Social Media Mining - Chapter 2 (Graph Essentials)

Graph
Essentials
SOCIAL
MEDIA
MINING

2Social Media Mining Measures and Metrics 2Social Media Mining Graph Essentialshttp://socialmediamining.info/
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Bridges of Konigsberg
• There are 2 islands and 7 bridges that connect
the islands and the mainland
• Find a path that crosses each bridge exactly once
City Map (From Wikipedia) Graph Representation

Modeling the Problem by Graph Theory
• The key to solve this problem is an ingenious
graph representation
• Euler proved that since except for the starting
and ending point of a walk, one has to enter
and leave all other nodes, thus these nodes
should have an even number of bridges
connected to them
• This property does not hold in
this problem

Networks
• A network is a graph.
– Elements of the network have meanings
• Network problems can usually be represented in
terms of graph theory
Twitter example:
• Given a piece of information, a
network of individuals, and the
cost to propagate information
among any connected pair, find
the minimum cost to disseminate
the information to all individuals.

Food Web

Network are Pervasive
CitationNetworks
TwitterNetworks

Internet

Network of the US Interstate Highways

NY State Road Network

Social Networks and Social Network Analysis
• A social network
– A network where elements have a social structure
• A set of actors (such as individuals or organizations)
• A set of ties (connections between individuals)
• Social networks examples:
– your family network, your friend network, your
colleagues ,etc.
• To analyze these networks we can use Social
Network Analysis (SNA)
• Social Network Analysis is an interdisciplinary
field from social sciences, statistics, graph theory,
complex networks, and now computer science

Social Networks: Examples
High school friendship
High school dating

Graph Basics

Nodes and Edges
A network is a graph, or a collection of points
connected by lines
• Points are referred to as nodes, actors, or
vertices (plural of vertex)
• Connections are referred to as edges or ties
Node
Edge

Nodes or Actors
• In a friendship social graph, nodes are people
and any pair of people connected denotes the
friendship between them
• Depending on the context, these nodes are
called nodes, or actors
– In a web graph, “nodes” represent sites and the
connection between nodes indicates web-links
between them
– In a social setting, these nodes are called actors
– The size of the graph is

Edges
• Edges connect nodes and are also known as
ties or relationships
• In a social setting, where nodes represent
social entities such as people, edges indicate
internode relationships and are therefore
known as relationships or (social) ties
• Number is edges (size of the edge-set) is
denoted as

Directed Edges and Directed Graphs
• Edges can have directions. A directed edge is sometimes
called an arc
• Edges are represented using their end-points .
• In undirected graphs both representations are the same

Neighborhood and Degree (In-degree, out-degree)
For any node 𝑣, in an undirected graph, the set of nodes it is
connected to via an edge is called its neighborhood and is
represented as 𝑁 𝑣
– In directed graphs we have incoming neighbors 𝑁𝑖𝑛 𝑣 (nodes that
connect to 𝑣) and outgoing neighbors 𝑁𝑜𝑢𝑡 𝑣 .
The number of edges connected to one node is the degree
of that node (the size of its neighborhood)
– Degree of a node 𝑖 is usually presented using notation 𝑑𝑖
In Directed graphs:
– In-degrees is the number of edges pointing towards a node
– Out-degree is the number of edges pointing away from a node

Degree and Degree Distribution
• Theorem 1. The summation of degrees in an
undirected graph is twice the number of
edges
• Lemma 1. The number of nodes with odd
degree is even
• Lemma 2. In any directed graph, the
summation of in-degrees is equal to the
summation of out-degrees,

Degree Distribution
When dealing with very large graphs, how nodes’
degrees are distributed is an important concept to
analyze and is called Degree Distribution
is the number of
nodes with degree 𝑑
(Degree sequence)

Degree Distribution Plot
The 𝑥-axis represents the degree and the 𝑦-axis
represents the fraction of nodes having that
degree
– On social networking sites
There exist many users with few
connections and there exist a
handful of users with very large
numbers of friends.
(Power-law degree distribution)
Facebook
Degree Distribution

