Small World Modeling for Urban Street Networks: A Topological Perspective

Bin Jiang

University of Gävle, Sweden

http://fromto.hig.se/~bjg/

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Topics to cover

What are small worlds? Why is the concept so important? Is the real world random? Why small worlds are both good and bad?

What are scale-free networks? How vulnerable are scale-free networks? Is the real world networks (e.g., the internet and the web) are scale free?

What is PageRank algorithm? The Google legacy? A word of mouth? How to find a needle in a haystack?

What are the far reaching implications of the concepts to urban street networks, or urban systems in general?

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It’s a small world

It's a world of laughter, A world of tears. …

It's a small world after all.

There is just one moon, And one golden sun. …

Friendship to every one. Though the mountains divide, And the oceans are wide,

It's a small world after all.

(by Richard M. Sherman and Robert B. Sherman)

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Roadmap of my talk

A bit background on small world networks Stanley Milgram’s experiments Kevin Bacon Game The Erdös Number Project

Seminal papers Small world network (Watts and Strogatz 1998) Scale free network (Barabási and Albert 1999) Google’s PageRank algorithm (Page and Brin 1998)

Modeling and visualizing urban structures Why topology matters? A universal pattern of urban street networks Ranking spaces for predicting human movement A minority of streets account for a majority of traffic

Summary

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Stanley Milgram’s experiments

Six degrees of separation (individuals in Kansas and Nebraska, to one target in Boston)

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Kevin Bacon Game

The Game follows the rule: If you have acted in a movie with

Kevin Bacon, your have Bacon number one (Bacon himself has a Bacon number zero);

If you haven’t ever acted with him, but if you have acted with somebody else who has, then you have a Bacon number two

... then Bacon number three etc..Bacon is a famous movie star, and has acted in over fifty movies.

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Kevin Bacon Game (http://www.cs.virginia.edu/oracle/center.html)

Bacon number

Number of actors Cumulative total number of actors

0 1 1

1 1806 1807

2 145 024 146 831

3 395 126 541 957

4 95 497 637 454 (68% of 800 000 people)

7 106 645 944

8 13 645 957

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Is Kevin Bacon really the center of the movie universe?

The answer is NO.

Kevin Bacon is not the most important hub of the movie network, not even in the top one thousand.

Toppest is Rod Steiger (average Steiger Number is 2.652), followed by Christopher Lee, Dennis Hopper, Donald Pleasence, and Donald Sutherland….

Do you know how to calculate the average Bacon number?

Why was Kevin Bacon picked for this game?

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The Erdös number

http://www.oakland.edu/enp/

Authored or coauthored over 1500 papers

Theory of random graphs

The least number of roads that keep 50 villages interconnected is 98, (but everyone to everyother, it needs 1225 roads)

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The collaboration graph

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Six degrees of Monica Lewinsky

”Clinton number one”

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Small world network (Watts and Strogatz 1998)

A small-world network is a network with a small-separation and whose nodes are highly clustered.

Separation Six degrees of separation

Diameter of WWW is 19 clicks

Clustering Friends of a friend are being

friends

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FireFlies

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Small world metrics

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Emegent properteis of small world neworks

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Small world, a BIG idea

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Scale free networks (Barabási and Albert 1999)

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What is a power law?

A power law relationship between two variables xand y is one where the relationship can be written as (to the right)

where a (the constant of proportionality) and k (the exponent of the power law) are constants.

Power laws can be seen as a straight line on a log-log graph since, taking logs of both sides, the above equation becomes

which has the same form as the equation for a line

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What does the power law say?

A power-law implies that small occurrences are extremely common, whereas large instances are extremely rare.

Many man made and naturally occurring phenomena, including city sizes, incomes, word frequencies, and earthquake magnitudes, are distributed according to a power-law distribution.

Examples of power law probability distributions: The Pareto distribution, for example, the distribution of wealth in

capitalist economies Zipf's law, for example, the frequency of unique words in large

texts Scale-free networks, where the distribution of links is given by a

power law

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Pareto distribution

The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations.

Outside the field of economics it is at times referred to as the Bradford distribution.

