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7/27/2019 Influence Maximization in Social Network - Using A New Centrality Measure Diffusion Degree
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Influence Maximization in Social NetworkUsing A New Centrality Measure Diffusion Degree
Suman Kundu
Center for Soft Computing ResearchIndian Statistical Institute
Kolkata - 700108
June 21, 2011
7/27/2019 Influence Maximization in Social Network - Using A New Centrality Measure Diffusion Degree
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Outline
Social Networks Overview
Centrality Measures
Problem Definition, Application and Challenges
Available Solutions
Proposed Solution
Diffusion ModelDiffusion DegreeAssumption & Experimental Set UpAlgorithmResults
Conclusion
References
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Social Networks
S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 3 / 36
7/27/2019 Influence Maximization in Social Network - Using A New Centrality Measure Diffusion Degree
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Overview
What is Social Networks?
A Social Structure made up of individual or organization called nodeNodes are tied by one or more interdependence (e.g. friendship,common interest, financial exchange etc.)Many kind of ties between nodesOperates on many levels
From family upto national level
Example
Online Social Networks - Facebook, Twitter, Orkut, LinkedIn etc.Who-talks-to-Whom Networks - Telephonic Communication, EmailCommunication etc.Collaboration Networks - Co-Authorship Networks, Co-Appearance in amovie etc.Natural World Networks - Food webs are representations of thepredator-prey relationships between species, Biological Network of Neural Connections.
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Example
Figure: HEPph Citation Network
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Centrality Measures
S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 6 / 36
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Overview
What is centrality?Relative importance of a vertex within a graph
e.g. How important a person is within a social network
Measures of centralityDegree Centrality
Number of edges incedent on a vertex
Betweenness Centrality
Ratio of number of shortest path passing through the vertex and total
number of shortest paths between all pairs in the network
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Degree Centrality
Nieminen introduce a simple, natural and perfectly general measure of
centrality based upon degree 1
Count of the degree or number of adjacencies for a point
For a Graph G (V , E ) with n vertex (Where V is the set of vertex andE is the set of edges) degree centrality C D (v ) for vertex v is:
C D (v ) =
ni =1
σ(u i , v )
Where σ(u i , v ) = 1 if and only if u i and v are connected by a link
= 0 Otherwise
A point v , can at most be adjacent to n − 1 other points in thegraph. So,
C ∗D (v ) =
ni =1 σ(u i , v )
n − 1
1[Nieminen, 1974]S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 8 / 36
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Betweenness Centrality
For a Graph G (V , E ) with n vertex, betweenness of vertex v is:
C B (v ) =
s =v =t
σst (v )
σst
Where σst is the number of shortest path between s and t and σst (v )
is the number of shortest path from s
to t
passing through v
Freeman2 proved that the max value taken by C B (v ) is achieved onlyby the central point in a star; and it is equal to
n2− 3n + 2
2
Therefore, the relative betweenness centrality of any point in a graphmay be expressed as a ratio of
C ∗B (v ) = 2C B (v )
n2− 3n + 2
2[Freeman, 1977]S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 9 / 36
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Problem Definition, Applications and
Challenges
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Influence Maximization in Social Network
Social Network - A Medium of Information Spread
Opinions, Ideas, Information, Innovations and more
Influence in form of Word-of-MouthSignificant increase of profit
One of the major problem to achieve the above target is
How to select the influential individuals quickly, to target in informationspreading? That is selecting the initial seed set for influence spreading.
