• Instructors: Prof. Alan Mislove and Prof. Christo Wilson
• Contact: amislove@ccs.neu.edu and cbw@ccs.neu.edu
  • Use cs5750f13-staff@ccs.neu.edu for everything other than personal issues
• Office: 250 WVH (Mislove) and 348 WVH (Wilson)
• Office Hours: 4:30-6:00 Mondays (Mislove) and TBA (Wilson)
• Teaching Assistant: Arash Molavi Kakhki
• Contact: cs5750f13-staff@ccs.neu.edu
• Office: 266 WVH
• Office Hours: TBA

• Web page: http://www.ccs.neu.edu/~cbw/socialnetworks.html
• Forum: https://piazza.com/northeastern/fall2013/cs5750‎
• Textbook(s):
  • Recommended: Easley, David and Kleinberg, Jon. Networks, Crowds, and Markets: Reasoning About a Highly Connected World. University Press, Cambridge, UK. 2010.
  • Optional: Watts, Duncan J. Six Degrees: The Science of a Connected Age. W. W. Norton & Company, 2009.
  • Optional:Shirky, Clay. Here Comes Everybody: The Power of Organizing Without Organizations. Penguin Press, 2008.

• Course is primarily research-oriented; you must be prepared and participate
• Require 500-word summary of readings by midnight before class
  • Represents 10% of your grade, in aggregate
  • Must be ASCII text
• Students will be presenting the papers we discuss
• High-quality presentation, approx. 15-20 minutes
• Also bring a list of discussion questions
  • You are in charge of the class!
  • Represents 10% of your grade, in aggregate
• Sign-up sheet linked from Piazza

• There are four homeworks throughout the semester
  • Practical implementations of concepts in class
  • Hands-on experience with real data, systems, etc
• Will be completed in teams of two
  • You will pick your teammate
  • You can switch teammates between projects
  • Both teammates must put in equal effort
• You are given four slip days to use on homeworks
  • Can turn in late with no penalty
  • All team members must have a slip day to use it
• First homework handed out today!

• There is a course research project you must complete with a partner
  • Topic of your own choosing, in consultation with course staff
  • Research project, not a programming project
  • 40% of your overall grade!
• Example ideas of course projects:
  • Building an iOS/Android app to conduct an experiment
  • Using Amazon Mechanical Turk to solve a problem
  • Analysis of large-scale Twitter data
• Upcoming deadlines for course project:
  • Meet with instructors: Sept. 30
  • Project proposal (~5 pages): Oct. 9
  • Interim report (~10 pages): Nov. 6
  • Project presentation (~20 min): Nov. 20 - Dec. 4
  • Final report (~20 pages): TBA

• All summaries and homeworks will be submitted using our turnin system
• You must have a CCIS Linux account
  • If not, register for one immediately
• Emailed submissions will get no credit
• You must first register with the system:

bash$ /course/cs5750f13/bin/register-student ID
About to register user 'USER' with student ID 'ID'.  Is this correct? [yn]

• Then, you can submit assignments:

bash$ /course/cs5750f13/bin/turnin summary02 ~/cs5750/summary02.txt
  Added file summary02.txt (28392 bytes)
Successfully submitted summary02 for user amislove (confirmation ZiwKE5).
Submitted a total of 1 files (28392 bytes) in 0 directories.

• Graph theory born with Euler's bridges problem
• No way to cross all bridges exactly once and return to starting point
  • Convert city map into graph; prove no complete cycle exists

• A network (graph) is a set of vertices (nodes) connected by edges (links)
• Typically represented as \(G=(V,E)\)
  • \(V=\{v_1, v_2, ... v_n\}\) is the set of vertices
  • \(E=\{e_1, e_2, ... e_n\}\) is the set of edges, with \(e_i=(v_{from}, v_{to})\)
• Graphs can be directed or undirected
  • Links in directed graphs have a direction:
  • Links in undirected graphs have no direction:
  • When are directed/undirected graphs appropriate?

• Edges may have a weight attached to them
  • Can represent distance, current usage, cost, strength
  • Often used to find "low-cost" paths in the network
• Multiple edges may be allowed between nodes
• An edge may go from a node to itself
  • This is known as a self-loop

• Recall a walk is a series of sequential edges \(\{v_i,v_j\},\{v_j,v_k\},...\{v_m,v_n\}\)
• Often use random walks to understand graph structure
  • Walk formed by starting a vertex \(v_i\)
  • At each step, select random outgoing edge from current vertex
• Probability of selecting each outgoing edge is $$\frac{1}{{\rm outdegree}(v)}$$
• Used in many algorithms (e.g., PageRank)

