influence maximization in social network - using a new centrality measure diffusion degree

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

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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

S. Kundu (CSCR, ISI Kolkata) Influence Maximization in Social Network June 21, 2011 2 / 36

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Social Networks


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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


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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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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


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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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Results - DBLP with 0.05 Diffusion Probability


R l HEP h i h 0 05 Diff i P b bili

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Results - HEPph with 0.05 Diffusion Probability


R l T i i h 0 05 Diff i P b bili

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Results - Twitter with 0.05 Diffusion Probability


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


Reference I

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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.


Reference II

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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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influence maximization in social network - using a new centrality measure diffusion degree

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