Class 4: Random Graphs Network Science: Random Graphs 2012 Prof. Albert-László Barabási Dr. Baruch Barzel, Dr. Mauro Martino

See: BOEING BATTERY FAILURE

HUMAN DISEASE NETWORK

FIGHTING TERRORISM AND MILITARY Network Science: Introduction 2012

FIGHTING TERRORISM AND MILITARY Network Science: Introduction 2012 Started in 2009 includes four interconnected centers: 1.Social Cognitive Networks anchored at RPI 2.Information Networks anchored at UIUC 3.Communication Networks anchored at Penn State 4.Integrative Research Center anchored at BBN 10 year funding of over $120 mln awarded through nation-wide competition.

RealProjected EPIDEMIC FORECAST Predicting the H1N1 pandemic Network Science: Introduction 2012

Thex In September 2010 the National Institutes of Health awarded $40 million to researchers at Harvard, Washington University in St. Louis, the University of Minnesota and UCLA, to develop the technologies that could systematically map out brain circuits. The Human Connectome Project (HCP) with the ambitious goal to construct a map of the complete structural and functional neural connections in vivo within and across individuals. BRAIN RESEARCH Network Science: Introduction 2012

National Research Council: Network Science: Introduction 2012

GENERAL AUDIENCE Network Science: Introduction 2012

BOOKS Handbook of Graphs and Networks: From the Genome to the Internet (Wiley-VCH, 2003). S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, 2003). S. Goldsmith, W. D. Eggers, Governing by Network: The New Shape of the Public Sector (Brookings Institution Press, 2004). P. Csermely, Weak Links: The Universal Key to the Stability of Networks and Complex Systems (The Frontiers Collection) (Springer, 2006), rst edn. M. Newman, A.-L. Barabasi, D. J. Watts, The Structure and Dynamics of Networks: (Princeton Studies in Complexity) (Princeton University Press, 2006), rst edn. L. L. F. Chung, Complex Graphs and Networks (CBMS Regional Conference Series in Mathematics) (American Mathematical Society, 2006). Network Science: Introduction 2012

BOOKS R. Pastor-Satorras, A. Vespignani, Evolution and Structure of the Internet: A Statistical Physics Approach (Cambridge University Press, 2007), rst edn. F. Kopos, Biological Networks (Complex Systems and Interdisciplinary Science) (World Scientic Publishing Company, 2007), rst edn. B. H. Junker, F. Schreiber, Analysis of Biological Networks (Wiley Series in Bioinformatics) (Wiley-Interscience, 2008). T. G. Lewis, Network Science: Theory and Applications (Wiley, 2009). E. Ben Naim, H. Frauenfelder, Z.Torotzai, Complex Networks (Lecture Notes in Physics) (Springer, 2010), rst edn. M. O. Jackson, Social and Economic Networks (Princeton University Press, 2010). Network Science: Introduction 2012

1998: Watts-Strogatz paper in the most cited Nature publication from 1998; highlighted by ISI as one of the ten most cited papers in physics in the decade after its publication. 1999: Barabasi and Albert paper is the most cited Science paper in 1999;highlighted by ISI as one of the ten most cited papers in physics in the decade after its publication. 2001: Pastor -Satorras and Vespignani is one of the two most cited papers among the papers published in 2001 by Physical Review Letters. 2002: Girvan-Newman is the most cited paper in 2002 Proceedings of the National Academy of Sciences. Original papers: Network Science: Introduction 2012

Thex If you were to understand the spread of diseases, can you do it without networks? If you were to understand the WWW structure, searchability, etc, hopeless without invoking the Web’s topology. If you want to understand human diseases, it is hopeless without considering the wiring diagram of the cell. MOST IMPORTANT Networks Really Matter Network Science: Introduction 2012

Thex NGRAMSNetworks Awareness Network Science: Introduction 2012

Degree distribution p k THREE CENTRAL QUANTITIES IN NETWORK SCIENCE Average path length Clustering coefficient C Network Science: Graph Theory 2012

Degree distribution P(k): probability that a randomly chosen vertex has degree k N k = # nodes with degree k P(k) = N k / N ➔ plot k P(k) DEGREE DISTRIBUTION Network Science: Graph Theory 2012

discrete representation: p k is the probability that a node has degree k. continuum description: p(k) is the pdf of the degrees, where represents the probability that a node’s degree is between k 1 and k 2. Normalization condition: where K min is the minimal degree in the network. DEGREE DISTRIBUTION Network Science: Graph Theory 2012

Clustering coefficient: what portion of your neighbors are connected to each other? Node i with degree k i, so the maximum number of edges is k i (k i - 1)/2 C i in [0,1] CLUSTERING COEFFICIENT Network Science: Graph Theory 2012

Degree distribution: P(k) Path length: Clustering coefficient: THREE CENTRAL QUANTITIES IN NETWORK SCIENCE Network Science: Graph Theory 2012

