an introduction to networks

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

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.”