Algorithm in Social network of graph and social network analysis

CS 43118 & CS 53118
ST: Graph and Social Network Analysis
Chapter 7: Algorithms in Social Network
Niranjan Rai
Department of Computer Science
Kent State University
1

In this chapter, we will
● Learn about different algorithms
● Community Detection in Social Networks
● Community Search in Social Networks
● Influence Maximization in Social Networks
2

Communities
● Real-life graphs are not random
○ E.g., in a social network people pick their friends based on their
common interests and activities
● We expect that the nodes in a graph will be organized in
communities
○ Group of vertices which probably share common properties
and/or play similar roles within the graph
● How do we find them?
○ Nodes in communities will be densely connected to each other,
and sparsely connected with other communities
3

Community
● Community in Social Network is a structure in where vertices are
densely connected internally
8

Community types
● Overlapping communities vs non-overlapping communities
9

Non-Overlapping Community
● Each node is put into one and only one community
10

Overlapping Community
● Vertices are in more than one community
11

Community Detection
● The problem of efficiently finding the communities in a given
network is called Community Detection
12

Why Detect Communities?
● Finding an underlying community structure in a network, if it
exists, is important for a number of reasons:
○ Communities allow us to create a large scale map of a network
which makes its study easier
○ Individual communities also shed light on the function of the
system represented by the network since communities often
correspond to functional units of the system
○ They often have very different properties than the average
properties of the networks
13

Algorithms for community detection
● Minimum-cut method
● Hierarchical clustering
● Girvan–Newman algorithm
● Modularity maximization
● Statistical inference
● Clique-based methods
14

Minimum-cut method
● One of the oldest algorithms for dividing networks
● Cuts the network into a predetermined number of parts
● Usually of same size
● Tries to minimize the number of edges between groups
15

Minimum-cut method
● Works well in many of the applications for which it was originally
intended
● But is less than ideal for finding community structure in general
networks
● Since it will find communities regardless of whether they are
implicit in the structure
16

Hierarchical Clustering
● Uses a similarity measure quantifying
some (usually topological) type of
similarity between node pairs
● Commonly used measures: cosine
similarity, Jaccard index/similarity, and
Hamming distance
● Then one groups similar nodes into
communities according to this measure
17

Hierarchical Clustering
● There are several common schemes for performing the grouping
● Two simplest hierarchical clustering methods are:
○ single-linkage clustering: two groups are considered separate
communities if and only if all pairs of nodes in different
groups have similarity lower than a given threshold
○ complete linkage clustering: all nodes within every group have
similarity greater than a threshold
18

● Detects communities by progressively removing edges from the
original network
● The algorithm's steps for community detection are summarized
below
1. Calculate betweenness of all existing edges in the network
2. Remove edge(s) with the highest betweenness
3. Calculate betweenness of all edges affected by the removal
4. Repeat steps 2 and 3 until no edges remain
19
Girvan-Newman Algorithm

● The Girvan–Newman algorithm returns results of reasonable quality
and is popular because it has been implemented in a number of
standard software packages
● But it also runs slowly, taking time O(m2n) on a network of n
vertices and m edges, making it impractical for networks of more
than a few thousand nodes
23
Girvan-Newman Algorithm

Modularity Maximization
● One of the most widely used methods
for community detection
● Modularity is a benefit function that
measures the quality of a particular
division of a network into communities
24

Modularity Maximization
● Detects communities by searching over possible divisions of a
network for one or more that have particularly high modularity
● Some practical algorithms are based on approximate optimization
methods such as greedy algorithms, simulated annealing, or
spectral optimization, with different approaches offering different
balances between speed and accuracy
● A popular modularity maximization approach is the Louvain
method
25

Louvain Method
● Greedy algorithm - O(n log2n) time
● Widely utilized because of:
○ Rapid convergence property
○ High modularity
○ Hierarchical partitioning
26

Modularity Function, M
● Aij represents the edge weight between nodes i and j
● ki and kj are the sum of the weights of the edges attached to nodes i
and j, respectively
● m is the sum of all of the edge weights in the graph
● ci and cj are the communities of nodes
● δ is Kronecker delta function δ(x,y) = 1 if x = y, 0 otherwise
27

Louvain Method
● Divided into two phases:
○ Modularity Optimization
○ Community Aggregation
28

Modularity Optimization Phase
● First, each node in the network is assigned to its own community
● Then for each node i, the change in modularity is calculated for
removing i from its own community and moving it into the
community of each neighbor j of i
● This value is easily calculated by two steps:
○ remove i from its original community
○ insert i to the community of j. The two equations are quite
similar, and the equation for step (2) is:
29

