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Graph Analyses with Python and NetworkX

Social networks are not new, even though websites like Facebook and Twitter might make you want to believe they are; and trust me- I’m not talking about Myspace! Social networks are extremely interesting models for human behavior, whose study dates back to the early twentieth century. However, because of those websites, data scientists have access to much more data than the anthropologists who studied the networks of tribes! Because networks take a relationship-centered view of the world, the data structures that we will analyze model real world behaviors and community. Through a suite of algorithms derived from mathematical Graph theory we are able to compute and predict behavior of individuals and communities through these types of analyses. Clearly this has a number of practical applications from recommendation to law enforcement to election prediction, and more.

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Graph Analysis with
Python and NetworkX
Graphs and Networks
Graph Analyses with Python and NetworkX
Graph Analyses with Python and NetworkX
Graph Theory
The Mathematical study of the application and
properties of graphs, originally motivated by the
study of games of chance.
Traced back to Euler’s work on the Konigsberg Bridges problem (1735), leading to the
concept of Eulerian graphs.
A Graph, G, consists of a finite set denoted by V or V(G) and a collection E or E
(G) of ordered or unordered pairs {u,v} where u and v ∈ V
vertices (nodes)
edges (links)

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Graph Analyses with Python and NetworkX

  • 5. Graph Theory The Mathematical study of the application and properties of graphs, originally motivated by the study of games of chance. Traced back to Euler’s work on the Konigsberg Bridges problem (1735), leading to the concept of Eulerian graphs.
  • 6. A Graph, G, consists of a finite set denoted by V or V(G) and a collection E or E (G) of ordered or unordered pairs {u,v} where u and v ∈ V vertices (nodes) edges (links)
  • 7. Graphs can be directed or undirected DiGraphs, the edges are ordered pairs: (u,v)
  • 9. Representing Graphs Adjacency Matrix [[0, 0, 1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0, 0, 0, 0, 1], [1, 0, 1, 0, 0, 0, 1, 1, 1, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 1, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0, 1, 0]]
  • 10. Undirected graphs have symmetric adjacency matrices. Representing Graphs Adjacency Matrix [[0, 0, 1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 1, 1, 0, 0, 0, 0, 1], [1, 0, 1, 0, 0, 0, 1, 1, 1, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 1, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0, 1, 0]]
  • 11. Describing Vertices Directed Networks Undirected Networks Node Neighbors Degree
  • 12. O(G) = ? S(G) = ? kY = ?
  • 13. O(G) = 14 S(G) = 21 kY = 9
  • 14. Describing Graphs by Structure
  • 15. a path from i to k Length(p) = # of nodes in path Paths(i,j) = set of paths from i to j Shortest (unweighted) path length Paths in a Network Traversal is about following paths between vertices
  • 17. Classes of Graph Algorithms Generally speaking CS Algorithms are designed to solve the classes of Math problems, but we can further categorize them into the following: 1. Traversal (flow, shortest distance) 2. Search (optimal node location) 3. Subgraphing (find minimum weighted spanning tree) 4. Clustering (group neighbors of nodes)
  • 18. For Reference Bellman-Ford Algorithm | Dijkstra's Algorithm Ford-Fulkerson Algorithm | Kruskai's Algorithm Nearest neighbor | Depth-First and Breadth- First
  • 19. So why are Graphs important?
  • 20. Ryan vs. Biden Debate (Twitter Reaction) http://thecaucus.blogs.nytimes.com/2012/10/16/who-won-presidential-debate-on-twitter/?_r=0
  • 21. Topics shifting over time http://informationandvisualization.de/blog/graphbased-visualization-topic-shifts
  • 23. Graph Analysis for Big Data (Uber Trips in San Francisco) http://radar.oreilly.com/2014/07/there-are-many-use-cases-for-graph-databases-and-analytics.html
  • 25. Why Graphs? 1. Graphs are abstractions of real life 2. Represent information flows that exist 3. Explicitly demonstrate relationships 4. Enable computations across large datasets 5. Allow us to compute locally to areas of interest with small traversals 6. Because everyone else is doing it (PageRank, SocialGraph)
  • 33. Machine Learning using Graphs - Machine Learning is iterative but iteration can also be seen as traversal.
