RANDOM-REAL NETWORKS 1
Random networks: model A random graph is a graph of N nodes where each pair of nodes is connected by probability p: G(N,p)
Random networks: model p=1/6 N=12 L=8 L=10 L=7 The number of links is variable N(N-1) 2 å < L >= LP(L) = p L= 0 N(N -1) 2 3
Degree distribution Real networks are scale-free
Random networks: degree distribution Binomial distribution: æ P(k) = N -1 ö ç è k ø p k (1- p) (N-1)-k < k >= p(n -1) p = < k > (N -1) For large N and small p, it can be approximated by a Poisson distribution: P(k) = e -< k> < k > k k! 5
Real networks are scale-free Degree distribution is power-law 6
Scale-free real networks 7
Degree distribution Random networks Poisson distribution P(k) = e -< k> < k > k k! < k >= p(n -1) Real networks Power-law distribution 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. Hubs: individuals that are highly popular 8
Connected components Real networks: giant component and power-law connected components size distribution
Connected components P=0 disconnected nodes, <k>=0, N_G=1 P=1 fully connected, <k>=n-1, N_G=N One would expect that that the largest component grows gradually from N_G=1 to N_G=N 10
Section 3.4
Connected components disconnected nodes NETWORK. <k> Erdos and Renyi (1959): the condition for the emergence of a giant component is <k>=1. It is evident that one link per node is necessary, but counterintuitive that it also sufficient.
N=100 I: Subcritical <k> < 1 II: Critical <k> = 1 III: Supercritical <k> > 1 IV: Connected <k> > ln N <k> <k>=0.5 <k>=1 <k>=3 <k>=5
I: Subcritical <k> < 1 p < p c =1/N <k> No giant component. Isolated clusters, cluster size distribution is exponential The largest cluster is a tree, its size ~ ln N
II: Critical <k> = 1 p=p c =1/N <k> Unique giant component: N G ~ N 2/3 contains a vanishing fraction of all nodes, N G /N~N -1/3 Small components are trees, GC has loops. Cluster size distribution: p(s)~s -3/2 A jump in the cluster size: <k>=1,000 ln N~ 6.9; N 2/3 ~95 N=7 10 9 ln N~ 22; N 2/3 ~3,659,250
III: Supercritical <k> > 1 p > p c =1/N <k> <k>=3 Unique giant component: N G ~ (p-p c )N GC has loops. Cluster size distribution: exponential p(s) ~ s -3 / 2 -( k -1)s+(s-1)ln k e
IV: Connected <k> > ln N p > (ln N)/N <k>=5 <k> Only one cluster: N G =N GC is dense. Cluster size distribution: None
Connected components: real networks 19
Connected components: real networks Supercritical: not fully connected Internet: we should have routers that, being disconnected from the giant component, are unable to communicate with other routers. Power grid: some consumers should not get powered Fully connected Social media: no individual disconnected 20
Diameter and path length Small World
SIX DEGREES 1967: Stanley Milgram HOW TO TAKE PART IN THIS STUDY 1. ADD YOUR NAME TO THE ROSTER AT THE BOTTOM OF THIS SHEET, so that the next person who receives this letter will know who it came from. 2. DETACH ONE POSTCARD. FILL IT AND RETURN IT TO HARVARD UNIVERSITY. No stamp is needed. The postcard is very important. It allows us to keep track of the progress of the folder as it moves toward the target person. 3. IF YOU KNOW THE TARGET PERSON ON A PERSONAL BASIS, MAIL THIS FOLDER DIRECTLY TO HIM (HER). Do this only if you have previously met the target person and know each other on a first name basis. 4. IF YOU DO NOT KNOW THE TARGET PERSON ON A PERSONAL BASIS, DO NOT TRY TO CONTACT HIM DIRECTLY. INSTEAD, MAIL THIS FOLDER (POST CARDS AND ALL) TO A PERSONAL ACQUAINTANCE WHO IS MORE LIKELY THAN YOU TO KNOW THE TARGET PERSON. You may send the folder to a friend, relative or acquaintance, but it must be someone you know on a first name basis. Network Science: Random Graphs
SIX DEGREES 1967: Stanley Milgram Network Science: Random Graphs
SIX DEGREES 1991: John Guare "Everybody on this planet is separated by only six other people. Six degrees of separation. Between us and everybody else on this planet. The president of the United States. A gondolier in Venice. It's not just the big names. It's anyone. A native in a rain forest. A Tierra del Fuegan. An Eskimo. I am bound to everyone on this planet by a trail of six people. It's a profound thought. How every person is a new door, opening up into other worlds." Network Science: Random Graphs
Random graph: diameter Random graphs tend to have a tree-like topology with almost constant node degrees. N =1+ k + k 2 +...+ k d max = k d max +1-1 k -1» k d max d max = logn log k 25
Small World In most networks this offers a better approximation to the average distance between two randomly chosen nodes, d, than to d max. < d >= log N log k Small world phenomena: the property that the average path length or the diameter depends logarithmically on the system size. Small means that d is proportional to log N 26
Given the huge differences in scope, size, and average degree, the agreement is excellent.
Social networks N=7*10^9, <k>=1000 < d >= ln(n) ln k = 3.28 Using Facebook s social graph of May 2011, consisting of 721 million active users and 68 billion symmetric friendship links, researchers found an average distance 4.74 between the users. Therefore, the study detected only four degrees of separation. 28
WWW: 19 DEGREES OF SEPARATION Image by Matthew Hurst Blogosphere Network Science: Random Graphs Smaller average degree and larger order than the social network [d=ln N/ln <k>]
Clustering coefficient Real networks: triadic closure
Random networks: clustering coefficient The clustering coefficient of random graphs is small. For fixed degree, C decreases with the system size N. C is independent of a node s degree k. 31
Clustering coefficient: real networks In May 2011, Facebook had an average clustering coefficient of 0.5 for individuals who had 2 friends. 32
Real networks: clustering coefficient Random : (a) Green line: random network. The average clustering coefficient decreases as 1/N Circles: real networks The average clustering coefficient is independent of N. (b) (d) Dependence of the local clustering coefficient on node s degree. Green line: average clustering coefficient in random networks 33
Clustering coefficient Random networks The clustering coefficient of random graphs is small. For fixed degree C decreases with the system size N. C is independent of a node s degree k. Real networks A much higher clustering coefficient than expected for a random network of similar N and L. Independent of N High-degree nodes tend to have a smaller clustering coefficient than lowdegree nodes. 34
Small World Model
Small-world Model Small-world Model also known as the Watts and Strogatz model is a special type of random graphs with small-world properties, including: Short average path length High clustering. It was proposed by Duncan J. Watts and Steven Strogatz in their joint 1998 Nature paper
Constructing Small World Networks regular ring lattice of degree c: nodes are connected to their previous c/2 and following c/2 neighbors. As in many network generating algorithms Disallow self-edges Disallow multiple edges
Real-World Network and Simulated Graphs
Small world model Random networks Low average path length Real networks Low average path length High clustering coefficient Clustering coefficient independent of k High clustering coefficient Clustering coefficient dependent on k Poisson-like degree distribution Power-law degree distribution How can human networks be both clumpy and have short distances? The strength of weak ties 39