## On the Convexity of Latent Social Network Inference

Citations: | 28 - 2 self |

### BibTeX

@MISC{Myers_onthe,

author = {Seth A. Myers and Jure Leskovec},

title = {On the Convexity of Latent Social Network Inference},

year = {}

}

### OpenURL

### Abstract

In many real-world scenarios, it is nearly impossible to collect explicit social network data. In such cases, whole networks must be inferred from underlying observations. Here, we formulate the problem of inferring latent social networks based on network diffusion or disease propagation data. We consider contagions propagating over the edges of an unobserved social network, where we only observe the times when nodes became infected, but not who infected them. Given such node infection times, we then identify the optimal network that best explains the observed data. We present a maximum likelihood approach based on convex programming with a l1-like penalty term that encourages sparsity. Experiments on real and synthetic data reveal that our method near-perfectly recovers the underlying network structure as well as the parameters of the contagion propagation model. Moreover, our approach scales well as it can infer optimal networks on thousand nodes in a matter of minutes. 1

### Citations

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Citation Context ...etic data Each of the synthetic data experiments begins with the construction of the network. We ran our algorithm on a directed scale-free network constructed using the preferential attachment model =-=[3]-=-, and also on a Erdős-Rényi random graph. Both networks have 512 nodes and 1024 edges. In each case, the networks were constructed as unweighted graphs, and then each edge(i,j) was assigned a uniforml... |

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Citation Context ...cess. Our work here is similar in a sense that we “regress” the infection times of a target node on infection times of other nodes. Additionally, our work is also related to a link prediction problem =-=[12, 23, 18, 24]-=- but different in a sense that this line of work assumes that part of the network is already visible to us. The work most closely related to ours, however, is [10], which also infers networks through ... |

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Citation Context ...irected graphical models [7] and probabilistic relational models [7]. However, these formulations are often intractable and one has to reside to heuristic solutions. Recently, graphical Lasso methods =-=[25, 21, 6, 19]-=- for static sparse graph estimation and extensions to time evolving graphical models [1, 8, 22] have been proposed with lots of success. Our work here is similar in a sense that we “regress” the infec... |

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Citation Context ...uded in the network if u and v interacted more than τ times in the dataset. Similarly, inferring networks of interactions between proteins in a cell usually reduces to determining the right threshold =-=[9, 20]-=-. We address the problem of inferring the structure of unobserved social networks in a much more ambitious setting. We consider a diffusion process where a contagion (e.g., disease, information, produ... |

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Citation Context ... the set of variables. This dramatically reduces the number of variables as in practice the true A does not induce large cascades, causing the cascades to be sparse in the number of nodes they infect =-=[14, 17]-=-. Towards the convex problem. The Hessian of the log-likelihood/likelihood functions are indefinite in general, and this could make finding the globally optimal MLE forAdifficult. Here, we derive a co... |

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Citation Context ...ation diffusion and epidemic models, like the independent contagion model, the Susceptible–Infected (SI), Susceptible–Infected–Susceptible (SIS) or even the Susceptible–Infected–Recovered (SIR) model =-=[2]-=-. We show that calculating the maximum likelihood estimator (MLE) of the latent network (under any of the above diffusion models) is equivalent to a convex problem that can be efficiently solved. Prob... |

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Citation Context ...irected graphical models [7] and probabilistic relational models [7]. However, these formulations are often intractable and one has to reside to heuristic solutions. Recently, graphical Lasso methods =-=[25, 21, 6, 19]-=- for static sparse graph estimation and extensions to time evolving graphical models [1, 8, 22] have been proposed with lots of success. Our work here is similar in a sense that we “regress” the infec... |

189 |
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Citation Context ...to deciding whether to include the interaction between a pair of nodes as an edge in the underlying network. For example, inferring networks from pairwise interactions of cell-phone call [5] or email =-=[4, 13]-=- records simply reduces down to selecting the right threshold τ such that an edge (u,v) is included in the network if u and v interacted more than τ times in the dataset. Similarly, inferring networks... |

171 |
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Citation Context ...icular nodes get “infected” but we do not observe who infected them. In case of information propagation, as bloggers discover new information, they write about it without explicitly citing the source =-=[15]-=-. Thus, we only observe the time when a blog gets “infected” but not where it got infected from. Similarly, in disease spreading, we observe people getting sick without usually knowing who infected th... |

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Citation Context ...e often intractable and one has to reside to heuristic solutions. Recently, graphical Lasso methods [25, 21, 6, 19] for static sparse graph estimation and extensions to time evolving graphical models =-=[1, 8, 22]-=- have been proposed with lots of success. Our work here is similar in a sense that we “regress” the infection times of a target node on infection times of other nodes. Additionally, our work is also r... |