Subgraph
• Graph 𝐺 can be represented as a pair
where 𝑉 is the node set and 𝐸 is the edge set
• is a subgraph of
1
23
546
1
2
3
5

• Adjacency Matrix
• Adjacency List
• Edge List
Graph Representation

Graph Representation
• Graph representation is straightforward
and intuitive, but it cannot be
effectively manipulated using
mathematical and computational tools
• We are seeking representations that can
store these two sets in a way such that
– Does not lose information
– Can be manipulated easily by computers
– Can have mathematical methods applied easily

Adjacency Matrix (a.k.a. sociomatrix)



ijA
0, otherwise
1, if there is an edge between nodes 𝑣𝑖 and 𝑣𝑗
Social media networks have
very sparse Adjacency matrices
Diagonal Entries are self-links or loops

Adjacency List
• In an adjacency list for every node, we maintain
a list of all the nodes that it is connected to
• The list is usually sorted based on the node
order or other preferences

Edge List
• In this representation, each element is an
edge and is usually represented as 𝑢, 𝑣 ,
denoting that node 𝑢 is connected to node
𝑣 via an edge

• Null, Empty,
Directed/Undirected/Mixed,
Simple/Multigraph, Weighted,
Signed Graph, Webgraph
Types of Graphs

Null Graph and Empty Graph
• A null graph is one where the node set is
empty (there are no nodes)
– Since there are no nodes, there are also no edges
• An empty graph or edge-less graph is one
where the edge set is empty,
• The node set can be non-empty.
– A null-graph is an empty graph.

Directed/Undirected/Mixed Graphs
The adjacency
matrix for
undirected graphs is
symmetric (𝑨 = 𝑨 𝑻)
• The adjacency matrix for
directed graphs is often
not symmetric (𝑨 ≠ 𝑨 𝑻)
– 𝑨𝒊𝒋  𝑨𝒋𝒊
– We can have equality
though

Simple Graphs and Multigraphs
• Simple graphs are graphs where only a single
edge can be between any pair of nodes
• Multigraphs are graphs where you can have
multiple edges between two nodes and loops
• The adjacency matrix for multigraphs can include
numbers larger than one, indicating multiple
edges between nodes
Simple graph Multigraph

Weighted Graph
• A weighted graph 𝑮(𝑽, 𝑬, 𝑾) is one
where edges are associated with
weights
– For example, a graph could
represent a map where
nodes are airports and edges
are routes between them
• The weight associated with
each edge could represent the
distance between the
corresponding cities


 

vandbetween vedgenoisThere0,
Rwj),or w(i,w
ji
ij
ijA

Signed Graph
• When weights are binary (0/1, -1/1, +/-) we
have a signed graph
• It is used to represent friends or foes
• It is also used to represent social status

Webgraph
• A webgraph is a way of representing how
internet sites are connected on the web
• In general, a web graph is a directed
multigraph
• Nodes represent sites and edges represent
links between sites.
• Two sites can have multiple links pointing to
each other and can have loops (links pointing
to themselves)

Webgraph
Government Agencies
Bow-tie structure
Broder et al –
200 million pages, 1.5 billion links

• Adjacent nodes/Edges,
Walk/Path/Trail/Tour/Cycle
Connectivity in Graphs

Adjacent nodes and Incident Edges
Two nodes are adjacent if they are connected
via an edge.
Two edges are incident, if they share on end-
point
When the graph is directed, edge directions
must match for edges to be incident
An edge in a graph can be traversed when one
starts at one of its end-nodes, moves along the
edge, and stops at its other end-node.

