1. Lecture 10
Measures and Metrics

2. Network Growth Patterns
Network Segmentation
Graph Densification
Diameter Shrinkage

3. 1. Network Segmentation
Often, in evolving networks, segmentation takes place, where the large network is decomposed over time into three parts
Giant Component: As network connections stabilize, a giant component of nodes is formed, with a large proportion of network nodes and edges falling into this component.
Stars: These are isolated parts of the network that form star structures. A star is a tree with one internal node and n leaves.
Singletons: These are orphan nodes disconnected from all nodes in the network.

4. 2. Graph Densification

5. Densification in Real Networks
1.69
1.66
Source: Leskovec et al. KDD 2005
V(t)
V(t)
Physics Citations
Patent Citations

6. 3. Diameter Shrinking
In networks diameter shrinks over time
ArXiv citation graph
Affiliation Network

7. Community Evolution
Communities also expand, shrink, or dissolve in dynamic networks

8. Evaluating the Communities
Evaluation with ground truth
Evaluation without ground truth

9. Evaluation with Ground Truth
When ground truth is available
We have partial knowledge of what communities should look like
We are given the correct community (clustering) assignments
Measures
Precision and Recall, or F-Measure
Purity
Normalized Mutual Information (NMI)
TP – the intersection of the two oval shapes; TN – the rectangle minus the two oval shapes; FP – the circle minus the blue part; FN – the blue oval minus the circle. Accuracy = (TP+TN)/(TP+FP+FN+FN); Precision = TP/(TP+FP); Recall = TP/(TP+FN)

10. Precision and Recall True Positive (TP) : False Negative (FN) :
True Positive (TP) :
When similar members are assigned to the same communities
A correct decision.
True Negative (TN) :
When dissimilar members are assigned to different communities
A correct decision
False Negative (FN) :
When similar members are assigned to different communities
An incorrect decision
False Positive (FP) :
When dissimilar members are assigned to the same communities

11. Precision and Recall: Example
TP+FP = C(6,2) + C(8,2) = = 43; FP counts the wrong pairs within each cluster; FN counts the similar pairs but wrongly put into different clusters; TN counts dissimilar pairs in different clusters

12. F-Measure

13. We can assume the majority of a community represents the community
Purity
We can assume the majority of a community represents the community
We use the label of the majority against the label of each member to evaluate the communities
Purity. The fraction of instances that have labels equal to the community’s majority label
N – the total number of data points.
Purity can be easily tampered by
Points being singleton communities (of size 1); or by
Very large communities

14. Mutual Information
Mutual information (MI). The amount of information that two random variables share.
By knowing one of the variables, it measures the amount of uncertainty reduced regarding the others

15. Normalizing Mutual Information (NMI)

16. Normalized Mutual Information

17. Normalized Mutual Information
NMI values close to one indicate high similarity between communities found and labels
Values close to zero indicate high dissimilarity between them

18. Normalized Mutual Information: Example
Found communities (H)
[1,1,1,1,1,1, 2,2,2,2,2,2,2,2]
Actual Labels (L)
[2,1,1,1,1,1, 2,2,2,2,2,2,1,1] nh
h=1 6
h=2 8 nl 7
nh,l
h=1 5 1
h=2 2 6

19. Evaluation without Ground Truth
Evaluation with Semantics
A simple way of analyzing detected communities is to analyze other attributes (posts, profile information, content generated, etc.) of community members to see if there is a coherency among community members
The coherency is often checked via human subjects.
Or through labor markets: Amazon Mechanical Turk
To help analyze these communities, one can use word frequencies. By generating a list of frequent keywords for each community, human subjects determine whether these keywords represent a coherent topic.
Evaluation Using Clustering Quality Measures
Use clustering quality measures (SSE)
Use more than two community detection algorithms and compare the results and pick the algorithm with better quality measure

