nips nips2012 nips2012-182 knowledge-graph by maker-knowledge-mining

182 nips-2012-Learning Networks of Heterogeneous Influence

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Author: Nan Du, Le Song, Ming Yuan, Alex J. Smola

Abstract: Information, disease, and inﬂuence diffuse over networks of entities in both natural systems and human society. Analyzing these transmission networks plays an important role in understanding the diffusion processes and predicting future events. However, the underlying transmission networks are often hidden and incomplete, and we observe only the time stamps when cascades of events happen. In this paper, we address the challenging problem of uncovering the hidden network only from the cascades. The structure discovery problem is complicated by the fact that the inﬂuence between networked entities is heterogeneous, which can not be described by a simple parametric model. Therefore, we propose a kernelbased method which can capture a diverse range of different types of inﬂuence without any prior assumption. In both synthetic and real cascade data, we show that our model can better recover the underlying diffusion network and drastically improve the estimation of the transmission functions among networked entities. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Information, disease, and inﬂuence diffuse over networks of entities in both natural systems and human society. [sent-7, score-0.108]

2 Analyzing these transmission networks plays an important role in understanding the diffusion processes and predicting future events. [sent-8, score-0.585]

3 However, the underlying transmission networks are often hidden and incomplete, and we observe only the time stamps when cascades of events happen. [sent-9, score-0.772]

4 In this paper, we address the challenging problem of uncovering the hidden network only from the cascades. [sent-10, score-0.141]

5 The structure discovery problem is complicated by the fact that the inﬂuence between networked entities is heterogeneous, which can not be described by a simple parametric model. [sent-11, score-0.242]

6 In both synthetic and real cascade data, we show that our model can better recover the underlying diffusion network and drastically improve the estimation of the transmission functions among networked entities. [sent-13, score-0.886]

7 1 Introduction Networks have been powerful abstractions for modeling a variety of natural and artiﬁcial systems that consist of a large collection of interacting entities. [sent-14, score-0.027]

8 Due to the recent increasing availability of large-scale networks, network modeling and analysis have been extensively applied to study the spreading and diffusion of information, ideas, and even virus in social and information networks (see e. [sent-15, score-0.512]

9 However, the process of inﬂuence and diffusion often occurs in a hidden network that might not be easily observed and identiﬁed directly. [sent-18, score-0.491]

10 For instance, when a disease spreads among people, epidemiologists can know only when a person gets sick, but they can hardly ever know where and from whom he (she) gets infected. [sent-19, score-0.179]

11 Similarly, when consumers rush to buy some particular products, marketers can know when purchases occurred, but they cannot track in further where the recommendations originally came from [12]. [sent-20, score-0.081]

12 In all such cases, we could observe only the time stamp when a piece of information has been received by a particular entity, but the exact path of diffusion is missing. [sent-21, score-0.479]

13 Therefore, it is an interesting and challenging question whether we can uncover the diffusion paths based just on the time stamps of the events. [sent-22, score-0.473]

14 However, in these models, time is treated as discrete index and not modeled as a random variable. [sent-26, score-0.063]

15 In the diffusion network discovery problem, time is treated explicitly as a continuous variable, and one is interested in capturing how the occurrence of event at one node affects the time for its occurence at other nodes. [sent-27, score-0.928]

16 Speciﬁcally, Meyers and Leskovec inferred the diffusion network by learning the infection probability between two nodes using a convex programming, called CONNIE [14]. [sent-29, score-0.541]

17 inferred the network connectivity using a submodular optimization, called NETINF [4]. [sent-31, score-0.129]

18 However, both CONNIE and NETINF assume that the transmission model for each pair of nodes is ﬁxed with predeﬁned transmission rate. [sent-32, score-0.567]

19 proposed an elegant method, called NETRATE [3], using continuous temporal dynamics model to allow variable diffusion rates across network edges. [sent-34, score-0.386]

20 1 histogram exp rayleigh KernelCascade pdf pdf 0. [sent-37, score-0.285]

21 02 0 0 (a) Pair 1 histogram exp rayleigh KernelCascade 0. [sent-47, score-0.167]

22 3 20 t(hours) 40 0 0 50 t(hours) (b) Pair 2 100 (c) Pair 3 Figure 1: The histograms of the interval between the time when a post appeared in one site and the time when a new post in another site links to it. [sent-49, score-0.47]

23 Dotted and dash lines are density ﬁtted by NETRATE. [sent-50, score-0.077]

24 ﬁxed parametric form, such as exponential, power-law, or Rayleigh distribution, although the model parameters learned from cascades could be different. [sent-52, score-0.246]

25 In practice, the patterns of information diffusion (or a spreading disease) among entities can be quite complicated and different from each other, going far beyond what a single family of parametric models can capture. [sent-53, score-0.562]

26 For example, in twitter, an active user can be online for more than 12 hours a day, and he may instantly respond to any interesting message. [sent-54, score-0.128]

