nips nips2012 nips2012-219 knowledge-graph by maker-knowledge-mining

219 nips-2012-Modelling Reciprocating Relationships with Hawkes Processes

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Author: Charles Blundell, Jeff Beck, Katherine A. Heller

Abstract: We present a Bayesian nonparametric model that discovers implicit social structure from interaction time-series data. Social groups are often formed implicitly, through actions among members of groups. Yet many models of social networks use explicitly declared relationships to infer social structure. We consider a particular class of Hawkes processes, a doubly stochastic point process, that is able to model reciprocity between groups of individuals. We then extend the Inﬁnite Relational Model by using these reciprocating Hawkes processes to parameterise its edges, making events associated with edges co-dependent through time. Our model outperforms general, unstructured Hawkes processes as well as structured Poisson process-based models at predicting verbal and email turn-taking, and military conﬂicts among nations. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We present a Bayesian nonparametric model that discovers implicit social structure from interaction time-series data. [sent-11, score-0.156]

2 Social groups are often formed implicitly, through actions among members of groups. [sent-12, score-0.098]

3 Yet many models of social networks use explicitly declared relationships to infer social structure. [sent-13, score-0.303]

4 We consider a particular class of Hawkes processes, a doubly stochastic point process, that is able to model reciprocity between groups of individuals. [sent-14, score-0.124]

5 We then extend the Inﬁnite Relational Model by using these reciprocating Hawkes processes to parameterise its edges, making events associated with edges co-dependent through time. [sent-15, score-0.426]

6 Our model outperforms general, unstructured Hawkes processes as well as structured Poisson process-based models at predicting verbal and email turn-taking, and military conﬂicts among nations. [sent-16, score-0.301]

7 1 Introduction As social animals, people constantly organise themselves into social groups. [sent-17, score-0.256]

8 These social groups can revolve around particular activities, such as sports teams, particular roles, such as store managers, or general social alliances, like gang members. [sent-18, score-0.271]

9 Understanding the dynamics of group interactions is a difﬁcult problem that social scientists strive to address. [sent-19, score-0.168]

10 One basic problem in understanding group behaviour is that groups are often not explicitly deﬁned, and the members must be inferred. [sent-20, score-0.12]

11 How can we predict future interactions among individuals based on these inferred groups? [sent-22, score-0.197]

12 A common approach is to infer groups, or clusters, of people based upon a declared relationship between pairs of individuals [1, 2, 3, 4]. [sent-23, score-0.237]

13 For example, data from social networks, where two people declare that they are “friends” or in each others’ social “neighbourhood”, can potentially be used. [sent-24, score-0.256]

14 However these declared relationships are not necessarily readily available, truthful, or pertinent to inferring the social group structure of interest. [sent-25, score-0.195]

15 In this paper we instead propose an approach to inferring social groups based directly on a set of real interactions between people. [sent-26, score-0.199]

16 If we are interested in capturing groups that best reﬂect human behaviour we should be determining the groups from instances of that same behaviour. [sent-28, score-0.131]

17 We develop a model which can learn social group structure based on interactions data. [sent-29, score-0.168]

18 As examples, the actions we consider are that of one person sending an email to another, one person speaking to another, or one country engaging in military action towards another. [sent-31, score-0.21]

19 The key property that we leverage to infer social groups is reciprocity. [sent-32, score-0.183]

20 Reciprocity is a common social norm, where one person’s actions towards another increases the probability of the same type of action being returned. [sent-33, score-0.109]

21 For example, if Bob emails Alice, it increases the probability that Alice will email Bob in the near future. [sent-34, score-0.093]

22 When multiple people show a similar pattern of reciprocity, our model will place these people in their own group. [sent-36, score-0.076]

23 The Bayesian nonparametric model we use on these time-series data is generative and accounts for the rate of events between clusters of individuals. [sent-37, score-0.334]

24 It is built upon mutually-exciting point processes, known as Hawkes processes [5, 6]. [sent-38, score-0.113]

25 Pairs of mutually-exciting Hawkes processes are able to capture the causal nature of reciprocal interactions. [sent-39, score-0.169]

26 Here the processes excite one another through their actualised events. [sent-40, score-0.113]

27 Since Poisson processes are a special case of Hawkes processes, our model is also able to capture simpler one-way, non-reciprocal, relationships as well. [sent-41, score-0.134]

