{"title": "Modelling Reciprocating Relationships with Hawkes Processes", "book": "Advances in Neural Information Processing Systems", "page_first": 2600, "page_last": 2608, "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.", "full_text": "Modelling Reciprocating Relationships\n\nwith Hawkes Processes\n\nCharles Blundell\n\nGatsby Computational Neuroscience Unit\n\nUniversity College London\nLondon, United Kingdom\n\nc.blundell@gatsby.ucl.ac.uk\n\nKatherine A. Heller\n\nDuke University\nDurham, NC, USA\n\nkheller@stat.duke.edu\n\nJeffrey M. Beck\n\nUniversity of Rochester\n\nRochester, NY, USA\n\njbeck@bcs.rochester.edu\n\nAbstract\n\nWe present a Bayesian nonparametric model that discovers implicit social struc-\nture from interaction time-series data. Social groups are often formed implicitly,\nthrough actions among members of groups. Yet many models of social networks\nuse explicitly declared relationships to infer social structure. We consider a par-\nticular class of Hawkes processes, a doubly stochastic point process, that is able\nto model reciprocity between groups of individuals. We then extend the In\ufb01nite\nRelational Model by using these reciprocating Hawkes processes to parameterise\nits edges, making events associated with edges co-dependent through time. Our\nmodel outperforms general, unstructured Hawkes processes as well as structured\nPoisson process-based models at predicting verbal and email turn-taking, and mil-\nitary con\ufb02icts among nations.\n\n1\n\nIntroduction\n\nAs social animals, people constantly organise themselves into social groups. These social groups can\nrevolve around particular activities, such as sports teams, particular roles, such as store managers,\nor general social alliances, like gang members. Understanding the dynamics of group interactions is\na dif\ufb01cult problem that social scientists strive to address.\nOne basic problem in understanding group behaviour is that groups are often not explicitly de\ufb01ned,\nand the members must be inferred. How might we infer these groups, and from what data? How can\nwe predict future interactions among individuals based on these inferred groups?\nA common approach is to infer groups, or clusters, of people based upon a declared relationship\nbetween pairs of individuals [1, 2, 3, 4]. For example, data from social networks, where two people\ndeclare that they are \u201cfriends\u201d or in each others\u2019 social \u201cneighbourhood\u201d, can potentially be used.\nHowever these declared relationships are not necessarily readily available, truthful, or pertinent to\ninferring the social group structure of interest.\nIn this paper we instead propose an approach to inferring social groups based directly on a set\nof real interactions between people. This approach re\ufb02ects an \u201cactions speak louder than words\u201d\nphilosophy. If we are interested in capturing groups that best re\ufb02ect human behaviour we should be\ndetermining the groups from instances of that same behaviour. We develop a model which can learn\nsocial group structure based on interactions data.\n\n1\n\n\fIn the work that we present, our data will consist of a sequence of many events, each event re\ufb02ecting\none person, the sender, performing some sort of an action towards another person, the recipient, at\nsome particular point in time. As examples, the actions we consider are that of one person sending\nan email to another, one person speaking to another, or one country engaging in military action\ntowards another.\nThe key property that we leverage to infer social groups is reciprocity. Reciprocity is a common\nsocial norm, where one person\u2019s actions towards another increases the probability of the same type\nof action being returned. For example, if Bob emails Alice, it increases the probability that Alice\nwill email Bob in the near future. Reciprocity widely manifests across many cultures, perhaps most\ncommonly as the golden rule and tit for tat retaliation. When multiple people show a similar pattern\nof reciprocity, our model will place these people in their own group.\nThe Bayesian nonparametric model we use on these time-series data is generative and accounts for\nthe rate of events between clusters of individuals. It is built upon mutually-exciting point processes,\nknown as Hawkes processes [5, 6]. Pairs of mutually-exciting Hawkes processes are able to capture\nthe causal nature of reciprocal interactions. Here the processes excite one another through their\nactualised events. Since Poisson processes are a special case of Hawkes processes, our model is also\nable to capture simpler one-way, non-reciprocal, relationships as well.\nOur model is also related to the In\ufb01nite Relational Model (IRM) [1, 2]. The IRM typically assumes\nthat there is a \ufb01xed graph, or social network, which is observed. Here we are interested in inferring\nthe implicit social structure based only on the occurrences of interactions between vertices in the\ngraph. We apply our model to reciprocal behaviour in verbal and email conversations and to military\ncon\ufb02icts among nations.\nThe remainder of the paper is organised as follows: section 2 discusses using Poisson processes\ntogether with the IRM. Section 3 describes our use of self-exciting and pairs of Hawkes processes,\nand section 4 speci\ufb01es how they are used to develop our reciprocity clustering model. Section\n5 presents an inference algorithm for our model, section 6 discusses related work, and section 7\npresents experimental results using our model on synthetic, email, speech and intercountry con\ufb02ict\ndata.