Walk, Path, Trail, Tour, and Cycle
Walk: A walk is a sequence of incident edges visited
one after another
– Open walk: A walk does not end where it starts
– Closed walk: A walk returns to where it starts
• Representing a walk:
– A sequence of edges: 𝑒1, 𝑒2, … , 𝑒𝑛
– A sequence of nodes: 𝑣1, 𝑣2, … , 𝑣𝑛
• Length of walk:
the number of visited edges
Length of walk= 8

Trail
• A trail is a walk where no edge is visited
more than once and all walk edges are
distinct
• A closed trail (one that ends where it starts) is
called a tour or circuit

Path
• A walk where nodes and edges are distinct is
called a path and a closed path is called a
cycle
• The length of a path or cycle is the number of
edges visited in the path or cycle
Length of path= 4

Examples
Eulerian Tour
• All edges are traversed only once
– Konigsberg bridges
Hamiltonian Cycle
• A cycle that visits all nodes

Random walk
• A walk that in each step the next node is
selected randomly among the neighbors
– The weight of an edge can be used to define the
probability of visiting it
– For all edges that start at 𝑣𝑖 the following equation
holds

Random Walk: Example
Mark a spot on the ground
– Stand on the spot and flip the coin (or more than one
coin depending on the number of choices such as left,
right, forward, and backward)
– If the coin comes up heads, turn to the right and take a
step
– If the coin comes up tails, turn to the left and take a step
– Keep doing this many times and see where you end up

Connectivity
• A node 𝒗𝒊 is connected to node 𝒗𝒋 (or reachable
from 𝑣𝑗) if it is adjacent to it or there exists a path
from 𝑣𝑖 to 𝑣𝑗.
• A graph is connected, if there exists a path
between any pair of nodes in it
– In a directed graph, a graph is strongly connected if
there exists a directed path between any pair of nodes
– In a directed graph, a graph is weakly connected if
there exists a path between any pair of nodes, without
following the edge directions
• A graph is disconnected, if it not connected.

Connectivity: Example

Component
• A component in an undirected graph is a
connected subgraph, i.e., there is a path between
every pair of nodes inside the component
• In directed graphs, we have a strongly
connected components when there is a path
from 𝑢 to 𝑣 and one from 𝑣 to 𝑢 for every pair of
nodes 𝑢 and 𝑣.
• The component is weakly connected if replacing
directed edges with undirected edges results in a
connected component

Component Examples:
3 components 3 Strongly-connected
components

Shortest Path
• Shortest Path is the path between two nodes
that has the shortest length.
– We denote the length of the shortest path between
nodes 𝑣𝑖 and 𝑣𝑗 as 𝑙𝑖,𝑗
• The concept of the neighborhood of a node
can be generalized using shortest paths. An
n-hop neighborhood of a node is the set of
nodes that are within n hops distance from
the node.

Diameter
The diameter of a graph is the length of the
longest shortest path between any pair of nodes
between any pairs of nodes in the graph
• How big is the diameter of the web?

Adjacency Matrix and Connectivity
• Consider the following adjacency matrix
• Number of Common neighbors between node
𝑖 and node 𝑗
• That’s element of [ij] of matrix 𝐴 × 𝐴 𝑇 = 𝐴2
• Common neighbors are paths of length 2
• Similarly, what is 𝐴3?
ji

Special Graphs

Trees and Forests
• Trees are special cases of undirected graphs
• A tree is a graph structure that has no cycle in it
• In a tree, there is exactly one path between any
pair of nodes
• In a tree: |𝑉| = |𝐸| + 1
• A set of disconnected
trees is called a forest
A forest containing 3 trees

Special Subgraphs

Spanning Trees
• For any connected graph, the spanning tree is a
subgraph and a tree that includes all the nodes
of the graph
• There may exist multiple spanning trees for a
graph.
• In a weighted graph, the weight of a spanning
tree is the summation of the edge weights in the
tree.
• Among the many spanning trees found for a
weighted graph, the one with the minimum
weight is called the
minimum spanning tree (MST)

Steiner Trees
Given a weighted graph G(V, E, W) and a subset
of nodes 𝑉’ ⊆ 𝑉 (terminal nodes ), the Steiner
tree problem aims to find a tree such that it
spans all the 𝑉’ nodes and the weight of this tree
is minimized
What can be the terminal
set here?

