Professor Yashar Ganjali Department of Computer Science University of Toronto

Announcements Class mailing list: if you haven’t received an from me, please let me know. Check out class web page for slides, and lecture notes. Volunteer for lecture notes? SII Computer Networks and Society2University of Toronto – Fall 2015

Announcements – Cont’d Assignment # 1 is posted. Submission deadline: 5PM on Friday Oct. 9 th your solutions (one PDF file) to me; or Slide them under my office door BA5238 SII Computer Networks and Society3University of Toronto – Fall 2015

SII Computer Networks and SocietyUniversity of Toronto – Fall 2015 The Story So Far … Last week: Introduction to Computer Networks Basics concepts and components An introduction to the mail system An introduction to the Internet This week: The Science of Network Introduction to graphs Random graphs Scale-free networks Six degrees of separation Diffusion of information 4

Konigsberg Bridge Problem Can one walk cross the 7 bridges and never pass the same bridge twice? Model as a graph Nodes: pieces of land Edges: bridges Euler proved that … Such a path does not exist Why? D A B C SII Computer Networks and Society5University of Toronto – Fall 2015

Science of Networks Society as a graph Nodes: people Edges or links: relationships Why? Removes unnecessary details Easier to focus, and … To generalize ideas Use ideas from one domain in another one Can be applied to other networks Examples: students in this class, computers at your home, … SII Computer Networks and Society6University of Toronto – Fall 2015

Some Terminology Edges can be directed or undirected Examples? Edges can also have weights Indication of the strength of the relationship between two nodes Degree of a node Number of links connected to that node A component of a graph is … Set of nodes that are connected by links A connected graph has only one component SII Computer Networks and Society7University of Toronto – Fall 2015

Terminology – Cont’d Path between node a and node b Start at node a. Go through a sequence of nodes (and links) to reach b. Path Length is the number of links on the path SII Computer Networks and Society8University of Toronto – Fall 2015

Definitions – Cont’d Shortest path between node a and node b Path with the minimum number of links We can have more than one shortest path Distance is the length of shortest path Diameter is the distance of the two furthest nodes SII Computer Networks and Society9University of Toronto – Fall 2015

Characteristics of Social Networks What do social networks look like today? Are they completely random? Is there a pattern we can find? Properties of social networks Average number of friends Average path length between any two people How does information propagate? News, fashion, … Who are the most influential people? Fastest to distribute info, best target for advertisement How can we detect communities? SII Computer Networks and Society10University of Toronto – Fall 2015

Random Graphs Random graphs are extremely simple ways of modeling networks How to build a random graph Consider all pairs of nodes (a, b) With probability p connect a and b Question: what is the average degree of each node? Answer: p(N-1) Question: what is the average number of links? Answer: [p(N-1)N]/2 SII Computer Networks and Society11University of Toronto – Fall 2015

Example SII Computer Networks and Society12University of Toronto – Fall 2015 No. of nodes12 Probability (p)00.11 Avg. degree Avg. no. links Avg. path length??1

Properties of Random Graphs All nodes are treated similarly Is this true in human networks? What about computer networks? Normal degree distribution Sum of multiple coin flips Let’s try it … SII Computer Networks and Society13University of Toronto – Fall 2015

In Class Exercise Flip a coin 11 times Assuming there are 12 students in class If the i-th flip is heads You and student number i are friends Else You and student number i are not friends Count the number of friends you have. SII Computer Networks and Society14University of Toronto – Fall 2015

Social Networks vs. Random Graphs Can we model social networks with random graphs? There are groups of nodes which are highly connected to each other But have less connectivity to the outside Non-uniform probability of friendship Not all nodes are similar Few people have many many friends Degree distribution is not normal SII Computer Networks and Society15University of Toronto – Fall 2015 Social networks are far from being random

Hubs in Social Networks In social networks not all nodes are equal. Hubs: nodes with extremely larger degrees People who know a lot of people They connect different communities to each other Degree distribution is not normal. Heavy-tailed SII Computer Networks and Society16University of Toronto – Fall 2015