Modularity Optimization Phase
● Then, once this value is calculated for all communities i is
connected to, i is placed into the community that resulted in the
greatest modularity increase.
● If no increase is possible, i remains in its original community. This
process is applied repeatedly and sequentially to all nodes until no
modularity increase can occur.
● Once this local maximum of modularity is hit, the first phase has
ended
30

Community Aggregation Phase
● Groups all of the nodes and builds a new network where nodes are
the communities from the previous phase
● Any links between nodes of the same community are now
represented by self-loops on the new community node
● Links from multiple nodes in the same community to a node in a
different community are represented by weighted edges between
communities
● Once the new network is created, modularity optimization phase
can be re-applied to the new network and so on until maximum
modularity is achieved 31

Statistical Inference
● Attempts to fit a generative model to the network data, which
encodes the community structure
● Advantage of this approach:
○ More principled nature
○ Has capacity to inherently address issues of statistical
significance
33

● SVINET in [5] is an algorithm based on Statistical Inference
● It is based on a Bayesian model for graphs with overlapping
communities
● In this model, each node is assigned a vector of community
memberships 𝜃 of length K, where K is the number of
communities in the graph (which is assumed to be known)
34

● First, a set of pairs of nodes is subsampled from the graph
● Next, this subsample is analyzed and the community structure of the
graph is updated accordingly
● This subsampling procedure is repeated for a given number of
iterations
● Hence, SVINET analyzes a subgraph of the graph at each iteration
35

Clique-based Method
● Cliques are complete subgraphs
● As nodes cannot be more tightly connected
than this
● A node can be a member of more than one
clique, a node can be a member of more
than one community in these methods
giving an "overlapping community
structure"
36

Clique-based Method
● One approach is to find the "maximal cliques", that is find the
cliques which are not the subgraph of any other clique
● The classic algorithm to find these is the Bron–Kerbosch algorithm
● The overlap of these can be used to define communities in several
ways
● The simplest is to consider only maximal cliques bigger than a
minimum size (number of nodes)
● One common algorithm is Clique Percolation Method[4]
37

Clique Percolation Method
● Normally use cliques as a core or a seed to find larger communities
● Input
○ A parameter k, and a network
● Procedure
○ Find out all cliques of size k in a given network
○ Construct a clique graph: two cliques are adjacent if they share
k-1 nodes
○ The nodes depicted in the labels of each connected components
in the clique graph form a community
38

Clique Percolation Method[4]
39
{6,7,8}

Community Search
● Aims to find the most likely community that contains the query
node q
● Has attracted great attention from both academic and industry areas
● Some applications of community search are:
○ Event organization
○ Friend recommendation
○ Advertisement in e-commerce
40

Community Search
● How to effectively and efficiently find communities from large
graphs is an important research topic in the era of big data
● Some of the factors that the community retrieval solutions should
satisfy are:
○ High Efficiency
○ High Scalability
○ High personalization
○ High quality
○ Support for dynamic graphs
41

Community Search
● High efficiency. For many real applications (e.g., event
organization), the communities often need to be retrieved in real-
time, based on query requests
● Thus, the community retrieval solutions should be able to respond
in real-time
● High scalability. Nowadays, many real networks contain millions
or billions of vertices. As a result, the solutions should be scalable
to real big graphs
42

Community Search
● High personalization. In practice, for large networks, people
usually are interested in communities of some specific users, rather
than all the users. Thus, the solutions should allow users to specify
query vertices. Moreover, some personalized requirements on
structures (and attributes) could be imposed
● High quality. The vertices in the communities retrieved should be
cohesively linked. Moreover, the communities should be easy for
interpretation
● Support for dynamic graphs. Since real networks often involve as
the time goes on, the solutions should be able to adapt for the
dynamic changes easily 43

Community Search
● Generally, the goal of CS is to search high-quality communities in
an online manner, based on a query request
● Specifically, given a vertex q of a graph G, it aims to find a
community, or a dense subgraph, which contains q and satisfies the
properties:
○ connectivity, i.e., vertices in the community are connected;
○ cohesiveness, i.e., vertices in the community are intensively
linked to each other w.r.t. a particular goodness metric1
44

