Synopsis of “Emergence
of Scaling in
Random Networks”*
Presentation for ENGS 112
Doug Madory
Wed, 27 APR 05
*Albert-Laszlo Barabasi and Reka Albert,
Science, Vol 286, 15 October 1999
Background

Traditional approach - random graph
theory of Erdos and Renyi
 Rarely

tested in real world
Current technology allows analysis of
large complex networks (Ex: WWW,
citation patterns in science, etc)
Barabasi’s Claim

Independent of system and identity of its
constituents, the probability P(k) that a vertex in
the network interacts with k other vertices
decays as a power law, following:
P(k) ~ k-g

Existing network models fail to incorporate
growth and preferential attachment, two key
features of real networks.
Complex network examples
Actor collaboration
WWW
Power grid data
Citations in published papers: gcite = 3
Problems with other theories

Erdos-Renyi & Watts-Strogatz theories suggest
probability of finding a highly connected vertex
(large k) decreases exponentially with k
 Vertices

with large k are practically absent
Barabasi - power-law tail characterizing P(k) for
networks studied indicates that highly connected
(large k) vertices have a large chance of
occurring and dominating the connectivity
Problems with other theories


Erdos-Renyi & Watts-Strogatz assume fixed
number (N) of vertices
Barabasi - real world networks form by
continuous addition of new vertices, thus N
increases throughout lifetime of network.
Problems with other theories


Erdos-Renyi & Watts-Strogatz - probability that
two vertices are connected is random and
uniform
Barabasi - real networks exhibit preferential
connectivity
 New
actor cast supporting established one
 New webpage will link to established pages
Barabasi’s Experiment



Start with small number of vertices: mO
At each time step, add new vertex with m(<=mO)
edges that link new vertex to m previous vertices
Probability P that a new vertex will be connected
to vertex i depends on connectivity ki of that
vertex
P(ki) = ki/Sjkj (Preferential attachment)

Demo in Matlab
The “rich get richer” theory

Similar mechanisms could explain the origin of social
and economic disparities governing competitive
systems, because scale-free inhomogeneities are the
inevitable consequence of self-organization due to local
decisions made by individual vertices, based on
information that is biased toward more visible (richer)
vertices, irrespective of the nature and origin of this
visibility.
Summary


Common property of many large networks is
vertex connectivities follow a scale-free powerlaw distribution.
Consequence of two generic mechanisms:
(i) networks expand continuously by the
addition of new vertices, and
(ii) new vertices attach preferentially to sites
that are already well connected.