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Matthew effect

"the rich get richer and the poor get poorer"

Matthew (13:12 and 25:29)

“For unto everyone that hath shall be given, and he shall have abundance; but from him that hath not shall be taken away even that which he have”

Unto = to

Hath = have

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Zipf’s law

Zipf's law states that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table.

So, the most frequent word will occur approximately twice as often as the second most frequent word, which occurs twice as often as the fourth most frequent word, etc.

The term has come to be used to refer to any of a family of related power law probability distributions.

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Word frequency

Hermetic Word Frequency Counter

(http://www.hermetic.ch/wfca/zipf.htm)

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http://www.wordcount.org/main.php

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Look at Barabási’s citation

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PageRank -to find a needle in a haystack

A treasure-hunt approachA treasure-hunt approachWeb graphWeb graph

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PageRank (Page and Brin 1998)

“An important page is one that MANYIMPORTANT pages point to.”

The beauty of PageRank lies in the fact that it considers not only popularity (how many?), but also prestigious (how important?).

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PageRank - a random surfer model

IF (a node has no successors) THEN

with probability 1 it jumps to a randomly chosen node

ELSE

with probability d it moves to one of its successors with a uniform probability, AND

with probability (1-d) it jumps to a randomly chosen node

END

dangling nodesdangling nodes

tiredtired

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Danny Sullivan wrote in Search Engine Report:

”When I speak about search engines to groups and mention Google, something unusual happens to some members of the audience. They smile and nod, in the way you do when you feel like you’ve found a secret little getaway that on one else knows about. And each time I speak, I see more and more people smiling and nodding this way, pleased to have discovered Google.”

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Again, see how significant PageRank is

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NetLogo-based random walkers

C:\Program Files\NetLogo 4.0.3\extensions\gis\Traffic-Simulation-AddLine.netlogo

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Ubiquity of networks - visualization

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A city as a complex network

Source: Emergence by Steven Johnson (2001) (Hamburg circa 1850)

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The image of the city

Portugali 1996

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London as a network

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From pixels to perceptual units

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From street segments to streets

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Perceptual grouping based on named streets

Network Topology

distance, tracing, pathfinding etc. structures, morphology, patterns etc.

disordered ordered

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Two views of space (absolute vs relative)

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London underground map

Geometrically corrected Topologically retained

Harry Beck

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A universal topological pattern

20%

80%1%

Cu

mu

lati

ve f

req

ue

nc

y

Degree/length

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 0.5 1 1.5 2 2.5 3

Los Angeles

Phoenix

Chicago

Houston

Pasadena

San Diego

Hollywood

Dallas

Arlington

Las Vegas

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80% trivial versus 1% vital

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Ranking spaces

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London axial map is a small world and scale free network

y = -3.17x + 2.19

R2 = 0.98

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 0.5 1 1.5 2 2.5

y = -3.11x + 2.28

R2 = 0.99

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 0.5 1 1.5 2 2.5

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PageRank predicting human movement

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A minority of streets account for a majority of traffic (1)

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A minority of streets account for a majority of traffic (2)

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Summary

Fundamental to this topic is a topological view for spatial analysis and modeling.

In this sense, it is closely linked to space syntax modeling.

On the other hand, in terms of self-organization and bottom up, it is closely related to agent-based and cellular automata modeling.

Complex networks constitute an essential part of complexity theory, which has far reaching implications to urban systems.

Beyond urban street networks ...

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Further readings

Jiang B. (2009), Street hierarchies: a minority of streets account for a majority of traffic flow, International Journal of Geographical Information Science, 23.8, 1033-1048.

Jiang B. (2009), Ranking spaces for predicting human movement in an urban environment, International Journal of Geographical Information Science, 23.7, 823–837.

Jiang B., Zhao S., and Yin J. (2008), Self-organized natural roads for predicting traffic flow: a sensitivity study, Journal of Statistical Mechanics: Theory and Experiment, July, P07008.

Jiang B. (2007), a topological pattern of urban street networks:universality and peculiarity, Physica A: Statistical Mechanics and its Applications, 384, 647 - 655.

Jiang B. and Claramunt C. (2004), Topological analysis of urban street networks, Environment and Planning B: Planning and Design, Pion Ltd., 31, 151-162.