CS224W: Analysis of Networks Lecture Notes PDF

CS224W: Analysis of Networks Jure Leskovec, Stanford University http://cs224w.stanford.edu  Roles Communities RolX Fa...

CS224W: Analysis of Networks Jure Leskovec, Stanford University http://cs224w.stanford.edu  Roles Communities RolX Fast Modularity Henderson, et al., KDD 2012 Clauset, et al., Phys. Rev. E 2004 Nodes with different structural roles Nodes belonging to the same (connector node, bridge node, etc.) cluster/community 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 2  Plan for Today: ¡ Structural role discovery in networks ¡ Community detection via Modularity optimization 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 3  ¡ Roles are “functions” of nodes in a network: § Roles of species in ecosystems § Roles of individuals in companies ¡ Roles are measured by structural behaviors: § Centers of stars § Members of cliques § Peripheral nodes, etc. 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 5  centers of stars members of cliques peripheral nodes Network Science Co-authorship network [Newman 2006] 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 6  ¡ Role: A collection of nodes which have similar positions in a network: ¡ Roles are based on the similarity of ties among subsets of nodes § Different from community (or cohesive subgroup) § Group is formed based on adjacency, proximity or reachability § This is typically adopted in current data mining Nodes with the same role need not be in direct, or even indirect interaction with each other 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 7  ¡ Roles: § A group of nodes with similar structural properties ¡ Communities: § A group of nodes that are well-connected to each other ¡ Roles and communities are complementary ¡ Consider the social network of a CS Dept: § Roles: Faculty, Staff, Students § Communities: AI Lab, Info Lab, Theory Lab 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 8  ¡ Structural equivalence: Nodes ! and " are structurally equivalent if they have the same relationships to all other nodes [Lorrain & White 1971] § Structurally equivalent nodes are likely to be similar in other ways – i.e., friendships in social networks a b c u v d e 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 9  ¡ Nodes ! and " are structurally equivalent: § For all the other nodes #, node ! has tie to # iff node " has tie to # Adjacency matrix ¡ Example: 1 2 1 2 3 4 5 1 - 0 1 1 0 2 0 - 1 1 0 3 4 3 0 0 - 0 1 4 0 0 0 - 1 5 5 0 0 0 0 - ¡ E.g., nodes 3 and 4 are structurally equivalent 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 48 Task Example Application Role query Identify individuals with similar behavior to a known target Role outliers Identify individuals with unusual behavior Role dynamics Identify unusual changes in behavior Identity resolution Identify/de-anonymize, individuals in a new network Role transfer Use knowledge of one network to make predictions in another Network comparison Compute similarity of networks, determine compatibility for knowledge transfer 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 12  ¡ RolX: Automatic discovery Role Discovery of nodes’ structural roles in Input networks [Henderson, et al. 2011b] Output § Unsupervised learning approach § No prior knowledge required § Assigns a mixed-membership of roles to each node üAutomated discovery § Scales linearly in #(edges) üRoles Behavioral roles ü generalize 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 13  Input Example: degree, mean Recursive weight, # of edges in Node × Node Node × Feature ego-network, mean Feature Adjacency Matrix Matrix clustering coefficient of Extraction neighbors, etc. Role Extraction Node × Role Role × Feature Matrix Matrix Output 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 14  ¡ Recursive feature extraction [Henderson, et al. 2011a] turns network connectivity into structural features Regional Neighborhood Local Egonet Recursive 1411# 0# 1# 2# 1# 0# 0# 0# 1# 1# 0# 1# 0# 0# 1# 1# 2# 2# Recursive 1410# 0# 1# 1# 1# 0# 1# 