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Application of Influence Maximization
Marketing
Give limited free sample of products and/or applications
Wait for spreading of the informationCreate potential buyer of the product and/or applications
Other Than Marketing
Spread of InnovationDetect Stories in Blog
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Challenges
Online Social Networks are obvious choice for marketing orinformation spreading3
Online Social Networks connects a huge number of peopleOnline Social Networks collects huge amount of information about the
Social Network Structure and Communication Dynamics
Challenges
The Social Networks are Large ScaleComplex Connection StructureDynamic NetworkSolution needs to be very efficient and scalable
3[Chen et al., 2009]S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 13 / 36
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Complex Structure
Fi ure: HEP h Citation NetworkS. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 14 / 36
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Available Solutions
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7/27/2019 Influence Maximization in Social Network - Using A New Centrality Measure Diffusion Degree
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Classic Approach
High Degree Heuristics
Nodes are selected according to their degree rank
Random SelectionSeeding nodes are selected randomly
Most Central Heuristics
Nodes are selected according to their Betweenness or other centralityrank
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Hill Climbing Greedy Approach
Kempe, Kleinberg, and Tardos4 are the first to proposed greedysolution for the problem
Present a greedy algorithm which guarantees that the influencemaximization is within (1 − 1/e ) of the optimal influence spreadAlso show through experiments that greedy algorithm outperforms theclassic degree and centrality based heuristics
Drawback of Greedy AlgorithmEfficiency - Simulation based approach, needs to simulate sufficientamounts to get accurate estimation; Unlikely to get results for onlinesocial networks contains millions of nodes.
Some researcher showed that even for a 15K vertics graph taking daysto compute the result
4
[Kempe et al., 2003]S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 17 / 36
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Improvements over Greedy Algorithm
Several Attempts to improve the greedy approach is proposed-
Pabli A. et al proposed a Set Covering Greedy Algorithm5 to improvethe performance and efficiency of greedy algorithmLeskovec et al.6 present an optimized greedy algorithm referred byCost Effective Lazy ForwardChen et al.7 proposed two efficient algorithm to further improving ongreedy algorithm. This algorithm is known as NewGreedy andMixedGreedy
Even after improvement, these greedy approach is not even closer to
the speed of centrality based heuristic model
5[Estevez et al., 2007]6[Leskovec et al., 2007]7
[Chen et al., 2009]S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 18 / 36
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Degree Discount Heuristics
In their paper Chen et al.8 also proposed a degree discount heuristic
General idea of degree discount algorithm of Chen et al. is that if onenode is considered as seed then the links connecting with the node will
not be counted as a degree of the other nodes i.e. when consideringthe next node, the links connecting with the nodes already in the seedset will be discounted.The running time of the algorithm is comparable with high degreeheuristics. However, in our experiments we did not foundimprovements over the classic high degree heuristics.
8
[Chen et al., 2009]S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 19 / 36
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Proposed Solution
S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 20 / 36
7/27/2019 Influence Maximization in Social Network - Using A New Centrality Measure Diffusion Degree
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Diffusion Model
Independent Cascade Model
Is a probabilistic information diffusion modelStarts with a set of initial active nodes
In step t an active node gets single chance to activate an inactiveneighbor with diffusion probability
Linear Threshold Model
In this model one node become active if the fraction of its activeneighbor is greater then the threshold value
Other Variants
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Diffusion Degree - Overview
Many of the available centrality measures considered only structuralproperty of a node
However, diffusion Probability plays a vital role in influence flow over
the networkAdditionally in a social network, neighborhood has a significantimpact on ones influence
We proposed a new centrality measurement for vertex namedDiffusion Degree considering the above points of social network anddiffusion method
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Diffusion Degree - Mathematical Explanation I
The general Degree Centrality9 measure of node v can be defined as
C D (v ) =
n
i =1
σ(u i , v ) (1)
where function σ(u i , v ) defined as,
σ(u i , v ) = 1 if and only if u i and v are connected
= 0 otherwise.
In a diffusion process, a node v with propagation probability λv , canactivate its neighbor u with probability λv . So, considerablecontribution of node v in the diffusion process is
C DD (v ) = λv ∗ C D (v ). (2)
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Diffusion Degree - Mathematical Explanation II
When the diffusion process propagates to the next level, active
neighbors of v will try to activate their inactive neighbors.
