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Random Graphs
Liang Li
April 9, 2014
OutlineObjectives
Internet TopologyMelting points [2]
History
DefinitionWith high probability (whp)Near cliqueScale free
ModelsErdős-Rényi Model
Random Graphs
Edgar Gilbert Model
Results with classical random graphsGiant component
Probability methodsRamsey Number BoundHamiltonian paths
Watts and Strogatz Small world Model"Kavin Bacon game" and "Erdős number"Small World Model
generatingclustering coefficient
Liang Li | The University of Tennessee — Department of EECS 3/35
Random Graphs
Barabási and Albert Preferential attachment Modelgeneratingproperties
Applications
open problems
Some open problems
Homework problemsThe average Clustering coefficientProve or Disprove
Liang Li | The University of Tennessee — Department of EECS 4/35
Random Graphs
Objectives
Internet TopologyWhen you send or receive data over the internet you computer doesn’t really givehow the data travels. The media (wire, optic fibre, ox cart) and route (via hongkong or Champaign-Urbana) are irrelevant so long as we don’t mind waiting.
Of course, we do mind so in general routers try to route packets over the fastestlink and shortest distance. A program called traceroute finds out where data isflowing by sending out suicidal packets of information that self-destruct after theyhave seen a set number of computers. Of course some computers don’t care ifthe packet dies, some respond with nonsense, some respond too quickly or tooslowly.
Liang Li | The University of Tennessee — Department of EECS 5/35
Random Graphs
Figure 1: The internet topology in 2001 taken from https://www.fractalus.com/steve/stuff/ipmap/
Liang Li | The University of Tennessee — Department of EECS 6/35
Random Graphs
https://www.fractalus.com/steve/stuff/ipmap/layout2.gifhttps://www.fractalus.com/steve/stuff/ipmap/net-anim.gif
Liang Li | The University of Tennessee — Department of EECS 7/35
Random Graphs
Melting points [2]
Think of a solid as a three-dimensional grid of molecules, with neighboringmolecules joined by bonds.1. Adding energy excites molecules and breaks bonds.2. Bonds break at random as the temperature (energy level) raises.3. Break off bonds make the molecules form others, like a liquid or gas.
Liang Li | The University of Tennessee — Department of EECS 8/35
Random Graphs
Figure 2: Melting points
Liang Li | The University of Tennessee — Department of EECS 9/35
Random Graphs
History
1943
SzeleHamilton path
1947
ErdősRamsey number
1959-1961
ErdősRandom graphs
1998
Watt.Small world model.
1999
Barabási.BA model
2002
AlbertSurvey articles
2003
Watt
2006
Newman
2014
Remco
The theory of random graphs was founded by Erdős and Rényi (1959, 1960,1961a,b) after Erdős (1947, 1959, 1961) had discovered that probabilistic meth-ods [6, 7] were often useful in tackling extremal problems in graph theory [3].The small world model [4] of Watts and strogatz(1998) and the preferentialattachment model [5] of Barabási and Albert (1999) [1] have led to an explosionof research [8].
Liang Li | The University of Tennessee — Department of EECS 10/35
Random Graphs
Definition
With high probability (whp)We say that a graph has a certain property Q, if limn→∞ Pr(Graph has Q) = 1.
Near cliqueAn undirected graph is a near clique if adding an additional edge would make it aclique.
Scale freeThe degree distribution is almost independent of the size of the graph, and theproportion of vertices with degree k is close to proportional to P (k) ∼ k−τ ,typically 2 < τ < 3 for real network [11]. Or Nk ∼ cnk−τ [12].
Liang Li | The University of Tennessee — Department of EECS 11/35
Random Graphs
Models
Erdős-Rényi Model
G(n,M) consists of all graphs with vertex set V = {1, 2, ..., n} having M edges,in which the graphs have the same probability.Thus with the notations N =
(n2), 0 ≤M ≤ N , G(n,M) has
(NM
)elements and
every element occurs with probability(NM
)−1.The random variable GM denotes a graph generated in this way.