Examples of Hubs In the World Wide Web, hubs might be websites such as Google, Facebook. In Hollywood, the hubs are the actors who have worked with the most people. In school, hubs are students who take many classes and interact with many classmates. Membership in different groups Class representatives … SII Computer Networks and Society17University of Toronto – Fall 2015

Scale-Free Networks Small subset of nodes have high degree of connectivity Node degrees are heavy-tailed Small number of nodes have very high degrees Majority of nodes have small degrees Not normal distribution (like random graphs) SII Computer Networks and Society18University of Toronto – Fall 2015

Barabasi-Albert Networks Start from a small number of nodes Add new nodes one after another Each node will connect to k previous nodes Preferential attachment Connect to high node degrees with higher probability Rich gets richer This approach creates hubs Few nodes in the network with very large degree of connectivity SII Computer Networks and Society19University of Toronto – Fall 2015

Example (k = 1) SII Computer Networks and Society20University of Toronto – Fall 2015

Rich Get Richer In Barabasi-Albert model, incoming node has a higher probability of connecting to a node with a larger degree than a node with a small degree. In other words, “rich get richer”. This is what leads to creation of hubs. Question. If you want to increase the size of your friends network, who is a better target: Someone who has a similar background? Or, Someone you know with a completely different background? SII Computer Networks and Society21University of Toronto – Fall 2015

Six Degrees of Separation Harvard Psychology Professor Study of obedience In 1967 performed an experiment Random people where asked to send a letter (through intermediaries) to someone in Boston. If they didn’t know the target, they could send it to someone who might know him. Only send to someone who you know on a first-name basis. The average path length was six Among the letters that were received Many were not SII Computer Networks and Society22University of Toronto – Fall 2015 Small World

It Is a Small World Same “small world” phenomena is seen in other networks Co-authorship network Nodes: paper authors Links: co-authorship relationship Network of web page Nodes: web pages Links: referrals Hollywood Nodes: actors and actresses Links: playing in the same movie SII Computer Networks and Society23University of Toronto – Fall 2015

The Kevin Bacon Game Consider any actor You can get to Kevin Bacon in 6 steps or less By traveling along the links connecting actors Two actors are connected if they played in the same movie together. SII Computer Networks and Society24University of Toronto – Fall 2015

Separation of Web Pages, Molecules, … You can get from a given web page to any other in a maximum of 19 clicks Barabasi et al. Molecules in a cell Connected by reactions between molecules Most pairs can be linked by a path of length three SII Computer Networks and Society25University of Toronto – Fall 2015

Diffusion in Social Networks Ideas, products, viruses, … can spread in networks Spreading rate: how fast the number of adopters grows Depends on how likely people are to adopt University of Toronto – Fall 2015 Innovation Late adopters Early adopters SII Computer Networks and Society26

Diffusion in Networks – Cont’d In random networks Either the entire network is infected, or It dies out Depends on spreading rate Above a threshold  all nodes will be infected Below that threshold  spread will die out In scale-free networks however No epidemic threshold Steady state of small persistence rate Hubs have an important role in the spread It is critical to protect or infect the hubs SII Computer Networks and Society27University of Toronto – Fall 2015

Connectivity in Scale-Free Networks Removing a few nodes randomly in scale-free networks does not make it disconnected. You need to remove many nodes for that to happen. Why? Randomly selecting nodes, chances are you will select one with a small degree. There are very few hubs after all. In an adversarial attack however, one can remove hubs. Very few removals can break the system into several parts. Think in terms of communication networks. SII Computer Networks and Society28University of Toronto – Fall 2015

Summary and Discussion We model social networks with graphs. Random graphs capture some properties of social networks, but not all. Scale-free networks are better ways of modeling social networks. We have heavy-tailed degree distribution in scale-free networks. Diffusion of information Random networks: all nodes infected if rate is above a threshold Scale-free networks: steady persistence regardless of the rate Can we take advantage of this? SII Computer Networks and Society29University of Toronto – Fall 2015