Random Graphs
Drew Masters
March 9th, 2016
What is a random graph?
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A graph in which properties such as the number of graph vertices and graph
edges between them are determined in some random way
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Can use some graph generating model (constructive) or by randomly
sampling from a collection of graphs (sampling)
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Consists of n vertices and m edges
History
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First papers published in 1957
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Paul Erdős and Alfré d Ré nyi
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Edgar Gilbert
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First papers that formally defined a random graph
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Papers independent of each other
Paul Erdős
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Hungarian
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Mathematician
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Known for his social practice of mathematics
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engaged more than 500 collaborators
Erdős number
Alfré d Ré nyi
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Hungarian
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Mathematician
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Founded Mathematics Research Institute
of the Hungarian Academy of Sciences
Edgar Gilbert
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American
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Mathematician
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Researcher at Bell Laboratories
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Known for his contributions to coding theory
Random Graph Models
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Obtained by starting with a set of n isolated vertices and add successive
edges between the vertices at random
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Random graph models
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Edgar Gilbert: G(n,p)
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Erdős-Ré nyi model: G(n,M)
Edgar Gilbert Model
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Denoted G(n,p)
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n = number of isolated vertices
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p = probability every possible edge occurs independent to each other
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m = number of edges
n
2
Probability
of obtaining a particular graph =
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N=
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A graph is constructed by connecting nodes randomly.
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As p increases from 0 to 1, the model is more likely to produce graphs with
more edges
Erdős-Ré nyi Model
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Denoted G(n,M)
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Assigns equal probability to all graphs with exactly M edges
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M = number of edges
Every element occurs with probability:
A graph is chosen uniformly at random from the collection of all graphs which
have n nodes and M edges.
Comparison of the main models
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G(n,p) model is the more commonly used model
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G(n,M) model is not as easy to deal with mathematically
Equivalence of G(n,M) and G(n,p) is done by setting M =
* p.
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As n approaches infinity G(n,p) should behave similarly to G(n,M)
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Law of large numbers says G(n,p) will contain approximating the same number of edges as G
(n,M)
Characteristics of G(n,p)
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Expected number of edges =
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Expected degree: z (also called c or k in some papers)
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*p
z = (n-1) * p
Clustering coefficient: cc(v)
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Average probability two neighbours of a given vertex are also neighbours of one another
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Expected cc(v) of = p
Evolution of G(n,p)
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np << 1
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np = c where c < 1
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cycles start appearing
almost all vertices connected in trees
np = 1
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G(n,p) most likely does not have any connected components larger than O(log(n)) in size
Components consist of trees
G(n,p) most likely has a largest component of the order n2/3
np > 1
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G(n,p) most likely has a unique giant component contain a fraction of the vertices
No other component will contain more than O(log(n)) vertices
Dual phase evolution
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Process that promotes the emergence of large-scale order in complex
systems
Arises because of the property that the connectivity avalanche occurs in
graphs as the number of edges increases
Features necessary for Dual phase evolution to occur
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underlying network
phase shifts
selection and variation
system memory
Probabilities of the whole graph
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Probability the graph is connected
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N = number of vertices
p = probability that an edge exists
q = probability that an edge does not exist=1-p
PN = probability N vertices are connected
Probability two vertices are connected in the graph
Table from Gilbert paper
Diameter of random graphs
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Diameter: longest shortest path between two vertices
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Expected diameter of a random graph =
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c is the average degree of a vertex
Rado Graph
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Unique countably infinite graph R such that for every finite graph G and every
vertex v of G, every embedding of G-v as an induced subgraph of R can be
extended to an embedding of G into R.