Liang Li | The University of Tennessee — Department of EECS 12/35
Random Graphs
Edgar Gilbert Model
G(n, P (edge) = p) consists of all graphs with vertex set V = {1, 2, ..., n} inwhich the edges are chosen independently and with probability p.In other worlds, if G0 is a graph with vertex set V and it has m edges, thenP ({G0}) = P (G = G0) = pm(1− p)N−m.The random variable Gp denotes a graph generated in this way.
For M ' pN , the these two models are almost interchangeable [8].
Liang Li | The University of Tennessee — Department of EECS 13/35
Random Graphs
Results with classical random graphs
Giant componentErdős and Rényi discovered that there was a sharp threshold for the appearanceof many properties [1]. Let c > 0 be a constant and set p = c/n.
• if c < 1 , most of the connected components of the graph are small, whichthe largest having only O(logn) vertices, where the O symbol means thatthere is a constant C <∞ so that the Probability (the largest componentis ≤ C logn) tends to 1 as n→∞.
• if c > 1 there is a constant θ(c) > 0, so that the largest component has∼ θ(c)n vertices and the second largest component is O(logn). HereXn ∼ bn means that Xn/bn converges to 1 in probability as n→∞.
Liang Li | The University of Tennessee — Department of EECS 14/35
Random Graphs
Probability methods
Ramsey Number Bound
The Ramsey number [13] R(m,n) gives the solution to the party problem, whichasks the minimum number of guests R(m,n) that must be invited so that atleast m will know each other or at least n will not know each other.
In the language of graph theory, the Ramsey number is the minimum number ofvertices v = R(m,n) such that all undirected simple graphs of order v contains aclique of order m or an independent set of order n.
Liang Li | The University of Tennessee — Department of EECS 15/35
Random Graphs
Using the observation that P (⋃iAi) ≤
∑i P (Ai).
Theorem (Erdős (1947))If(nm
)21−(m
2 ) < 1, then R(m,m) > n.
Liang Li | The University of Tennessee — Department of EECS 16/35
Random Graphs
Proof. [2]
Define a probability model on graphs with vertex set n by letting each edgeappear independently with probability 0.5. If the probability of the event Q=" nom-clique or independent m-set" is positive, then the desired graph exists.
Each possible p-clique occurs with probability 2−(m2 ), since obtaining the complete
graph requires obtaining all its edges, and they occur independently. Hence theprobability of having at least one m-clique is bounded by
(nm
)2−(m
2 ). The samebound holds for independent m-sets. Hence the probability of "not Q" is boundedby(nm
)21−(m
2 ), and the given inequality guarantees that P (Q) > 0.
Liang Li | The University of Tennessee — Department of EECS 17/35
Random Graphs
Hamiltonian paths
A random variable is a function assigning a real number to each element of aprobability space. We use X = k to denote the event consisting of all elementswhere variable X has the value k.The expection E(X) of a random variableX is the weighted average
∑k kP (X = k).
The pigeonhole property of the expectation is the statement that there existsan element of the probability space for which the value of X is as large as (or assmall as) E(X).
Liang Li | The University of Tennessee — Department of EECS 18/35
Random Graphs
Theorem (Szele (1943))Some n vertex tournament has at least n!/2n−1 Hamiltonianpaths.
Proof. [2] Generate tournament on n randomly by choosing i→ j or j → i withequal probability for each pair {i, j}. Let X be the number of Hamiltonian parts;X is the sums of n! indicator variables for the possible Hamiltonian paths. EachHamiltonian path occurs with probability 1/2n−1, so E(X) = n!/2n−1. In sometournament, X is at least as large as the expectation.
This simple bound using expectation gives almost the right answer for themaximum number of Hamiltonian paths in an n-vertex tournament; Alon[14]proved that it is at least n!/(2 + o(1))n.