Complete Graphs
• A complete graph is a graph where for a set of
nodes 𝑉, all possible edges exist in the graph
• In a complete graph, any pair of nodes are
connected via an edge

Planar Graphs
A graph that can be drawn in such a way that no
two edges cross each other (other than the
endpoints) is called planar
Planar Graph Non-planar Graph

Bipartite Graphs
A bipartite graph 𝐺(𝑉, 𝐸) is a graph where the
node set can be partitioned into two sets such
that, for all edges, one end-point is in one set
and the other end-point is in the other set.

Affiliation Networks
An affiliation network is a bipartite graph. If an
individual is associated with an affiliation, an
edge connects the corresponding nodes.

People Companies
Affiliation Networks: Membership
Affiliation of people on
corporate boards of
directors

Bipartite Representation / one-mode Projections
• We can save some space by keeping
membership matrix X
– What is 𝑋𝑋 𝑇?
– What is 𝑋 𝑇
𝑋?
Similarity between users - [Bibliographic Coupling]
Similarity between groups - [Co-citation]
Elements on the diagonal are number of groups
the user is a member of
OR
number of users in the group

Social-Affiliation Network
Social-Affiliation network is a combination of a
social network and an affiliation network

Regular Graphs
• A regular graph is one in which all
nodes have the same degree
• Regular graphs can be connected or
disconnected
• In a 𝑘-regular graph, all nodes have
degree 𝑘
• Complete graphs are examples of
regular graphs
Regular graph
With 𝑘 = 3

Egocentric Networks
• Egocentric network: A focal actor (ego) and a
set of alters who have ties with the ego
• Usually there are limitations for nodes to
connect to other nodes or have relation with
other nodes
– Example: In a network of mothers and their
children:
• Each mother only holds mother-children relations with her
own children
• Additional examples of egocentric networks are
Teacher-Student or Husband-Wife

Bridges (cut-edges)
• Bridges are edges whose removal will increase
the number of connected components

Graph Algorithms

Graph/Network
Traversal Algorithms

Graph/Tree Traversal
• We are interested in surveying a social media site
to computing the average age of its users
– Start from one user;
– Employ some traversal technique to reach her friends
and then friends’ friends, …
• The traversal technique guarantees that
1. All users are visited; and
2. No user is visited more than once.
• There are two main techniques:
– Depth-First Search (DFS)
– Breadth-First Search (BFS)

Depth-First Search (DFS)
• Depth-First Search (DFS) starts from a node 𝑣𝑖,
selects one of its neighbors 𝑣𝑗 from 𝑁(𝑣𝑖) and
performs Depth-First Search on 𝑣𝑗 before
visiting other neighbors in 𝑁(𝑣𝑖)
• The algorithm can be used both for trees and
graphs
– The algorithm can be implemented using a stack
structure

DFS Algorithm

Depth-First Search (DFS): An Example

Breadth-First Search (BFS)
• BFS starts from a node and visits all its
immediate neighbors first, and then moves to
the second level by traversing their neighbors.
• The algorithm can be used both for trees and
graphs
– The algorithm can be implemented using a queue
structure

BFS Algorithm

Breadth-First Search (BFS)

Finding Shortest Paths

Shortest Path
When a graph is connected, there is a chance that
multiple paths exist between any pair of nodes
– In many scenarios, we want the shortest path between
two nodes in a graph
• How fast can I disseminate information on social media?
Dijkstra’s Algorithm
– Designed for weighted graphs with non-negative edges
– It finds shortest paths that start from a provided node 𝑠
to all other nodes
– It finds both shortest paths and their respective lengths

Dijkstra’s Algorithm: Finding the shortest path
1. Initiation:
– Assign zero to the source node and infinity to all other nodes
– Mark all nodes as unvisited
– Set the source node as current
2. For the current node, consider all of its unvisited
neighbors and calculate their tentative distances
– If tentative distance is smaller than neighbor’s distance, then
Neighbor’s distance = tentative distance
3. After considering all of the neighbors of the current
node, mark the current node as visited and remove it
from the unvisited set
4. If the destination node has been marked visited or if
the smallest tentative distance among the nodes in
the unvisited set is infinity, then stop
5. Set the unvisited node marked with the smallest
tentative distance as the next "current node" and go
to step 2
A visited node will
never be checked
again and its
distance recorded
now is final and
minimal
Tentative distance =
current distance +
edge weight