Cohesiveness Metrics
● Some commonly used cohesiveness metrics are:
○ k-core
○ k-truss
○ k-clique
○ k-ECC (k-Edge Connected Component)
45

k-core
● A k-core is a maximal group of actors, all of whom are connected to
some number (k) of other members of the group
46

k-truss
● Given an integer k ≥ 2, the k-truss of G is the inclusion maximal
edge-induced subgraph H of G such that each edge e of H belongs
to at least k − 2 triangles of H
47
An example of 4-truss

k-clique
● A k-clique is a complete graph with k vertices where there is an
edge between every pair of vertices
● Example: {A, B, C, D} is a 4-clique, {A, B, E} is a 3-clique, any
edge is a 2-clique
48

k-ECC (k-edge connected component)
● A connected graph is k-edge-connected if it remains connected
whenever fewer than k edges are removed
● Example: The subgraph of {A, B, C, D} is the 3-ECC, because for
any pair of vertices in it, to disconnect them, we need to remove at
least 3 edges
● The 2-ECC has vertices {A, B, C, D, E}
49

Cohesiveness and Computational
Efficiency
50

K-Core-Based Community Search
● Uses k-core as structure cohesiveness metric
● In [1] Sozio et. al. defines the community search problem as:
○ Given an undirected simple graph G(V, E), a set of query vertices Q ⊆
V, and a goodness function f, return a subgraph H(VH, EH) of G such
that
■ VH contains Q;
■ H is connected;
■ f(H) is maximized among all feasible choices for H
○ f(H) is a general goodness function for measuring cohesiveness of the
community H
○ Higher f(H) means densely connected
51

● Many possible choices for f
● One outstanding method is based on the minimum degree i.e. f(H) =
min∀v∊HdegH(v)
● The minimum degree is a good metric for the community because:
○ it is the most fundamental characteristics of a graph
○ it is often used to measure the cohesiveness of user groups in social
network
○ it is better than other metrics, for community search
52

● To solve the problem of community search mentioned earlier, there
are two online algorithms which are based on global and local
search and one index-based algorithm
53

A global search algorithm
● Sozio et al. [1] proposed a greedy algorithm, Global
● Follows the peeling framework of computing the densest subgraphs
and removes vertices iteratively
● Specifically, let G0=G and Gt be the graph in t-th iteration (1 ≤ t <
n) step, it removes the vertex which has the minimum degree in Gt−1
and obtain an updated graph Gt
● The above operation iterates and stops at the T-th step, if either
1. at least one of the query vertices Q has minimum degree in the graph
GT −1, or
2. the query vertices Q are no longer connected
54

A local search algorithm
● In [7], Cui et al. proposed a local CS method named “Local”
● It works in a local expansion manner and finds a community that
may have smaller size than that of Global
● Specifically, it assumes that there is only one query vertex q (i.e.,
Q={q})
● Local consists of three steps:
● First, it expands the search space from q
● Second, it generates a candidate vertex set C in the search space
● Third, it finds the community from C
55

An index-based algorithm
● In [8], Barbieri et al. proposed an index structure, called ShellStruct,
which organizes all the connected k-cores in an offline manner
● Based on ShellStruct, k-core community search problem can be
answered in optimal time cost, i.e., O(|HV|), where HV is the set of
vertices in the returned community and it is the same with that of
Global
56

● The index is built based on the key observation that cores are nested
● That is, for any integer 0 < k ≤ kmax , the k-core is contained by the
(k–1)-core, where kmax is the maximum core number
● ShellStruct is a tree-like structure with kmax levels
● The root of the tree corresponds to the 1-core, and the k-th level
keeps track of the information about the k-th core
57

● In k-th level, each tree node, pk, corresponds to a connected
component Ck of the k-core, and it keeps:
● the set of “children” nodes, each of which corresponds to a connected
component that is in the (k+1)-core and contained by Ck;
● the set of vertices in Ck but not in (k+1)-core
58

K-Truss-Based Community Search
● Uses k-truss as structure cohesiveness metrics
● Some CS that are based on k-truss includes triangle connected truss
community [9,10], closest truss community [11], attribute-driven
truss community [12], and weighted truss community [13], etc.
59

● In [9] Huang et al. proposes a triangle connected k-truss community
model
● Finds all communities containing a query vertex
60
● Problem:
○ k-truss may be disconnected with several
components in a graph
○ For example, in the Figure, there exist two
shaded components to form the 4-truss of G,
which are disconnected
○ Disconnected graphs are insufficient to define
a cohesive and meaningful community
Example of 4-truss with 2
disconnected components

● To address the dis-connectivity problem, k-truss triangle
connectivity is imposed on top of the k-truss in [9]
● Given two triangles ∆ 1 and ∆2 in G, ∆1 and ∆2 are said to be
adjacent if they share a common edge
● Then, for two edges e1, e2 ∈ E, e1 and e2 are triangle connected if
they either belong to the same triangle, or are reachable from each
other through a series of adjacent triangles
61