Liang Li | The University of Tennessee — Department of EECS 19/35
Random Graphs
Watts and Strogatz Small world Model
"Kavin Bacon game" and "Erdős number"
0 1 2 3 4 5 6 7 8
1 1673 130,851 349,031 84,615 6,718 788 107 11
Table 1: Bacon number
Kevin Bacon number is 2.94; Erdős number is 4.7 with 337,454 authors and496,489 edges. Facebook released two papers in Nov.2011 that 721 million userswith 69 billion friendship links, average distance is 4.74.
Liang Li | The University of Tennessee — Department of EECS 20/35
Random Graphs
Small World Model
The Gp graphs have small diameters, but have very few triangles. (while in socialnetworks if A and B are friends and A and C are friends, it is fairly that B and Care also friends.)
To construct a network with small diameter and a positive density of K3, Wattsand Strogatz started a ring lattice with n vertices and k edges per vertex, wherethe construction interpolates between regularity (p = 0) and disorder (p = 1).
Liang Li | The University of Tennessee — Department of EECS 21/35
Random Graphs
generating
Figure 3: Generating small world graphs [15]
• Disallow self-edges.
• Disallow multiple edges.
Liang Li | The University of Tennessee — Department of EECS 22/35
Random Graphs
clustering coefficientDenote L(p) be the average distance between two randomly chosen vertices anddefine clustering coefficient C(p) to be the fraction of connections that existbetween the
(k2)pairs of neighbors of a site.
Local clustering coefficient of node V: |actual edges between neighbors of v|| possible edges between neighbors of v|
The clustering coefficient for the whole graph is the average of the local values.
Figure 4: C(v) = 46
Liang Li | The University of Tennessee — Department of EECS 23/35
Random Graphs
A graph is considered small world, if:
• its average clustering coefficient is significantly higher than the one of arandom graph constructed on the same vertex set, and
• it has approximately the same mean shortest path length as its correspond-ing random graph.
The regular graph has L(0) ∼ n/2k and C(0) ≈ 3/4 if k is large, which thedisorder one has L(1) ∼ (logn)(log k) and C(1) ∼ k/n. Here L(p) decreasesquickly near 0, which C(p) changes slowly so there is a broad interval of p overwhich L(p) is almost as small as L(1), yet C(p) is far from 0 [1].
• Small-world networks tend to contain cliques, and near-cliques.
• Most paris of nodes will be connected by at least one short path.
Liang Li | The University of Tennessee — Department of EECS 24/35
Random Graphs
Barabási and Albert Preferential attachmentModel
BA model an algorithm for generatingrandom scale-free networks using a pref-erential attachment mechanism.It incorporates two important generalconcepts:Growth means the number of nodes inthe network increases over time.Preferential attachment means thatthe more connected a node is, themore likely it is to receive new links.
Liang Li | The University of Tennessee — Department of EECS 25/35
Random Graphs
generating
The network begins with an initial connected network of m0 nodes.New nodes are added to the network one at a time with the probability that isproportional to the number of links that the existing nodes already have:pi = ki∑
jkj
where ki is the degree of node i and the sum is made over all pre-existing nodesj.
Liang Li | The University of Tennessee — Department of EECS 26/35
Random Graphs
properties
• BA model is scale free. Its power law of the form p(k) ∼ k−3
• The average path length increases approximately with the size of thenetwork l ∼ lnN
ln lnN
• The clustering coefficient with network size C ∼ N−0.75
For example, on the web, very well known sites such as Google or Wikipedia,rather than to pages that hardly anyone knows will be more likely to be linked. Ifsomeone selects a new page to link to by randomly choosing an existing link, theprobability of selecting a particular page would be proportional to its degree.
Liang Li | The University of Tennessee — Department of EECS 27/35
Random Graphs
ApplicationsWorld Wide Web...Internet...Movie actor collaboration network...Cellular networks....Ecological networks...Phone call network ...Citation network...Networks in linguistics...Power and neural networks........