Dijkstra’s Algorithm: Execution Example

Dijkstra’s Algorithm: Notes
• Dijkstra’s algorithm is source-dependent
– Finds the shortest paths between the source node and
all other nodes.
• To generate all-pair shortest paths,
– We can run Dijsktra’s algorithm 𝑛 times, or
– Use other algorithms such as Floyd-Warshall algorithm.
• If we want to compute the shortest path from
source 𝑣 to destination 𝑑,
– we can stop the algorithm once the shortest path to the
destination node has been determined

Finding
Minimum Spanning Tree

Prim’s Algorithm: Finding Minimum Spanning Tree
Finds MST in a weighted graph
1. Selecting a random node and add it to the MST
2. Grows the spanning tree by selecting edges which
have one endpoint in the existing spanning tree and
one endpoint among the nodes that are not selected
yet. Among the possible edges, the one with the
minimum weight is added to the set (along with its
end-point).
3. This process is iterated until the graph is fully
spanned

Prim’s Algorithm Execution Example

Network Flow

Network Flow
• Consider a network of pipes that connects an
infinite water source to a water sink.
– Given the capacity of these pipes, what is the maximum
flow that can be sent from the source to the sink?
• Parallel in Social Media:
– Users have daily cognitive/time limits (the capacity,
here) of sending messages (the flow) to others,
– What is the maximum number of messages the
network should be prepared to handle at any time?

Flow Network
• A Flow network G(V,E,C) is a directed weighted
graph, where we have the following:
– ∀ (𝑢, 𝑣) ∈ 𝐸, 𝑐(𝑢, 𝑣) ≥ 0 defines the edge capacity.
– When 𝑢, 𝑣 ∈ 𝐸, 𝑣, 𝑢 ∉ 𝐸 (opposite flow is impossible)
– 𝑠 defines the source node and 𝑡 defines the sink node.
An infinite supply of flow is connected to the source.

Flow
• Given edges with certain capacities, we can fill
these edges with the flow up to their capacities
(capacity constraint)
• The flow that enters any node other than source
𝑠 and sink 𝑡 is equal to the flow that exits it so
that no flow is lost (flow conservation constraint)
• ∀ (𝑢, 𝑣) ∈ 𝐸, 𝑓(𝑢, 𝑣) ≥ 0 defines the flow passing
through the edge.
• ∀ (𝑢, 𝑣) ∈ 𝐸, 0 ≤ 𝑓(𝑢, 𝑣) ≤ 𝑐(𝑢, 𝑣)
• ∀𝑣 ∈ 𝑉 − 𝑠, 𝑡 , 𝑘: 𝑘,𝑣 ∈𝐸 𝑓 𝑘, 𝑣 = 𝑙:(𝑣,𝑙)∈𝐸 𝑓 𝑣, 𝑙
(capacity constraint)
(flow conservation constraint)

A Sample Flow Network
• Commonly, to visualize an edge with capacity
𝑐 and flow 𝑓 , we use the notation 𝑓/𝑐.

Flow Quantity
• The flow quantity (or value of the flow) in any
network is the amount of
– Outgoing flow from the source minus the incoming
flow to the source.
– Alternatively, one can compute this value by
subtracting the outgoing flow from the sink from
its incoming value

What is the flow value?
• 19
– 11+8 from s, or
– 4+15 to t

Ford-Fulkerson Algorithm
• Find a path from source to sink such that
there is unused capacity for all edges in the
path.
• Use that capacity (the minimum capacity
unused among all edges on the path) to
increase the flow.
• Iterate until no other path is available.

Residual Network
• Given a flow network 𝐺(𝑉, 𝐸, 𝐶), we define
another network 𝐺(𝑉, 𝐸 𝑅, 𝐶 𝑅)
• This network defines how much capacity
remains in the original network.
• The residual network has an edge between
nodes 𝑢 and 𝑣 if and only if either (𝑢, 𝑣) or
(𝑣, 𝑢) exists in the original graph.
– If one of these two exists in the original network,
we would have two edges in the residual network:
one from (𝑢, 𝑣) and one from (𝑣, 𝑢).

