![Page 1: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/1.jpg)
Forward Looking Session
Small and Large Graphs
Bojan Mohar
Simon Fraser University & IMFM
Austin TX – June 16, 2010
B. Mohar FLS
![Page 2: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/2.jpg)
Signs of global warming
Observation. When I was young, there was much more snow thantoday. I remember that the snow level was always well above my head.
B. Mohar FLS
![Page 3: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/3.jpg)
Some trends in modern graph theory
Observation. When I was young, all graphs were very small(or infinite).
Observation. Today, the mainstream seems to be about large graphs.(Also evident from several talks at this conference.)
Is this a sign of a global change?
B. Mohar FLS
![Page 4: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/4.jpg)
Some trends in modern graph theory
Observation. When I was young, all graphs were very small(or infinite).
Observation. Today, the mainstream seems to be about large graphs.(Also evident from several talks at this conference.)
Is this a sign of a global change?
B. Mohar FLS
![Page 5: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/5.jpg)
Main reasons and some examples
I Applications• (Theoretical) computers science (e.g. computational
complexity, expanders)• Emergence of large networks and large data sets• Internet graph• Social networks• Bioinformatics (Evolution trees, genomics, protein folding)• Biomedicine (living cells, brain network)• Mathematics (e.g. number theory)• Theoretical physics (Sir Michael Atiyah contraversy)
I Tools• Probabilistic method• Excluded minor structure• Regularity lemma• Graph limits
B. Mohar FLS
![Page 6: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/6.jpg)
Main reasons and some examples
I Applications• (Theoretical) computers science (e.g. computational
complexity, expanders)• Emergence of large networks and large data sets• Internet graph• Social networks• Bioinformatics (Evolution trees, genomics, protein folding)• Biomedicine (living cells, brain network)• Mathematics (e.g. number theory)• Theoretical physics (Sir Michael Atiyah contraversy)
I Tools• Probabilistic method
• Excluded minor structure• Regularity lemma• Graph limits
B. Mohar FLS
![Page 7: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/7.jpg)
Main reasons and some examples
I Applications• (Theoretical) computers science (e.g. computational
complexity, expanders)• Emergence of large networks and large data sets• Internet graph• Social networks• Bioinformatics (Evolution trees, genomics, protein folding)• Biomedicine (living cells, brain network)• Mathematics (e.g. number theory)• Theoretical physics (Sir Michael Atiyah contraversy)
I Tools• Probabilistic method• Excluded minor structure
• Regularity lemma• Graph limits
B. Mohar FLS
![Page 8: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/8.jpg)
Main reasons and some examples
I Applications• (Theoretical) computers science (e.g. computational
complexity, expanders)• Emergence of large networks and large data sets• Internet graph• Social networks• Bioinformatics (Evolution trees, genomics, protein folding)• Biomedicine (living cells, brain network)• Mathematics (e.g. number theory)• Theoretical physics (Sir Michael Atiyah contraversy)
I Tools• Probabilistic method• Excluded minor structure• Regularity lemma
• Graph limits
B. Mohar FLS
![Page 9: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/9.jpg)
Main reasons and some examples
I Applications• (Theoretical) computers science (e.g. computational
complexity, expanders)• Emergence of large networks and large data sets• Internet graph• Social networks• Bioinformatics (Evolution trees, genomics, protein folding)• Biomedicine (living cells, brain network)• Mathematics (e.g. number theory)• Theoretical physics (Sir Michael Atiyah contraversy)
I Tools• Probabilistic method• Excluded minor structure• Regularity lemma• Graph limits
B. Mohar FLS
![Page 10: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/10.jpg)
Chromatic number
Theorem (Molloy and Reed 1998). χ′′T (G ) ≤ ∆(G ) + 1026.
Tools: Probabilistic method.
Theorem (Dvorak, Kral, Thomas 2010+). G planar and the distancebetween any two triangles of G is ≥ 10100, then G is 3-colorable.
For practical reasons this seems like a result about infinite graphs (orthe Grotzsch theorem with at most one triangle allowed).
B. Mohar FLS
![Page 11: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/11.jpg)
Chromatic number
Theorem (Molloy and Reed 1998). χ′′T (G ) ≤ ∆(G ) + 1026.
Tools: Probabilistic method.
Theorem (Dvorak, Kral, Thomas 2010+). G planar and the distancebetween any two triangles of G is ≥ 10100, then G is 3-colorable.
For practical reasons this seems like a result about infinite graphs (orthe Grotzsch theorem with at most one triangle allowed).
B. Mohar FLS
![Page 12: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/12.jpg)
Regularity lemma and its success
Theorem (Szemeredi Regularity lemma).∀m, ε > 0 : ∃M = M(m, ε) with the following property:
∀G with |G | ≥ M has an ε-regular partition V1, . . . ,Vk , wherem ≤ k ≤ M.
