A connected simple graph is Eulerian iff every graph vertex has even degree. The numbers of Eulerian graphs with, 2,... nodes are 1, 1, 2, 3, 7, 16, 54, 243, 243, 2038,... (Sloane's A002854; Robinson 1969; Mallows and Sloane 1975; Buekenhout 1995, p. 881; Colbourn and Dinitz 1996, p. 687). There is an explicit formula giving these numbers.connectedsimple graphiffgraph vertex evendegreeA002854

Everything you've learned in school as "obvious" becomes less and less obvious as you begin to study the universe. For example, there are no solids in the universe. There's not even a suggestion of a solid. There are no absolute continuums. There are no surfaces. There are no straight lines. R. Buckminster Fuller

Power-Law Distribution

The longest shortest path from v5 (5) to v7 (7). Diameter is 5. N=20 Av degree = 4 Paper I Fig 1 Random Net

Resilience

N=8 Av degree = 3 The longest shortest path from v1 (1) to v7 (7). Diameter is 3. Random Net “Diameter”

Bell-Shaped Distribution

Power-Law Distribution

Network Resilience

N=20 M0=3 Rewire lines =1 P=0.2 Q=0.4 Paper SW-C

Scale Free Net Resilience

N=20 M0=3 Rewirelines =1 P=0.2 Q=0.4 Paper C ScaleFree1

N=8 M0=3 Rewirelines =1 P=0.2 Q=0.4 Paper C

v1v2v3v4v5v6v7V8 v1 v2 v3 v4 v5 v6 v7 v8

The longest shortest path from v2 (2) to v3 (3). Diameter is 4.

The longest shortest path from v13 (13) to v19 (19). Diameter is 9. Number of unreachable pairs: 212 Average distance among reachable pairs:

P(k) prob that node is connected to k other nodes Web

P(k) prob that node is connected to k other nodes Web

Diameter of the Web Albert, Jeong & Barabasi (1999) Nature. For Web searching, the important quantity is l the smallest number of URL links that connect document A to document B. If we have N web-pages then we find (by experiment) the average of l over all pairs of links is given by

Scale-Free Model Barabasi et al. (2000) Growth Preferential Attachment Matthew Effect

Scale-Free Model Barabasi et al. (2000) Results of Model

Classification of Networks Newman (2003) Social Networks Friendships, business relationships, sexual contacts Miligram’s “small world” networks, “6 degrees of separation” Information Networks Citations between academic papers. World Wide Web Peer to Peer Networks Technological Networks Electric Power Grid Communication: Airline, Road, Rail Telephone Internet

Classification of Networks Newman (2003) Biological Networks Metabolic Pathways Physical Interaction between proteins Gene regulatory network Food Web Neural Networks Blood vessels and vascular system

Scientific Collaboration Girvan & Newman (2002)

Sexual Contact Potterat et al. (2002)

Stats for several published networks Newman

Community Structure Moody (2001)

Community Structure

NetworkType P(k)CDcommunityNotes Real World (RW) Long right tailC prop k^-1 RandomPoisson bell- shape Small C = O(N^-1) K^D=N D=logN/logk noShows SW effect RegularN/4k2k/N Scale Free (SF) Log(N)P(K) prop k^- gam C prop k^-1Log(N) or log(log(N)) No SW effect Small World (SW) (i) Log(N) for fixed k (ii) << N D=logN/logkP(k) does not match RW

Small-World Network Average path length small compared to size of network Net has many tightly connected groups, small tieds to outsiders Clustering. Friends: friends of a person are also likely to be friends.

Small-World Network Power law distribution – vast number of nodes have only few connexions, but small number are highly connected and therefore play significant role in networking. “Clustering coefficient”.. See numerical graph. Explained by “growth” – connect to nodes that are well connected!

WWW (2000)

Measurement and Analysis of Online Social Networks Mislove et al. (2007) How users browse Flickr 81% of viewers came from within Flickr 6% of viewers used the Flickr photo search 13% of viewers came in from an external link

Network Resilience Homogeneous net Internet net

Interconnect Topologies Bus Mesh Ring Complete Point to Point Star

Internet Mapping Project by Hal Burch and Bill Cheswick (LUMETA)

Networks

Some Definitions Degree kNumber of edges connected to a vertex Diameter DThe longest of the shortest path from one node to another Average of all shortest paths Cluster Coefficient C Measures the density of triangles in the network

High School Dating

Politics

Given a network, there are a number of structural questions we may ask: 1. How many connections does the average node have? 2. Are some nodes more connected than others? 3. Is the entire network connected? 4. On average, how many links are there between nodes? 5. Are there clusters or groupings within which the connections are particularly strong? 6. What is the best way to characterize a complex network? 7. How can we tell if two networks are “different”? 8. Are there useful ways of classifying or categorizing networks?

Network Questions: Communities 1. Are there clusters or groupings within which the connections are particularly strong? 2. What is the best way to discover communities, especially in large networks? 3. How can we tell if these communities are statistically significant? 4. What do these clusters tell us in specific applications?

Network Questions: Dynamics on 1. How do diseases/computer viruses/innovations/rumors/revolutions propagate on networks? 2. What properties of networks are relevant to the answer of the above question? 3. If you wanted to prevent (or encourage) spread of something on a network, what should you do? 4. What types of networks are robust to random attack or failure? 5. What types of networks are robust to directed attack? 6. How are dynamics of and dynamics on coupled?

Network Questions: Algorithms 1. What types of networks are searchable or navigable? 2. What are good ways to visualize complex networks? 3. How does google page rank work? 4. If the internet were to double in size, would it still work? There are also many domain-specific questions: 1. Are networks a sensible way to think about gene regulation or protein interactions or food webs? 2. What can social networks tell us about how people interact and form communities and make friends and enemies? 3. Lots and lots of other theoretical and methodological questions What else can be viewed as a network? Many applications await.

Some Questions Technological Networks (eg Internet) Information Networks (eg WWW) Social Networks

A network model treats all nodes the same and all links or edges the same In a picture of a network, the spatial location of nodes is arbitrary Networks are abstractions of connection and relation Networks have been used to model a vast array of stuff Vertex / node Edge / link

1736 Leonhard Euler wonders, can I walk through the city of Konigsberg and cross each bridge once and only once?

RegularRandomSmall World

Probability (chance) of reconnection

Draw The Internet