Randomness and Complexity in Networks

Abstract

I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key idea is that in many cases the process involves copying of properties of near neighbours in the network and this is a type of short random walk which in turn produce a natural preferential attachment mechanism. Applying this to networks of fixed size I show that copying and innovation are processes with special mathematical properties which include the ability to solve a simple model exactly for any parameter values and at any time. I finish by looking at variations of this basic model.

Are Copying and Innovation Enough?

“…This is equivalent to using a complete graph with self loops for the social network at this stage but these preferential attachment forms emerge naturally when using a random walk on a general network [7]. This choice for Π A has two other special properties: one involves the scaling properties [5] and the second is that these exact equations can be solved analytically [3,5,6,4]. The generating function G(z, t)…”

Diophantine Networks

“…For example, in many situations it is not realistic to assume that a node has the complete information about the degree distribution it would need in order to know where to attach preferentially. From the other hand this information is just what is needed to normalise properly [34] the attachment probabilities as defined in most of the analytic models introduced in literature so far [1,6]). Alternative mechanisms generating scale-free topologies have then been introduced, as for example the static model [15], the varying fitness model [10,11,12,13,14] and random walk models incorporating a copying mechanism [34].…”