Random graphs and complex networks

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1 Random graphs and complex networks Julia Komjathy, Remco van der Hofstad Random Graphs and Complex Networks (2WS12) course

2 Complex networks Figure 2 Ye a s t p ro te in in te ra c tio n n e tw o rk. A m a p o f p ro tein p ro tein in tera c tio n s 18 in Yeast protein interaction network Internet topology in 2001

3 Complex networks 2. Friendship network on facebook Dec 2010

4 Network functions Internet: WWW: Information gathering Friendship networks: gossiping, spread of information and disease Online friendship networks: gossiping, spread of information, advertising Power grids: reliability

5 Network functions Internet: Routing on networks, congestion, network failure WWW: Information gathering: advertisement Crawling networks, motion on networks, Friendship networks: gossiping, spread of information and disease Spread of diseases, motion on networks, consensus reaching Online friendship networks: gossiping, spread of information, advertising Spread of memes, gossips, targeted advertising, etc. Power grids: reliability Robustness to (random and deliberate) attacks

6 Network functions Internet: Routing on networks, congestion, network failure WWW: Information gathering Crawling networks, motion on networks Friendship networks: gossiping, spread of information and disease Spread of diseases, motion on networks, consensus reaching Online friendship networks: gossiping, spread of information, advertising Spread of memes, gossips, targeted advertising, etc. Power grids: reliability Robustness to (random and deliberate) attacks Fundamental properties networks still ill understood!

7 Internet and WWW Vertices Internet are Internet routers, edges physical links between routers. Edges have weights. is sent along path with smallest sum of weights. Vertices WWW are web pages, edges are (directed) hyper links between pages. Properties: Internet is large, chaotic, (fairly) homogeneous, connected,... WWW is huge, directed, hard to measure, non-connected,... Router (IP) and autonomous system (AS) lev- Levels Internet: els.

8 Scale-free paradigm " out" exp( ) * x ** ( ) ¾ À Á Â Ã Ä ÅÅÄ ÆÇ 1 1 ÌÍ ÎÏÐ Ñ=Ò ÓÔ Õ$Ñ7ÖÏ Ã5 Ñ-ÎÐ Ñ-Ñ7Ø$Ù Ö Ã Ú ÓÛ/ÖÎ Ä Ù ÖÎXØ$Ù Ö Ã ÖÜÜ Ð Ñ@Ý Ï$Ñ Â$ÞßJ Loglog plot of degree sequences in Internet Movie Data Base (2007) and in the AS graph (FFF97) " out" exp( ) * x ** ( )

9 Scale-free paradigm Degree sequence (N 1, N 2, N 3,...) of graph: N 1 is number of elements with degree 1, N 2 is number of elements with degree 2,... N k is number of elements with degree k. Then precisely when N k Ck τ, log N k log C τ log k.

10 Small-world paradigm Frequency Gay.eu 8 1e8 histogram for user distance in ( values) Density LiveJournal histogram for user distance in 2007 ( values) Distance Distance Distances in social networks gay.eu on December 2008 and livejournal in 2007.

Six degrees of separation Everybody on this planet is separated only by six other people. Six degrees of separation. Between us and everybody else on this planet. The president of the United States.

11 Six degrees of separation Everybody on this planet is separated only by 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 the rain forest. (...) An Eskimo. I am bound to everyone on this planet by a trail of six people. It is a profound thought.

12 Six degrees of separation In a 1967 experiment by Milgram, various persons in the Mid-West were asked to send a package to someone in Massachusetts, via intermediaries they know on first name basis. Experimental fact: (almost always) at most six intermediaries: Six degrees of separation. Experiment received enormous attention, but was later heavily criticized due to the small data set that it was based upon. In 2000, Duncan Watts performed an version of the experiment on a huge scale, with roughly the same outcome: Small-world phenomenon.

13 Collaboration network, Erdos number vertices are mathematicians: N two mathematicians share an edge if they wrote a paper together: E Distance from Paul Erdos Graph has a power law with τ 2.97

14 Networks and Graphs Ann Barbara Frank Chris Emma Dave

15 Networks and Graphs Ann Barbara Frank Chris Emma Dave

16 Networks and Graphs A B F C E D

17 Modeling complex networks Inhomogeneous Random Graphs: Static random graph, independent edges with inhomogeneous edge occupation probabilities, yielding scale-free graphs. (BJR07, CL02, CL03, BDM-L05, CL06, NR06, EHH06,...) Configuration Model: Static random graph with prescribed degree sequence. (MR95, MR98, RN04, HHV05, EHHZ06, HHZ07, JL07, FR07,...) Preferential Attachment Model: Dynamic random graph, attachment proportional to degree plus constant. (BA99, BRST01, BR03, BR04, M05, B07, HH07,...)

18 New aspect: role of communities weighted community model from Kertesz et al.

19 New trends: Networks with communities Random intersection graphs: static random graph, projection of a bipartite graph to one partition set modification of the configuration model yielding graphs with cliques Dynamically evolving graphs with communities Try to explain how communities are formed, mostly triangles are added Aim: prove Granovetter s hypothesis of The strength of weak ties Algorithmic aspects: community detection various methods: spectral methods, min-cut, hierarchical clustering, modularity maximization, clique-based methods

20 overlapping community detection from G. Palla Processes and communities How does communities affect information spread? do they slow down, or accelerate the spread? Targeted advertising: PageRank, link farms, etc. Overlapping communities:

21 Configuration model Let n be number of vertices. Consider sequence of degrees d 1, d 2,..., d N. Often will take d i = D i, where (D i ) n i=1 is sequence of independent and identically distributed (i.i.d.) random variables with a certain distribution. Special attention for power-law degrees, i.e., when where c τ is constant and τ > 1. P(D 1 k) = c τ k τ+1 (1 + o(1)),

22 Power-law degree sequence CM Loglog plot of degree sequence CM with i.i.d. degrees n = 1, 000, 000 and τ = 2.5 and τ = 3.5, respectively.

