Network Science: A Short Introduction i3 Workshop

1. Network Science: A Short Introductioni3 Workshop Konstantinos Pelechrinis Summer 2014 Figures are taken from: M.E.J. Newman, “Networks: An Introduction”

2. Network models • We want to have formal processes which can give rise to networks with specific properties • E.g., degree distribution, transitivity, diameters etc. • These models and their features can help us understand how the properties of a network (network structure) arise • By growing networks according to a variety of different rules/models and comparing the results with real networks, we can get a feel for which growth processes are plausible and which can be ruled out • Random graphs represent the “simplest” model

3. Erdos-Renyi random network model • Simplest model • Basic idea: nodes are connected completely at random • Given: number of nodes n • Number of edge: m • In this case for each edge m we pick uniformly at random a pair of nodes • Probability that any two nodes in the network are connected: p • In this case, we go over all possible pairs of the n nodes and connect each one of them with a probability p • Both processes/models are equivalent

4. Properties of random networks • The random network model can generate networks with: • Short paths • Giant components • It cannot generate networks with: • High clustering • Skewed degree distribution

5. Small-world model • Random graphs exhibit small paths but not clustering • If we consider an ordered network (lattice) exhibits high clustering but large paths • Why not combine both these models ?

6. Small-world models • The small-world model (Watts and Strogatz 1998) tries to do exactly this • We start with a circle model of n vertices in which every vertex has a degree of c • We go through each of the edges and with some probability p we rewire it • Remove this edge and pick two vertices uniformly at random and connect them with a new edge • Shortcut edge

7. Small-world models • The parameter p controls the interpolation between the circle model and the random graph • p=0  ordered situation/circle model • p=1  random graph • Intermediate values of p give networks somewhere in between • The crucial and interesting point is that small paths appear even for small values of p as we increase from p=0, while the high clustering remains until fairly large values of p • Hence, there is a regime for values of p where both small paths as well as high clustering exists!

8. Small-world models

9. Small world models Small-world regime • For c=6 and n=600 Cannot generate: Skewed degree distribution

10. Preferential attachment • Both previous models cannot generate skewed degree distributions • How can we have networks where there are a few nodes with a large number of edges, while the majority of them has few edges only? • A simple growth process can provide insights! • Until now we have fixed topology models • Given number of nodes and edges from the beginning • In other words, nodes do not appear one-by-one in time • A growth process refers to the evolution of the network by the addition of nodes (and edges for these nodes)

11. Preferential attachment • Nodes prefer to attach to existing nodes that have high degree! • At every point of time a new node is created and this node generates b edges • Each of this edges is connected to the existing nodes randomly • NOT UNIFORMLY AT RANDOM BUT WITH A PROBABILITY PROPORTIONAL TO THE NUMBER OF EDGES AN EXISTING NODE ALREADY HAS! • Rich-gets-richer, cumulative advantage, Matthew effect etc.