1. Analysis of complex networks Philippe Giabbanelli, MOPS&ISNO projects

2. Intro When to think about networks Main course Properties of complex networks (small-world, scale-free…) Side dish Using a software: Visone (not the only one!) Dessert MoCSSy’s special: tea & cookies

3. Analysis of complex networks When to think about networks Let’s start from something we know…cellular automata.

4. Analysis of complex networks When to think about networks Here is a neighborhood. Lets say we study some changes in houses. It’s easy to think of that space as a nice grid. Then you can design rules such « if most neighbours of X do barbecues then so does X ». You can still create a mapping between this place and an automaton. It may be a hassle…

5. Analysis of complex networks You want to study roads? Sure, nice grid, get a cellular automaton ! or not…  You still can model these roads as cellular automata using a CA with a dimension greater than 2. A network, or graph, has a more general structure than a CA. node edge The question is not « can we model everything using one tool », but is it really the most convenient? What about the analysis of the model? Use as a graph when you want to be able to analyze the structure. What is the interchange through the highest number of roads (i.e. a vulnerable point)? Which roads are the most central to travel between two places?

6. Analysis of complex networks When to think about networks Lets look at a second example: populations in epidemiology. SI R susceptibleinfectedremoved αβ In a compartment model, you consider that people are in three possible states, and that there are some probabilities to move between these states. This ignores the topology of the population. People are all connected to each other: regardless of who you are, you have a probability α to get infected (even if you don’t see anybody…).

7. Analysis of complex networks When to think about networks Lets look at a second example: populations in epidemiology. If we want to bring the population in, we represent it as a graph: nodes are people, and edges exist between 2 persons if they know each other. Who should we vaccinate first? We can answer that question only if we can analyze the structure (graph).

8. Analysis of complex networks Properties of complex networks slides reused from a presentation in CMPT880

9. Basic Properties – Small World – Scale Free – Other measurements 2 We analyze the structure of (big) networks from the real-world to understand which properties are underlying them. If a general class of network has a given property then we can use it to reason about any unknown network of this class. What do we do in complex network? Social Network Biological network BlogsFacebookPopulation Property: some individuals are very social compared to other ones. Goal: spread a saucy rumor! Idea: whatever the network as long as it’s a social network, try to target the social individuals. There is a tremendous number of applications and since the main two properties were discovered in 1998, there has been hundreds if not thousands of papers on complex networks. How strong is the connection from A to C? What is the influence of A over C in the social network? How likely is it that if A is infected by a virus then C will get infected?

10. Basic Properties – Small World – Scale Free – Other measurements 3 Transitivity « There are high chances that a husband knows the family of his wife. » 1 1 1 1 2 2 2 ►Transitivity measures the probability that if A is connected to B and B is connected to C, then A is connected to C. 1 triangle 8 connected triples C = 3/8 = 0,375

11. Basic Properties – Small World – Scale Free – Other measurements 4 Network Motifs When we looked for transitivity, we basically counted the number of subgraphs of a particular type (triangles and triples). We can generalize this approach to see which patterns are ‘very frequent’ in the network. Those patterns are called network motifs. To measure the frequency, we compare with how expected it is to see such patterns in a random network. The significance profile (SP) of the network is a vector of those frequencies. For each subgraph, we measure its relative frequency in the network. As we are measuring for the 13 possible directed connected graphs of 3 vertices, it is called a triad significance profile (TSP). 4 networks of different micro-organisms are shown to have very similar TSPs, and in particular the triad 7 called « feed-forward loop ».

12. Basic Properties – Small World – Scale Free – Other measurements 5 Let’s play the Kevin Bacon Game. Think of an actor or an actress… → If they’ve been in a film with him, they have Bacon Number one. → Otherwise, if they have been in a film with somebody who has Bacon Number one, then they have Bacon Number two, etc. Hollywood’s world is pretty large. What do you think is the average Bacon Number an american actor will get? Only 4 ! Laurence Fishburne (alias Morpheus in Matrix) Played with Kevin Bacon in Mystic Rivers ! Mos Def (in The Italian Job) played with Kevin Bacon in The Woodsman

13. Basic Properties – Small World – Scale Free – Other measurements 6 The Small-World property Through this Kevin Bacon’s experiment, we know that although the network of actors is quite big, the average distance is very small. A network is said to have the small-world property if the average shortest path L is at most logarithmically on the network size N. → An e-mail network of 59 812 nodes… L = 4.95 ! → Actor network or 225 226 actors… L = 3.65 ! It tells you that transmitting information in small-world networks will be very fast. And so, transmitting viruses will be fast too… Some authors defined the small-world property with an additional constraint with the presence of a high clustering. It’s a choice… At a local level, we have strongly connected communities. Efficient to exchange information at a local scale. Efficient to exchange information at a global scale. Global efficiency of small-world networks.

14. This value is the typical size. Basic Properties – Small World – Scale Free – Other measurements Many of the things we measure are centered around a particular value. However, there are things that have an enormous variation in the distribution. If we plot this histogram with logarithmic horizontal and vertical axis, a pattern will clearly emerge: a line. In a normal histogram, this line is p(x) = -αx + c. Here it’s log-log, so: 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. 7 The Scale-Free property A power law is the only distribution that is the same whatever scale we look at it on, i.e. p(bx) = g(b)p(x). So, it’s also called scale-free.

15. Basic Properties – Small World – Scale Free – Other measurements 8 We found that the population has the scale-free property! In 1955, Herbert Simon already showed that many systems follow a power law distribution, so that’s neither new nor unique. Sizes of earthquakes Moon craters Solar flares Computer files Wars Number of citations received / paper Number of hits on web pages People’s annual incomes The Scale-Free property It has been found that the distribution of the degree of nodes follows a power-law in many networks, i.e. many networks are scale-free… What is important is not so much to find a power-law as it’s common, but to understand why and which other structural parameters can be there.

16. Basic Properties – Small World – Scale Free – Other measurements 9 The Scale-Free property Myth and reality Scaling distributions are a subset of a larger family of heavy-tailed distributions that exhibit high variability. One important claim of the litterature for scale-free networks was the presence of highly connected central hubs. However, it only requires high variability and not strict scaling… It was said that « the most highly connected nodes represent an Achilles’ heel »: delete them and the graph breaks into pieces. Recent research have shown that complex networks that claimed to be scale-free have a power-law but not this Achilles’ heel. One mechanism was used to build scale-free networks, called preferential attachment, or « the rich get richer ». It is only one of several, and not less than 7 other mechanisms give the same result, so preferential attachment gives little or no insight in the process.

17. Basic Properties – Small World – Scale Free – Other measurements 10 Other measurements We have the clustering, distribution of degree, etc. Are there other global characteristics relevant to the performances of the network, in term of searchability or stability? Rozenfeld has proposed in his PhD thesis to study the cycles, with algorithms to approximate their counting (as it’s exponential otherwise). Using cycles as a measure for complex networks has received attention: Inhomogeneous evolution of subgraphs and cycles in complex networks (Vazquez, Oliveira, Barabasi. Phys. Rev E71, 2005). Degree-dependent intervertex separation in complex networks (Dorogovtsev, Mendes, Oliveira. Phys Rev. E73 2006) See also studies on the correlation of degree (i.e. assortativity).