● Based on the k-truss and triangle connectivity, the problem of triangle-
connected truss community (TTC) search is formulated as follows
● (TTC search) Given an undirected simple graph G(V, E), a query
vertex q ∈ V, and an integer k ≥ 2, return all subgraphs H ⊆ G satisfies
the following three properties:
1. Structure Cohesiveness. H contains the query vertex q such that ∀e ∈
E(H), sup(e, H) ≥ (k − 2);
2. Triangle Connectivity. ∀e1, e2 ∈ E(H), e1 and e2 are triangle connected;
3. Maximal Subgraph. H is the maximal subgraph of G satisfying
Properties 1 and 2
62

● To tackle the problem of TTC search, in [9] Huang et al. proposes
an online search algorithm
63

Online Search Algorithm
● The algorithm firstly applies the truss decomposition on graph G to
compute the trussness of all edges in G
● By the community definition, it starts from the query vertex q and
checks an incident edge of (q, v) ∈ E with trussness τ ((q, v)) ≥ k to
search triangle-connected truss communities
● It explores all edges that are triangle-connected to (q, v) and having
trussness no less than k in a BFS manner
● This process iterates until all incident edges of q have been
processed
● Finally, a set of k-truss communities containing q are returned
64

K-Clique-Based Community Search
● Uses k-clique or its variants to capture the structure cohesiveness
● In [6], Cui et al. showed that there are two shortcomings for k-
clique community model:
1. there are overwhelming number of k-cliques communities in real life
graphs; and
2. the k-clique constraint and the definition of adjacent (i.e., sharing k-1
common vertices) are not flexible in practice
65

● To address these two shortcomings, they proposed an online community
search (OCS) problem
● Instead of enumerating all communities, they focused on the search of the
communities containing a given query vertex q
● They relaxed the k clique adjacent from k−1 common vertices to α vertices,
namely α-adjacency
66

67
● They also relaxed k-clique model to γ-quasi-k-clique model
● By doing this, the k-clique components in the k-clique communities
are relaxed to the γ-quasi-k-clique components

68
● A naive algorithm for exact solution is to enumerate all γ quasi-k-
cliques containing the query vertex q, and then compute the γ-quasi-
k-clique components based on the α-adjacency
● To avoid enumerating cliques belonging to none of the valid
communities, a new computing framework is proposed to check the
adjacency when a clique is discovered
● By carefully maintaining the visit status of each clique, authors
further optimize the searching cost
● Authors also proposed an approximate solution

69
● To reduce the search space, the approximate algorithm only
enumerates an unvisited clique which contains at least one new
vertex not contained by any existing community
● A heuristic is proposed to choose a vertex sequence such that the
resulting clique sequence is short, leading to a good approximation
solution

K-ECC-Based Community Search
● Uses k-ECC as the community structure cohesiveness
● Given a graph G and a set Q of vertices, the goals is to find a
subgraph H of G, which contains Q and has the maximum edge-
connectivity, also called the Steiner Maximum-Connected Subgraph
(SMCS)
70

● In [14], Chang et al. computed the maximum SMCS for a set of
query vertices Q, which is defined as follows:
● Problem Definition. Given an undirected simple graph G(V, E), and
a set of query vertices Q ⊆ V , return a subgraph H(VH, EH) of G,
such that
1. VH contains Q;
2. Connectivity, λ(H), is maximized;
3. There exists no other subgraph H0 satisfying the above properties,
such that H ⊂ H’
71

● For example, in the graph below. Let Q = {v1, v4} then the query
will return the subgraph g1, and its connectivity is λ(g1) = 4
72

● To improve the query efficiency, Chang et al. [14] proposed a novel
compact index structure, which allows the query can be answered in
optimal time cost, i.e., the time cost is linear to the size of H
● The index is built based on a key observation that for any pair of
vertices u and v in H, their connectivity λ(u, v) is at least λ(H)
73

● To preserve all the connectivity information of G, Chang et al.
developed the concept of connectivity graph Gc for the graph G,
which has the same sets of vertices and edges with G, and for each
edge (u, v) ∈ Gc, it is associated with a connectivity value denoting
the edge connectivity between vertices u and v in G
● Then, the maximum spanning tree (MST) of Gc is the index
structure built for G
● For example, figure b presents the index structure for the graph in
Fig. (a). The index can be built by first constructing the connectivity
graph Gc and then computing the MST from Gc
74

