
k closest elements.
Slide 10.1.34
In this case, we've chosen k to be 3. The three closest points consist of two "no"s and a "yes", so our
answer would be "no".
Slide 10.1.35
It's not entirely obvious how to choose k. The smaller the k, the more noise-sensitive your
hypothesis is. The larger the k, the more "smear... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
high-dimensional spaces. There
are two ways to handle this problem. One is to do "feature selection", and try to reduce the problem
back down to a lower-dimensional one. The other is to fit hypotheses from a much smaller
hypothesis class, such as linear separators, which we will see in the next chapter.
6.034 Arti... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
. You can see here that it makes a noticable increase in
performance.
Slide 10.1.45
We ran nearest neighbor with both normalized and un-normalized inputs on the auto-MPG data. It
seems to perform pretty well in all cases. It is still relatively insensitive to k, and normalization only
seems to help a tiny amount. ... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
a value greater than 2. If not, then the output is 1.
6.034 Artificial Intelligence. Copyright © 2004 by Massachusetts Institute of Technology.
Slide 10.2.5
If f1 is greater than 2, then we have another split, this time on whether f2 is greater than 4. If it is,
the answer is 0, otherwise, it is 1. You can see the... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
ting in the L dimension at 1.5 will do the best job of reducing entropy, so we pick that split.
6.034 Artificial Intelligence. Copyright © 2004 by Massachusetts Institute of Technology.
Slide 10.2.13
And we see that, conveniently, all the points with L not greater than 1.5 are of class 0, so we can
make a leaf the... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
10.2.20
But performance isn't everything. One of the nice things about the decision tree algorithm is that we
can interpret the hypothesis we get out. Here is an example decision tree resulting from the learning
algorithm.
I'm not a doctor (and I don't even play one on TV), but the tree at least kind of makes sense... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
than 57.5, then
we declare them to be heart-disease free. Whew!
Slide 10.2.27
If they're older than 57.5, then we examine some technical feature of the cardiogram, and let that
determine the output.
Hypotheses like this are very important in real domains. A hospital would be much more likely to
base or change the... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
case where the y's are numeric values. We'll see how
to extend nearest neighbor and decision trees to solve regression problems.
Slide 10.3.2
The simplest method for doing regression is based on nearest neighbor. As in nearest neighbor, you
remember all your data.
6.034 Artificial Intelligence. Copyright © 2004 b... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
hard to go back and reformulate it to depend on the k nearest.
Slide 10.3.9
Rather than committing to the details of the weighting function right now, let's just assume that we
have a "kernel" function K, which takes the query point and a training point, and returns a weight,
which indicates how much influence the ... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
decision trees, but which have
numeric constants at the leaves rather than booleans.
Slide 10.3.13
Here's an example regression tree. It has the same kinds of splits as a regular tree (in this case, with
numeric features), but what's different are the labels of the leaves.
Slide 10.3.14
Let's start by thinking ab... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
6.034 Artificial Intelligence. Copyright © 2004 by Massachusetts Institute of Technology.
Slide 10.3.19
We're going to use the average variance of the children to evaluate the quality of splitting on a
particular feature. Here we have a data set, for which I've just indicated the y values. It currently has
a varian... | https://ocw.mit.edu/courses/6-034-artificial-intelligence-spring-2005/3effa3b9e955738f0fb9775c8f578d69_ch6_mach2.pdf |
L8a Markov Decision Processes:
Reactive Planning to Maximize Reward
Brian C. Williams
16.410 / 13
October 5th, 2015
Slides adapted from:
Manuela Veloso,
Reid Simmons, &
Tom Mitchell, CMU
Assignments
• Reading:
• Today: Markov Decision Processes: AIMA 17.1-3.
• Wednesday: Hidden Markov Models: AIMA 15.... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
Learns to play Backgammon
States:
• Board configurations (1020)
Actions:
• Moves
Rewards:
•
•
•
+100 if win
- 100 if lose
0 for all other states
• Trained by playing 1.5 million games against self.
è Became roughly equal to best human player.
16.410/13 F15: Markov Decision Processes
5
... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
that maximizes lifetime reward.
16.410/13 F15: Markov Decision Processes
9
Markov Decision Processes (MDPs)
Model:
Process:
• Finite set of states, S
• Finite set of actions, A
• (Probabilistic) state
transitions, δ(s,a)
• Reward for each state
and action, R(s,a)
1. ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
9
R
0.1
S2
Industry
D
1.0
0.9
S3
Grad School
D
R
0.1
S4
Academia
D
0.1
R
0.1
0.9
1.0
16.410/13 F15: Markov Decision Processes
12
MDP Problem: Model
Agent
State
Reward
Action
Environment
a0
s0
a1
s1
a2
s2
s3
r2
r1
r0
Given an environment model a... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
0/13 F15: Markov Decision Processes
15
MDP Problem: Lifetime Reward
Agent
State
Reward
Action
Environment
a0
s0
a1
s1
a2
s2
s3
r2
r1
r0
Given an environment model as a MDP, create a policy for acting
... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
10
G
10
10
10
G
10
10
10
G
10
10
16.410/13 F15: Markov Decision Processes
19
Markov Decision Processes
• Motivation
• What are Markov Decision Processes (MDPs)?