28 The IRM typically assumes that there is a ﬁxed graph, or social network, which is observed. [sent-43, score-0.109]

29 Here we are interested in inferring the implicit social structure based only on the occurrences of interactions between vertices in the graph. [sent-44, score-0.167]

30 We apply our model to reciprocal behaviour in verbal and email conversations and to military conﬂicts among nations. [sent-45, score-0.27]

31 The remainder of the paper is organised as follows: section 2 discusses using Poisson processes together with the IRM. [sent-46, score-0.113]

32 2 Poisson processes with the Inﬁnite Relational Model The Inﬁnite Relational Model (IRM) [1, 2] was developed to model relationships among entities as graphs, based upon previously declared relationships. [sent-49, score-0.202]

33 Let V denote the vertices of the graph, corresponding to individuals, and let euv denote the presence or absence of a relationship between vertices u and v, corresponding to an edge in the graph. [sent-50, score-0.091]

34 Hence vertex u belongs to the cluster given by π(u), and consequently, the clusters in π are given by range(π). [sent-52, score-0.125]

35 Often in interaction data there are many instances of interactions between the same pair of individuals–this cannot be modelled by the IRM. [sent-54, score-0.117]

36 Unfortunately, a vanilla Gamma-Poisson observation model does not allow us to predict events into the future, outside the observed time window. [sent-56, score-0.256]

37 We shall consider Poisson processes on [0, ∞), such that the number of events in any interval [s, s ) of the real-half line, denoted N [s, s ), is Poisson distributed with rate λ(s − s). [sent-60, score-0.397]

38 The graph in the top left shows the clusters and edge weights learned by our model from the data in the bottom right plot. [sent-68, score-0.097]

39 The top right plot shows the rates of interaction events between clusters. [sent-69, score-0.303]

40 In the graph, the width and temperature (how red the colour is) denotes the expected rate of events between pairs of clusters (using equations (9) and (10)). [sent-71, score-0.356]

41 While in plots on the right, line colours indicates the identity of cluster pairs, and box colours indicate the originator of the event: Alice (red), Bob (blue), Mallory (black). [sent-72, score-0.121]

42 Only after many events caused by Mallory do Alice or Bob respond, and when they do respond they both, similarly, respond more sparsely. [sent-75, score-0.306]

43 Inference proceeds by conditioning on, Nuv [0, T ) = nuv where nuv is the total number of events directed from u to v in the given time interval. [sent-77, score-0.568]

44 There are two notable deﬁciencies of this model: the rate of events on each edge is independent of every other edge, and conditioned on the time interval containing all observed events, the times of these events are uniformly distributed. [sent-79, score-0.564]

45 If I send an email to someone, it is more likely that I will receive an email from them than had I not sent an email, and the probability of receiving a reply decreases as time advances. [sent-81, score-0.186]

46 These processes are intuitively similar to Poisson processes, but unlike Poisson processes, the rates of Hawkes processes depend upon their own historic events and those of other processes in an excitatory fashion. [sent-84, score-0.595]

47 We shall consider an array of K × K Hawkes processes, where K is the number of clusters in a partition drawn from a CRP restricted to the individuals V . [sent-85, score-0.185]

48 As in the IRM, the CRP allows the 3 number of processes to grow in an unconstrained manner as the number of individuals in the graph grows. [sent-86, score-0.271]

49 However, unlike the IRM, these Hawkes processes will be pairwise-dependent: the Hawkes process governing events from cluster p to cluster q, will depend upon the Hawkes process governing events from cluster q to cluster p. [sent-87, score-1.041]

50 Each Hawkes process is a point process whose rate at time t is given by: t gpq (t − s)dNqp (s) λpq (t) = γpq np nq + (7) −∞ where γpq is the base rate of the counting measure of the Hawkes, process, Npq . [sent-89, score-0.315]

51 np and nq are the number of individuals in cluster p and q respectively, and gpq is a non-negative function such that ∞ gpq (s)ds < 1, ensuring that Npq is stationary. [sent-90, score-0.45]

