1. Complex Networks Measures and deterministic models Philippe Giabbanelli

2. Intro Measures (clustering, degree distribution) Main course Deterministic models Side dish Generalizing fractal graphs Leftover Discussion ∙ clustering augmentation ….∙ fractal graphs

3. Complex networks Motifs – Clustering – Average distance – Degree distribution Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Given a graph G… and a set S of random graphs of the same size and average degree, a motif is a subgraph that appears at a ‘very’ different frequence in G than in S. 1 2 3 2 1 0 0 0

4. Complex networks Motifs – Clustering – Average distance – Degree distribution Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Milo et al., Science, 303, 2004 Milo et al., Science, 298, 2002

5. Complex networks Motifs – Clustering – Average distance – Degree distribution Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 For a given node i, we denote its neighborhood by N i. The clustering coefficient C i of i is the edge density of its neighborhood. Here, there are two edges between nodes in N i. At most, it’d be a complete graph with N i.(N i -1) edges. C i = 2.2/(5.4) = 0.2

6. Complex networks Motifs – Clustering – Average distance – Degree distribution Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 For a given node i, we denote its neighborhood by N i. The clustering coefficient C i of i is the edge density of its neighborhood. If a graph has high clustering coefficient, then there are communities (i.e., cliques) in this graph. People tend to form communities so it is common in social networks.

7. Complex networks Motifs – Clustering – Average distance – Degree distribution Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Average distance: average length of shortest path between all pairs of nodes The average distance l is: ∙ small if l ∝ ln(n) ∙ ultrasmall if l ∝ ln(ln(n)) ← M.E.J Newman, The structure and function of complex networks

8. Complex networks Motifs – Clustering – Average distance – Degree distribution Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Many measured phenomena are centered around a particular value.

9. Complex networks Motifs – Clustering – Average distance – Degree distribution Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Many measured phenomena are centered around a particular value. There also exists numerous phenomena with a heavy-tailed distribution. lets plot it on a log-log scale

10. Complex networks Motifs – Clustering – Average distance – Degree distribution Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 There also exists numerous phenomena with a heavy-tailed distribution. The equation of a line is p(x) = -αx + c. Here we have a line on a log-log scale: ln p(x) = -α ln x + c apply exponent e p(x) = ecx c -α We say that this distribution follows a power-law, with exponent α.

11. Complex networks Motifs – Clustering – Average distance – Degree distribution Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 We say that this distribution follows a power-law, with exponent α. Keep in mind that this is quite common. people’s incomes computer files moon craters visits on web pages

12. Complex networks Motifs – Clustering – Average distance – Degree distribution A network with high clustering and low average distance is small-world. fast communications locallyand globally There are other definitions (e.g., network that you can navigate easily). A network with power-law degree distribution is scale-free. (Luckily, there aren’t other definitions, we’re already messy enough.) See: Efficient measurement of complex networks using link queries (Tarissan, NetSciCom’09), aaaaReverse centrality queries in complex networks (Nielsen, MSc Thesis SFU dec. ’09)

13. Complex networks Small world models (Watts Strogatz, Comellas et al., Giabbanelli et al.) Ref.: Watts & Strogatz, « Collective dynamics of ‘small-world’ networks », Nature 393, 1998 Almost all examples you will find use a simplified version. Get n nodes labelled from 0 to n. A node i is connected to (i+1, i+2, …, i Δ/2) mod n. Lets use Δ = 4. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Change one endpoint for an edge with probability p This scheme yields ‘good’ values for 0.01 < p 0.1 Small average distance Large clustering coefficient

14. Complex networks Small world models (Watts Strogatz, Comellas et al., Giabbanelli et al.) Ref.: Comellas, Ozon, Peters « Deterministic small-world communication networks », 2000 Get n nodes labelled from 0 to n. A node i is connected to (i+1, i+2, …, i Δ/2) mod n. Lets use Δ = 6. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A double step graph C(h; a,b) has h nodes, and i is connected to i a (mod h), i b (mod h) Select h equidistant nodes, and connect them as C(h;a,b). h=6, a=1, b=2 0 1 2 3 4 5 Then, some deterministic fiddling to keep the degree unchanged…

