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

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

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.

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

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...

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

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

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

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

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

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

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).