## Inferring Networks of Diffusion and Influence

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Citations: | 58 - 6 self |

### BibTeX

@MISC{Leskovec_inferringnetworks,

author = {Jure Leskovec and Andreas Krause},

title = {Inferring Networks of Diffusion and Influence},

year = {}

}

### OpenURL

### Abstract

Information diffusion and virus propagation are fundamental processes talking place in networks. While it is often possible to directly observe when nodes become infected, observing individual transmissions (i.e., who infects whom or who influences whom) is typically very difficult. Furthermore, in many applications, the underlying network over which the diffusions and propagations spread is actually unobserved. We tackle these challenges by developing a method for tracing paths of diffusion and influence through networks and inferring the networks over which contagions propagate. Given the times when nodes adopt pieces of information or become infected, we identify the optimal network that best explains the observed infection times. Since the optimization problem is NP-hard to solve exactly, we develop an efficient approximation algorithm that scales to large datasets and in practice gives provably near-optimal performance. We demonstrate the effectiveness of our approach by tracing information cascades in a set of 170 million blogs and news articles over a one year period to infer how information flows through the online media space. We find that the diffusion network of news tends to have a core-periphery structure with a small set of core media sites that diffuse information to the rest of the Web. These sites tend to have stable circles of influence with more general news media sites acting as connectors between them.

### Citations

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Citation Context ...onential in the size of G but perhaps surprisingly, this super-exponential sum can be performed in time polynomial in the number n of nodes in the graph G, by applying Kirchhoff’s matrix tree theorem =-=[15]-=-: THEOREM 1 (TUTTE (1948)). If we construct a matrix A such that ai,j = P wk,j if i = j and ai,j = −wi,j if i ̸= j and if Ak,m is the matrix created by removing any row k and column m from A such that... |

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Citation Context ...7% of NP-hard to compute optimal. structure [5] ([0.962, 0.107; 0.107, 0.962]) and a core-periphery network [22] ([0.962, 0.535; 0.535, 0.107]). Notice that Forest Fire generates a scale free network =-=[4]-=-. We then simulate cascades on G ∗ using the generative model defined in Section 2.1 that is parameterized by α, which controls how quickly a cascade spreads, and β, that controls how far a cascade sp... |

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Citation Context ... Fire model [20] and the Kronecker Graphs model [19] to generate G ∗ . For Kronecker graphs, we consider three sets of parameters that produce networks with a very different structure: a random graph =-=[8]-=- (Kronecker parameter matrix [0.5, 0.5; 0.5, 0.5]), a network with hierarchical community Figure 4: Score achieved by NETINF in comparison with the online upper bound from Theorem 4. In practice NETIN... |

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Citation Context ...ing, Social networks. 1. INTRODUCTION Cascading behavior, diffusion and spreading of ideas, innovation, information, influence, viruses and diseases are fundamental processes taking place in networks =-=[12, 28, 30]-=-. In order to study network diffusion there are two fundamental challenges one has to address. First, to be able to track cascading processes taking place in a network, one needs to identify the conta... |

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Citation Context ...ch nodes influence which other nodes. Last, we will define P(c|G), which is the probability that cascade c occurs in a network G. Cascade transmission model. We build on the independent cascade model =-=[13]-=- which posits that an infected node infects each of its neighbors independently with some chosen probability. As in this model the time is modeled only implicitly through the epochs of the propagation... |

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Citation Context ...erformed poorly while our method (c) almost recovered G ∗ perfectly by making only two errors. Experimental setup. We consider two models of directed realworld networks, namely, the Forest Fire model =-=[20]-=- and the Kronecker Graphs model [19] to generate G ∗ . For Kronecker graphs, we consider three sets of parameters that produce networks with a very different structure: a random graph [8] (Kronecker p... |

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Citation Context ...e choosing a more complex transmission model. 5. RELATED WORK There are several lines of work we build upon. Although the information diffusion in on-line settings has received considerable attention =-=[2, 11, 16, 17, 23, 24, 25]-=-, only a few studies were able to study the actual shapes of cascades [23, 25]. The problem of inferring links of diffusion was first studied by Adar and Adamic [2], who formulated it as a supervised ... |

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Citation Context ...e choosing a more complex transmission model. 5. RELATED WORK There are several lines of work we build upon. Although the information diffusion in on-line settings has received considerable attention =-=[2, 11, 16, 17, 23, 24, 25]-=-, only a few studies were able to study the actual shapes of cascades [23, 25]. The problem of inferring links of diffusion was first studied by Adar and Adamic [2], who formulated it as a supervised ... |

