An Introduction to Networks
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This document provides an introduction to networks. It discusses how networks are used to model relationships between entities in various domains, including social networks, protein interactions, and infrastructure networks. It also describes some key concepts in network analysis, such as degree distribution, shortest paths, centrality measures, and topological properties like the small-world and scale-free networks commonly seen in real-world systems.
An Introduction to Networks
- 1. An Introduction to Networks Francesco Gadaleta, PhD.
- 2. Networks are around us
- 3. Spreading consensus The model • few peers spreading a message (advertising) • others sharing to their friends (if they don’t already know)
- 8. The network of relationships
- 12. Not-so-recent graph of the Internet
- 13. Solving the problem of DNA sequencing (c) 2014 www.worldofpiggy.com 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
- 14. Being a freeloader with networks Homeless Visit a place Doesn’t repeat a node
- 15. Networks today • Social relationships • Professional networks (boss, employees) • Power grids • Internet • Biology (cells, genes, proteins, diseases…)
- 16. Power grids
- 18. Facebook at 10am
- 19. Graph Theory (Mathematics) Social Network Analysis 1920 economic transactions trades among nations communications between groups
- 20. Complexity of networks • irregular structure • evolution in time • dimension
- 21. Complexity of networks • irregular structure • evolution in time • dimension time = t0 time = t
- 22. Nature(1998) Small-world networks Watts D., Strogatz S.
- 23. Science(1999) Scale-free networks Barabasi, Albert
- 24. Topology Related to the structure of the network eg. how nodes are connected
- 25. Topology: Modules Subnetworks with specific properties
- 26. Some definitions graph: N nodes E edges directed undirected
- 27. Neighbours of node i (of order k) neigh(i,k) neigh(3,1) = ? neigh(2,2) = ? {2,4} {1,3,4,5} Example
- 28. 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)
- 30. Connectivity matrix (also known as adjacency matrix) A = Size binary or weighted
- 33. Node degree d(4) = ? d(6) = ? 3 1
- 34. Degree distribution Determines the statistical properties of uncorrelated networks
- 35. Degree distribution Determines the statistical properties of uncorrelated networks
- 36. Degree distribution Determines the statistical properties of uncorrelated networks
- 37. Shortest path • Indicates the distance between i and j in terms of geodesics (unweighted) • Can define the structure of a network Transport and communication p(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)
- 38. Diameter • Indicates the maximum number of hops between i and j (unweighted) • global property of a network
- 39. Average Shortest Path - ASP • • global property of a network Problem? i and j are disconnected Solution (efficiency)
- 40. Betweenness centrality # SPs from j to k via i # SPs from j to k
- 41. Which node is the most important?
- 42. Communities/Clusters • Local properties are shared only by a subset of the nodes
- 43. Facebook
- 44. Facebook (again)
- 46. Network components • define the topology • locally • globally (how many triads/pendants/dyads…)
- 47. example: count the number of triads in a network for comparison
- 49. Topologies: small-world Random shortcuts ASP each node is connected to any other node in only log(N) steps
- 50. Topologies: scale-free Degree distribution follows power-law Fact! most real networks follow a power-law
- 51. 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…
- 52. Topologies: random Nodes are statistically independent
- 53. Networks static (1) given a degree distrib. -> connect dynamic (2) 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
- 54. Weighted networks A = 3 1 2 5 8 2 0 5 0 0 1 0 5 0 8 0 6 0 6 0 8 0 2 0 0 0 0 2 0 3 2 1 6 0 3 0 0 0 0 0 2 0 0
- 55. Weighted networks node strength strength of nodes of degree k (independence between weight and topology) average weight with correlation
- 56. “The richest people in the world look for and build networks. Everyone else looks for work.” –Robert Kiyosaki