Random Graphs Liang Li April 9, 2014 Outline Objectives Internet - PowerPoint PPT Presentation

Random Graphs Liang Li April 9, 2014

Outline Objectives Internet Topology Melting points [2] History Definition With high probability ( whp ) Near clique Scale free Models Erdős-Rényi Model

Random Graphs Edgar Gilbert Model Results with classical random graphs Giant component Probability methods Ramsey Number Bound Hamiltonian paths Watts and Strogatz Small world Model "Kavin Bacon game" and "Erdős number" Small World Model generating clustering coefficient Liang Li | The University of Tennessee — Department of EECS 3/35

Random Graphs Barabási and Albert Preferential attachment Model generating properties Applications open problems Some open problems Homework problems The average Clustering coefficient Prove or Disprove Liang Li | The University of Tennessee — Department of EECS 4/35

Random Graphs Objectives Internet Topology When you send or receive data over the internet you computer doesn’t really give how the data travels. The media (wire, optic fibre, ox cart) and route (via hong kong 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 fastest link and shortest distance. A program called traceroute finds out where data is flowing by sending out suicidal packets of information that self-destruct after they have seen a set number of computers. Of course some computers don’t care if the packet dies, some respond with nonsense, some respond too quickly or too slowly. 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.gif https://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 neighboring molecules 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 Small world model. Ramsey number Survey articles Watt. Erdős Albert Hamilton path Random graphs BA model Szele Erdős Barabási. Watt Newman Remco 1943 1947 1959-1961 1998 1999 2002 2003 2006 2014 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 preferential attachment model [ 5 ] of Barabási and Albert (1999) [ 1 ] have led to an explosion of 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 lim n →∞ Pr(Graph has Q ) = 1 . Near clique An undirected graph is a near clique if adding an additional edge would make it a clique. Scale free The degree distribution is almost independent of the size of the graph, and the proportion of vertices with degree k is close to proportional to P ( k ) ∼ k − τ , typically 2 < τ < 3 for real network [11]. Or N k ∼ c n k − 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. � n � N � � Thus with the notations N = , 0 ≤ M ≤ N , G ( n, M ) has elements and 2 M � − 1 . � N every element occurs with probability M The random variable G M 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 } in which the edges are chosen independently and with probability p . In other worlds, if G 0 is a graph with vertex set V and it has m edges, then P ( { G 0 } ) = P ( G = G 0 ) = p m (1 − p ) N − m . The random variable G p 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 component Erdős and Rényi discovered that there was a sharp threshold for the appearance of 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, which the largest having only O ( log n ) vertices, where the O symbol means that there is a constant C < ∞ so that the Probability (the largest component is ≤ C log n ) 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 ( log n ) . Here X n ∼ b n means that X n /b n 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, which asks the minimum number of guests R ( m, n ) that must be invited so that at least 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 of vertices v = R ( m, n ) such that all undirected simple graphs of order v contains a clique 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 ( � i A i ) ≤ � i P ( A i ) . Theorem (Erdős (1947)) � n 2 1 − ( m 2 ) < 1 , then R ( m, m ) > n. � If m 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 edge appear independently with probability 0 . 5 . If the probability of the event Q=" no m -clique or independent m -set" is positive, then the desired graph exists. Each possible p -clique occurs with probability 2 − ( m 2 ) , since obtaining the complete graph requires obtaining all its edges, and they occur independently. Hence the � n 2 − ( m 2 ) . The same � probability of having at least one m -clique is bounded by m bound holds for independent m -sets. Hence the probability of "not Q" is bounded � n 2 1 − ( m 2 ) , and the given inequality guarantees that P ( Q ) > 0 . � by m 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 a probability space. We use X = k to denote the event consisting of all elements where variable X has the value k . The expection E ( X ) of a random variable X is the weighted average � k kP ( X = k ) . The pigeonhole property of the expectation is the statement that there exists an element of the probability space for which the value of X is as large as (or as small 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 ! / 2 n − 1 Hamiltonian paths. Proof. [ 2 ] Generate tournament on n randomly by choosing i → j or j → i with equal 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. Each Hamiltonian path occurs with probability 1 / 2 n − 1 , so E ( X ) = n ! / 2 n − 1 . In some tournament, X is at least as large as the expectation. This simple bound using expectation gives almost the right answer for the maximum 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 and 496,489 edges. Facebook released two papers in Nov.2011 that 721 million users with 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 G p graphs have small diameters, but have very few triangles. (while in social networks if A and B are friends and A and C are friends, it is fairly that B and C are also friends.) To construct a network with small diameter and a positive density of K 3 , Watts and Strogatz started a ring lattice with n vertices and k edges per vertex, where the construction interpolates between regularity ( p = 0) and disorder ( p = 1) . Liang Li | The University of Tennessee — Department of EECS 21/35