Information diffusion on Graphs Gennaro Cordasco gennaro cordascounicampania
![How to model the influence in social network? • Linear Threshold model [Kempe+03] •](https://slidetodoc.com/presentation_image_h2/61c4e3830822835a41fb39ce350c0ea1/image-12.jpg "How to model the influence in social network? • Linear Threshold model [Kempe+03] •")
![Example IE[S, ℓ] = IE[S, ℓ− 1] ∪ {e ∈ E : |V(e)∩ I](https://slidetodoc.com/presentation_image_h2/61c4e3830822835a41fb39ce350c0ea1/image-66.jpg "Example IE[S, ℓ] = IE[S, ℓ− 1] ∪ {e ∈ E : |V(e)∩ I")
![Example IE[S, ℓ] = IE[S, ℓ− 1] ∪ {e ∈ E : |V(e)∩ I](https://slidetodoc.com/presentation_image_h2/61c4e3830822835a41fb39ce350c0ea1/image-67.jpg "Example IE[S, ℓ] = IE[S, ℓ− 1] ∪ {e ∈ E : |V(e)∩ I")
- Slides: 72
Information diffusion on Graphs Gennaro Cordasco gennaro. cordasco@unicampania. it DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI
Outline • Motivation • Influence Propagation in Social Networks • Target Selection Problem on Graphs • Heuristics for the TSS Problem • Results and Experiments • Target Selection Problem on Hypergraphs • A novel model to emulate the diffusion • Conclusion 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 2
Motivation 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 3
Ice Bucket diffusion Challenge Dynamics of information Social Network plays a fundamental role as a medium for spreading new ideas, beliefs, opinions, behaviors, … • Social Influence: Process in which individuals adjust their opinions, emotions, or behaviors affected by others (intentionally or unintentionally according to interactions with other people) 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 5
Spread of Influence • People decisions are influenced by their neighbor’s (friends) decisions • as individuals become aware of new ideas/products, they have the potential to pass them to their friends • People have a tendency to conform: • following the crowd • the individual’s need to be accepted by others • the ‘‘direct benefit’’ effect : your benefits in adopting a new behavior increase more and more as your friends embrace it • Effects of group pressure seem to be observable even when the group is represented by a virtual community • Salganik, Dodds, and Watts (2006) investigated the effect of social influence on the success of market products like books, songs or movies. 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 6
Dynamics of information diffusion The natural human tendency to conform can be exploited in viral marketing Buy Some. Thing™! Followers VIP 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 7
Dynamics of information diffusion The Word of Mouth effect…. • Firms are increasingly recruiting customers to take part in word-ofmouth marketing campaigns. • Network targeting seeks to convert some influential individuals (seeds) who are expected to use their social network position, personal influence, and broad peer contacts to trigger cascades of product adoption. 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 8
Gaming Console Viral Market • A giant company is about to sell a new console, the Xstation IV® • Viral Market campaign, an initial set of influential people should be targeted and receive a complimentary personalized Xstation (or a discount) • Social Conformity • the ‘‘direct-benefit’’ effect • The Network is known, users play on the “free” XStation Network • Desiderata: • maximize the spread of influence $$$$$, or • minimize the marketing expences 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 9
Similar models for a variety of phenomena • • • Cascading failures in physical infrastructure networks Information cascades in social and economic systems Spread of infectious diseases Spread of computer viruses Spreading of ideas, fashion and behaviors among people … Corresponding dynamics are studied with tools and concepts from Epidemiology, Social Sciences, Physics, Mathematics, and Computer Science. 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 10
Viral Marketing 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 11
Dynamic Processes on Networks 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 12