• Models
• Lifetime Reward
• Policies
• Computing Policies From a Model
• Summary
16.410/13 F15: Markov D... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
An Optimal Policy π* Given Value Function V*
Idea: Given state s
1. Examine all possible actions ai in state s.
2. Select action ai with greatest lifetime reward.
Lifetime reward Q(s, ai) is:
the immediate reward for taking action r(s,a) …
•
• plus life time reward starting in target state V( δ(s, a) ) …
• disc... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
0 + 0.9 x 100 = 90
• b: 0 + 0.9 x 81 = 72.9
Ø select a
Model + V:
a
100
90
100
b
G
0
100
81
90
100
π:
G
16.410/13 F15: Markov Decision Processes
25
Example: Mapping Value Function to Policy
• Agent selects optimal action from V:... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
Markov Decision Processes
• Motivation
• Markov Decision Processes
• Computing Policies From a Model
• Value Functions
• Mapping Value Functions to Policies
• Computing Value Functions through Value Iteration
• An Alternative: Policy Iteration
• Summary
16.410/13 F15: Markov Decision Processes
28
... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
Value Function V* for an optimal policy π*
Example
RA
A
B
SA
RB
SB
RA
A
RB
B
• Optimal value function for a one step horizon:
V*1(s) = maxai [r(s,ai)]
• Optimal value function for a two step horizon:
V*2(s) = maxai [r(s,ai) + γV 1
*(δ(s, ai))]
• Optimal value function for an n step horizon:
V*n(s) = ma... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
Iteration
Insight: Calculate optimal values iteratively using Dyn Prog
Algorithm:
• Iteratively calculate value using Bellman’s Equation:
V*t+1(s) ← maxa [r(s,a) + γV*
t(δ(s, a))]
• Terminate when values are “close enough”
|V*t+1(s) - V*
t (s) | < ε
• Agent selects optimal action by one step lookahead on V* : ... | https://ocw.mit.edu/courses/16-412j-cognitive-robotics-spring-2016/3f0739aa808805e51c445195485a7ebb_16-412s16ResourceFile.pdf |
6.092: Thursday
Lecture
Lucy Mendel
MIT EECS
MIT 6.092
IAP 2006
1
Topics
z Interfaces, abstract classes
z Exceptions
z Inner classes
MIT 6.092
IAP 2006
2
Abstract Classes
z Use when subclasses have some code in common
abstract class Person {
private String name = “”;
public String getName() { return name; ... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/3f3161bffccbbbc2376e56def7119d7a_lecture4.pdf |
click() {
SOP("clicking..."); }
public void drag() {
SOP("dragging..."); }
MIT 6.092
IAP 2006
7
Subtyping
class Square {
public int width;
}
class Rectangle {
Should:
Square extend Rectangle?
Rectangle extend Square?
public int width,height;
}
…
int calculateArea (Square x) {
return (x.width)*(x.width); }... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/3f3161bffccbbbc2376e56def7119d7a_lecture4.pdf |
classes) re-uses code
z A true subtype will behave the right way when used
by code expecting its supertype.
class B {
Bicycle myMethod(Bicycle arg) {…}
}
class A {
RacingBicycle myMethod(Vehicle arg) {…}
}
MIT 6.092
IAP 2006
12
Composite
z Contain a class, rather than extend it
class ListSet { // might want ... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/3f3161bffccbbbc2376e56def7119d7a_lecture4.pdf |
2006
17
Nested (or Inner) Classes
class EnclosingClass {
...
class ANestedClass {
...
}
…
}
Why use nested
classes?
MIT 6.092
IAP 2006
18
Nested Classes
MIT 6.092
IAP 2006
19
Nested Class Properties
z Have access to all members of the enclosing class,
even private members
z can be declared static (and... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/3f3161bffccbbbc2376e56def7119d7a_lecture4.pdf |
.6100000000000000001
System.out.println(1.00 - 9*.10) ;
// 0.0999999999999999995
z BigDecimal
z
z Pain of using Objects
int or long
z keep track of decimal yourself, eg, put money in terms of
pennies
MIT 6.092
IAP 2006
24
Defensive Programming
MIT 6.092
IAP 2006
25
public class Man {
private Wallet myWalle... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/3f3161bffccbbbc2376e56def7119d7a_lecture4.pdf |
Lecture 6: Quantitative Aspects Of
Networks III: Outline
• Network Analysis Terminology notated
• Connectivity
• Some Social Network Concepts-intuition and calculation
transitivity (clustering)
•
• centrality
• degree, closeness, betweenness, information, eigenvector
• prestige and acquaintance
• degree distributions
... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
, is the degree.
