Small-World Models and Network Growth Models Anastassia Semjonova Roman Tekhov
Small world 6 billion small world? 1960s Stanley Milgram Six degree of separation Small world effect
Motivation Not only friends: - Movie actors - Baseball players Epidemiology - Spread of disease - Flu, HIV, SARS.. Culture formation and reservation - Spread of news, jokes, fashion etc Other - Airplane traffic - Online traffic Virus spread on internet
Small-world network Definition: is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps. clustered networks with small diameters highly regular and thoroughly random networks
Random graph N people z average number of acquitances (coordination number); z=100...1000 ½ Nz connections A has z 2 second neighbours, z 3, z 4,.. D - degrees of separation/diameter z D = N D = log N/log z => N big, D small SIR (susceptible/infectious/recovered) models
Problems Circles of acquitances overlap B has z 2 second neighbours => clustering property C clustering coefficient C=1 fully connected C = z/n random graph 1>>C>>O(N -1 ) real world networks d Poisson (mean) vs Gaussian distribution
Example N number of vertexes l average distance l between pairs of nodes C clustering coefficient C rand clustering coefficient on random graph
Watts and Strogatz model (1998) Clustering + small world properties d(node) ~= const (+social networks, -www) Lattice connections to z neighbour vertexes C can calculate exactly z < 2/3 N N = Ld tends to 3/4 C= 3(z-2d )/4(z-1d) Low dimension lattice with some degree of randomness =>Move one of the ends Animation: http://projects.si.umich.edu/netlearn/netlogo4/smallworldws.html
ER vs SW: clustering coefficient
ER vs SW: average path length
Examples: transmission of SARS Watts-Strogatz K - compactness T the number of infective
Examples: transmission of SARS infodiaphaneity the ability of public media to propagandize the epidemic situation so that people may insulate themselves in time
Newman and Watts Extra links (shortcuts) No links are removed! => no disconnection Easier to analyze
Kasturirangan Problem: a few nodes in the network that have unusually high coordination number or widely distributed set of neighbours 1-d periodic lattice a small number of vertices are added to the center each is connected randomly to a large number of vertices of the original lattice
Kleinberg Note: people are surprisingly good at finding short paths between pairs of individuals given only local information about the structure of the network 2-d lattice (standard grid) + shortcuts between vertices with a probability varying inversely propotional to the distance between them d ij r = (Manhattan distance) x i - x j + y i - y j
Example: routing distributed routing FreeNet peer-to-peer system Agent search simple greedy routing - finds routes - O(log 2 (n)) greedy routing - always picking the neighbor which is closest to the destination, in terms of the lattice distance d, as the next step. //the results are far from what could be hoped for
Properties of SWM Large p Random graph shortest distance s between two points on diametrically opposite sides of the graph 10 shortcuts reduce l by a factor of 2, 100 to reduce it by a factor of 10
Network Growth Models Networks tend to grow as time passes Understanding this process can help to understand and predict the way network properties change in time
α-model Idea: if I introduce my 2 friends to each other then they will also become friends
α-model If I introduce 2 of my friends to each other then they will also become friends Initial edges (friendships) are added randomly
α-model If I introduce 2 of my friends to each other then they will also become friends C introduces B and D F introcudes A and E
α-model If I introduce 2 of my friends to each other then they will also become friends A and D are common friends of B and E, so they are very likely to meet
α-model The process stops when the average number of friendships reaches a predefined limit Leads to isolated communities
Preferential Attachment Model The "rich get richer" principle o It is more likely that a new site will contain a link to Wikipedia rather than some less popular location o Model: o Start with some initial graph o At each stage add a new vertex of degree m o Probability that an old vertex v gets connected to the new vertex is
Preferential Attachment Model m = 2 Initial graph
Preferential Attachment Model Stage 1, add new vertex E m = 2 deg(a) = 3 deg(b) = 2 deg(c) = 2 deg(d) = 1
Preferential Attachment Model Stage 1, add new vertex E m = 2 deg(a) = 3 deg(b) = 2 deg(c) = 2 deg(d) = 1 A, B and C have a higher probability of being connected to E
Preferential Attachment Model Stage 2, add new vertex F m = 2 deg(a) = 4 deg(b) = 3 deg(c) = 2 deg(d) = 1 deg(e) = 2
Preferential Attachment Model Stage 2, add new vertex F m = 2 deg(a) = 4 deg(b) = 3 deg(c) = 2 deg(d) = 1 deg(e) = 2 A and B have a higher probability of being connected to F
Preferential Attachment Model Resulting graph properties o Power-law degree distribution that tends to deg-3 o Low average path length but small clustering coefficient Don't exactly fit the small world Older vertexes have more connections than the new ones, in the real world that is not allways true
Copying Models Start with some initial graph
Copying Models Stage 1, add new vertex E Choose a random old vertex, say A
Copying Models Stage 1, add new vertex E Choose a random old vertex, say A Create edges between E and each neighbor of A with some predefined probability p
Copying Models Power-law degree distribution deg-α α satisfies the equations p(α - 1) = 1 - pα - 1
Copying Models - 2 Stage 1, add new vertex E m = 2 Choose a random old vertex, say C
Copying Models - 2 Stage 1, add new vertex E m = 2 Choose a random old vertex, say C For each i = 1..m create an edge either to some random vertex (with probability β) or some neighbor of C (with probability1 - β)
Copying Models - 2 Also has a power-law degree distribution deg-α α =(2 - β) / (1 - β)