0# 0# 1# 0# 1# 0# 1# 0# 1# 1# 1# 338# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 1# 0# 0# 0# 339# 1# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 1# 0# 1# 0# 0# 0# 1415# 0# 1# 1# 2# 0# 1# 0# 0# 0# 0# 0# 0# 1# 1# 1# 1# 1# feature 941# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1414# 0# 1# 1# 1# 0# 1# 0# 0# 0# 0# 0# 0# 1# 1# 0# 1# 1# 942# 0# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1413# 0# 1# 1# 1# 0# 1# 1# 0# 0# 0# 0# 0# 1# 1# 0# 1# 1# extraction 1412# 0# 0# 0# 0# 0# 0# 0# 1# 2# 0# 1# 1# 0# 0# 1# 2# 0# 940# 0# 0# 1# 0# 0# 0# 0# 1# 0# 0# 0# 1# 1# 0# 1# 1# 1# ReFeX 1419# 0# 0# 1# 0# 0# 1# 0# 1# 1# 0# 1# 1# 1# 0# 1# 1# 1# Nodes 945# 0# 1# 4# 3# 0# 0# 0# 0# 2# 0# 1# 0# 0# 2# 1# 3# 1# 332# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1418# 0# 0# 1# 0# 0# 0# 0# 1# 0# 0# 0# 1# 2# 0# 1# 0# 1# 946# 0# 1# 1# 0# 0# 1# 0# 1# 0# 0# 0# 1# 4# 0# 1# 1# 2# 333# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1417# 0# 1# 1# 1# 0# 2# 0# 0# 1# 0# 1# 0# 1# 0# 1# 1# 1# 943# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1# 0# 0# 330# 1# 3# 2# 0# 1# 2# 2# 0# 2# 2# 2# 0# 3# 1# 0# 2# 5# 1416# 0# 1# 1# 1# 1# 2# 0# 0# 1# 0# 1# 0# 1# 0# 0# 1# 1# 944# 0# 1# 4# 2# 0# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 3# 1# 331# 0# 3# 2# 1# 0# 1# 0# 0# 2# 0# 2# 0# 2# 0# 1# 2# 5# 949# 0# 0# 0# 0# 2# 0# 0# 1# 0# 1# 0# 1# 0# 0# 0# 0# 0# 336# 0# 0# 0# 0# 2# 0# 0# 1# 1# 1# 1# 1# 0# 0# 0# 1# 0# 337# 1# 1# 1# 0# 0# 1# 2# 0# 1# 1# 1# 0# 1# 1# 1# 1# 1# 947# 1# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 1# 0# 1# 0# 0# 0# 334# 0# 0# 0# 1# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 948# 0# 0# 0# 0# 0# 1# 0# 1# 1# 0# 1# 1# 1# 0# 1# 1# 0# 335# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1# 0# 0# 531# 1# 0# 0# 0# 1# 0# 2# 0# 0# 2# 0# 0# 0# 2# 0# 0# 0# ¡ Neighborhood features: What is a node’s connectivity pattern? ¡ Recursive features: To what kinds of nodes is a node connected? 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 15  ¡ Idea: Aggregate features of a node and use them to generate new recursive features ¡ Base set of a node’s neighborhood features: § Local features: All measures of the node degree: § If network is directed, include in- and out-degree, total degree § If network is weighted, include weighted feature versions § Egonetwork features: Computed on the node’s egonet: § Egonet includes the node, its neighbors, and any edges in the induced subgraph on these nodes § #(within-egonet edges), #(edges entering/leaving egonet) Egonet for red node 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 16  ¡ Start with the base set of node features ¡ Use the set of current node features to generate additional features: § Two types of aggregate functions: means and sums § E.g., mean value of “unweighted degree” feature among all neighbors of a node § Compute means and sums over all current features, including other recursive features Features § Repeat 1411# 1410# 338# 0# 0# 0# 1# 1# 0# 2# 1# 0# 1# 1# 0# 0# 0# 1# 0# 1# 0# 0# 0# 1# 1# 0# 0# 1# 1# 0# 0# 0# 1# 1# 1# 0# 0# 0# 0# 0# 1# 0# 1# 0# 1# 1# 1# 0# 2# 1# 0# 2# 1# 0# The number of possible recursive 339# 1# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 1# 0# 1# 0# 0# 0# 1415# 0# 1# 1# 2# 0# 1# 0# 0# 0# 0# 0# 0# 1# 1# 1# 1# 1# ¡ 941# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1414# 0# 1# 1# 1# 0# 1# 0# 0# 0# 0# 0# 0# 1# 1# 0# 1# 1# 942# 0# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1413# 0# 1# 1# 1# 0# 1# 1# 0# 0# 0# 0# 0# 1# 1# 0# 1# 1# 1412# 0# 0# 0# 0# 0# 0# 0# 1# 2# 0# 1# 1# 0# 0# 1# 2# 0# features grows exponentially with 940# 0# 0# 1# 0# 0# 0# 0# 1# 0# 0# 0# 1# 1# 0# 1# 1# 1# Nodes 1419# 0# 0# 1# 0# 0# 1# 0# 1# 1# 0# 1# 1# 1# 0# 1# 1# 1# Output 945# 0# 1# 4# 3# 0# 0# 0# 0# 2# 0# 1# 0# 0# 2# 1# 3# 1# 332# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1418# 0# 0# 1# 0# 0# 0# 0# 1# 0# 0# 0# 1# 2# 0# 1# 0# 1# 946# 0# 1# 1# 0# 0# 1# 0# 1# 0# 0# 0# 1# 4# 0# 1# 1# 2# 333# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# each recursive iteration: 1417# 0# 1# 1# 