an introduction to networks
TRANSCRIPT
An Introduction to NetworksFrancesco Gadaleta, PhD.
Networks are around us
Spreading consensus
The model
• few peers spreading a message(advertising)
• others sharing to their friends(if they don’t already know)
The network of relationships
Economy network
Protein-protein interaction network
Political network
Not-so-recent graph of the Internet
(c) 2014 www.worldofpiggy.com
Solving the problem of DNA sequencing
Definition: Each read is an edge
Nodes are prefix and suffix of the string that connects them
Solution: Find a cycle in such a graph: reading the superstring that contains all reads with maximum overlap.
Hey! That’s an Eulerian cycle
Being a freeloader with networks
Homeless Visit a place Doesn’t repeat a node
• Social relationships
• Professional networks (boss, employees)
• Power grids
• Internet
• Biology (cells, genes, proteins, diseases…)
Networks today
Power grids
Facebook at 10am
Graph Theory(Mathematics)
Social Network Analysis1920
economic transactions
trades among nations
communications between groups
Complexity of networks
• irregular structure • evolution in time • dimension
Complexity of networks • irregular structure • evolution in time • dimension
time = ttime = t0
Nature(1998) Small-world networks Watts D., Strogatz S.
Science(1999) Scale-free networks Barabasi, Albert
TopologyRelated to the structure of the network eg. how nodes are connected
Topology: Modules
Subnetworks with specific properties
Some definitions
N nodes E edges
graph:
directed
undirected
Neighbours of node i (of order k) neigh(i,k)
neigh(3,1) = ?neigh(2,2) = ?
{2,4}{1,3,4,5}
Example
Reachability of two nodes i and j
walk: alternating sequence of nodes and edges from i to j eg. (1-2-3-4-3)
trail: a walk with no repeated edges eg. (1-2-3-4-5-2)
path: a walk with no repeated nodes eg. (1-2-3-4-6)
Connectivity matrix(also known as adjacency matrix)
A =
Sizebinary or weighted
Node degree
d(4) = ? d(6) = ?
31
Degree distributionDetermines the statistical properties of uncorrelated networks
Degree distributionDetermines the statistical properties of uncorrelated networks
Degree distributionDetermines the statistical properties of uncorrelated networks
Shortest path• Indicates the distance between i and j in terms of geodesics
(unweighted)
• Can define the structure of a network
Transport and communicationp(1,3) = {1-5-4-3} {1-5-2-3} {1-2-5-4-3} {1-2-3}
Warning: the “longest” path can be the shortest (weighted graph)
Diameter• Indicates the maximum number of hops between i and j
(unweighted)
• global property of a network
Average Shortest Path - ASP
•
• global property of a network
Problem?i and j are disconnected
Solution (efficiency)
Betweenness centrality
# SPs from j to k via i
# SPs from j to k
Which node is the most important?
Communities/Clusters
• Local properties are shared only by a subset of the nodes
Facebook(again)
Network components
Network components
• define the topology • locally • globally
(how many triads/pendants/dyads…)
example: count the number of triads in a network for comparison
Topologies: small-world
Random shortcuts
ASP
each node is connected to any other node in only log(N) steps
Topologies: scale-free
Degree distribution follows power-law
Fact! most real networks follow a power-law
Topologies: scale-free
Degree distribution follows power-law
• the sizes of earthquakes • craters on the moon • solar flares • the foraging pattern of various species • the sizes of activity patterns of
neuronal populations • the frequencies of words in
most languages • frequencies of family names • sizes of power outages • wars • criminal charges per convict • and many more…
Topologies: random
Nodes are statistically independent
Networks
static (1)
dynamic (2)
given a degree distrib. -> connect
structural changes are governed by evolution of the system (gene-gene, web, social net.)
(1) given , assign uniform prob. to all random graphs with a number of nodes with degree k (Aiello et. al) N and k are fully determined
(2) Prob. of link j connected to existing node i is proportional to
Weighted networks
A =3 1
852
20 5 0 0 1 05 0 8 0 6 0
60 8 0 2 0 00 0 2 0 3 21 6 0 3 0 00 0 0 2 0 0
Weighted networks
node strength
strength of nodes of degree k(independence between weight and topology)
average weight
with correlation
–Robert Kiyosaki
“The richest people in the world look for and build networks. Everyone else looks for work.”