238 | The dynamics of viral marketing
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Citation Context ...d cascade can be created. Similarly, in viral marketing, a person may purchase a product due to the influence of peers (i.e., network effect) or for some other reason (e.g., seing a commercial on TV) =-=[17]-=-. In order to account for such phenomena when a cascade “jumps” across the network, we introduce an additional node m that represents an external source that can infect any node u. We connect the exte... |

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Citation Context ...hyperplane in case of SVMs) and predicts individually the most probable links. Network structure learning has been considered for estimating the dependency structure of probabilistic graphical models =-=[9, 10]-=- and for estimating epidemiological networks [29]. In both cases, the problem is formulated in a probabilistic framework. However, since the problem is intractable, heuristic greedy hill-climbing or s... |

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165 | Cost-effective Outbreak Detection in Networks
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Citation Context ...s property of the model, we prove that NETINF infers near-optimal networks. We also speed-up NETINF algorithm by exploiting the local structure of the objective function and by using lazy evaluations =-=[21]-=-. Our results on synthetic datasets show that we can reliably infer the underlying propagation and influence networks, regardless of the overall network structure. Validation on real and synthetic dat... |

150 |
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Citation Context ...tween blog posts to trace the flow of information [23]. As the use of hyperlinks to refer to the source of information is relatively rare (especially in mainstream media), we also use the MemeTracker =-=[18]-=- methodology to extract more than 343 million short textual phrases (like, “Joe, the plumber” or “lipstick on a pig”). Out of these, 8 million distinct phrases appeared more than 10 times, with the cu... |

123 | Statistical properties of community structure in large social and information networks
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Citation Context ... with the online upper bound from Theorem 4. In practice NETINF finds networks that are at 97% of NP-hard to compute optimal. structure [5] ([0.962, 0.107; 0.107, 0.962]) and a core-periphery network =-=[22]-=- ([0.962, 0.535; 0.535, 0.107]). Notice that Forest Fire generates a scale free network [4]. We then simulate cascades on G ∗ using the generative model defined in Section 2.1 that is parameterized by... |

122 |
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Citation Context ...hical community Figure 4: Score achieved by NETINF in comparison with the online upper bound from Theorem 4. In practice NETINF finds networks that are at 97% of NP-hard to compute optimal. structure =-=[5]-=- ([0.962, 0.107; 0.107, 0.962]) and a core-periphery network [22] ([0.962, 0.535; 0.535, 0.107]). Notice that Forest Fire generates a scale free network [4]. We then simulate cascades on G ∗ using the... |

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Citation Context ...) is NP-hard, so we cannot expect to find the optimal solution: THEOREM 2. The diffusion network inference problem defined by equation (6) is NP-hard. PROOF. By reduction from the MAX-k-COVER problem =-=[14]-=-. In MAX-k-COVER, we are given a finite set W , |W | = n and a collection of subsets S1, . . . , Sm ⊆ W . The function FMC(A) = | ∪i∈A Si| counts the number of elements of W covered by sets indexed by... |

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Citation Context ...and a power-law model, each with parameter α: P(∆) ∝ e −∆ α and P(∆) ∝ 1 . ∆α We consider both the power-law and exponential waiting time models since they have both been argued for in the literature =-=[3, 23, 26]-=-. In the end, our algorithm does not depend on the particular choice of the waiting time distribution and more complicated functions can easily be chosen [6]. Also, we interpret ∞ + ∆ = ∞, i.e., if tu... |

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Citation Context ...ing, Social networks. 1. INTRODUCTION Cascading behavior, diffusion and spreading of ideas, innovation, information, influence, viruses and diseases are fundamental processes taking place in networks =-=[12, 28, 30]-=-. In order to study network diffusion there are two fundamental challenges one has to address. First, to be able to track cascading processes taking place in a network, one needs to identify the conta... |

78 | Cascading Behavior in Large Blog Graphs
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Citation Context ... or the purchase. These challenges are especially pronounced in information diffusion on the Web, where there have been relatively few large scale studies of information propagation in large networks =-=[2, 23, 24, 25]-=-. In order to study paths of diffusion over networks, one essentially requires to have complete information about who influences whom, as a single missing link in a sequence of propagations can lead t... |

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Citation Context ... or the purchase. These challenges are especially pronounced in information diffusion on the Web, where there have been relatively few large scale studies of information propagation in large networks =-=[2, 23, 24, 25]-=-. In order to study paths of diffusion over networks, one essentially requires to have complete information about who influences whom, as a single missing link in a sequence of propagations can lead t... |