20. Cocitation and Bibliographic coupling
Cocitation of two vertices i and j is the number of vertices that have outgoing edges to both
𝐶 𝑖𝑗 = 𝑘=1 𝑛 𝐴 𝑖𝑘 𝐴 𝑗𝑘 = 𝑘=1 𝑛 𝐴 𝑖𝑘 𝐴 𝑘𝑗 𝑇
𝐶=𝐴 𝐴 𝑇
Bibliographic coupling is the number of vertices to which both point
𝐵= 𝑘=1 𝑛 𝐴 𝑘𝑖 𝐴 𝑘𝑗 = 𝑘=1 𝑛 𝐴 𝑖𝑘 𝑇 𝐴 𝑘𝑗
𝐵= 𝐴 𝑇 𝐴

21. Edge independent paths: if they share no common edge
Vertex independent paths: if they share no common vertex except start and end vertices
Vertex-independent => Edge-independent
Also called disjoint paths
These set of paths are not necessarily unique
Connectivity of vertices: the maximal number of independent paths between a pair of vertices
Used to identify bottlenecks and resiliency to failures

22. Cut Sets and Maximum Flow
A minimum cut set is the smallest cut set that will disconnect a specified pair of vertices
Need not to be unique
Menger’s theorem: If there is no cut set of size less than n between a pair of vertices, then there are at least n independent paths between the same vertices.
Implies that the size of min cut set is equal to maximum number of independent paths
for both edge and vertex independence
Maximum Flow between a pair of vertices is the number of edge independent paths times the edge capacity.

24. Transitivity
 is said to be transitive if a  b and b  c together imply a  c
Perfect transitivity in network → cliques
Partial transitivity
u knows v and v knows w →
𝐶= 𝑐𝑙𝑜𝑠𝑒𝑑 𝑝𝑎𝑡ℎ𝑠 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑤𝑜 𝑝𝑎𝑡ℎ𝑠 𝑜𝑓 𝑙𝑒𝑛𝑔𝑡ℎ 𝑡𝑤𝑜 = 3 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑡𝑟𝑖𝑝𝑙𝑒𝑡𝑠

25. Clustering Coefficient and Triples
Triple: an ordered set of three nodes,
connected by two (open triple) edges or
three edges (closed triple)
A triangle can miss any of its three edges
A triangle has 3 Triples
𝑣 𝑖 𝑣 𝑗 𝑣 𝑘 and 𝑣 𝑗 𝑣 𝑘 𝑣 𝑖 are different triples
The same members
First missing edge 𝑒(𝑣 𝑘 ,𝑣 𝑖 ) and second missing 𝑒(𝑣 𝑖 ,𝑣 𝑗 )
𝑣 𝑖 𝑣 𝑗 𝑣 𝑘 and 𝑣 𝑘 𝑣 𝑗 𝑣 𝑖 are the same triple

26. [Global] Clustering Coefficient
Clustering coefficient measures transitivity in undirected graphs
Count paths of length two and check whether the third edge exists
When counting triangles, since every triangle has 6 closed paths of length 2
Or we can rewrite it as

27. [Global] Clustering Coefficient: Example

28. Local Clustering Coefficient
Local clustering coefficient measures transitivity at the node level
Commonly employed for undirected graphs
Computes how strongly neighbors of a node 𝑣 (nodes adjacent to 𝑣) are themselves connected
In an undirected graph, the denominator can be rewritten as:

29. Local Clustering Coefficient: Example
Thin lines depict connections to neighbors
Dashed lines are the missing connections among neighbors
Solid lines indicate connected neighbors
When all neighbors are connected 𝐶=1
When none of neighbors are connected 𝐶=0

30. Structural Metrics: Clustering coefficient

31. Local Clustering and Redundancy
𝐶 𝑖 = 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑝𝑎𝑖𝑟𝑠 𝑜𝑓 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟𝑠 𝑜𝑓 𝑖 𝑝𝑎𝑖𝑟𝑠 𝑜𝑓 𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟𝑠 𝑜𝑓 𝑖
𝐶 𝑊𝑆 = 1 𝑛 𝑖=1 𝑛 𝐶 𝑖
Redundancy
𝐶 𝑖 = 𝑅 𝑖 𝑘 𝑖 −1
𝑅 𝑖 = 𝐶 𝑖 ( 𝑘 𝑖 −1)