27 However, an inactive user may just log in and respond once a day. [sent-55, score-0.115]

28 As a result, the spreading pattern of the messages between the active user and his friends can be quite different from that of the inactive user. [sent-56, score-0.157]

29 Another example is from the information diffusion in a blogsphere: the hyperlinks between posts can be viewed as some kind of information ﬂow from one media site to another, and the time difference between two linked posts reveal the pattern of diffusion. [sent-57, score-0.641]

30 In Figure 1, we examined three pairs of media sites from the MemeTracker dataset [3, 9], and plotted the histograms of the intervals between the the moment when a post ﬁrst appeared in one site and the moment when it was linked by a new post in another site. [sent-58, score-0.471]

31 We can observe that information can have very different transmission patterns for these pairs. [sent-59, score-0.284]

32 Parametric models ﬁtted by NETRATE may capture the simple pattern in Figure 1(a), but they might miss the multimodal patterns in Figure 1(b) and Figure 1(c). [sent-60, score-0.043]

33 Our key idea is to model the continuous information diffusion process using survival analysis by kernelizing the hazard function. [sent-63, score-0.956]

34 We obtain a convex optimization problem with grouped lasso type of regularization and develop a fast block-coordinate descent algorithm for solving the problem. [sent-64, score-0.029]

35 The sparsity patterns of the coefﬁcients provide us the structure of the diffusion network. [sent-65, score-0.341]

36 In both synthetic and real world data, our method can better recover the underlying diffusion networks and drastically improve the estimation of the transmission functions among networked entities. [sent-66, score-0.722]

37 2 Preliminary In this section, we will present some basic concepts from survival analysis [7, 8], which are essential for our later modeling. [sent-67, score-0.309]

38 Given a nonnegative random variable T corresponding to the time when an t event happens, let f (t) be the probability density function of T and F (t) = P r(T ≤ t) = 0 f (x)dx be its cumulative distribution function. [sent-68, score-0.219]

39 The probability that an event does not happen up to time t is thus given by the survival function S(t) = P r(T ≥ t) = 1 − F (t). [sent-69, score-0.562]

40 The survival function is a continuous and monotonically decreasing function with S(0) = 1 and S(∞) = limt→∞ S(t) = 0. [sent-70, score-0.309]

41 Given f (t) and S(t), we can deﬁne the instantaneous risk (or rate) that an event has not happened yet up to time t but happens at time t by the hazard function P r(t ≤ T ≤ t + ∆t|T ≥ t) f (t) h(t) = lim = . [sent-71, score-0.688]

42 (1) ∆t→0 ∆t S(t) With this deﬁnition, h(t)∆t will be the approximate probability that an event happens in [t, t + ∆t) given that the event has not happened yet up to t. [sent-72, score-0.497]

43 Furthermore, the hazard function h(t) is also d related to the survival function S(t) via the differential equation h(t) = − dt log S(t), where we have used f (t) = −S (t). [sent-73, score-0.655]

44 Solving the differential equation with boundary condition S(0) = 1, we can recover the survival function S(t) and the density function f (t) based on the hazard function h(t), i. [sent-74, score-0.655]

45 , t S(t) = exp − t h(x) dx f (t) = h(t) exp − and 0 h(x) dx . [sent-76, score-0.07]

46 0 2 (2) a b c t0 t1 a b t2 t0 t3 c t1 a b t3 c t2 t4 d e d e d e (a) Hidden network (b) Node e gets infected at time t4 (c) Node e survives Figure 2: Cascades over a hidden network. [sent-77, score-0.453]

47 Solid lines in panel(a) represent connections in a hidden network. [sent-78, score-0.08]

48 In panel (b) and (c), ﬁlled circles indicate infected nodes while empty circles represent uninfected ones. [sent-79, score-0.422]

49 Node a, b, c and d are the parents of node e which got infected at t0 < t1 < t2 < t3 respectively and tended to infect node e. [sent-80, score-0.781]

50 In panel (b), node e survives given node a, b and c shown in green dash lines. [sent-81, score-0.65]

51 In panel (c), node e survives even though all its parents got infected. [sent-83, score-0.488]

52 3 Modeling Cascades using Survival Analysis We use survival analysis to model information diffusion for networked entities. [sent-84, score-0.708]

53 We assume that there is a ﬁxed population of N nodes connected in a directed network G = (V, E). [sent-87, score-0.231]

54 Neighboring nodes are allowed to directly inﬂuence each other. [sent-88, score-0.085]

55 Nodes along a directed path may inﬂuence each other only through a diffusion process. [sent-89, score-0.356]

56 Because the true underlying network is unknown, our observations are only the time stamps when events occur to each node in the network. [sent-90, score-0.554]

57 The time stamps are then organized as cascades, each of which corresponds to a particular event. [sent-91, score-0.175]