Intuition
• When there is no flow going through an edge
in the original network, a flow of as much as
the capacity of the edge remains in the
residual.
• In the residual network, one has the ability to
send flow in the opposite direction to cancel
some amount of flow in the original network.

Residual Network (Example)
• Edges that have zero capacity in the residual
are not shown

Augmentation / Augmenting Paths
1. In the residual graph, when edges are in the
same direction as the original graph,
– Their capacity shows how much more flow can be
pushed along that edge in the original graph.
2. When edges are in the opposite direction,
– their capacities show how much flow can be
pushed back on the original graph edge.
• By finding a flow in the residual, we can
augment the flow in the original graph.

Augmentation / Augmenting Paths
• Any simple path from 𝑠 to 𝑡 in the residual graph
is an augmenting path.
– All capacities in the residual are positive,
• These paths can augment flows in the original, thus increasing
the flow.
– The amount of flow that can be pushed along this path
is equal to the minimum capacity along the path
• The edge with the minimum capacity limits the amount of flow
being pushed
• We call the edge the Weak link

How do we augment?
• Given flow 𝑓 (𝑢, 𝑣) in the original graph and
flow 𝑓𝑅(𝑢, 𝑣) and 𝑓𝑅(𝑣, 𝑢) in the residual graph,
we can augment the flow as follows:
Flow Quantity: 1

Augmenting

The Ford-Fulkerson Algorithm

Maximum Bipartite Matching

Example
• Given 𝑛 products and
𝑚 users
– Some users are only
interested in certain
products
– We have only one copy
of each product.
– Can be represented as a
bipartite graph
– Find the maximum
number of products
that can be bought by
users
• No two edges selected
share a node
Matching Maximum
Matching

Matching Solved with Max-Flow
• Create a flow graph
𝐺(𝑉’, 𝐸’, 𝐶) from our
bipartite graph 𝐺(𝑉, 𝐸)
1. Set 𝑉’ = 𝑉 ∪ 𝑠 ∪ 𝑡
2. Connect all nodes in 𝑉𝐿
to 𝑠 and all nodes in 𝑉𝑅
to 𝑡
3. Set 𝑐(𝑢, 𝑣) = 1, for all
edges in 𝐸’

Bridges, Weak Ties, and
Bridge Detection

Bridge and a Local Bridge
• Bridge: Bridges are edges
whose removal will increase
the number of connected
components
– Bridges are extremely rare in
real-world social networks.
• Local Bridge: when the
endpoints have no friend in
common
– the removal increases the
length of shortest path to
more than 2
– Span of the local bridge:
How much the distance
between the endpoints would
become if the edge is
removed
• Large span is desirable to find
communities
Source: Easley and Kleinberg – Networks, Crowds, and Markets

Strength of Ties
• Assume that you can
divide connections
into two categories:
– Strong tie (S):
• friends
– Weak ties (W):
• acquaintances
• Strong Triadic Closure:
– Consider a node 𝒖 that has two strong ties to nodes 𝒗 and 𝒘
– If there is no edge between 𝒗 and 𝒘 (weak or strong tie) then 𝒖
does not exhibit a strong triadic closure

Connection between Bridges and Tie Strength
Why?
If a node exhibits Strong Triadic Closure and
has at least two strong ties, then if it part of
a local bridge, that bridge must be a weak tie
Source: Easley and Kleinberg – Networks, Crowds, and Markets

Generalizing to Real-World Networks
• Consider a cell-phone network
– We have an edge if both end points
call each other
– Tie Strength: it does not have to be
weak/strong
• For (𝑢, 𝑣), the number of minutes
spent 𝑢 and 𝑣 spent talking to each
other on the phone
– Local Bridge: can be generalized
using neighborhood overlap:
The numerator is
called embeddedness
of an edge
When numerator is zero we have a local bridge
Tie Strength
Neighborhood
Overlap

Bridge Detection

Social Media Mining - Chapter 2 (Graph Essentials)

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