Network Science: A Short Introduction i3 Workshop Konstantinos Pelechrinis Summer 2014 Figures are taken from: M.E.J. Newman, “Networks: An Introduction”

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

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

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 http://www.ladamic.com/netlearn/NetLogo501/ErdosRenyiDegDist.html

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 ?

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

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!

Small-world models

Skewed degree distribution Small world models Small-world regime For c=6 and n=600 http://www.ladamic.com/netlearn/NetLogo4/SmallWorldWS.html Cannot generate: Skewed degree distribution

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)

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. http://www.ladamic.com/netlearn/NetLogo501/RAndPrefAttachment.html