Drawback: M is a tower of exponents of height O(ε−5).Known lower bounds (Gowers 1997) are also towers of exponents.
Applications in algebra and in extremal combinatoric.
B. Mohar FLS
![Page 13: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/13.jpg)
Regularity lemma and its success
Theorem (Szemeredi Regularity lemma).∀m, ε > 0 : ∃M = M(m, ε) with the following property:
∀G with |G | ≥ M has an ε-regular partition V1, . . . ,Vk , wherem ≤ k ≤ M.
Drawback: M is a tower of exponents of height O(ε−5).Known lower bounds (Gowers 1997) are also towers of exponents.
Applications in algebra and in extremal combinatoric.
B. Mohar FLS
![Page 14: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/14.jpg)
Examples of large finite graphs
I The internet graph (> 1010 nodes)
I Social networks (up to almost 1010 nodes)
I Biological networks (genomes, protein interactions, human brain)
I Some very concrete networks (VLSI)
I The Universe (elementary particle interaction),1060 < N < 10100.
We will need new methods to deal with large graphs. Probabilisticapproach has been used throughout.
B. Mohar FLS
![Page 15: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/15.jpg)
Examples of large finite graphs
I The internet graph (> 1010 nodes)
I Social networks (up to almost 1010 nodes)
I Biological networks (genomes, protein interactions, human brain)
I Some very concrete networks (VLSI)
I The Universe (elementary particle interaction),1060 < N < 10100.
We will need new methods to deal with large graphs. Probabilisticapproach has been used throughout.
B. Mohar FLS
![Page 16: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/16.jpg)
Examples of large finite graphs
I The internet graph (> 1010 nodes)
I Social networks (up to almost 1010 nodes)
I Biological networks (genomes, protein interactions, human brain)
I Some very concrete networks (VLSI)
I The Universe (elementary particle interaction),1060 < N < 10100.
We will need new methods to deal with large graphs. Probabilisticapproach has been used throughout.
B. Mohar FLS
![Page 17: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/17.jpg)
Graph limits and graphons
Theorem. A large (dense) graph can be approximated by a graphon,a symmetric measurable function W : [0, 1]× [0, 1]→ [0, 1].
Example: A graphon for G (n, p) satisfies W (x , y) = p.
B. Mohar FLS
![Page 18: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/18.jpg)
Graph limits and graphons
Theorem. A large (dense) graph can be approximated by a graphon,a symmetric measurable function W : [0, 1]× [0, 1]→ [0, 1].
Example: A graphon for G (n, p) satisfies W (x , y) = p.
B. Mohar FLS
![Page 19: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/19.jpg)
Summary
Some conclusions with which you (and the speaker) may or may notagree.
I There is increased interest in large graphs.
I New tools (theory and algorithms) are needed.
I Physics is no longer the only driving force for mathematicalfrontiers (CS, bio).
I Theoretical physics may turn out to be a subarea of graph theory(the study of a single very large graph).
Thank you . . .
B. Mohar FLS
![Page 20: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/20.jpg)
Summary
Some conclusions with which you (and the speaker) may or may notagree.
I There is increased interest in large graphs.
I New tools (theory and algorithms) are needed.
I Physics is no longer the only driving force for mathematicalfrontiers (CS, bio).
I Theoretical physics may turn out to be a subarea of graph theory(the study of a single very large graph).
Thank you . . .
B. Mohar FLS
![Page 21: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/21.jpg)
Summary
Some conclusions with which you (and the speaker) may or may notagree.
I There is increased interest in large graphs.
I New tools (theory and algorithms) are needed.
I Physics is no longer the only driving force for mathematicalfrontiers (CS, bio).
I Theoretical physics may turn out to be a subarea of graph theory(the study of a single very large graph).
Thank you . . .
B. Mohar FLS
![Page 22: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/22.jpg)
Summary
Some conclusions with which you (and the speaker) may or may notagree.
I There is increased interest in large graphs.
I New tools (theory and algorithms) are needed.
I Physics is no longer the only driving force for mathematicalfrontiers (CS, bio).
I Theoretical physics may turn out to be a subarea of graph theory(the study of a single very large graph).
Thank you . . .
B. Mohar FLS
![Page 23: Forward Looking Session [8mm] Small and Large Graphsmohar/Conf/ForwardLookingSession... · Emergence of large networks and large data sets Internet graph Social networks Bioinformatics](https://reader034.vdocuments.net/reader034/viewer/2022042219/5ec55b289e7020370409bb7f/html5/thumbnails/23.jpg)
Summary
Some conclusions with which you (and the speaker) may or may notagree.
I There is increased interest in large graphs.
I New tools (theory and algorithms) are needed.
I Physics is no longer the only driving force for mathematicalfrontiers (CS, bio).
I Theoretical physics may turn out to be a subarea of graph theory(the study of a single very large graph).
Thank you . . .
B. Mohar FLS