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Citation Context ...Further related work. There are several different lines of work connected to our research. First is the network structure learning for estimating the dependency structure of directed graphical models =-=[7]-=- and probabilistic relational models [7]. However, these formulations are often intractable and one has to reside to heuristic solutions. Recently, graphical Lasso methods [25, 21, 6, 19] for static s... |

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Citation Context ...onal social network studies typically involved a very limited number of people (usually less than 100). The emergence of large scale social computing applications has made massive social network data =-=[16]-=- available, but there are important settings where network data is hard to obtain and thus the whole network must thus be inferred from the data. For example, populations, like drug injection users or... |

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2009) Inferring friendship network structure by using mobile phone data
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Citation Context ...works have to be inferred from the observational data. Even though inferring social networks has been attempted in the past, it usually assumes that the pairwise interaction data is already available =-=[5]-=-. In this case, the problem of network inference reduces to deciding whether to include the interaction between a pair of nodes as an edge in the underlying network. For example, inferring networks fr... |

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Citation Context ...irected graphical models [7] and probabilistic relational models [7]. However, these formulations are often intractable and one has to reside to heuristic solutions. Recently, graphical Lasso methods =-=[25, 21, 6, 19]-=- for static sparse graph estimation and extensions to time evolving graphical models [1, 8, 22] have been proposed with lots of success. Our work here is similar in a sense that we “regress” the infec... |

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Citation Context ...]. And, in a viral marketing setting, we observe people purchasing products or adopting particular behaviors without explicitly knowing who was the influencer that caused the adoption or the purchase =-=[11]-=-. Thus, the question is, if we assume that the network is static over time, is it possible to reconstruct the unobserved social network over which diffusions took place? What is the structure of such ... |

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Citation Context ... the set of variables. This dramatically reduces the number of variables as in practice the true A does not induce large cascades, causing the cascades to be sparse in the number of nodes they infect =-=[14, 17]-=-. Towards the convex problem. The Hessian of the log-likelihood/likelihood functions are indefinite in general, and this could make finding the globally optimal MLE forAdifficult. Here, we derive a co... |

61 | Inferring networks of diffusion and influence
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Citation Context ... link prediction problem [12, 23, 18, 24] but different in a sense that this line of work assumes that part of the network is already visible to us. The work most closely related to ours, however, is =-=[10]-=-, which also infers networks through cascade data. The algorithm proposed (called NetInf) assumes that the weights of the edges in latent network are homogeneous, i.e. all connected nodes in the netwo... |

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Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures
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Citation Context ...hus, we only observe the time when a blog gets “infected” but not where it got infected from. Similarly, in disease spreading, we observe people getting sick without usually knowing who infected them =-=[26]-=-. And, in a viral marketing setting, we observe people purchasing products or adopting particular behaviors without explicitly knowing who was the influencer that caused the adoption or the purchase [... |

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Citation Context ...uded in the network if u and v interacted more than τ times in the dataset. Similarly, inferring networks of interactions between proteins in a cell usually reduces to determining the right threshold =-=[9, 20]-=-. We address the problem of inferring the structure of unobserved social networks in a much more ambitious setting. We consider a diffusion process where a contagion (e.g., disease, information, produ... |

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Citation Context ...e often intractable and one has to reside to heuristic solutions. Recently, graphical Lasso methods [25, 21, 6, 19] for static sparse graph estimation and extensions to time evolving graphical models =-=[1, 8, 22]-=- have been proposed with lots of success. Our work here is similar in a sense that we “regress” the infection times of a target node on infection times of other nodes. Additionally, our work is also r... |

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Citation Context ...cess. Our work here is similar in a sense that we “regress” the infection times of a target node on infection times of other nodes. Additionally, our work is also related to a link prediction problem =-=[12, 23, 18, 24]-=- but different in a sense that this line of work assumes that part of the network is already visible to us. The work most closely related to ours, however, is [10], which also infers networks through ... |

17 |
Inferring relevant social networks from interpersonal communication
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Citation Context ...to deciding whether to include the interaction between a pair of nodes as an edge in the underlying network. For example, inferring networks from pairwise interactions of cell-phone call [5] or email =-=[4, 13]-=- records simply reduces down to selecting the right threshold τ such that an edge (u,v) is included in the network if u and v interacted more than τ times in the dataset. Similarly, inferring networks... |

4 | Timevarying dynamic bayesian networks
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Citation Context ...e often intractable and one has to reside to heuristic solutions. Recently, graphical Lasso methods [25, 21, 6, 19] for static sparse graph estimation and extensions to time evolving graphical models =-=[1, 8, 22]-=- have been proposed with lots of success. Our work here is similar in a sense that we “regress” the infection times of a target node on infection times of other nodes. Additionally, our work is also r... |