• One application of random walks: stationary distribution
• Suppose we take longer and longer random walks....
  • Eventually probability of being at any node stabilizes
  • This distribution is called stationary distribution
• Can be used to measure mixing time: how long to reach stationary distribution?
  • Example of a graph with high mixing time?
  • Low mixing time?
• Can also define:
  • Hit time \(h(u,v)\): expected random walk length from \(u\) to \(v\)
  • Commute time \(c(u,v) = h(u,v) + h(v,u)\)

• A geodesic is also known as a shortest path
  • Path with minimal hops from \(A\) to \(B\)
• Often, we want to find such a path
  • Djikstra's algorithm is useful for this
  • Gives shortest paths to all destinations from single source
• Can also measure average path length of a graph: $$\frac{\sum_{u,v\in V; u\neq v} {\rm path\_length}(u,v)}{N * (N-1)}$$ (pairs of nodes with no path between them are ignored)

• Often interested in "how far" nodes are in a graph
  • Average path length on doesn't say much about worst off/best off nodes
• Define eccentricity of a node to be $$e(v) = \max_{w\in V, w\neq v} {\rm path\_length}(v,w)$$
• Then can define:
  • The radius of a graph is $$\min_{v\in V} e(v)$$
  • The diameter of a graph is $$\max_{v\in V} e(v)$$

• Two graphs \(G\) and \(H\) are isomorphic if there exists a bijection $$f : V_G \rightarrow V_H$$ and $$(v_i, v_j) \in E_G \iff (f(v_i), f(v_j)) \in E_H$$
• Graph isomorphism problem is determining whether such a bijection exists
  • Does it sound hard or easy?

• The neighborhood of node \(i\) is the set of nodes \(i\) is connected to $$N(i) = \{v\ \ {\rm s.t.}\ \ (i, v)\in E\}$$
• The degree of \(i\) is the size of the neighborhood \(|N(i)|\)
• For directed graphs, slightly more complicated
  • Define outdegree as the number of outgoing edges from \(i\)
  • Define indegree as the number of incoming edges to \(i\)
• Define average [out,in]degree as the average across all nodes

• The degree distribution \(p(d)\) is the probability distribution of degrees
  • \(p(1) = \frac{1}{6}\), \(p(2) = \frac{1}{2}\), \(p(3) = \frac{1}{6}\), \(p(d>4) = 0\)
• The \(k_{nn}\) distribution is average neighbors' degree based on degree
  • \(k_{nn}(1) = 3\)
  • \(k_{nn}(2) = \frac{\frac{2+3}{2} + \frac{2+3}{2} + \frac{2+2}{2}}{3} = \frac{7}{3}\)
  • \(k_{nn}(3) = \frac{2+2+1}{3} = \frac{5}{3}\)
  • \(k_{nn}(d>4)\ {\rm undefined}\)

• Captures linking behavior of nodes
  • Can be used to measure "homophily" of any node property
  • Typically used to measure degree correlation (i.e., do high degree nodes link to other high degree nodes?)
• Defined as Pearson correlation coefficient of node degrees
  • Range is \(r\in [-1,1]\), 0 represents no correlation
  • Graph with \(r<0\) is disassortative, \(r>0\) is assortative
• For directed networks, can define \(r(in,out)\), \(r(in,in)\), etc
• Why would assortativity be useful to understand?

• Often interested in "importance" of nodes; referred to as centrality
  • Many ways to measure; most accepted is betweenness centrality
  • Essentially, how much does this node connect others?
• Defined as $$g(v) = \frac{\sum_{s\neq v\neq t} \frac{\sigma_{st}(v)}{\sigma_{st}}}{(N-1)(N-2)}$$ where
  • \(N\) is the number of nodes
  • \(\sigma_{st}\) is number of shortest paths from \(s\) to \(t\)
  • \(\sigma_{st}(v)\) is the number of these that pass through \(v\)

• Often interested in how graphs break down (i.e., how resilient is a graph?)
• Can define \(k\)-core of a graph as
  • A maximal connected subgraph (i.e., the largest subgraph)
  • Where all vertices have degree \(k\)
• How to determine if a \(k\)-core exists? (for a fixed \(k\))
  • Recursively remove all vertices with degree < \(k\)
  • If you are left with no vertices, no \(k\)-core exists
• What kind of graph would have a large \(k\)-core? A small \(k\)-core?

Slides and material are borrowed from many places, including:

• Jon Crowcroft's Priciples of Communication course, University of Cambridge
• Daron Acemoglu's and Asu Ozdaglar's OpenCourseWare 6.207 Networks course, MIT
• Dragomir Radev's Network Theory class, Columbia University

Slides were developed using Panic's Coda web-editing software, Google's I/O HTML5 slide code, and sigma.js.