58 For instance, a piece of news posted on CNN website about “Facebook went public” can be treated as an event. [sent-92, score-0.16]

59 It can spread across the blogsphere and trigger a sequence of posts from other sites referring to it. [sent-93, score-0.175]

60 Each site will have a time stamp when this particular piece of news is being discussed and cited. [sent-94, score-0.322]

61 The goal of the model is to capture the interplay between the hidden diffusion network and the cascades of observed event time stamps. [sent-95, score-0.857]

62 More formally, a directed edge, j → i, is associated with an transmission function fji (ti |tj ), which is the conditional likelihood of an event happening to node i at time ti given that the same event has already happened to node j at time tj . [sent-96, score-1.732]

63 The transmission function attempts to capture the temporal dependency between the two successive events for node i and j. [sent-97, score-0.532]

64 In addition, we focus on shiftinvariant transmission functions whose value only depends on the time difference, i. [sent-98, score-0.3]

65 , fji (ti |tj ) = fji (ti − tj ) = fji (∆ji ) where ∆ji := ti − tj . [sent-100, score-0.884]

66 Given the likelihood function, we can compute the corresponding survival function Sji (∆ji ) and hazard function hji (∆ji ). [sent-101, score-0.73]

67 When there is no directed edge j → i, the transmission function and hazard function are both identically zeros, i. [sent-102, score-0.646]

68 , fji (∆ji ) = 0 and hji (∆ji ) = 0, but the survival function is identically one, i. [sent-104, score-0.592]

69 Therefore, the structure of the diffusion network is reﬂected in the non-zero patterns of a collection of transmission functions (or hazard functions). [sent-107, score-0.99]

70 A cascade is an N -dimensional vector tc := (tc , . [sent-108, score-0.374]

71 , tc ) with i-th dimension recording the time 1 N stamp when event c occurs to node i. [sent-111, score-0.843]

72 Furthermore, tc ∈ [0, T c ] ∪ {∞}, and the symbol ∞ labels i nodes that have not been inﬂuenced during observation window [0, T c ] — it does not imply that nodes are never inﬂuenced. [sent-112, score-0.422]

73 A dataset can contain a collection, C, of cascades t1 , . [sent-114, score-0.199]

74 The time stamps assigned to nodes by a cascade induce a directed acyclic graph (DAG) by deﬁning node j as the parent of i if tj < ti . [sent-118, score-0.944]

75 Thus, it is meaningful to refer to parents and children within a cascade [3], which is different from the parent-child structural relation on the true underlying diffusion network. [sent-119, score-0.493]

76 Since the true network is inferred from many cascades (each of which imposes its own DAG structure), the inferred network is typically not a DAG. [sent-120, score-0.457]

77 The likelihood (tc ) of a cascade induced by event c is then simply a product of all individual likelihood i (tc ) that event c occurs to each node i. [sent-121, score-0.871]

78 Depending on whether event c actually occurs to node i in the data, we can compute this individual likelihood as: Event c did occur at node i. [sent-122, score-0.75]

79 We assume that once an event occurs at node i under the inﬂuence of a particular parent j in a cascade, the same event will not happen again. [sent-123, score-0.783]

80 In Figure 2(b), node e is susceptible given its parent a, b, c and d. [sent-124, score-0.319]

81 However, only node d is the ﬁrst parent who infects node e. [sent-125, score-0.552]

82 Because each parent could be equally likely to ﬁrst inﬂuence node i, the likelihood is just a simple sum over the likelihoods of the mutually disjoint events that node i has survived from the inﬂuence of all the other parents except the ﬁrst parent j, i. [sent-126, score-0.873]

83 , + c i (t ) Ski (∆c ) = ki fji (∆c ) ji = j:tc T c i + c i (t ) . [sent-128, score-0.345]

84 (5) tc ≤T c i uninfected nodes infected nodes Therefore, the likelihood of all cascades is a product of the these individual cascade likelihoods, i. [sent-129, score-1.017]

85 In the end, we take the negative log of this likelihood function and regroup all terms associated with edges pointing to node i together to derive   L({t1 , . [sent-138, score-0.276]

86 , t|C| }) = − log S(∆c ) + ji   i j  h(∆c ) ji log (6) } } { { } { There are two interesting implications from this negative log likelihood function. [sent-141, score-0.337]

87 First, the function can be expressed using only the hazard and the survival function. [sent-142, score-0.629]

88 Second, the function is decomposed into additive contribution from each node i. [sent-143, score-0.233]

89 We can therefore estimate the hazard and survival function for each node separately. [sent-144, score-0.862]

90 [3] used parametric hazard and survival functions, and they estimated the model parameters using the convex programming. [sent-146, score-0.676]

91 In contrast, we will instead formulate an algorithm using kernels and grouped parameter regularization, which allows us to estimate complicated hazard and survival functions without overﬁtting. [sent-147, score-0.69]

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