32. Reciprocity
How likely is it that the node you point to will point to you as well.
𝑟= 1 𝑚 𝑖𝑗 𝐴 𝑖𝑗 𝐴 𝑗𝑖 = 1 𝑚 Tr 𝐴 2

33. If you become my friend, I’ll be yours
Reciprocity
If you become my friend, I’ll be yours
Reciprocity is simplified version of transitivity
It considers closed loops of length 2
If node 𝑣 is connected to node 𝑢,
𝑢 by connecting to 𝑣, exhibits reciprocity

34. Reciprocity: Example
Reciprocal nodes: 𝑣1, 𝑣2

35. Signed Edges and Structural balance
Friends / Enemies
Friend of friend →
Enemy of my enemy →
Structural balance: only loops of even number of “negative links”
Structurally balanced → partitioned into groups where internal links are positive and between group links are negative

36. Triangle of nodes 𝑖, 𝑗, and 𝑘, is balanced, if and only if
Social Balance Theory
Consistency in friend/foe relationships among individuals
Informally, friend/foe relationships are consistent when
In the network
Positive edges demonstrate friendships (𝑤𝑖𝑗=1)
Negative edges demonstrate being enemies (𝑤𝑖𝑗=−1)
Triangle of nodes 𝑖, 𝑗, and 𝑘, is balanced, if and only if
𝑤𝑖𝑗 denotes the value of the edge between nodes 𝑖 and 𝑗

37. Social Balance Theory: Possible Combinations
For any cycle if the multiplication of edge values become positive, then the cycle is socially balanced

38. Structural Equivalence: share many of the same neighbors
Similarity
Structural Equivalence: share many of the same neighbors
Jaccard Similarity: 𝜎 𝑖𝑗 = 𝑛 𝑖𝑗 | 𝑛 𝑖 ∪ 𝑛 𝑗 |
Cosine Similarity: 𝜎 𝑖𝑗 = 𝑛 𝑖𝑗 𝑘 𝑖 𝑘 𝑗
Pearson Coefficient: Given degree of two nodes, how many common neighbors they have ( 𝑟 𝑖𝑗 )
Euclidian Distance: 𝑑 𝑖𝑗 = 𝑘 ( 𝐴 𝑖𝑘 − 𝐴 𝑗𝑘 ) 2
Regular Equivalence: neighbors are the same
Katz Similarity: 𝜎 𝑖𝑗 =𝛼 𝑘𝑙 𝐴 𝑖𝑘 𝐴 𝑗𝑙 𝜎 𝑘𝑙
𝝈=𝛼𝑨𝝈+𝑰

39. Homophily and Assortative Mixing
Assortativity: Tendency to be linked with nodes that are similar in some way
Humans: age, race, nationality, language, income, education level, etc.
Citations: similar fields than others
Web-pages: Language
Disassortativity: Tendency to be linked with nodes that are different in some way
Network providers: End users vs other providers
Assortative mixing can be based on
Enumerative characteristic
Scalar characteristic

40. Assortativity: An Example
The friendship network in a US high school in 1994
Colors represent races,
White: whites
Grey: blacks
Light Grey: hispanics
Black: others
High assortativity between individuals of the same race

41. Assortativity Significance
The difference between measured assortativity and expected assortativity
The higher this difference, the more significant the assortativity observed
Example
In a school, half the population is white and the other half is Hispanic.
We expected 50% of the connections to be between members of different races.
If all connections are between members of different races, then we have a significant finding

42. Modularity (enumerative)
Extend to which a node is connected to a like in network
+ if there are more edges between nodes of the same type than expected value
- otherwise
𝑄= 1 2𝑚 𝑖𝑗 𝐴 𝑖𝑗 − 𝑘 𝑖 𝑘 𝑗 2𝑚 𝛿 𝑐 𝑖 , 𝑐 𝑗
𝛿 𝑐 𝑖 , 𝑐 𝑗 is 1 if ci and cj are of same type, and 0 otherwise
𝑄= 𝑟 𝑒 𝑟𝑟 − 𝑎 𝑟 2
err is fraction of edges that join same type of vertices
ar is fraction of ends of edges attached to vertices type r