How to model the influence in social network? • Linear Threshold model [Kempe+03] • Each node is labelled with a threshold t( ) • We start with an active set of nodes (0 th step) • In ith step a node v that have at least t(v) neighbours active become active Threshold=3 Inactive Active • Independent Cascade [Kempe+03] 0. 1 0. 8 0. 3 0. 5 0. 1 0. 8 0. 1 0. 3 0. 5 08/06/2021 0. 5 • Each edge is labelled with a probability • We start with an active set of nodes (0 th step) • In ith step those nodes that became active in the(i -1)th step may activate their neighbours • Node u activate its neighbour v with the probability assigned to their linking edge • Each node has only one chance to activate its neighbours Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 13
Viral Market • A Network coordinate game: • 2 possible behaviors (named A and B) • Initially every node uses B as behavior • Then a set of initial adopters switch to A (for some reason) • The diffusion of information is modeled via a linear threshold model Threshold=3 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 14
Viral Market Threshold=3 • Given a fixed network • We want to flood the network with a new product (the new behavior) • We want to advertise the product as less as possible • Which is the smallest number of early adopters? 08/06/2021 Target Selection problem (TSS) Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 15
Target Selection Problem on Graphs 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 16
Problem Setting (LTM) • Given • a social network • estimates for influence between individuals • Goal • Trigger a large cascade of influence • Question • Which set of individuals should be target at? 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 17
Target Selection Problem • Given • a social network represented by a graph G=(V, E) • threshold t(u) assigned to each node u • easy-to-influence individuals have ‘’low" threshold values, • hard-to-influence individuals have ‘’high" threshold • Goal • Trigger a cascade of influence that involves the whole network • Question • Which set of individuals should be target at? • Find a small set of individuals (Target Set), who when convinced about a new product will influence all the others by WOM 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 18
Target Selection Problem • 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 19
Target Selection Problem • 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 20
An example: Target Selection Problem 1 2 5 3 5 5 Node: User in a social network Edge: Friendship among users 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 21
An example: Target Selection Problem 1 2 5 3 5 5 Node: User in a social network (red – Target Set) Edge: Friendship among users 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 22
An example: Target Selection Problem 1 1 st round 2 5 3 5 5 Node: User in a social network (red – Target Set) (orange – influenced node) Edge: Friendship among users 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 23
An example: Target Selection Problem 1 2 nd round 2 5 3 5 5 Node: User in a social network (red – Target Set) (orange– influenced node) Edge: Friendship among users 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 24
An example: Target Selection Problem 1 3 rd round 2 5 3 5 5 Node: User in a social network (red – Target Set) (orange – influenced node) Edge: Friendship among users 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 25
An example: Target Selection Problem 1 4 th round 2 5 3 5 The whole network is influenced 5 Node: User in a social network (red – Target Set) (orange – influenced node) Edge: Friendship among users 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 26
Heuristics for the TSS Problem 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 27
TSS (what is already known) • Optimal solution for simple graphs • • Path Tree Complete Graphs Dense Graphs • Some very simple heuristics have also been presented • Additive greedy strategies (based on some measure of node influence) 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 28
Greedy TSS: an additive algorithm S is not sufficient to influence all the nodes Greedy choice 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 29
