an introduction to networks
TRANSCRIPT
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An Introduction to NetworksFrancesco Gadaleta, PhD.
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Networks are around us
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Spreading consensus
The model
• few peers spreading a message(advertising)
• others sharing to their friends(if they don’t already know)
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The network of relationships
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Economy network
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Protein-protein interaction network
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Political network
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Not-so-recent graph of the Internet
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(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
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Being a freeloader with networks
Homeless Visit a place Doesn’t repeat a node
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• Social relationships
• Professional networks (boss, employees)
• Power grids
• Internet
• Biology (cells, genes, proteins, diseases…)
Networks today
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Power grids
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Facebook at 10am
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Graph Theory(Mathematics)
Social Network Analysis1920
economic transactions
trades among nations
communications between groups
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Complexity of networks
• irregular structure • evolution in time • dimension
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Complexity of networks • irregular structure • evolution in time • dimension
time = ttime = t0
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Nature(1998) Small-world networks Watts D., Strogatz S.
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Science(1999) Scale-free networks Barabasi, Albert
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TopologyRelated to the structure of the network eg. how nodes are connected
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Topology: Modules
Subnetworks with specific properties
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Some definitions
N nodes E edges
graph:
directed
undirected
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Neighbours of node i (of order k) neigh(i,k)
neigh(3,1) = ?neigh(2,2) = ?
{2,4}{1,3,4,5}
Example
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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)
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Connectivity matrix(also known as adjacency matrix)
A =
Sizebinary or weighted
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Node degree
d(4) = ? d(6) = ?
31
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Degree distributionDetermines the statistical properties of uncorrelated networks
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Degree distributionDetermines the statistical properties of uncorrelated networks
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Degree distributionDetermines the statistical properties of uncorrelated networks
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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)
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Diameter• Indicates the maximum number of hops between i and j
(unweighted)
• global property of a network
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Average Shortest Path - ASP
•
• global property of a network
Problem?i and j are disconnected
Solution (efficiency)
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Betweenness centrality
# SPs from j to k via i
# SPs from j to k
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Which node is the most important?
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Communities/Clusters
• Local properties are shared only by a subset of the nodes
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Facebook(again)
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Network components
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Network components
• define the topology • locally • globally
(how many triads/pendants/dyads…)
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example: count the number of triads in a network for comparison
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Topologies: small-world
Random shortcuts
ASP
each node is connected to any other node in only log(N) steps
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Topologies: scale-free
Degree distribution follows power-law
Fact! most real networks follow a power-law
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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…
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Topologies: random
Nodes are statistically independent
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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
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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
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Weighted networks
node strength
strength of nodes of degree k(independence between weight and topology)
average weight
with correlation
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–Robert Kiyosaki
“The richest people in the world look for and build networks. Everyone else looks for work.”