15. Complex networks Small world models (Watts Strogatz, Comellas et al., Giabbanelli et al.) Ref.: Giabbanelli & Peters, submitted to AlgoTel’10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Intuition Consider that we start with the cycle C n. The added edges should provide a good coverage of distances. When we connect i to i 1,…,i (Δ/2), we create lots of short-range edges Adding edges from a double- step graph mainly provides medium-range edges

16. Complex networks Small world models (Watts Strogatz, Comellas et al., Giabbanelli et al.) Ref.: Giabbanelli & Peters, submitted to AlgoTel’10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Intuition Consider that we start with the cycle C n. The added edges should provide a good coverage of distances. As long as d(i)≠Δ, connect i to i 2, …, i 2 0k +1 +2 +4 +8

17. Complex networks Small world models (Watts Strogatz, Comellas et al., Giabbanelli et al.) Ref.: Giabbanelli & Peters, submitted to AlgoTel’10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Intuition We want high clustering coefficient. What’s the graph with the highest clustering coefficient? → complete graph If a node has degree Δ-1, we add to it a K Δ Pretty artificial… but has the values required for small-world.

18. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) We’ll just introduce as much as we need. Starting graph A dotted edge is said to be active. At each time step, all dotted edges get replaced by a pattern graph. Pattern graph

19. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) We’ll just introduce as much as we need. Starting graph A dotted edge is said to be active. At each time step, all dotted edges get replaced by a pattern graph. Pattern graph

20. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) Ref.: Zhang, Rong, Guo, Physica A: Stat. Mech. And Appl., 363, 2006 Starting graphPattern graph Here’s the definition of ZRG using our graph grammar. t This generates a (planar) small-world graph. Giabbanelli, Mazauric, Pérennes, submitted to AlgoTel’10 L 0L 1L It also has a simple labelling scheme.

21. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) Ref.: Miralles, Comellas, Chen, Zhang, Physica A, 389, 2010 Starting graphPattern graph Here’s the definition of M using our graph grammar. d,t d Example for d = 2 There is no triangle so the clustering coefficient is 0. The result is scale-free, planar, with small average distance. For d=1: Comellas, Mirales, Physica A, 388, 2009

22. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) Ref.: Comellas, Zhang, Chen, J. Physc. A., 42, 2009 Starting graphPattern graph We’re not limited to active edges. For example, lets have active cycles. Given the active cycle and the pattern, how do we know which edge of the cycle gets replaced by which edge of the pattern? We use a function that maps the active cycle in the pattern (= morphism) a b c d a b c d

23. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) Ref.: Comellas, Zhang, Chen, J. Physc. A., 42, 2009 Starting graphPattern graph We’re not limited to active edges. For example, lets have active cycles. How many active cycles are there in the pattern graph?

24. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) Ref.: Comellas, Zhang, Chen, J. Physc. A., 42, 2009 Starting graphPattern graph We’re not limited to active edges. For example, lets have active cycles. How many active cycles are there in the pattern graph? This is NOT an active cycle. 4

25. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) Ref.: Comellas, Zhang, Chen, J. Physc. A., 42, 2009 Starting graphPattern graph We’re not limited to active edges. For example, lets have active cycles. How many active cycles are there in the pattern graph?4 The result is scale-free, planar, with small average distance.

26. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) So maybe we could discuss their properties a bit more generally. We can define tons of patterns (people actually did and published them). Lets have a look at active edges.

27. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) Pattern graph Black box Diameter D N nodes P We start from a triangle with a pattern having two active edges. P For each box, we add N p -2 nodes. We start with 3 nodes, add 3 boxes: 3(N p -2)+3 nodes Diameter at most D p 2 Step t = 1

28. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) Pattern graph Black box Diameter D N nodes P We start from a triangle with a pattern having two active edges. P The number of added nodes doubles at each step: we now add 3.2(N p -2) nodes. The longest path is through 2t boxes Diameter at most 2tD p

29. Complex networks Fractal graphs (Graph grammar, Zhang et al., Perspectives) Pattern graph Black box Diameter D N nodes P We start from a triangle with a pattern having two active edges. P The average distance is small regardless of the pattern you choose. The same conclusion holds for a pattern graph with at least two active edges, and any starting graph. P. Giabbanelli, Properties of fractal network models, submitted to Physica A

30. Complex networks