62 | Patterns of influence in a recommendation network
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Citation Context ... or the purchase. These challenges are especially pronounced in information diffusion on the Web, where there have been relatively few large scale studies of information propagation in large networks =-=[2, 23, 24, 25]-=-. In order to study paths of diffusion over networks, one essentially requires to have complete information about who influences whom, as a single missing link in a sequence of propagations can lead t... |

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52 | Scalable modeling of real graphs using kronecker multiplication
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Citation Context ... almost recovered G ∗ perfectly by making only two errors. Experimental setup. We consider two models of directed realworld networks, namely, the Forest Fire model [20] and the Kronecker Graphs model =-=[19]-=- to generate G ∗ . For Kronecker graphs, we consider three sets of parameters that produce networks with a very different structure: a random graph [8] (Kronecker parameter matrix [0.5, 0.5; 0.5, 0.5]... |

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Citation Context ... been argued for in the literature [3, 23, 26]. In the end, our algorithm does not depend on the particular choice of the waiting time distribution and more complicated functions can easily be chosen =-=[6]-=-. Also, we interpret ∞ + ∆ = ∞, i.e., if tu = ∞, then tv = ∞ with probability 1. Now that we specified the probability Pc(u, v) that node u influences node v, we define the probability of observing ca... |

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Citation Context ...y the most probable links. Network structure learning has been considered for estimating the dependency structure of probabilistic graphical models [9, 10] and for estimating epidemiological networks =-=[29]-=-. In both cases, the problem is formulated in a probabilistic framework. However, since the problem is intractable, heuristic greedy hill-climbing or stochastic search that offer no performance guaran... |

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Citation Context ...and a power-law model, each with parameter α: P(∆) ∝ e −∆ α and P(∆) ∝ 1 . ∆α We consider both the power-law and exponential waiting time models since they have both been argued for in the literature =-=[3, 23, 26]-=-. In the end, our algorithm does not depend on the particular choice of the waiting time distribution and more complicated functions can easily be chosen [6]. Also, we interpret ∞ + ∆ = ∞, i.e., if tu... |

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Citation Context ...lyspecifieswhichnodesinfluence whichothernodes. Last,wewilldefine P(c|G),whichistheprobabilitythat cascade c occurs inanetwork G. Cascade transmission model. We build on the independent cascade model =-=[13]-=- which posits that an infected node infects each of its neighbors independently with some chosen probability. As in this model the time is modeled only implicitly through the epochs ofthepropagationwe... |

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Citation Context ...) performed poorly while our method (c) almost recovered G ∗ perfectlybymaking only twoerrors. Experimental setup. We consider two models of directed realworld networks, namely, the Forest Fire model =-=[20]-=- and the Kronecker Graphs model [19] to generate G ∗ . For Kronecker graphs, we consider three sets of parameters that produce networks with a very different structure: a random graph [8] (Kronecker p... |

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Citation Context ...returns property of the model, we prove that NETINF infersnear-optimalnetworks. Wealsospeed-up NETINFalgorithm by exploiting the local structure of the objective function and by using lazyevaluations =-=[21]-=-. Our results on synthetic datasets show that we can reliably infer the underlying propagation and influence networks, regardless of the overall network structure. Validation on real and synthetic dat... |

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Citation Context ...tic. Considering the hardness of the problem, we might expect the greedy algorithm to perform arbitrarily bad. However, we will see that this is not the case. A fundamental result of Nemhauser et al. =-=[27]-=- proves that for monotonic submodular functions, the set ˆ G returnedbythegreedyalgorithmobtainsatleastaconstantfraction of (1−1/e) ≈ 63%oftheoptimalvalueachievable using k edges. Moreover, we can acq... |

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Citation Context ...time ∆, an exponential andapower-law model, each withparameter α: P(∆) ∝ e −∆ α and P(∆) ∝ 1 . ∆α Weconsiderboththepower-lawandexponentialwaitingtimemodelssincetheyhavebothbeenarguedforintheliterature=-=[3,23,26]-=-. In the end, our algorithm does not depend on the particular choice of the waiting time distribution and more complicated functions can easily be chosen [6]. Also, we interpret ∞ + ∆ = ∞, i.e., if tu... |

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Citation Context ... withhierarchical community Figure 4: Score achieved by NETINF in comparison with the onlineupperboundfromTheorem4. Inpractice NETINFfinds networks thatare at 97% of NP-hardtocomputeoptimal. structure=-=[5]-=-([0.962, 0.107; 0.107, 0.962])andacore-peripherynetwork [22] ([0.962, 0.535; 0.535, 0.107]). Notice that Forest Fire generates a scale free network[4]. Wethensimulatecascadeson G ∗ usingthegenerativem... |

