1. Lecture 1: Introduction CS 765: Complex Networks
Slides are adapted from
Constantine Dovrolis, Eileen Kraemer, Peter Dodds, and Sergei Maslov

2. Network: (net + work, 1500’s) Noun:
Basic definitions
Network: (net + work, 1500’s)
Noun:
Any interconnected group or system
Multiple computers and other devices connected together to share information
Verb:
To interact socially for the purpose of getting connections or personal advancement
To connect two or more computers or other computerized devices

3. Links = Connections between nodes
Basic definitions
Nodes = A collection of entities which have properties that are somehow related to each other
e.g., people, forks in rivers, proteins, webpages, organisms,...
Links = Connections between nodes
may be real and fixed (rivers),
real and dynamic (airline routes),
abstract with physical impact (hyperlinks),
purely abstract (semantic connections between concepts).
Links may be directed or undirected.
Links may be binary or weighted.

4. Complex: (Latin = with + fold/weave (com + plex)) Adjective
Basic definitions
Complex: (Latin = with + fold/weave (com + plex))
Adjective
Made up of multiple parts; intricate or detailed
Not simple or straightforward
Complex System—Basic ingredients:
Relationships are nonlinear
Relationships contain feedback loops
Complex systems are open (out of equilibrium)
Modular (nested)/multiscale structure
Opaque boundaries
May result in emergent phenomena
Many complex systems can be regarded as complex networks of physical or abstract interactions
Opens door to mathematical and numerical analysis

5. What passes for a complex network?
Complex networks are large (in node number)
Complex networks are sparse (low edge to node ratio)
Complex networks are usually dynamic and evolving
Complex networks can be social, economic, natural, informational, abstract, ...
Isn’t this graph theory?
Yes, but emphasis is on data and mechanistic explanations...

6. What is a Network? Network is a mathematical structure
composed of points connected by lines
Network Theory <-> Graph Theory
Network  Graph
Nodes  Vertices (points)
Links  Edges (Lines)
A network can be build for any functional system
System vs. Parts = Networks vs. Nodes

7. Networks As Graphs
Networks can be undirected or directed, depending on whether
the interaction between two neighboring nodes proceeds in both
directions or in only one of them, respectively. 1 2 3 4 5 6 
The specificity of network nodes and links can be quantitatively
characterized by weights
2.5
7.3
3.3
12.7
8.1
5.4
Vertex-Weighted
Edge-Weighted

8. Networks As Graphs - 2
A network can be connected (presented by a single component) or
disconnected (presented by several disjoint components).
connected
disconnected
Networks having no cycles are termed trees. The more cycles the
network has, the more complex it is.
trees
cyclic graphs

9. Networks As Graphs - 3 Some Basic Types of Graphs Paths Stars Cycles
Complete Graphs
Bipartite Graphs

10. Networks in complex systems
Large number of components interacting with each other
All components and/or interactions are different from each other
Paradigms:
104 types of proteins in an organism,
106 routers in the Internet
109 web pages in the WWW
1011 neurons in a human brain
The simplest property:
who interacts with whom?
can be visualized as a network
Complex networks are just a backbone for complex dynamical systems

11. Why study the topology of Complex Networks?
Lots of easily available data
Large networks may contain information about basic design principles and/or evolutionary history of the complex system
This is similar to paleontology:
learning about an animal from its backbone

12. Early social network analysis
1933 Moreno displays first sociogram at meeting of the Medical Society of the state of New York
article in NYT
interests: effect of networks on e.g. disease propagation
Preceded by studies of (pre)school children in the 1920’s
Source: The New York Times (April 3, 1933, page 17).