23 Configuration model: graph construction How to construct graph with above degree sequence? Assign to vertex j degree d j. Let l n = n i=1 d i be total degree. Assume l n is even. Incident to vertex i have d i half edges.

24 Configuration model: graph construction How to construct graph with above degree sequence? Assign to vertex j degree d j. Let l n = n i=1 d i be total degree. Assume l n is even. Incident to vertex i have d i half edges. Connect half edges to create edges as follows: Number half edges from 1 to l n in any order. First connect first half edge at random with one of other l n 1 half edges. Continue with second half edge (when not connected to first) and so on, until all half edges are connected...

25 Properties configuration model Total number of edges is l n, and average degree is l n /n E[D], where D is degree of uniformly chosen vertex. CM has giant component when ν = E[D(D 1)] E[D] > 1. [This is weaker than E[D] > 2.] Local neighborhood of vertex in CM is close to tree. Cycles occur, but are quite long.

26 Graph distances in configuration model H n is graph distance between uniformly chosen pair of connected vertices in graph.

27 Graph distances in configuration model H n is graph distance between uniformly chosen pair of connected vertices in graph. Theorem 1. (HHVM03). When τ > 3 and ν = E[D(D 1)]/E[D] > 1 and fluctuations are bounded. H n log ν n P 1,

28 Graph distances in configuration model H n is graph distance between uniformly chosen pair of connected vertices in graph. Theorem 1. (HHVM03). When τ > 3 and ν = E[D(D 1)]/E[D] > 1 and fluctuations are bounded. H n log ν n P 1, Theorem 2. (HHZ07, Norros+Reittu 04). When τ (2, 3), H n log log n and fluctuations are bounded. P 2 log (τ 2),

29 x log log x grows extremely slowly Plots of x log x and x log log x.

30 Discussion Theorems 1-2 Proof relies on coupling of vertex-neighborhoods to branching process. Diameter graph is maximal distance between pairs of connected vertices. Diameter CM is Θ(log n) when P(D i 3) < 1 (FR07, HHZ07), while of order log log n when τ (2, 3) and P(D i 3) = 1 (HHZ07). More information Erdős-Rényi + power-law degree random graphs: rhofstad/notesrgcn.pdf

31 Comparison Internet data Number of AS traversed in hopcount data (blue) compared to the model (purple) with τ = 2.25, N = 10, 940.

32 Preferential attachment In preferential attachment models, network is growing in time, in such a way that new vertices are more likely to be connected to vertices that already have high degree.

33 Preferential attachment In preferential attachment models, network is growing in time, in such a way that new vertices are more likely to be connected to vertices that already have high degree. At time t, a single vertex is added to the graph with m edges emanating from it. Probability that an edge connects to the i th vertex is proportional to D i (t 1) + δ, where D i (t) is degree vertex i at time t, δ > m is parameter model.

34 Preferential attachment In preferential attachment models, network is growing in time, in such a way that new vertices are more likely to be connected to vertices that already have high degree. At time t, a single vertex is added to the graph with m edges emanating from it. Probability that an edge connects to the i th vertex is proportional to D i (t 1) + δ, where D i (t) is degree vertex i at time t, δ > m is parameter model. Different edges can attach with different updating rules: (a) intermediate updating degrees with self-loops (BA99, BR04, BRST01) (b) intermediate updating degrees without self-loops; (c) without intermediate updating degrees. i.e., independently.

35 Properties PA model Total number of edges is 2mt, so average degree is 2m. PA model is with high probability connected when m 2. When m = 1, have a tree. Structure of graph is quite insensitive to precise construction graph: Attachment rules; Fixed number of edges per vertex or random.

36 Scale-free nature PA Yields power-law degree sequence with exponent τ = 3 + δ/m (2, ) (m = 2, δ = 0, τ = 3 + δ m = 3)

37 ...the scale-free topology is evidence of organizing principles acting at each stage of the network formation. (...) No matter how large and complex a network becomes, as long as preferential attachment and growth are present it will maintain its hub-dominated scale-free topology.

38 Distances PA models of size n Diameter PA bounded above by C log n for all m 1 and δ > m (HH07); Diameter PA Θ(log log n) for m 2 and δ ( m, 0) for which powerlaw exponent τ (2, 3) (DHH09); Diameter PA bounded above by (1 + ε) log t log log n which power-law exponent τ = 3 (a) (BR04); for m 2 and δ = 0 for Diameter PA bounded below by (1 + ε) log t for m 2 and δ = 0 for log log n which power-law exponent τ = 3 (BR04, DHH09). Diameter PA bounded below by ε log n for m 2 and δ > 0 for which power-law exponent τ (3, ) (DHH09).

39 Universality PA models First evidence of strong form of universality: random graphs with similar degree structure share similar behavior. For random graphs, universality predicted by physics community. Universality is leading paradigm in statistical physics. Only few examples where universality can be rigorously proved. More information on random graphs: rhofstad/notesrgcn.pdf

41 Small-world phenomenon Distances in social network (Small-World Project Watts (2003))

RANDOM-REAL NETWORKS

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