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Citation Context ...hyperplane in case of SVMs) and predicts individually the most probable links. Network structure learning has been considered for estimating the dependency structure of probabilistic graphical models =-=[9, 10]-=- and for estimating epidemiological networks [29]. In both cases, the problem is formulated in a probabilistic framework. However, since the problem is intractable, heuristic greedy hill-climbing or s... |

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Citation Context ...ne choosing a more complextransmission model. 5. RELATED WORK There are several lines of work we build upon. Although the information diffusion in on-line settings has received considerable attention =-=[2, 11, 16, 17, 23, 24, 25]-=-, only a few studies were able tostudy the actual shapes of cascades [23, 25]. The problem of inferringlinksofdiffusionwasfirststudiedbyAdarandAdamic[2], who formulated it as a supervised classificati... |

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Citation Context ...ne choosing a more complextransmission model. 5. RELATED WORK There are several lines of work we build upon. Although the information diffusion in on-line settings has received considerable attention =-=[2, 11, 16, 17, 23, 24, 25]-=-, only a few studies were able tostudy the actual shapes of cascades [23, 25]. The problem of inferringlinksofdiffusionwasfirststudiedbyAdarandAdamic[2], who formulated it as a supervised classificati... |

1 |
Meme-tracking and the dynamics of thenews cycle
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Citation Context ...tween blog posts to trace the flow of information [23]. As the use of hyperlinks to refer to the source of information is relatively rare (especially in mainstream media), we also use the MemeTracker =-=[18]-=- methodology to extract more than 343 million short textual phrases (like, “Joe, the plumber” or “lipstick on a pig”). Out of these, 8 million distinct phrases appeared more than 10 times, with the cu... |

1 |
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Citation Context ...(c) almost recovered G ∗ perfectlybymaking only twoerrors. Experimental setup. We consider two models of directed realworld networks, namely, the Forest Fire model [20] and the Kronecker Graphs model =-=[19]-=- to generate G ∗ . For Kronecker graphs, we consider three sets of parameters that produce networks with a very different structure: a random graph [8] (Kronecker parameter matrix [0.5, 0.5; 0.5, 0.5]... |

1 |
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Citation Context ...INF in comparison with the onlineupperboundfromTheorem4. Inpractice NETINFfinds networks thatare at 97% of NP-hardtocomputeoptimal. structure[5]([0.962, 0.107; 0.107, 0.962])andacore-peripherynetwork =-=[22]-=- ([0.962, 0.535; 0.535, 0.107]). Notice that Forest Fire generates a scale free network[4]. Wethensimulatecascadeson G ∗ usingthegenerativemodeldefinedinSection2.1thatisparameterizedby α,whichcontrols... |

1 |
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Citation Context ...d the adoption or the purchase. Thesechallengesareespeciallypronouncedininformationdiffusion on the Web, where there have been relatively few large scale studiesofinformationpropagationinlargenetworks=-=[2,23,24,25]-=-. In order to study paths of diffusion over networks, one essentially requirestohavecompleteinformationaboutwhoinfluenceswhom, as a single missing link in a sequence of propagations can lead to wrong ... |

1 |
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Citation Context ...d the adoption or the purchase. Thesechallengesareespeciallypronouncedininformationdiffusion on the Web, where there have been relatively few large scale studiesofinformationpropagationinlargenetworks=-=[2,23,24,25]-=-. In order to study paths of diffusion over networks, one essentially requirestohavecompleteinformationaboutwhoinfluenceswhom, as a single missing link in a sequence of propagations can lead to wrong ... |

1 |
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Citation Context ...d the adoption or the purchase. Thesechallengesareespeciallypronouncedininformationdiffusion on the Web, where there have been relatively few large scale studiesofinformationpropagationinlargenetworks=-=[2,23,24,25]-=-. In order to study paths of diffusion over networks, one essentially requirestohavecompleteinformationaboutwhoinfluenceswhom, as a single missing link in a sequence of propagations can lead to wrong ... |

1 |
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Citation Context ...time ∆, an exponential andapower-law model, each withparameter α: P(∆) ∝ e −∆ α and P(∆) ∝ 1 . ∆α Weconsiderboththepower-lawandexponentialwaitingtimemodelssincetheyhavebothbeenarguedforintheliterature=-=[3,23,26]-=-. In the end, our algorithm does not depend on the particular choice of the waiting time distribution and more complicated functions can easily be chosen [6]. Also, we interpret ∞ + ∆ = ∞, i.e., if tu... |