A. Degree distribution: p k B. Path length: C. Clustering coefficient: THREE CENTRAL QUANTITIES IN NETWORK SCIENCE Network Science: Graph Theory 2012

ANOTHER EXAMPLE -THREE CENTRAL QUANTITIES IN NETWORK SCIENCE Both networks have two parts, each a symmetric mirror image of the other, so we consider just nodes of left square in each network, its nodes identified by position as UL, LL, UR, LR. Degrees: blue UL,LL,UR 2, LR 3, so =2.25, red UL,LL,UR 3, LR 4, so =3.25. Distributions are show to the right of graphs. Blue paths are UL=(2,1,1,2,1), LL,UR=(2,2,2,1) and LR=(3,3,1), d max =5, = 2.28 Red paths are UL,LL,UR=(3,1,3) and LR(4,3), d max =3, = 1. Clustering coefficients are: blue all 0, red UL,LL,UR=1 and LR=1/2 Blue network Red network /2 P(k) k

The above assumes end nodes connected so, largest distance is N/2 nodes. The average path-length varies as Constant degree, constant clustering coefficient. ONE DIMENSIONAL LATTICE: nodes on a line Network Science: Graph Theory 2012

RANDOM NETWORK MODEL Network Science: Random Graphs 2012

Erdös-Rényi model (1960) Connect with probability p p=1/6 N=10  k  ~ 1.5 Pál Erdös ( ) Alfréd Rényi ( ) RANDOM NETWORK MODEL Network Science: Random Graphs 2012 Here = (6+2*2+4)/10 = 1.4

RANDOM NETWORK MODEL Definition: A random graph is a graph of N labeled nodes where each pair of nodes is connected by a preset probability p. We will call is G(N, p). Network Science: Random Graphs 2012

RANDOM NETWORK MODEL p=1/6 N=12 Network Science: Random Graphs 2012

RANDOM NETWORK MODEL p=0.03 N=100 Network Science: Random Graphs 2012

RANDOM NETWORK MODEL N and p do not uniquely define the network– we can have many different realizations of it. How many? N=10 p=1/6 The probability to form a particular graph G(N,p) isThat is, each graph G(N,p) appears with probability P(G(N,p)). Network Science: Random Graphs 2012

RANDOM NETWORK MODEL P(L): the probability to have exactly L links in a network of N nodes and probability p: The maximum number of links in a network of N nodes. Number of different ways we can choose L links among all potential links. Binomial distribution... Network Science: Random Graphs 2012

MATH TUTORIAL the mean of a binomial distribution There is a faster way using generating functions, see: Network Science: Random Graphs 2012

MATH TUTORIAL the variance of a binomial distribution Network Science: Random Graphs 2012

MATH TUTORIAL the variance of a binomial distribution Network Science: Random Graphs 2012

MATH TUTORIAL Binomian Distribution: The bottom line Network Science: Random Graphs 2012

RANDOM NETWORK MODEL P(L): the probability to have a network of exactly L links The average number of links in a random graph The standard deviation Network Science: Random Graphs 2012

DEGREE DISTRIBUTION OF A RANDOM GRAPH As the network size increases, the distribution becomes increasingly narrow—we are increasingly confident that the degree of a node is in the vicinity of. Select k nodes from N-1 probability of having k edges probability of missing N-1-k edges Network Science: Random Graphs 2012

DEGREE DISTRIBUTION OF A RANDOM GRAPH For large N and small k, we can use the following approximations: for Network Science: Random Graphs 2012

DEGREE DISTRIBUTION OF A RANDOM GRAPH P(k) k Network Science: Random Graphs 2012

DEGREE DISTRIBUTION OF A RANDOM NETWORK Exact Result -binomial distribution- Large N limit -Poisson distribution- Probability Distribution Function (PDF) Network Science: Random Graphs 2012

What does it mean? Continuum formalism: If we consider a network with average degree then the probability to have a node whose degree exceeds a degree k 0 is: For example, with =10, the probability to find a node whose degree is at least twice the average degree is the probability to find a node whose degree is at least ten times the average degree is × the probability to find a node whose degree is less than a tenth of the average degree is See The probability of seeing a node with very high of very low degree is exponentially small. Most nodes have comparable degrees. The larger the size of a random network, the more similar are the node degrees What does it mean? Discrete formalism: NODES HAVE COMPARABLE DEGREES IN RANDOM NETWORKS

NO OUTLIERS IN A RANDOM SOCIETY According to sociological research, for a typical individual k ~1,000 The probability to find an individual with degree k>2,000 is Given that N ~10 9, the chance of finding an individual with 2,000 acquaintances is so tiny that such nodes are virtually inexistent in a random society.  a random society would consist of mainly average individuals, with everyone with roughly the same number of friends.  It would lack outliers, individuals that are either highly popular or recluse. Network Science: Random Graphs 2012