1# 0# 2# 0# 0# 1# 0# 1# 0# 1# 0# 1# 1# 1# 943# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1# 0# 0# 330# 1# 3# 2# 0# 1# 2# 2# 0# 2# 2# 2# 0# 3# 1# 0# 2# 5# 1416# 0# 1# 1# 1# 1# 2# 0# 0# 1# 0# 1# 0# 1# 0# 0# 1# 1# 944# 0# 1# 4# 2# 0# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 3# 1# 331# 0# 3# 2# 1# 0# 1# 0# 0# 2# 0# 2# 0# 2# 0# 1# 2# 5# 949# 0# 0# 0# 0# 2# 0# 0# 1# 0# 1# 0# 1# 0# 0# 0# 0# 0# 336# 0# 0# 0# 0# 2# 0# 0# 1# 1# 1# 1# 1# 0# 0# 0# 1# 0# 337# 1# 1# 1# 0# 0# 1# 2# 0# 1# 1# 1# 0# 1# 1# 1# 1# 1# § Reduce the number of features using a 947# 1# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 1# 0# 1# 0# 0# 0# 334# 0# 0# 0# 1# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 948# 0# 0# 0# 0# 0# 1# 0# 1# 1# 0# 1# 1# 1# 0# 1# 1# 0# 335# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1# 0# 0# 531# 1# 0# 0# 0# 1# 0# 2# 0# 0# 2# 0# 0# 0# 2# 0# 0# 0# pruning technique: § Look for pairs of features that are highly correlated § Eliminate one of the features whenever two features are correlated above a user-defined threshold 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 17 Input 1411# 0# 1# 2# 1# 0# Features 0# 0# 1# 1# 0# 1# 0# 0# 1# 1# 2# 2# 1410# 0# 1# 1# 1# 0# 1# 0# 0# 1# 0# 1# 0# 1# 0# 1# 1# 1# 338# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 1# 0# 0# 0# 339# 1# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 1# 0# 1# 0# 0# 0# 1415# 0# 1# 1# 2# 0# 1# 0# 0# 0# 0# 0# 0# 1# 1# 1# 1# 1# 941# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1414# 0# 1# 1# 1# 0# 1# 0# 0# 0# 0# 0# 0# 1# 1# 0# 1# 1# 942# 0# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# Recursively 1413# 0# 1# 1# 1# 0# 1# 1# 0# 0# 0# 0# 0# 1# 1# 0# 1# 1# 1412# 0# 0# 0# 0# 0# 0# 0# 1# 2# 0# 1# 1# 0# 0# 1# 2# 0# 940# 0# 0# 1# 0# 0# 0# 0# 1# 0# 0# 0# 1# 1# 0# 1# 1# 1# Nodes 1419# 0# 0# 1# 0# 0# 1# 0# 1# 1# 0# 1# 1# 1# 0# 1# 1# 1# 945# 0# 1# 4# 3# 0# 0# 0# 0# 2# 0# 1# 0# 0# 2# 1# 3# 1# extract features 332# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1418# 0# 0# 1# 0# 0# 0# 0# 1# 0# 0# 0# 1# 2# 0# 1# 0# 1# 946# 0# 1# 1# 0# 0# 1# 0# 1# 0# 0# 0# 1# 4# 0# 1# 1# 2# 333# 0# 0# 0# 0# 1# 0# 1# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 1417# 0# 1# 1# 1# 0# 2# 0# 0# 1# 0# 1# 0# 1# 0# 1# 1# 1# 943# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1# 0# 0# 330# 1# 3# 2# 0# 1# 2# 2# 0# 2# 2# 2# 0# 3# 1# 0# 2# 5# 1416# 0# 1# 1# 1# 1# 2# 0# 0# 1# 0# 1# 0# 1# 0# 0# 1# 1# 944# 0# 1# 4# 2# 0# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 3# 1# 331# 0# 3# 2# 1# 0# 1# 0# 0# 2# 0# 2# 0# 2# 0# 1# 2# 5# 949# 0# 0# 0# 0# 2# 0# 0# 1# 0# 1# 0# 1# 0# 0# 0# 0# 0# 336# 0# 0# 0# 0# 2# 0# 0# 1# 1# 1# 1# 1# 0# 0# 0# 1# 0# 337# 1# 1# 1# 0# 0# 1# 2# 0# 1# 1# 1# 0# 1# 1# 1# 1# 1# 947# 1# 0# 0# 0# 2# 0# 1# 0# 0# 2# 0# 1# 0# 1# 0# 0# 0# 334# 0# 0# 0# 1# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 948# 0# 0# 0# 0# 0# 1# 0# 1# 1# 0# 1# 1# 1# 0# 1# 1# 0# 335# 0# 0# 0# 1# 0# 0# 0# 0# 0# 0# 0# 0# 0# 0# 1# 0# 0# 531# 1# 0# 0# 0# 1# 0# 2# 0# 0# 2# 0# 0# 0# 2# 0# 0# 0# 1) Can compare nodes based on their structural similarity 2) Can cluster nodes to identify different Output structural roles e.g, RolX uses a clustering technique called non-negative matrix factorization 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 18  ¡ Task: Cluster nodes based on their structural similarity ¡ Two networks: § Network science co-authorship network: § Nodes: Network scientists; Edges: The number of co-authored papers § Political books co-purchasing network: § Nodes: Political books on Amazon; Edges: Frequent co-purchasing of books by the same buyers ¡ Setup: For each network: § Use RolX to assign each node a distribution over the set of discovered, structural roles § Determine similarity between nodes by comparing their role distributions 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 19 IP traffic classes are well-separated in role space” with as few as 3 roles. (a) t showing the degree of membership of P2P, and Web host in each of three roles. density plot obtained by adding uniform (a) Role-colored Visualization of the Network to reveal overlapping points. DEVICE RolX Baseline in a) of Time s. DEVICE (a) m (a) Business Role-colored Role-colored Student Visualization graph: vs. Resteach nodeof the Networkby is colored Role affinity heat-map the primary role that RolX finds (b) Role Affinity Heat Map RolX Baseline Figure 9: RolX e↵ectively discovers roles in the Making sense of roles: Network Science Co-authorship Graph. (a) Author ¡ Blue circle: Tightly knit, nodes that participate in tightly-coupled groups network RolX discovered four roles, like the het- ¡ Red diamond: Bridge nodes, that connectbridges erophilous groups (red of nodes diamond ), as well as the ho- ¡ Gray rectangle: Main-stream, mostmophilous “pathy” nodesclique, of nodes, neither a (green nor a chain(b) Affin- triangle) ¡ Green triangle: Pathy, nodes thatity belong matrix to elongated (red clustersblue is low) - strong is high score, homophily for roles #1 and #4. 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 20  Roles Communities RolX Fast Modularity Henderson, et al., KDD 2012 Clauset, et al., Phys. Rev. E 2004 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 22  ¡ We often think of networks “looking” like this: ¡ What led to such a conceptual picture? 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 23  ¡ How does information flow through the network? § What structurally distinct roles do nodes play? § What roles do different links (“short” vs. “long”) play? ¡ How do people find out about new jobs? § Mark Granovetter, part of his PhD in 1960s § People find the information through personal contacts ¡ But: Contacts were often acquaintances rather than close friends § This is surprising: One would expect your friends to help you out more than casual acquaintances ¡ Why is it that acquaintances are most helpful? 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 24  [Granovetter ‘73] ¡ Two perspectives on friendships: § Structural: Friendships span different parts of the network § Interpersonal: Friendship between two people is either strong or weak ¡ Structural role: Triadic Closure a If two people in a network have a friend in common, then there is an increased likelihood b they will become friends c themselves. Which edge is more likely, a-b or a-c? 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 25  ¡ Granovetter makes a connection between social and structural role of an edge ¡ First point: Structure § Structurally embedded edges are also socially strong § Long-range edges spanning different parts of the network are socially weak ¡ Second point: Information § Long-range edges allow you to gather information from different parts of the network and get a job § Structurally embedded edges are S Weak Strong heavily redundant in terms of S a W b S information access S 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 26  ¡ Triadic closure == High clustering coefficient Reasons for triadic closure: ¡ If ! and " have a friend # in common, then: § ! is more likely to meet " B § (since they both spend time with #) § ! and " trust each other A C § (since they have a friend in common) § # has incentive to bring ! and " together § (since it is hard for # to maintain two disjoint relationships) ¡ Empirical study by Bearman and Moody: § Teenage girls with low clustering coefficient are more likely to contemplate suicide 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 27  ¡ For many years Granovetter’s theory was not tested ¡ But, today we have large who-talks-to-whom graphs: § Email, Messenger, Cell phones, Facebook ¡ Onnela et al. 2007: § Cell-phone network of 20% of country’s population § Edge strength: # phone calls 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 28  ¡ Edge overlap: |&(() ⋂ & + | !"