Networks. CS4423. Prof. Götz Pfeiffer Semester 2 (2018/2019)

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1 .. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway Semester 2 (2018/2019)

2 .. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway Semester 2 (2018/2019)

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5 Useful Books. Vito Latora, Vincenzo Nicosia and Giovanni Russo. Complex. Cambridge UP Albert-László Barabási. Network Science. Cambridge UP Ernesto Estrada and Philip A. Knight. A first course in network theory. Oxford UP Mark Newman. : an. Oxford UP David Easley and Jon Kleinberg., Crowds and Markets. Cambridge UP 2010.

6 Connections. Modern societies are in many ways highly connected. Certain aspects of this phenomenon are frequently described as networks: social networks (of people linked by friendship or social interaction) computer networks (the Internet) document networks (World Wide Web) academic networks (collaborations, citations of scientific publications) transportation networks (road, rail, airline, subway) networks in biology (e.g., biochemical, ecological, or neural networks) financial networks (transactions between traders)... What do these structures have in common? How can they be distinguished?

7 Simple Forms of. In its simplest form, a network is just a collection of points (called vertices or nodes), some of which are joined in pairs (called edges or links). Many systems of interest are composed of individual parts that are in some way linked together. Such systems can be regarded as networks, and thinking about them in this way can often lead to new and useful insights. Network science studies the pattern of connections between the components of a system Naturally, the structure of the network can have a big impact on the behavior of a system. The connections in a social network, for example, affect how people learn, form opinions, or spread a disease.

8 Network Science A network is a simplified representation of a complex system by vertices and edges. The scientific study of networks is an interdisciplinary undertaking that combines ideas from mathematics, physics, biology, computer science, the social sciences and many other areas. Between these scientific fields, many tools have been developed for analyzing, modeling and understanding networks.

9 Network An example of a useful and important type of network measures is that of centrality. It is concerned with the question of how important a particular vertex or edge is in a networked system. Different concepts have suggested for capturing mathematically what it means to be central. A simple measure of the centrality of a vertex is its degree, i.e., the number of edges it is part of. Which measurements and calculations give meaningful answers for a particular system depends of course on the specific nature of the system and the questions one wants to ask.

10 Network Concepts Another interesting network concept is the small-world effect. It is concerned with the question of how far apart two randomly chosen points in a network typically are. Here,distance is usually measured by the number of edges one would need to cross over, when travelling along a path from one vertex to the other. In real world social networks the distance between people tends to be rather small. This observation and is known as the six degrees of separation in popular culture. This course is meant as an introduction to Network Science where, within the given time and space constraints, some but certainly not all of its interesting aspects will be discussed.

11 Examples of. Newman broadly divides the most commonly studied real world networks into four classes: technological networks, social networks, information networks and biological networks. There is of course some overlap between these classes. But it is an interesting exercise to list important examples within each class, and to describe their general structure, and the techniques used to discover and measure this structure in each example.

12 Technological. Technological networks rely on a physical infrastructure. In many cases, this infrastructure has been built over many decades It forms part of the backbone of modern societies. This includes road and other transportation networks, power grids, the telephone network, and more recently the Internet.

13 The Internet (Internet Mapping Project, 1998)

14 Power Grids TRANSMISSION SYSTEM 400, 275, 220 AND 110kV JANUARY 2016 HVDC SCREEBE SALTHILL KILLHILL CLOGHBOOLA WOODHOUSE

