Evolution of Networks

1. Evolution of Networks Notes from Lectures of J.Mendes CNR, Pisa, Italy, December 2007 Eva Jaho Advanced Networking Research Group National and Kapodistrian University of Athens February 2008

2. Presentation Outline • Basic graph notions • Tree • Loops • Degree distribution • Clustering • Average length • k-core • Equilibrium random networks • Small-world networks • Watts-Strogatz model • Non-equilibrium random networks • A citation graph • Growing exponential networks

3. Presentation Outline (cntd.) • Non-equilibrium random networks (cntd.) • Barabasi-Albert model • Linear preference - general case • How does the general case relates to the Barabasi-Albert model? • Mixture of random and preferential attachments • Growth of the WWW • Accelerated growth of networks • Example of an accelerated growth • Evolution of language • Failures and attacks

4. Tree • A tree is a particular kind of graphs without loops. A connected tree is a tree with no separate parts. • In a tree: N = L+1 (N vertices, L edges)

5. Loops • In a connected undirected graph, the number I of loops is related to N and L. • I = L+1-N • Two loops in the figure (cannot be reduced to smaller ones)

6. ki ko Degree distribution • In a directed network (directed edges), • in-degree ki: number of incoming edges of a vertex • out-degree ko: number of outgoing edges of a vertex • k = ki+ko: degree, the total number of connections.

7. Degree distribution (cntd.) • p(k,s,N): probability that the vertex s in the network of size N has k connections (k nearest neighbours) • The degree distribution of a randomly chosen node is: • The average degree of a randomly chosen node in a network of size N is:

8. Degree distribution (cntd.) • Poisson: , e.g. a classical random equilibrium graph of Erdos and Renyi when the total number of vertices is infinite • Exponential: , e.g. a citation graph with attachment of new vertices to randomly chosen old ones • Power-law: , e.g. a citation graph with attachment of new vertices to preferentially chosen old ones

9. i Clustering • The clustering coefficient characterizes the “connectedness” of the environment close to a vertex. • ni: number of connections among the neighbors • ki(ki-1)/2: number of possible connections among the neighbors.

10. Clustering (cntd.) • The average value reflects how connected are the neighboring nodes • also shows the “density” of small loops of length 3 • of a tree is 0, of a fully connected graph (clique) is 1

11. Average length • Distance between two vertices = length of the shortest path between them • Distances l are distributed with some distribution function P(l) • P(l): the probability that the length of the shortest path between two randomly chosen vertices is l • Average length of the shortest path

12. Average length (cntd.) • In a tree-like network about vertices are at distance l or closer from a vertex. The average length is: • Tree-like networks have the small-world effect: average length l is still small for large size of networks N

13. Clique of 4 nodes: 3-core Tree: 1-core Cycle: 2-core k-core • k-core: maximum sub-graph of a graph whose vertices have degree at least k within this sub-graph

14. k-core (cntd.) • Pruning rule: • Remove from the graph all vertices with degree smaller than k • Some vertices may have now less than k degree • Prune again the vertices • Repeat until no more pruning is possible • Example of 2-core:

15. Equilibrium random networks • Example: The classical random graph (the Erdos-Renyi model) • Rules: (a) The total number of vertices is fixed (b) Randomly chosen vertices are connected

16. Small-world networks • Small-world networks combine • high clustering of regular lattices • “small-world effect” (small average shortest-path length) of random networks • Construction • From regular lattices by rewiring bonds (Watts and Strogatz (1998)) • by making shortcuts between randomly chosen vertices

17. p=0 p small p large Watts-Strogatz model • All the edges of a regular lattice are rewired with a probability p to randomly chosen vertices • p small, similar to original regular lattice • p large, similar to the classical random graph

18. Watts-Strogatz model (cntd.) • Normalized average shortest-path length L(p)/L(0) and clustering coefficient C(p)/C(0) vs the fraction p of rewired connecions small-world

19. Non-equilibrium random networks • A random graph growing through the simultaneous addition of vertices and edges • Rules: (a) At each time step, a new vertex is added to the graph (b) Simultaneously, a pair of randomly chosen vertices (or several pairs) is connected by an edge

20. Non-equilibrium random networks (cntd.) • An example: (a) (b) (c) (d) (e) • The “oldest” vertices are the most connected and degrees of new nodes are the smallest

21. A citation graph • At each time step, a new vertex is added to the graph • It is connected with some old node via an edge (a) (b) (c) (d) (e)

22. s=2 Old net s=1 New vertex Randomly chosen vertex t=2 Growing exponential networks • Initial network • s: vertex, t: time step • At each time step t • A new vertex is attached to a randomly chosen vertex • There are t vertices and t edges