  • 34. Machine Learning using Graphs - Machine Learning is iterative but iteration can also be seen as traversal. - Machine Learning requires many instances with which to fit a model to make predictions.
  • 35. Machine Learning using Graphs - Machine Learning is iterative but iteration can also be seen as traversal. - Machine Learning requires many instances with which to fit a model to make predictions. - Important analyses are graph algorithms: clusters, influence propagation, centrality.
  • 36. Machine Learning using Graphs - Machine Learning is iterative but iteration can also be seen as traversal. - Machine Learning requires many instances with which to fit a model to make predictions. - Important analyses are graph algorithms: clusters, influence propagation, centrality. - Performance benefits on sparse data
  • 37. Machine Learning using Graphs - Machine Learning is iterative but iteration can also be seen as traversal. - Many domains have structures already modeled as graphs (health records, finance) - Important analyses are graph algorithms: clusters, influence propagation, centrality. - Performance benefits on sparse data - More understandable implementation
  • 38. Iterative PageRank in Python def pageRank(G, s = .85, maxerr = .001): n = G.shape[0] # transform G into markov matrix M M = csc_matrix(G,dtype=np.float) rsums = np.array(M.sum(1))[:,0] ri, ci = M.nonzero() M.data /= rsums[ri] sink = rsums==0 # bool array of sink states # Compute pagerank r until we converge ro, r = np.zeros(n), np.ones(n) while np.sum(np.abs(r-ro)) > maxerr: ro = r.copy() for i in xrange(0,n): Ii = np.array(M[:,i].todense())[:,0] # inlinks of state i Si = sink / float(n) # account for sink states Ti = np.ones(n) / float(n) # account for teleportation r[i] = ro.dot( Ii*s + Si*s + Ti*(1-s) ) return r/sum(r) # return normalized pagerank
  • 39. Graph-Based PageRank in Gremlin pagerank = [:].withDefault{0} size = uris.size(); uris.each{ count = it.outE.count(); if(count == 0 || rand.nextDouble() > 0.85) { rank = pagerank[it] uris.each { pagerank[it] = pagerank[it] / uris.size() } } rank = pagerank[it] / it.outE.count(); it.out.each{ pagerank[it] = pagerank[it] + rank; } }
  • 41. ● Existence: Does there exist [a path, a vertex, a set] within [constraints]? ● Construction: Given a set of [paths, vertices] is a [constraint] graph construction possible? ● Enumeration: How many [vertices, edges] exist with [constraints], is it possible to list them? ● Optimization: Given several [paths, etc.] is one the best? Classes of Graph Analyses
  • 44. A social network is a data structure whose nodes are composed of actors (proper nouns except places) that transmit information to each other according to their relationships (links) with other actors. Semantic definitions of both actors and relationships are illustrative: Actor: person, organization, place, role Relationship: friends, acquaintance, penpal, correspondent Social networks are complex - they have non-trivial topological features that do not occur in simple networks. Almost any system humans participate and communicate in can be modeled as a social network (hence rich semantic relevance)
  • 45. - Scale Free Networks http://en.wikipedia.org/wiki/Scale-free_network - Degree distribution follows a power law - Significant topological features (not random) - Commonness of vertices with a degree that greatly exceeds the average degree (“hubs”) which serve some purpose - Small World Networks http://en.wikipedia.org/wiki/Small-world_network - Most nodes aren’t neighbors but can be reached quickly - Typical distance between two nodes grows proportionally to the logarithm of the order of the network. - Exhibits many specific clusters Attributes of a Social Network
  • 46. Degree Distribution - We’ve looked so far at per node properties (degree, etc) and averaging them gives us some information. - Instead, let’s look at the entire distribution - Degree Distribution: Power Law distribution:
  • 50. Graphs contain semantically relevant information - “Property Graph” https://github.com/tinkerpop/blueprints/wiki/Property-Graph-Model
  • 51. Property Graph Model The primary data model for Graphs, containing these elements: 1. a set of vertices ○ each vertex has a unique identifier. ○ each vertex has a set of outgoing edges. ○ each vertex has a set of incoming edges. ○ each vertex has a collection of properties defined by a map from key to value. 2. a set of edges ○ each edge has a unique identifier. ○ each edge has an outgoing tail vertex. ○ each edge has an incoming head vertex. ○ each edge has a label that denotes the type of relationship between its two vertices. ○ each edge has a collection of properties defined by a map from key to value.