• m/[(n)(n-1)] or <k>/2(n-1)is the “sparseness” or normalized
interconnection “density”
• Path length, l
1
(
nn
−
)1
l
=
1
2
∑
i
≥
j
ijd
Professor C. Magee, 2006
Page 3
Connectivity
• Fraction of nodes connected in a network
• Of interest in resilience/robustness which we will cover in a
later lect... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
Magee, 2006
Page 5
Social Network Analysis
• Many structural metrics have been invented and used by Social
Scientists studying social networks over the past 70+ years.
• These are well-covered in Wasserman and Faust –Social
Network Analysis (1994) The following slides cover a few
selected examples in on... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
. Magee, 2006
Page 8
Transitivity or Clustering coefficient, II
•
(Almost) always > than expected from random networks thus
offering some support for earlier assertions that real networks
have some non-random “structure” (more later)
• Thus, assessing transitivity is a quick check whether you
have a random graph wh... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
.214
Centralization
0.257
Professor C. Magee, 2006
Page 13
Closeness Centrality
• Actor
• Closest is shortest
(geodesic) distance from
other nodes =1 for max
closeness and 0 for min
'
(
nC
C
i
)
n
−
1
nnd
i
(
,
)
j
= n
∑
j
1
=
• Group
• = 0 for circle graph or full
network
• = 1 for star graph
• 0.277 for line (7 n... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
0.400
0.389
0.333
0.467
0.326
0.560
0.286
0.368
---
0.500
0.389
0.438
0.483
Centralization
0.257
0.322
Professor C. Magee, 2006
Page 15
Betweeness Centrality I
• Actor
• Power or influence comes
•
from being an intermediary
z is the number of
geodesics between two
points
('
nC
B
i
)
=
∑ <
kj
n
[(
−
z
(
n
i
/)
z
j... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
:Bischeri
n3:Barbadori
n13:Ridolfi
n16:Tornabuoni
n9:Medici
n14:Salvati
n10:Pazzi
n7:Guadagni
n8:Lamberteschi
n2:Ablizzi
n1:Acciaiuoli
n6:Genori
Professor C. Magee, 2006
Page 19
Betweeness Centrality II
• Actor
• Power or influence comes
•
from being an intermediary
z is the number of
geodesics between two
points ... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
0.043
0.095
0.033
0.069
---
0.080
0.050
0.070
0.080
Centralization
0.257
0.322
0.437
---
Professor C. Magee, 2006
Page 22
Eigenvector Centrality-UCINET
• UCINET-help, help topics, index (on toolbar), eigenvector centrality
• Given an adjacency matrix A, the centrality of vertex i (denoted ci), is given by
ci =aSAijc... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
the node (i)
x
i
• And
xA
ij
∑−= 1λ
j
j
• And then Ax = x
• Thus the weights are an eigenvector of the
λ
adjacency matrix (A) with eigenvalue
λ
Professor C. Magee, 2006
Page 24
Florentine Families Centrality Metrics
(with Eigenvector Centrality)
Acciaiuoli
Ablizzi
Barbadori
Bischeri
Castellani
Genori
Guadagni
Lam... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
3
0.092
)
I nC′
( i
0.049
0.074
0.068
0.074
0.070
0.043
0.081
0.043
0.095
0.033
0.069
---
0.080
0.050
0.070
0.080
Centralization
0.257
0.322
0.437
---
.19
.35
.30
.40
.37
.11
.41
.12
.61
.06
.39
0
.48
.20
.50
.46
.43
Professor C. Magee, 2006
Page 25
Centrality II
• Numerous metrics exist in the Social Networks Literat... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
rality-like metrics
Jon Kleinberg (Computer Science at Cornell) has done much
of the leading work in search and navigation (more later).
In some of his earliest work on this topic (1997-1999), he
“invented” some useful new metrics for looking at important
nodes (particularly on directed networks and probably most
u... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
• truncation
• Structural implications and growth assumptions
• Examples of some metrics from broad Assign. # 3 “systems”
• Project Discussion
Professor C. Magee, 2006
Page 30
Degree Distributions
kp
• Define as the fraction of nodes in a network with degree k. This is
equivalent to the probability of randomly pickin... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
4x105
Population of City
10-2
10-4
10-6
10-8
104
105
106
107
Population of City
Figure by MIT OCW.
See
Newman, M. E. J. cond-mat/0412004v2
Professor C. Magee, 2006
Page 33
Degree Distributions II
kp
• Define as the fraction of nodes in a network with degree k. This is
equivalent to the probability of randomly pic... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
Newman, M. E. J. cond-mat/0412004v2
Professor C. Magee, 2006
Page 35
P(k)
10-1
10-2
10-3
10-4
10-5
10-6
100
A
100
10-2
10-4
10-6
10-8
C
B
100
10-1
10-2
10-3
10-4
100
104
101
k
101
102
103
100
k
102
k
Figure by MIT OCW.
Barabasi and Albert(1999) A is actor collaboration, B is www
and C is the Western Power Gri... | https://ocw.mit.edu/courses/esd-342-advanced-system-architecture-spring-2006/3f52837eed9bbdee37893c79dbe7c514_lec6.pdf |
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