15 Chesham Amersham Uxbridge Heathrow Terminals 2 & 3 Heathrow Terminal 5 Interchange stations Under a 10 minute walk between stations Step-free access from street to train Step-free access from street to platform National Rail Airport Hillingdon Hayes & Harlington Chalfont & Latimer West Ruislip Riverboat services Chorleywood Ickenham Ruislip Gardens South Ruislip Northolt Victoria Coach Station Rickmansworth Greenford Ruislip Emirates Air Line cable car Transport for London December 2018 Rayners Lane West Ealing Osterley Hounslow Central Hounslow West Hatton Cross Heathrow Terminal 4 Hanwell Southall Perivale Watford Croxley Moor Park Ruislip Manor Eastcote Northfields Boston Manor Hounslow East South Harrow Sudbury Hill Sudbury Town Hanger Lane Alperton Park Royal North Ealing Ealing Broadway Ealing Common South Ealing Northwood Northwood Hills Pinner West Harrow Acton Main Line West Acton North Harrow Acton Central South Acton Acton Town Chiswick Park Bromleyby-Bow Harrowon-the-Hill Turnham Green Gunnersbury Kew Gardens Richmond Station in Zone 9 Station in Zone 8 Station in Zone 7 Station in both zones Station in Zone 6 Station in Zone 5 Station in Zone 4 Station in both zones Station in Zone 3 Station in Zone 2 Station in both zones Station in Zone 1 Northwick Park South Kenton North Wembley Wembley Central Stonebridge Park Harlesden Willesden Junction North Acton Stamford Brook Harrow & Wealdstone Kilburn Park East Acton Maida Vale White City Wood Lane Shepherd s Bush Market Goldhawk Road Hammersmith Ravenscourt Park Watford Junction Watford High Street Bushey Carpenders Park Hatch End Headstone Lane Kenton Preston Road Wembley Park Kensal Rise Kensal Green Queen s Park Warwick Avenue Paddington Shepherd s Bush Kensington (Olympia) Barons Court Royal Oak West Kensington West Brompton Fulham Broadway Parsons Green Putney Bridge East Putney Southfields Wimbledon Park Wimbledon Stanmore Canons Park Queensbury Kingsbury Holland Park Neasden 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Enfield Town Crystal Palace Bush Hill Park Harringay Green Lanes Finsbury Park Canonbury Seven Sisters Dalston Kingsland Dalston Junction Bermondsey Reeves Corner Wandle Park Haggerston Hoxton Shoreditch High Street Aldgate East Tower Gateway Rotherhithe Canada Water Surrey Quays New Cross Gate Brockley Honor Oak Park Forest Hill Sydenham Centrale Church Street Southbury Edmonton Green Silver Street White Hart Lane Bruce Grove South Tottenham Tottenham Hale Stamford Hill Hackney Downs Wapping Anerley London Fields Cambridge Heath Bethnal Green Bethnal Green Whitechapel Shadwell Rectory Road New Cross Penge West Turkey Street Stoke Newington Hackney Central Norwood Junction Cheshunt Theobalds Grove Clapton Stepney Green Limehouse West Croydon Walthamstow Queen s Road Wellesley Road George Street Blackhorse Road Homerton Mile End Chingford Highams Park Wood Street Leyton Midland Road St James Street Devons Road Langdon Park All Saints Westferry Poplar West India Quay Canary Wharf Heron Quays South Quay Crossharbour Mudchute Island Gardens East Croydon Lebanon Road Walthamstow Central Hackney Wick Bow Road Bow Church Blackwall Greenwich Leytonstone High Road Stratford International Cutty Sark for Maritime Greenwich Deptford Bridge Elverson Road Lewisham Sandilands East India North Greenwich Lloyd Park Pudding Mill Lane Coombe Lane Epping Theydon Bois Debden Loughton Buckhurst Hill Roding Valley Woodford South Woodford Snaresbrook Wanstead Leytonstone Leyton Chigwell Redbridge Stratford West Ham Emirates Greenwich Peninsula Star Lane Canning Town Grange Hill Gants Hill Maryland Stratford High Street Abbey Road Arena Woodside Blackhorse Lane Addiscombe Wanstead Park Gravel Hill Forest Gate Plaistow Hainault Fairlop Barkingside Newbury Park Upton Park Royal Victoria Ilford Manor Park Woodgrange Park Goodmayes Seven Kings Barking East Ham Emirates Royal Docks Pontoon Dock Upney West Silvertown London City Airport Harrington Road Addington Village Elmers End King George V Birkbeck Fieldway Chadwell Heath Becontree Romford Dagenham East Gidea Park Dagenham Heathway Upminster Upminster Bridge Hornchurch Elm Park Custom House for ExCeL Prince Regent Avenue Road King Henry s Drive Royal Albert Harold Wood Emerson Park Beckton Park Cyprus Woolwich Arsenal Beckenham Road New Addington Shenfield Brentwood Gallions Reach Beckton Beckenham Junction Earl s Court No step-free access to the District line from late January until late June East Ham No step-free access to the westbound platform from early January until early July Heathrow TfL Rail customers should change at Terminals 2 & 3 for free rail transfer to Terminal Hounslow West Step-free access for manual wheelchairs only Services or access at these stations are subject to variation. 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16 Mediaval Trade Routes

17 Social. The vertices of a social network are people, with edges representing some sort of social interaction. A social network s existence does not depend on social networking sites like facebook or twitter. Sociologists have studied social networks long before people started exhibiting their relations to others online. An affiliation network has two types of nodes, one type representing people and another type representing groups of people. In mathematical terms, an affiliation network is an example of a bipartite graph. The affiliation network can be projected onto the set of people, defining a new network which has only the people as vertices...

18 Information. An information network consists of data items which are linked to each other in some way. Example: relational databases. Sets of information (like scientific publications) have been linking to each other (e.g., through citations) long before computers were invented. However, links in digital form are easier to follow The (World Wide) Web. The WWW is probably the most wide spread and best known examples of an information network. Its nodes are web pages containing information in form of text and pictures, and its edges are the hyperlinks. Hyperlinks run in one direction only, from the page that contains the hyperlink to the page that is referenced. Therefore, the WWW is a directed network, a graph where each edge has a direction.

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20 Biological. Biochemical networks represent molecular level patterns of interaction and control mechanisms in the biological cell: metabolic networks, protein-protein interaction networks and genetic regulatory networks. A neural network can be represented as a set of vertices, the neurons, connected by two types of directed edges, one for excitatory inputs and one for inhibiting inputs. Ecological networks are networks of ecological interactions between species.

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22 Human Brain as Neural Network

23 Food Web

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25 Graphs. A graph can serve as a mathematical model of a network. Example As far as the network structure is concerned, this diagram contains the same information, without the distracting details of the US geography. The internet in December Nodes are computers, connected by a link if they can directly communicate with each other. At the time, only 13 computers participated in that network. STAN SRI UCLA UCSB UTAH SDC RAND MIT BBN LINC HARV CASE CARN

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