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Contains all finite and countably infinite graphs as induced subgraphs
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For any finite disjoint sets of vertices U and V, there exists a vertex x
connected to everything in U, and to nothing in V
Random Regular Graph
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Graph selected from Gn,r
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Gn,r denotes the probability space of all r-regular graphs on n vertices
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3 ≤ r < n and nr is even
Possible to prove that certain properties of random r-regular graph almost
surely hold
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Non-trivial to implement
Threshold Functions
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Function, m*(n), for a monotone increasing property P in random graph G
such that as n approaches infinity
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Every non-trivial monotone graph property has a threshold
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Proved by Bollobá s and Thomason in 1987
Other Models
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Barabá si-Albert
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Model for generating random scale-free networks using a preferential attachment mechanism
Watts-Strogatz
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Model produces graphs with small-world properties including short average path lengths and
high clustering
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Stochastic Block
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Model produces graphs containing communities
Barabá si-Albert Model
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Scale-free networks are widely observed in natural and human-made systems
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network whose degree distribution follows a power law
Incorporates two concepts
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growth: number of nodes in network increases over time
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preferential attachment: the more connected a node is, the more likely it is to receive new
edges
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As nodes are added to network, it is connected to m vertices where m is less
than the beginning number of vertices in graph with probability that is
proportional to the number of links that existing nodes already have.
Watts-Strogatz Model
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Created because Erdős-Ré nyi graphs do not have two properties that are in
many real-world networks
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Do not generate local clustering and triadic closures
Do not account for formation of hubs
Algorithm
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Given N nodes, average degree z, and a special parameter
■ 0≤ ≤1
■ N >> K >> ln(N) >> 1
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edges
Construct regular ring lattice
Rewire the edges
■ choose end point with probability
■ avoid self-loops and link duplication
Stochastic Block Model
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Communities
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Subsets that are connected with one another with particular edge densities
Parameters
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n=number of vertices
Partition of vertex set into disjoint subsets C1,...,Cr
Symmetric r x r matrix P of inter-community edge probabilities
Network Probability Matrix
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Probability structure of a network based on the historical presence or absence
of edges in a network
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Example: individuals in a social network
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Simulated by varying the probabilities that certain nodes will communicate
Percolation theory
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Describes the behavior of connected clusters in a random graph
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Consider some liquid is poured on top of some porous material. Will the liquid
be able to make its way from hole to hole and reach the bottom?
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Modelled as 3D network of n x n x n vertices
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Edges is open (allows liquid through) with probability p or closed with probability q
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Problem is called bond percolation
Question: For a given p, what is the probability that a path exists between top
and bottom?
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Erdős and Ré nyi process in a unweighted link percolation on complete graph
Other Applications
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Answer questions about properties of typical graphs
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Property A almost always implies property B
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Study small world phenomena
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Used to model various types of networks
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social networks
internet
power grid
networks of collaborators
neural networks
food web
Open questions
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Chromatic polynomial of a random graph
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number of proper coloring of random graphs given a number of q colors
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Community structure
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Link prediction
Homework
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What is the expected diameter of a random graph with 7 billion vertices and
an average degree of 1000?
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Consider G(1000000, 0.00000001). What kind of components should you
expect? What about G(1000000, 0.001)?
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Convert G(1000000, 0.00000001) (G(n,p)) to G(n,M).
References
[1] https://en.wikipedia.org/wiki/Random_graph
[2] https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model
[3] https://en.wikipedia.org/wiki/Watts_and_Strogatz_model
[4] https://en.wikipedia.org/wiki/Network_probability_matrix
[5] https://en.wikipedia.org/wiki/Percolation_theory
[6] https://en.wikipedia.org/wiki/Community_structure
[7] https://en.wikipedia.org/wiki/Chromatic_polynomial
References
[8] https://en.wikipedia.org/wiki/Rado_graph
[9] https://en.wikipedia.org/wiki/Random_regular_graph
[10] https://en.wikipedia.org/wiki/Paul_Erd%C5%91s
[11] https://en.wikipedia.org/wiki/Edgar_Gilbert
[12] https://en.wikipedia.org/wiki/Alfr%C3%A9d_R%C3%A9nyi
[13] https://en.wikipedia.org/wiki/Barab%C3%A1si%E2%80%93Albert_model
[14] https://en.wikipedia.org/wiki/Stochastic_block_model
References
[15] https://en.wikipedia.org/wiki/Dual-phase_evolution
[16] “The Link Prediction Problem for Social Networks” by David Liben-Nowell and
Jon Kleinberg
[17] “Random Graphs as Models of Networks” by M. E. J. Newman
[18] “Random Graphs” by Edgar Gilbert
[19] “On random graphs I.” by Paul Erdős and Alfré d Ré nyi
[20] “Lecture 4” put together by Sriram Pemmaraju
[21] “The Rado graph and the Urysohn space” by Peter J. Cameron