network measures
DESCRIPTION
Social Media Mining. Network Measures. Klout. Why Do We Need Measures?. Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the like-minded users and how can we find these similar individuals? - PowerPoint PPT PresentationTRANSCRIPT
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Network Measures
Social Media Mining
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Measures and Metrics 2Social Media Mining
Network Measures
Klout
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Measures and Metrics 3Social Media Mining
Network Measures
Why Do We Need Measures?
• Who are the central figures (influential individuals) in the network?
• What interaction patterns are common in friends?
• Who are the like-minded users and how can we find these similar individuals?
• To answer these and similar questions, one first needs to define measures for quantifying centrality, level of interactions, and similarity, among others.
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Network Measures
Centrality defines how important a node is within a network.
Centrality
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Network Measures
Degree Centrality
• The degree centrality measure ranks nodes with more connections higher in terms of centrality
• di is the degree (number of adjacent edges) for vertex vi
In this graph degree centrality for vertex v1 is d1 = 8 and for all others is dj = 1, j 1
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Degree Centrality in Directed Graphs
• In directed graphs, we can either use the in-degree, the out-degree, or the combination as the degree centrality value:
douti is the number of outgoing links for
vertex vi
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Network Measures
Normalized Degree Centrality
• Normalized by the maximum possible degree
• Normalized by the maximum degree
• Normalized by the degree sum
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Network Measures
Eigenvector Centrality
• Having more friends does not by itself guarantee that someone is more important, but having more important friends provides a stronger signal
• Eigenvector centrality tries to generalize degree centrality by incorporating the importance of the neighbors (undirected)
• For directed graphs, we can use incoming or outgoing edges
Ce(vi): the eigenvector centrality of node vi
: some fixed constant
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Eigenvector Centrality, cont.
• Let T
• This means that Ce is an eigenvector of adjacency matrix A and is the corresponding eigenvalue
• Which eigenvalue-eigenvector pair should we choose?
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Eigenvector Centrality, cont.
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Eigenvector Centrality: Example
= (2.68, -1.74, -1.27, 0.33, 0.00)Eigenvalues Vector
max = 2.68
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Katz Centrality
• A major problem with eigenvector centrality arises when it deals with directed graphs
• Centrality only passes over outgoing edges and in special cases such as when a node is in a weakly connected component centrality becomes zero even though the node can have many edge connected to it
• To resolve this problem we add bias term to the centrality values for all nodes
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Katz Centrality, cont.
Bias termControlling term
Rewriting equation in a vector form
vector of all 1’s
Katz centrality:
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Katz Centrality, cont.
• When =0, the eigenvector centrality is removed and all nodes get the same centrality value ᵦ
• As gets larger the effect of ᵦ is reduced• For the matrix (I- AT) to be invertible, we must
have – det(I- AT) !=0 – By rearranging we get det(AT- -1I)=0– This is basically the characteristic equation, which first
becomes zero when the largest eigenvalue equals -1
• In practice we select < 1/λ, where λ is the largest eigenvalue of AT
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Katz Centrality Example
• The Eigenvalues are -1.68, -1.0, 0.35, 3.32• We assume =0.25 < 1/3.32 ᵦ=0.2
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Network Measures
PageRank
• Problem with Katz Centrality: in directed graphs, once a node becomes an authority (high centrality), it passes all its centrality along all of its out-links
• This is less desirable since not everyone known by a well-known person is well-known
• To mitigate this problem we can divide the value of passed centrality by the number of outgoing links, i.e., out-degree of that node such that each connected neighbor gets a fraction of the source node’s centrality
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PageRank, cont.
What if the degree is
zero?
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PageRank Example
• We assume =0.95 and ᵦ=0.1
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Network Measures
Betweenness Centrality
Another way of looking at centrality is by considering how important nodes are in connecting other nodes
the number of shortest paths from s to t that pass through vi
the number of shortest paths from vertex s to t – a.k.a. information pathways
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Normalizing Betweenness Centrality
• In the best case, node vi is on all shortest paths from s to t, hence,
Therefore, the maximum value is 2
Betweenness centrality:
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Betweenness Centrality Example
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Closeness Centrality
• The intuition is that influential and central nodes can quickly reach other nodes
• These nodes should have a smaller average shortest path length to other nodes
Closeness centrality:
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Compute Closeness Centrality
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Group Centrality
• All centrality measures defined so far measure centrality for a single node. These measures can be generalized for a group of nodes.
• A simple approach is to replace all nodes in a group with a super node– The group structure is disregarded.