The TSS Algorithm: a subtractive algorithm Case 1: v is influenced by the node in S Case 2: v is added to S Case 3: a vertex is selected to be discarded 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 30
An Example 1 v 4 3 v 1 1 1 v 2 v 3 2 v 5 1 v 6 v 1 v 2 v 3 v 4 v 5 v 6 U 08/06/2021 S V-S Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 31
An Example 1 v 4 3 v 1 2 v 5 1 v 6 1 v 3 Case 2: (v 1)<k(v 1) v 1 v 2 v 3 v 4 v 5 v 6 U 08/06/2021 S V-S Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 32
An Example 0 v 4 2 v 5 1 v 6 0 v 3 Case 1: k(v 3)=0 v 1 v 2 v 3 v 4 v 5 v 6 U 08/06/2021 S V-S Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 33
An Example 0 v 4 2 v 5 1 v 6 Case 1: k(v 4)=0 v 1 v 2 v 3 v 4 v 5 v 6 U 08/06/2021 S V-S Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 34
An Example 1 v 5 1 v 6 v 1 v 2 v 3 v 4 v 5 v 6 U 08/06/2021 S V-S Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 35
An Example 1 v 6 Case 2: (v 2)<k(v 2) v 1 v 2 v 3 v 4 v 5 v 6 U 08/06/2021 S V-S Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 36
1 v 4 An Example 3 v 1 1 1 v 2 v 3 2 v 5 1 v 6 • S 0={v 1, v 6} • S 1={v 1, v 2, v 3, v 4, v 6} • S 2 ={v 1, v 2, v 3, v 4, v 5, v 6}=V 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 37
Results • 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 38
Experiments 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 39
Computational Experiments • Random Graphs G(n, p) • n {30, 50} , p {1/10, 2/10, …, 9/10} • t(v)= random [1, d(v)] 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 40
Computational Experiments • Large Real-Life Networks • Stanford Large Network Data set Collection (SNAP) • Social Computing Data Repository at Arizona State University 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 41
Computational Experiments • Competing algorithm • Enhanced Greedy • TIP-DECOMP • Vir. Ads • Thresholds (fixed between 1 and 10) • Let t the fixed threshod, for each node v, t(v)= min (t, d(v)) 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 42
Computational Experiments 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 43
Computational Experiments 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 44
Another heuristic for TSS 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 45
Another heuristic for TSS • At each iteration, a vertex is selected and: • Case 3) it is moved into a limbo, represented by the set L. • Case 2) a vertex v remain with less neighbors than its thresholds, such vertex is added to the target set and deleted from the graph while their outgoing neighbors’ thresholds are decreased by 1 (since they receive v’s influence) • Case 1) a vertex threshold is decreased down to 0, such vertex is deleted from the graph and its outgoing neighbors’ thresholds are decreased by 1 (since once v activates, they will receive v’s influence). 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 46
1 v 4 An Example U 2 v 1 1 2 v 3 L 2 v 5 2 v 6 S v 1 v 2 v 3 v 4 v 5 v 6 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 47
1 v 4 An Example 2 v 1 1 2 v 3 2 v 5 2 v 6 Case 2: in(v 1)<k(v 1) U v 1 v 2 v 3 v 4 v 5 v 6 08/06/2021 L S v 2 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 48
1 v 4 An Example 2 v 5 0 1 v 2 v 3 2 v 6 Case 1: k(v 2)=0 U L S v 1 v 2 v 3 v 4 v 5 v 6 08/06/2021 v 2 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 49
1 v 4 An Example 2 v 5 2 v 6 1 v 3 U L S v 1 v 3 v 4 v 5 v 6 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 50
1 v 4 An Example 2 v 5 2 v 6 1 v 3 Case 2: in(v 5)<k(v 5) U L S v 1 v 3 v 4 v 5 v 6 08/06/2021 v 3 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 51
0 v 4 An Example 1 v 6 1 v 3 Case 1: k(v 4)=0 U L S v 1 v 3 v 4 v 3 v 5 v 6 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 52
An Example 1 v 6 1 v 3 Case 2: in(v 6)<k(v 6) U L S v 1 v 3 v 5 v 6 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 53
An Example 0 v 3 Case 1: k(v 3)=0 U L S v 1 v 3 v 5 v 6 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 54
An Example U L S v 1 v 5 v 6 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 55