43. Assortative coefficient (enumerative)
Modularity is almost always less than 1, hence we can normalize it with the Qmax value
𝑟= 𝑄 𝑄 𝑚𝑎𝑥 = 𝑖𝑗 𝐴 𝑖𝑗 − 𝑘 𝑖 𝑘 𝑗 2𝑚 𝛿 𝑐 𝑖 , 𝑐 𝑗 2𝑚 − 𝑖𝑗 𝑘 𝑖 𝑘 𝑗 2𝑚 𝛿 𝑐 𝑖 , 𝑐 𝑗

44. Assortative coefficient (scalar)
𝑟= 𝑖𝑗 𝐴 𝑖𝑗 − 𝑘 𝑖 𝑘 𝑗 2𝑚 𝑥 𝑖 . 𝑥 𝑗 𝑖𝑗 𝑘 𝑖 𝛿 𝑖𝑗 − 𝑘 𝑖 𝑘 𝑗 2𝑚 𝑥 𝑖 . 𝑥 𝑗
r=1, perfectly assortative
r=-1, perfectly disassortative
r=0, non-assortative
Usually node degree is used as scale
𝑟= 𝑖𝑗 𝐴 𝑖𝑗 − 𝑘 𝑖 𝑘 𝑗 2𝑚 𝑘 𝑖 . 𝑘 𝑗 𝑖𝑗 𝑘 𝑖 𝛿 𝑖𝑗 − 𝑘 𝑖 𝑘 𝑗 2𝑚 𝑘 𝑖 . 𝑘 𝑗

45. Modularity: Matrix Form
Let denote the indicator matrix and let 𝑘 denote the number of types
The Kronecker delta function can be reformulated using the indicator matrix
Therefore,

46. Normalized Modularity: Matrix Form
Let Modularity matrix be
𝑑∈ ℝ 𝑛 ×1 is the degree vector
Modularity can be reformulated as

47. Modularity Example
The number of edges between nodes of the same color is less than the expected number of edges between them

48. Assortativity Coefficient of Various Networks
M.E.J. Newman. Assortative mixing in networks

49. Measuring Assortativity for Ordinal Attributes
A common measure for analyzing the relationship between ordinal values is covariance
It describes how two variables change together
In our case, we have a network
We are interested in how values assigned to nodes that are connected (via edges) are correlated

50. Covariance Variables
The value assigned to node 𝑣𝑖 is 𝑥𝑖
We construct two variables 𝑋𝐿 and 𝑋𝑅
For any edge (𝑣𝑖,𝑣𝑗), we assume that 𝑥𝑖 is observed from variable 𝑋𝐿 and 𝑥𝑗 is observed from variable 𝑋𝑅
𝑋𝐿 represents the ordinal values associated with the left-node (the first node) of the edges and 𝑋𝑅 represents the values associated with the right-node (the second node) of the edges
We need to compute the covariance between variables 𝑋𝐿 and 𝑋𝑅

51. Covariance Variables: Example
List of edges:
(A, C)
(C, A)
(C, B)
(B, C)
𝑋𝐿 : (18, 21, 21, 20) 𝑋𝑅 : (21, 18, 20, 21)

52. Covariance
For two given column variables 𝑋𝐿 and 𝑋𝑅 the covariance is
𝐸(𝑋𝐿) is the mean of the variable and 𝐸(𝑋𝐿 𝑋𝑅) is the mean of the multiplication 𝑋𝐿 and 𝑋𝑅

53. Covariance

54. Normalizing Covariance
Pearson correlation 𝜌(𝑋,𝑌) is the normalized version of covariance
In our case:
\sigma = E(X-E(X))^2

55. Correlation Example