# = |&(() ⋃ & + | § &(() … a set of neighbors of node ( ¡ Overlap = - when an edge is a local bridge 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 29  ¡ Cell-phone network ¡ Observation: § Highly used links True Neighborhood overlap have high overlap! Permuted strengths ¡ Legend: § True: The data § Permuted strengths: Keep the network structure but randomly reassign edge strengths Edge strength (#calls) 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 30  ¡ Real edge strengths in mobile call graph § Strong ties are more embedded (have higher overlap) 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 31  ¡ Same network, same set of edge strengths but now strengths are randomly shuffled 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 32  Low Size of largest component disconnects the network sooner Fraction of removed links ¡ Removing links by strength (#calls) § Low to high § High to low Conceptual picture of network structure 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 33  Low Size of largest component disconnects the network sooner Fraction of removed links ¡ Removing links based on overlap § Low to high § High to low Conceptual picture of network structure 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 34  ¡ Granovetter’s theory leads to the following conceptual picture of networks Strong ties Weak ties 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 35  ¡ Granovetter’s theory suggest that networks are composed of tightly connected sets of nodes Communities, clusters, ¡ Network communities: groups, modules § Sets of nodes with lots of internal connections and few external ones (to the rest of the network). 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 37  ¡ How to automatically find such densely connected groups of nodes? ¡ Ideally such automatically detected clusters would then correspond to real groups Communities, clusters, ¡ For example: groups, modules 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 38 ¡ Zachary’s Karate club network: § Observe social ties and rivalries in a university karate club § During his observation, conflicts led the group to split § Split could be explained by a minimum cut in the network 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 39  Find micro-markets by partitioning the “query-to-advertiser” graph in web search: query advertiser Nodes: advertisers and queries/keywords; Edges: Advertiser advertising on a keyword. 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 40  Can we identify node groups? (communities, modules, clusters) Nodes: Teams Edges: Games played 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 41  NCAA conferences Nodes: Teams Edges: Games played 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 42  Can we identify social communities? Nodes: Users Edges: Friendships 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 43 High school Company Stanford (Basketball) Stanford (Squash) Nodes: Users Social communities Edges: Friendships 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 44  Can we identify functional modules? Nodes: Proteins Edges: Interactions 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 45 Functional modules Nodes: Proteins Edges: Interactions 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 46  ¡ Communities: sets of tightly connected nodes ¡ Define: Modularity ! § A measure of how well a network is partitioned into communities § Given a partitioning of the network into groups " ∈ $: Q µ ∑sÎ S [ (# edges within group s) – (expected # edges within group s) ] Need a null model! 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 47  ¡ Given real ! on " nodes and # edges, construct rewired network !’ § Same degree distribution but i random connections j § Consider !’ as a multigraph § The expected number of edges between nodes '& '% '& % and & of degrees '% and '& equals to: '% ⋅ )# = )# § The expected number of edges in (multigraph) G’: + '% '& + + § = ∑ ∑ = ⋅ ∑ ' ∑&∈. '& = ) %∈. &∈. )# ) )# %∈. % Note: + § = )# ⋅ )# = # 0 31 = 25 /# 1∈2 10/11/18 Jure Leskovec, Stanford CS224W: Analysis of Networks 48  ¡ Modularity of partitioning S of graph G: § Q µ ∑sÎ S [ (# edges within group s) – (expected # edges within group s) ] & 0, 0- § ! ", $ = '( ∑*∈$ ∑,∈* ∑-∈*.,- − '( Aij = 1 if i®j, Normalizing const.: -1