● Based on the index MST, Chang et al. proposed an efficient query
algorithm to solve k-ECC based community search
● Specifically, it first computes λ(H) by using the MST, and then
finds the maximum SMCS by collecting the subtree of MST, whose
edges have connectivity values being at least λ(H)
● By using the technique of lowest common ancestor (LCA), the
query can achieve a time cost of O(|HV|), which is optimal since
outputting the vertex set of H takes O(VH) time
75

Influence Maximization
● Social network plays a fundamental role as a medium for the
spread of INFLUENCE among its members
○ opinions, ideas, information, innovation…
● Word of Mouth (WOM) effect
76

Problem Setting
● Given
○ a limited budget B for initial advertising (e.g. give away free samples
of product)
○ estimates for influence between individuals
● Goal
○ trigger a large cascade of influence (e.g. further adoptions of a
product)
● Question
○ Which set of individuals should B target at?
● Application besides product marketing
○ spread an innovation
○ detect stories in blogs
77

Influence Models
● Linear Threshold Model
● Independent Cascade Model
78

Linear Threshold Model
● A node v has random threshold θv ~ U[0,1]
● A node v is influenced by each neighbor w according to a weight
bvw such that
● A node v becomes active when at least (weighted) θv fraction of its
neighbors are active
79

Linear Threshold Model
● A node v has random threshold θv ~ U[0,1]
● A node v is influenced by each neighbor w according to a weight
bvw such that
● A node v becomes active when at least (weighted) θv fraction of its
neighbors are active
80

Example
Inactive Node
Active Node
Threshold
Active neighbors
v
w
0.5
0.3
0.2
0.5
0.1
0.4
0.3 0.2
0.6
0.2
Stop!
U
X

Independent Cascade Model
● When node v becomes active, it has a single chance of activating
each currently inactive neighbor w
● The activation attempt succeeds with probability pvw
82

Example
v
w
0.5
0.3 0.2
0.5
0.1
0.4
0.3 0.2
0.6
0.2
Inactive Node
Active Node
Newly active
node
Successful
attempt
Unsuccessful
attempt
Stop!
U
X

References
1. M. Sozio and A. Gionis. The community-search problem and how to plan a successful cocktail
party. In SIGKDD, pages 939–948, 2010.
2. https://slideplayer.com/slide/14253033/
3. V. D. Blondel, J.-L. Guillaume, R. Lambiotte and E. Lefebvre, "Fast unfolding of communities
in large networks," J. Stat. Mech. (2008) P10008, p. 12, 2008.
4. Uncovering the overlapping community structure of complex networks in nature and society.
5. Gopalan PK, Blei DM. Efficient discovery of overlapping communities in massive networks.
Proc Natl Acad Sci 2013, 110:14534–14539
6. W. Cui, Y. Xiao, H. Wang, Y. Lu, and W. Wang. Online search of overlapping communities.
In SIGMOD, pages 277–288, 2013.
7. W. Cui, Y. Xiao, H. Wang, and W. Wang. Local search of communities in large graphs. In
SIGMOD, pages 991– 1002, 2014.
8. N. Barbieri, F. Bonchi, E. Galimberti, and F. Gullo. Efficient and effective community search.
DMKD, 29(5):1406–1433, 2015.
84

References
9. X. Huang, H. Cheng, L. Qin, W. Tian, and J. X. Yu. Querying k-truss community in large and
dynamic graphs. In SIGMOD, pages 1311–1322, 2014.
10. E. Akbas and P. Zhao. Truss-based community search: A truss-equivalence based indexing
approach. PVLDB, 10(11):1298–1309, Aug. 2017.
11. X. Huang, L. V. Lakshmanan, J. X. Yu, and H. Cheng. Approximate closest community search
in networks. PVLDB, 9(4):276–287, 2015.
12. X. Huang and L. V. S. Lakshmanan. Attribute-driven community search. PVLDB, 10(9):949–
960, May 2017.
13. Z. Zheng, F. Ye, R.-H. Li, G. Ling, and T. Jin. Finding weighted k-truss communities in large
networks. Information Science, 417(C):344–360, Nov. 2017.
14. L. Chang, X. Lin, L. Qin, J. X. Yu, and W. Zhang. Index-based optimal algorithms for
computing steiner components with maximum connectivity. In SIGMOD, pages 459–474,
2015.
85

Algorithm in Social network of graph and social network analysis

More Related Content

What's hot

Similar to Algorithm in Social network of graph and social network analysis

Recently uploaded

Algorithm in Social network of graph and social network analysis