23. Growing exponential networks (cntd.) • p(k,s,t): probability that vertex s has degree k at time t • Evolution: • Initial condition: • Boundary condition:

24. Growing exponential networks (cntd.) • P(k,t): probability that a randomly chosen vertex has degree k at time t • Initial condition: • : stationary degree distribution • Stationary equation:

25. Old net New vertex Preferentially chosen vertex The Barabasi-Albert model • Barabasi and Albert (1999) combined the growth and preferential linking • At each time step t • A new vertex is attached to an old one with a probability proportional to its degree

26. The Barabasi-Albert model (cntd.) • k/(2t): probability that vertex s of degree k gets a new connection at time t (2t, total degree) • Initial condition: • Boundary condition: • Stationary equation: (γ=3)

27. m New vertex Old net Linear preference - general case • Directed growing network, considering only in-degree distributions q=ki • At each time step, a new vertex is added to the network • m: incoming links (don’t care about source ends) • The probability that a new edge becomes attached to some vertex of in-degree q is proportional to q+A (A, positive constant)

28. Linear preference – general case (cntd.) • Equation for the average in-degree of individual vertices: • where • Solution: • Power-law:

29. Linear preference – general case (cntd.) • While parameter A increases from 0 to ∞ • γ increases from 2 to ∞ • β decreases from 1 to 0 • If A → 0 • Most connections will be to the oldest vertex • If A → ∞ • Preference is absent • Similar to exponential growing network

30. Old net Preferentially chosen vertices New vertex How does the general case relate to the Barabasi-Albert model? • At each time step a new vertex has m outgoing connections • Old vertices obtain only incoming links • Degree of an arbitrary vertex: k=q+m • In general case if A=m(β=1/2 and γ=3) • the probability of attaching a new edge - proportional to the degree of a vertex (q+A=q+m=k), that is Barabasi-Albert model

31. n m nr New vertex Old net Mixture of random and preferential attachments • At each time step a new vertex with n incoming links is added • Simultaneously, m new edges become attached to preferentially chosen vertices • And in addition, nr new edges become attached to randomly chosen vertices

32. Mixture of random and preferential attachments (cntd.) • Equation for the average in-degree of individual vertices: • where • The degree distribution of the network will remain power-law • β=m/(m+n+nr+A) • γ=2+(n+nr+A)/m

33. WWW New Web document Growth of the WWW • WWW is an array of documents (pages) connected by hyperlinks, which are mutual references in these documents • A new Web document must have at least one incoming hyperlink to be accessible • It has several (or none) references to existing documents of the WWW • Old pages can be updated, and so new hyperlinks can appear between them

34. Growth of the WWW (cntd.) • β=m/(m+n+nr+A) • γ=2+(n+nr+A)/m • From experimental data • (m+n+nr)≈10 • n=1 and if A+nr=0 • γ≈2.1

35. Accelerated growth of networks • Accelerated growth ≡ average degree grows • Assume (growth exponent a>0) • ( ) • Degree distribution is power-law with k0(t)≤k≤kcut(t), z>0: • Normalization: • To converge:

36. Accelerated growth of networks (cntd.) • Cut-off position: • for any γ>1 -> • 1<γ<2

37. Accelerated growth of networks (cntd.) • For γ<2, z<a<1 • If z -> 0 • Only kcut increases with time, P(k) stationary • γ>2 • For γ>2 • z>a>0 • P(k) not stationary

38. Old net New vertex Example of an accelerated growth • Undirected citation graph • A new vertex is attached to a randomly chosen old one • plus to some of its nearest neighbors, to each one of them with probability p

39. Example of an accelerated growth (cntd.) • For the total number of edges L(t): • If p<1/2 • Stationary degree distribution • If p>1/2 • Non-stationary degree distribution

40. Old word web New word Evolution of language • The total number of connections between words grows more rapidly than the number of words in the language • Accelerated growth of Word Web

41. Failures and attacks • The integrity of a network depends on the giant (largest connected) component • A network is damaged by eliminating its giant component • Simulations of the random damage (failure) and attack on exponential and power-law networks • Cluster size distribution for different f (fraction of deleted vertices) • For small f under failure or attack • There is still a large cluster in exponential and scale-free networks

42. Failures and attacks (cntd.) • For large f under failure • Elimination of the giant component in exponential network • There is still a large cluster in scale-free network • Scale-free nets under attack behave similarly as exponential nets under attack or failures • Scale-free nets are robust to random failures