  • 52. Graph: An object that contains vertices and edges. Element: An object that can have any number of key/value pairs associated with it (i.e. properties) Vertex: An object that has incoming and outgoing edges. Edge: An object that has a tail and head vertex.
  • 53. Modeling property graphs with labels, relationships, and properties. http://neo4j.com/developer/guide-data-modeling/
  • 54. Getting out of Memory
  • 55. File Based Serialization <xml /> GraphML {json} GraphSon NetworkX Gephi
  • 57. Neo4j: Querying with Cypher and a visual interface.
  • 58. Relational Data: Use GraphGen http://konstantinosx.github.io/graphgen-project/ NetworkX GraphGen
  • 59. Relational Data: Use GraphGen http://konstantinosx.github.io/graphgen-project/
  • 61. Sample graph - note the shortest paths from D to F and A to E.
  • 62. What is the most important vertex?
  • 63. Identification of vertices that play the most important role in a particular network (e.g. how close to the center of the core is the vertex?)
  • 64. A measure of popularity, determines nodes that can quickly spread information to a localized area.
  • 65. Degree centrality simply ranks nodes by their degree. k=4 k=4 k=1 k=1 k=1 k=2k=3
  • 66. Shows which nodes are likely pathways of information and can be used to determine how a graph will break apart of nodes are removed.
  • 67. Betweenness: the sum of the fraction of all the pairs of shortest paths that pass through that particular node. Can also be normalized by the number of nodes or an edge weight.
  • 68. A measure of reach; how fast information will spread to all other nodes from a single node.
  • 69. Closeness: average number of hops to reach any other node in the network. The reciprocal of the mean distance: n-1 / size(G) - 1 for a neighborhood, n
  • 70. A measure of related influence, who is closest to the most important people in the Graph? Kind of like “power behind the scenes” or influence beyond popularity.
  • 71. Eigenvector: proportional to the sum of centrality scores of the neighborhood. (PageRank is a stochastic eigenvector scoring)
  • 72. Detection of communities or groups that exist in a network by counting triangles. Measures “transitivity” - tripartite relationships that indicate clusters T(i) = # of triangles with i as a vertex Local Clustering Coefficient Graph Clustering Coefficient Counting the number of triangles is a start towards making inferences about “transitive closures” - e.g. predictions or inferences about relationships
  • 73. Green lines are connections from the node, black are the other connections
  • 74. ki = 6 T(i) = 4 Ci = (2*4) / (6*(6-1)) = 0.266
  • 75. Partitive classification utilizes subgraphing techniques to find the minimum number of splits required to divide a graph into two classes. Laplacian Matrices are often used to count the number of spanning trees.
  • 76. Distance based techniques like k-Nearest Neighbors embed distances in graphical links, allowing for very fast computation and blocking of pairwise distance computations.
  • 77. Just add probability! Bayesian Networks are directed, acyclic graphs that encode conditional dependencies and can be trained from data, then used to make inferences.
  • 79. Layouts - Open Ord - http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=731088 - Draws large scale undirected graphs with visual clusters - Yifan Hu - http://yifanhu.net/PUB/graph_draw_small.pdf - Force Directed Layout with multiple levels and quadtree - Force Atlas - http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0098679 - A continuous force directed layout (default of Gephi) - Fruchterman Reingold - http://cs.brown.edu/~rt/gdhandbook/chapters/force-directed.pdf - Graph as a system of mass particles (nodes are particles, edges are springs) This is the basis for force directed layouts Others: circular, shell, neato, spectral, dot, twopi …
  • 82. Lane Harrison, The Links that Bind Us: Network Visualizations http://blog.visual.ly/network-visualizations
  • 83. Lane Harrison, The Links that Bind Us: Network Visualizations http://blog.visual.ly/network-visualizations
  • 84. Lane Harrison, The Links that Bind Us: Network Visualizations http://blog.visual.ly/network-visualizations
  • 85. Lane Harrison, The Links that Bind Us: Network Visualizations http://blog.visual.ly/network-visualizations
  • 89. Tools for Graph Visualization
  • 90. Plan of Study Extraction of Network from Email Introduction to NetworkX Analyzing our Email Networks Visualizing our Email Network Relief from Gephi
  • 91. Graph Extraction from an email MBox