If the propagation probability of i th neighbor of v is λi , considerablecontribution of i th neighbor in the diffusion process is
C
DD (i ) = λi ∗
C D (i ). (3)
Thus the cumulative contribution in the diffusion process byneighbors of v will be maximized when all of its neighbors will beactivated in the previous step.
In this scenario, the total contribution of neighbors of v is
C DD (v ) =
i ∈neighbors (v )
C DD (i ). (4)
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Diffusion Degree - Mathematical Explanation III
The diffusion degree of a node is defined as the cumulativecontribution score of the node itself and its neighbors.
So, from the equations 2 and 4 we can define the diffusion degreeC DD as
C DD = C DD + C DD (5)
= λi ∗ C D (i ) +
i ∈neighbors (v )
C DD (i ) (6)
= λi ∗ C D (i ) + i ∈neighbors (v )
λi ∗ C D (i ). (7)
9
[Nieminen, 1974]S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 25 / 36
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Assumption & Experimental Data Set
Assumption
Independent Cascade Model for experiment and result comparisonDirected Social Networks
Only the in degree is contributing towards influencingExperimental Data Set
Large Scale Citation Network - DBLP(4.47Lac+ nodes & 23.27Lac+Links) and HEPph(35K+ nodes & 4Lac+ links)Large Scale Online Social Network - Twitter(4.15Lac+ Nodes & 8.2Lac
links)
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Algorithm
Step 1: Compute diffusion degree for all vertexes
Step 2: Order the vertexes based on the diffusion degree
Step 3: Select top k nodes for top k-influence maximization problem
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Results
We compare our result with other centrality based heuristics model
like high degree and degree discount algorithmTo obtain approximate result of influenced nodes we simulate 100times
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ff
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Results - DBLP with 0.05 Diffusion Probability
S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 29 / 36
R l HEP h i h 0 05 Diff i P b bili
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Results - HEPph with 0.05 Diffusion Probability
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R l T i i h 0 05 Diff i P b bili
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Results - Twitter with 0.05 Diffusion Probability
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C l i
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Conclusion
Online Social Networks attracts marketing teams for exploring andincreasing market over the globe. Thus the demand of fast algorithmwith satisfying results are in demand
Based on our study so far and primary results, we believe that finetuned centrality based heuristics may provide truly scalable solutionsto the influence maximization problem with satisfying influence spreadand fast running time.
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Your Question?
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Thank You
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Reference I
7/27/2019 Influence Maximization in Social Network - Using A New Centrality Measure Diffusion Degree
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Reference I
Chen, W., Wang, Y., and Yang, S. (2009).Efficient influence maximization in social networks.In KDD ’09: Proceedings of the 15th ACM SIGKDD international
conference on Knowledge discovery and data mining , pages 199–208,New York, NY, USA. ACM.
Estevez, P. a., Vera, P., and Saito, K. (2007).Selecting the Most Influential Nodes in Social Networks.In 2007 International Joint Conference on Neural Networks , pages2397–2402. Ieee.
Freeman, L. (1977).A set of measures of centrality based on betweenness.Sociometry , pages 35–41.
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Reference II
7/27/2019 Influence Maximization in Social Network - Using A New Centrality Measure Diffusion Degree
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Reference II
Kempe, D., Kleinberg, J., and Tardos, E. (2003).Maximizing the spread of influence through a social network.In KDD ’03: Proceedings of the ninth ACM SIGKDD international
conference on Knowledge discovery and data mining , pages 137–146,New York, NY, USA. ACM.
Leskovec, J., Krause, A., Guestrin, C., Faloutsos, C., VanBriesen, J.,and Glance, N. (2007).Cost-effective outbreak detection in networks.In Proceedings of the 13th ACM SIGKDD international conference on
Knowledge discovery and data mining , pages 420–429. ACM.
Nieminen, J. (1974).On centrality in a graph.Scandinavian Journal of Psychology , 15:322–336.
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