52 Nqp is the counting measure of the reciprocating 0 Hawkes process of Npq . [sent-91, score-0.124]

53 Intuitively, if Npq governs events from cluster p to cluster q, then Nqp governs events from cluster q to cluster p. [sent-92, score-0.812]

54 Equation (7) shows how the rates of events in these two processes are intimately intertwined. [sent-93, score-0.369]

55 Since Nqp is an atomic measure, whose atoms correspond to the times of events, we can express the rate of Npq given in (7), by conditioning on the events of its reciprocating processes Nqp , as: gpq (t − tqp ) i λpq (t) = γpq np nq + (8) i:tqp < 1 for the process to be stationary. [sent-94, score-0.703]

56 Henceforth we will use uniform thinning—each event in Npq (·) is assigned uniformly at random among all Nuv (·) where p = π(u) and q = π(v)—but in principle any thinning scheme may be used. [sent-97, score-0.094]

57 For a Hawkes process Npq , the rate at which no events occurs in the interval [s, s ) is: e− s s λpq (t)dt (15) Suppose we observe the times of all the events in [0, T ), {tuv }nuv for process Nuv (nuv being the i i=1 total number of events from u to v in [0, T )). [sent-98, score-0.87]

58 Suppose that individual u is in cluster p and that individual v is in cluster q. [sent-99, score-0.15]

59 Furthermore, assume there are no events before time 0. [sent-100, score-0.256]

60 The likelihood of each edge between individuals u and v is thus: n qp p({tuv }nuv |θpq , {tqp }i=1 ) = e i i=1 i − np1 q n T 0 nuv λpq (t)dt i=1 λpq (tuv ) i np nq (16) n qp where θpq = (γpq , βpq , τpq ), {tqp }i=1 are the times of the reciprocal events. [sent-101, score-0.419]

61 To infer the partition of individuals π, the concentration parameter α, and the parameters of each Hawkes process θpq = (γpq , βpq , τpq ), we use Algorithm 5 [11] adapted to the IRM and slice sampling [12] to draw samples from the posterior. [sent-106, score-0.22]

62 pq pq 6 Related work Several authors have considered modelling occurrence events [13, 14, 15] using piecewise constant rate Markov point processes for known number of event types. [sent-110, score-1.194]

63 Our work directly models interaction 5 events (where an event is structured to have a sender and recipient) and the number of possible events types is not limited. [sent-111, score-0.63]

64 [16] describes a model of occurrence events as a discrete time-series using a latent ﬁrst-order Markov model. [sent-112, score-0.256]

65 Our model differs in that it considers interaction events in continuous time and requires no ﬁrst-order assumption. [sent-113, score-0.303]

66 The model in Section 2 relates the work of [17] to the IRM [1], yielding a version of their model that learns the number of clusters whilst maintaining conjugacy. [sent-114, score-0.078]

67 However our model does not use a Poisson process to model event times, instead using processes which have a time-varying rate. [sent-115, score-0.186]

68 Hawkes processes are also the basis of this work, however our work does not use side-channel information to group individuals by imposing ﬁxed marks on the process; instead we learn structure among several co-dependent Hawkes processes and use Bayesian inference for the parameters and structure. [sent-117, score-0.408]

69 The interest is in modelling the activation and co-activation of neurons and as such they do not directly model cluster structure among the neurons, while our model does model this structure. [sent-119, score-0.121]

70 We compared these models quantitatively by comparing their log predictive densities (with respect to the space of ordered sequences of events) on events falling in the ﬁnal 10% of the total time of the data (Table 2). [sent-127, score-0.256]

71 We normalised the times of all events such that the ﬁrst 90% of total time lay in the interval [0, 1]. [sent-128, score-0.256]

72 The data involves three individuals and is plotted in Figure 1. [sent-131, score-0.135]

73 The Poisson IRM is uncertain how to cluster individuals as it cannot model the temporal dependence between individuals, while the Hawkes IRM can and so performs better at prediction as well. [sent-133, score-0.21]

74 A single Hawkes process does not model the structure among individuals and so performs worse than the Hawkes IRM, although it is able to model dependence among events. [sent-134, score-0.222]

75 Enron email threads We took the ﬁve longest threads from the Enron 2009 data set [21]. [sent-135, score-0.171]

76 All of these threads involve two different people so there is little scope for learning much group structure in these data: either both people are in the same cluster, or they are in two separate clusters. [sent-137, score-0.137]