An Example 1 v 4 2 v 1 1 2 v 3 2 v 5 2 v 6 • S 0={v 1, v 5, v 6} • S 1={v 1, v 2, v 3, v 4, v 5, v 6}=V 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 56
MTS Results • Correcteness • Lemma For any digraph G=(V, E), the number of iterations of the while loop of algorithm MTS(G) is at most 2|V|. • Theorem For any digraph G=(V, E) and threshold function t, the algorithm MTS(G) outputs a target set for G. The algorithm can be implemented so to run in O(|E| log |V|) time. 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 57
Experiments • Test Setting • 20 real network from • Stanford Large Network Dataset Collection (SNAP) • the Social Computing Data Repository at Arizona State University • Newman’s Network data • 4 Competiting Algorithms • • TSS [2014] Greedy[Chen et al 2009] TIP_DECOMP[Demaine et al 2014] Discount. Frac[Shakarian et al 2013] • Thresholds • Random • Degree Proportional 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 58
Results: Random 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 59
Results: Higgs-twitter, Degree Proportional Thresholds 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 60
Results: Higgs-twitter, Degree Proportional Thresholds 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 61
Correlation between Network Modularity and Normalized Target Size 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 62
Target Selection Problem on Hypergraphs 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 63
Target Selection Problem on Hypergraphs • Given • a social network represented by a hypergraph H=(V, E) • threshold t. V(u) assigned to each node u V • threshold t. E(e) assigned to each edge e E • Goal • Trigger a cascade of influence that involves the whole network • Question • Which set of individuals should be target at? • Find a small set of individuals (Target Set), who when convinced about a new product will influence all the others by WOM 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 64
Target Selection Problem on Hypergraphs • 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 65
Target Selection Problem on Hypergraphs • A target set for H is a seed set S that will influence the whole network • IV[S, ℓ]=V, for some ℓ ≥ 0. Target Selection (TSSH) Instance: An hypergraph H = (V, E) with thresholds t. V : V → N and t. E : E → N. Problem: Find a target set S ⊆ V of minimum size for H. • It is NP-hard • Efficient heuristics are needed 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 66
Example IE[S, ℓ] = IE[S, ℓ− 1] ∪ {e ∈ E : |V(e)∩ I ≥ t. E(u)} V[S, ℓ− 1]| IV[S, ℓ] = IV[S, ℓ− 1] ∪ {v ∈ V : |E(v)E[S, ℓ− 1]| ∩ I ≥ t. V(u)} Target Selection Problem on Hypergraphs IE[S, 1] = {P 2} IV[S, 1] ={v 1, v 3, v 4} IE[S, 0] = ∅ IV[S, 0] = S ={v 1, v 4} P 2 P 2 2 v 1 2 2 v 3 1 v 5 P 1 2 v 3 2 1 v 2 P 3 2 v 5 2 v 3 1 2 v 2 2 2 1 P 3 2 v 1 v 4 v 5 P 1 2 2 v 2 P 3 2 2 1 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI v 4
Example IE[S, ℓ] = IE[S, ℓ− 1] ∪ {e ∈ E : |V(e)∩ I ≥ t. E(u)} V[S, ℓ− 1]| IV[S, ℓ] = IV[S, ℓ− 1] ∪ {v ∈ V : |E(v)E[S, ℓ− 1]| ∩ I ≥ t. V(u)} IE[S, 1] = {P 2} IV[S, 1] ={v 1, v 3, v 4} IE[S, 2] = {P 2, P 3} IV[S, 2] ={v 1, v 3, v 4, v 5} P 2 2 1 P 2 2 v 1 2 v 3 IE[S, 3] = {P 1, P 2, P 3} IV[S, 3] ={v 1, v 2, v 3, v 4, v 5} v 1 2 2 v 5 P 1 2 v 3 2 1 v 2 P 3 2 2 1 v 4 v 1 2 v 5 P 1 2 v 3 2 1 v 2 P 3 2 2 v 2 2 2 1 P 3 2 2 v 5 v 4 1 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI v 4
A subtractive algorithm for TSS on Hypergraphs Not clear yet!!!! 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 69
A subtractive algorithm for TSS on Hypergraphs 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 70
A subtractive algorithm for TSS on Hypergraphs 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 71
Conclusion • Information diffusion is an interesting and challenging problem • An Information diffusion model on Hypergraphs (based on LTM) • An idea of a subtractive algorithm (to be tuned) • Is it still true that the proposed subtractive algorithm is better than an additive one? 08/06/2021 Hy. TAC 2019 DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI 72
Thanks for your attention gennaro. cordasco@unicampania. it DIPARTIMENTO DI PSICOLOGIA UNIVERSITÀ DEGLI STUDI DELLA CAMPANIA LUIGI VANVITELLI