• Let S denote the set of nodes in the group and V-S the set of outsiders
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Group Centrality
• Group Degree Centrality
– We can normalize it by dividing it by |V-S|• Group Betweeness Centrality
• We can normalize it by dividing it by 2
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Group Centrality
• Group Closeness Centrality
– It is the average distance from non-members to the group
• One can also utilize the maximum distance or the average distance
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Group Centrality Example
• Consider S={v2,v3}
• Group degree centrality=3• Group betweenness centrality = 3• Group closeness centrality = 1
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Transitivity and Reciprocity
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Transitivity
• Mathematic representation:– For a transitive relation R:
• In a social network:– Transitivity is when a friend of my friend is
my friend– Transitivity in a social network leads to a denser
graph, which in turn is closer to a complete graph
– We can determine how close graphs are to the complete graph by measuring transitivity
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[Global] Clustering Coefficient
• Clustering coefficient analyzes transitivity in an undirected graph– We measure it by counting paths of length two
and check whether the third edge exists
When counting triangles, since every triangle has 6 closed paths of length 2:
In undirected networks:
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[Global] Clustering Coefficient: Example
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Local Clustering Coefficient
• Local clustering coefficient measures transitivity at the node level
• Commonly employed for undirected graphs, it 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:
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Local Clustering Coefficient: Example
• Thin lines depict connections to neighbors• Dashed lines are the missing connections among
neighbors• Solid lines indicate connected neighbors
– When none of neighbors are connected C=0– When all neighbors are connected C=1
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Reciprocity
If you become my friend, I’ll be yours• Reciprocity is a more simplified version
of transitivity as it considers closed loops of length 2
• If node v is connected to node u, u by connecting to v, exhibits reciprocity
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Reciprocity: Example
Reciprocal nodes: v1, v2
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• Measuring stability based on an observed network
Balance and Status
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Social Balance Theory• Social balance theory discusses consistency in
friend/foe relationships among individuals. Informally, social balance theory says 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
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Social Balance Theory: Possible Combinations
For any cycle if the multiplication of edge values become positive, then the cycle is socially balanced
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Social Status Theory
• Status defines how prestigious an individual is ranked within a society
• Social status theory measures how consistent individuals are in assigning status to their neighbors
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Social Status Theory: Example
• A directed ‘+’ edge from node X to node Y shows that Y has a higher status than X and a ‘-’ one shows vice versa
Unstable configuration Stable configuration
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• How similar are two nodes in a network?
Similarity
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Structural Equivalence
• In structural equivalence, we look at the neighborhood shared by two nodes; the size of this neighborhood defines how similar two nodes are.
For instance, two brothers have in common sisters, mother, father,
grandparents, etc. This shows that they are similar, whereas two random male
or female individuals do not have much in common and are not similar.
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Structural Equivalence: Definitions
• Vertex similarity
• In general, the definition of neighborhood N(v) excludes the node itself v. – Nodes that are connected and do not share a neighbor will be
assigned zero similarity– This can be rectified by assuming nodes to be included in their
neighborhoods
Jaccard Similarity:
Cosine Similarity:
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Similarity: Example
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Normalized Similarity
Comparing the calculated similarity value with its expected value where vertices pick their neighbors at random• For vertices i and j with degrees di and dj this
expectation is didj/n
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Normalized Similarity, cont.
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Normalized Similarity, cont.
Covariance between Ai and Aj
Covariance can be normalized by the multiplication of Standard deviations: Pearson correlation coefficient: [-1,1]
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Regular Equivalence
• In regular equivalence, we do not look at neighborhoods shared between individuals, but how neighborhoods themselves are similar
For instance, athletes are similar not because they know each other in
person, but since they know similar individuals, such as coaches, trainers,
other players, etc.
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• vi, vj are similar when their neighbors vk and vl
are similar
• The equation (left figure) is hard to solve since it is self referential so we relax our definition using the right figure
Regular Equivalence
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Regular Equivalence
• vi, vj are similar when vj is similar to vi’s neighbors vk
• In vector formatA vertex is highly similar to itself, we guarantee this by adding an identity matrix to the equation
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Regular Equivalence
• vi, vj are similar when vj is similar to vi’s neighbors vk
• In vector formatA vertex is highly similar to itself, we guarantee this by adding an identity matrix to the equation
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Regular Equivalence: Example
• Any row/column of this matrix shows the similarity to other vertices
• We can see that vertex 1 is most similar (other than itself) to vertices 2 and 3
• Nodes 2 and 3 have the highest similarity
The largest eigenvalue of A is 2.43Set α = 0.4 < 1/2.43