77 However as can be seen in Table 2 these data suggest a predictive advantage to using mutually-exciting Hawkes processes, as automatically determined by our model, instead of a single self-exciting Hawkes process and of both of these approaches over their corresponding Poisson processes model. [sent-138, score-0.15]

78 A self-exciting Hawkes process is unable to mark the sender and receiver of events as differing, whilst Poisson process-based models are unable to model the causal structure of events. [sent-139, score-0.378]

79 These con6 Gran$ Rose$ Dan$ KUW$ USA$ Pa,$ Bern$ X$ Cher$ Paul$$ Many$ Dere$ Kare$ AFG$ TAW$ IRQ$ RUS$ CHN$ Figure 2: Graphs of clusters of individuals inferred by our model. [sent-142, score-0.185]

80 Edge width and temperature (how red the colour is) denotes the expected rate of events between pairs of clusters (using equations (9) and (10); edges whose marginal rate is below 1 are not included). [sent-143, score-0.384]

81 On the left is the graph inferred on the “SB conv 26” data set. [sent-144, score-0.127]

82 versations cover a variety of social situations: questions during a university lecture (12), a book discussion group (23), a meeting among city ofﬁcials (26), a family argument/discussion (33), and a conversation at a family birthday party (49). [sent-146, score-0.237]

83 We modelled the turn-taking behaviour of these groups by taking the times of when one speaker switched to the next. [sent-147, score-0.111]

84 In Figure 2(left) we show the cluster graph found by our model for conversation 26, involving city ofﬁcials discussing a grant application. [sent-148, score-0.156]

85 Incidents vary from diplomatic threats of military force to the actual deployment of military force against another state. [sent-154, score-0.098]

86 For exposition purposes, we show the graph (in Figure 2(right)) on part of the MID data set, by restricting to events among the USA, Kuwait, Afghanistan, Taiwan, Russia, China, and Iraq. [sent-158, score-0.304]

87 Thicker and redder lines between clusters (computed from equations 9 and 10) reﬂect a higher rate of incidents directed between the countries along the edge. [sent-159, score-0.225]

88 There were three main conﬂicts involving the countries we modelled during the time period this data covers. [sent-161, score-0.109]

89 1) Revolved mostly around border disputes coming out of the Soviet war in Afghanistan, and incidents sometimes involved using former Soviet countries as proxies. [sent-163, score-0.246]

90 It is interesting to note that groups involving smaller countries were found to be more likely to initiate incidents with larger countries in a dispute (e. [sent-166, score-0.304]

91 7 Synthetic Small MID Full MID Enron 0 Enron 1 Enron 2 Enron 3 Enron 4 SB conv 23 SB conv 26 SB conv 12 SB conv 49 SB conv 33 N 3 7 82 2 2 2 2 2 18 11 12 11 10 T 239 57 412 896 204 122 117 85 832 95 133 620 499 Hawkes IRM E[K] log probability 2. [sent-170, score-0.52]

92 N denotes the number of individuals in the data set. [sent-249, score-0.135]

93 T denotes the total number of events in the data set. [sent-250, score-0.256]

94 Synthetic Small MID Full MID Enron 0 Enron 1 Enron 2 Enron 3 Enron 4 SB conv 23 SB conv 26 SB conv 12 SB conv 49 SB conv 33 Hawkes IRM 43. [sent-253, score-0.52]

95 The intuition behind why our model works well is that it captures part of the reciprocal nature of interactions among individuals in social situations, which in turn requires modelling some of the causal relationship of events. [sent-358, score-0.383]

96 For example, individuals might contribute to groups differently to one another. [sent-361, score-0.188]

97 There may be different kinds of events between individuals and other side-channel information. [sent-362, score-0.391]

98 It would be interesting to consider other parameterisations of gpq (·) that, for example, include periods of delay between reciprocation; the exponential parameterisation lends itself to efﬁcient computation [10] whilst other parameterisations do not necessarily have this property. [sent-364, score-0.189]

99 But different choices of gpq (·) may yield better statistical models. [sent-365, score-0.085]

100 Another interesting avenue is to explore other structure amongst interaction events using Hawkes processes, beyond reciprocity. [sent-366, score-0.303]

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