Liang Li | The University of Tennessee — Department of EECS 28/35
Random Graphs
open problemsRandom Structures: a model of real world networks, such as Internet, socialnetwork or biological networks it leaves a lot to be desired.
Figure 5: The internet topology in 2001 taken from https://www.fractalus.com/steve/stuff/ipmap/
Liang Li | The University of Tennessee — Department of EECS 29/35
Random Graphs
Some open problemsIs that true that who Gm has δ(Gm)/2 Hamilton cycles?[19]It is known to be true as long as δ(Gm)/2 = o(average degree).
What is the expected time to for a random walk to get within distance d for everyvertex?
More problems[20]:Ramsey theory...Graph coloring problems.....
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Random Graphs
Homework problems
The average Clustering coefficient
Figure 6:
Liang Li | The University of Tennessee — Department of EECS 31/35
Random Graphs
Prove or Disprove
When p is constant, then almost every Gp is has diameter 2 (and Gp is connected).
Liang Li | The University of Tennessee — Department of EECS 32/35
Random Graphs
References[1] Rick Durrett, Random Graph Dynamics (Cambridge Series in Statistical and Probabilistic Mathematics), Cambridge
University Press, New York, NY, 2006
[2] D. B. West. Introduction to Graph Theory (2nd Edition). Edited by Prentice Hall. Prentice Hall, 2001.
[3] http://en.wikipedia.org/wiki/Random_graph
[4] D. J.Watts, S. H. Strogatz(1998). "Collective dynamics of ’small-world’ networks". Nature 393 (6684): 440Ð442.
[5] R. Albert, A.-L. Barabási (2002). "Statistical mechanics of complex networks". Reviews of Modern Physics 74: 47Ð97.
[6] Erdös, P.; Rényi, A. (1959). "On Random Graphs. I". Publicationes Mathematicae 6: 290Ð297
[7] Erdös, P.; Rényi, A. (1960). "The Evolution of Random Graphs". Magyar Tud. Akad. Mat. Kutató Int. Közl. 5: 17Ð61.
[8] Bollobas, B. and Riordan, O.M.(2003) "Mathematical results on scale-free random graphs" in "Handbook of Graphs andNetworks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed.
[9] http://en.wikipedia.org/wiki/Small-world_network#Properties_of_small-world_networks
[10] http://en.wikipedia.org/wiki/Barab%C3%A1si%E2%80%93Albert_model
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Random Graphs
[11] http://en.wikipedia.org/wiki/Scale-free_network
[12] R. V. D. Hofstad. (2014) "Random Graphs and Complex NetworksÓ. Department of Mathematics and Computer ScieneEindhoven University of Technology.
[13] http://mathworld.wolfram.com/RamseyNumber.html
[14] Alon, Noga. (1990)"The maximum number of Hamiltonian paths in tournaments." Combinatorica VOL.10. NO 4. 319-324.
[15] http://cs.brynmawr.edu/Courses/cs380/spring2013/section02/slides/06_SmallWorldNetworks.pdf
[16] http://en.wikipedia.org/wiki/Barab%C3%A1si%E2%80%93Albert_model#Clustering_coefficient
[17] Albert, RŐka, and Albert-LĞszlŮ BarabĞsi. (2002) "Statistical mechanics of complex networks." Reviews of modernphysics VOL. 74.NO. 1:47-93.
[18] Watts, Duncan J.; Strogatz, Steven H. (June 1998). "Collective dynamics of ’small-world’ networks". Nature 393 (6684):440Ð442.
[19] http://www.math.cmu.edu/~af1p/Talks/RandomGraphs/rgtalk.pdf
[20] Chung, F. R. K. "Open problems of Paul Erdos in graph theory." Journal of Graph Theory 25.1 (1997): 3-36.
Liang Li | The University of Tennessee — Department of EECS 34/35
Questions?