\n\n2 Poisson processes with the In\ufb01nite Relational Model\n\nThe In\ufb01nite Relational Model (IRM) [1, 2] was developed to model relationships among entities\nas graphs, based upon previously declared relationships. Let V denote the vertices of the graph,\ncorresponding to individuals, and let euv denote the presence or absence of a relationship between\nvertices u and v, corresponding to an edge in the graph. The generative process of the IRM is:\n\n\u03c0 \u223c CRP(\u03b1)\n\u03bbpq \u223c Beta(\u03b3, \u03b3)\neuv \u223c Bernoulli(\u03bb\u03c0(u)\u03c0(v))\n\n\u2200p, q \u2208 range(\u03c0)\n\u2200u, v \u2208 V\n\n(1)\n(2)\n(3)\n\nwhere \u03c0 is a partition of the vertices V , distributed according to the Chinese restaurant process (CRP)\nwith concentration parameter \u03b1, with p and q indexing clusters of \u03c0. Hence vertex u belongs to the\ncluster given by \u03c0(u), and consequently, the clusters in \u03c0 are given by range(\u03c0). The probability\nof an edge between vertex u and vertex v is then the parameter \u03bbpq associated with their pair of\nclusters.\nOften in interaction data there are many instances of interactions between the same pair of\nindividuals\u2013this cannot be modelled by the IRM. A straightforward way to modify the IRM to ac-\ncount for this is to use a Gamma-Poisson observation model instead of this usual Beta-Bernoulli\nmodel. Unfortunately, a vanilla Gamma-Poisson observation model does not allow us to predict\nevents into the future, outside the observed time window. Therefore we consider using a Poisson\nprocess instead.\nPoisson processes are stochastic counting processes. For an introduction see [7]. We shall consider\nPoisson processes on [0,\u221e), such that the number of events in any interval [s, s(cid:48)) of the real-half\nline, denoted N [s, s(cid:48)), is Poisson distributed with rate \u03bb(s(cid:48) \u2212 s).\n\n2\n\n\fFigure 1: A simple example. The graph in the top left shows the clusters and edge weights learned by our\nmodel from the data in the bottom right plot. The top right plot shows the rates of interaction events between\nclusters. The bottom right plot shows the interaction events. In the graph, the width and temperature (how red\nthe colour is) denotes the expected rate of events between pairs of clusters (using equations (9) and (10)). While\nin plots on the right, line colours indicates the identity of cluster pairs, and box colours indicate the originator\nof the event: Alice (red), Bob (blue), Mallory (black). Alice and Bob interact with each other such that they\npositively reciprocate each others\u2019 actions. Mallory, however, has an asymmetric relationship with both Alice\nand Bob. Only after many events caused by Mallory do Alice or Bob respond, and when they do respond they\nboth, similarly, respond more sparsely.\n\n\u03c0 \u223c CRP(\u03b1)\n\u03bbpq \u223c Gamma(\u03b4, \u03b2)\n\n\u2200p, q \u2208 range(\u03c0)\n\u2200u, v \u2208 V\n\nWith Gamma priors on the rate parameter, the full Poisson Process IRM model is:\n\nNuv(\u00b7) \u223c PoissonProcess(\u03bb\u03c0(u)\u03c0(v))\n\n(4)\n(5)\n(6)\nwhere Nuv(\u00b7) is the random counting measure of the Poisson process, and \u03b4 and \u03b2 are respectively\nthe shape and inverse scale parameters of the Gamma prior on the rate of the Poisson processes, \u03bbpq.\nInference proceeds by conditioning on, Nuv[0, T ) = nuv where nuv is the total number of events\ndirected from u to v in the given time interval. Since conjugacy can be maintained, due to the\nsuperposition property of Poisson processes, inference in this model is possible in much the same\nway as in the original IRM [2, 1].\nThere are two notable de\ufb01ciencies of this model: the rate of events on each edge is independent of\nevery other edge, and conditioned on the time interval containing all observed events, the times of\nthese events are uniformly distributed. This is not the typical pattern we observe in interaction data.\nIf I send an email to someone, it is more likely that I will receive an email from them than had I not\nsent an email, and the probability of receiving a reply decreases as time advances. In the following\nsections we will introduce and utilise mutually-exciting Hawkes processes, which are able to exactly\nmodel these phenomena.\n\n3 Self-Exciting and Pairs of Mutually-Exciting Hawkes Processes\n\nHawkes [5, 6] introduced a family of self- and mutually-exciting Markov point processes, often\ncalled Hawkes processes. These processes are intuitively similar to Poisson processes, but unlike\nPoisson processes, the rates of Hawkes processes depend upon their own historic events and those\nof other processes in an excitatory fashion.\nWe shall consider an array of K \u00d7 K Hawkes processes, where K is the number of clusters in a\npartition drawn from a CRP restricted to the individuals V . As in the IRM, the CRP allows the\n\n3\n\nMallory'Bob'Alice'050100150200250rate\u03bbpq(t)Alice,Bob\u2192Alice,BobMallory\u2192Alice,BobAlice,Bob\u2192MalloryMallory\u2192Mallory0.00.20.40.60.81.0timetMallory\u2192MalloryAlice,Bob\u2192MalloryMallory\u2192Alice,BobAlice,Bob\u2192Alice,Bob\f(cid:90) t\n\n\u2212\u221e\n\n(cid:88)\n\nnumber of processes to grow in an unconstrained manner as the number of individuals in the graph\ngrows. However, unlike the IRM, these Hawkes processes will be pairwise-dependent: the Hawkes\nprocess governing events from cluster p to cluster q, will depend upon the Hawkes process governing\nevents from cluster q to cluster p.\nLet Npq be the counting measure of the (p, q)th Hawkes process. Each Hawkes process is a point\nprocess whose rate at time t is given by:\n\n\u03bbpq(t) = \u03b3pqnpnq +\n\ngpq(t \u2212 s)dNqp(s)\n\n(7)\n\n(cid:82) \u221e\n\nwhere \u03b3pq is the base rate of the counting measure of the Hawkes, process, Npq. np and nq are the\nnumber of individuals in cluster p and q respectively, and gpq is a non-negative function such that\n0 gpq(s)ds < 1, ensuring that Npq is stationary. Nqp is the counting measure of the reciprocating\nHawkes process of Npq. Intuitively, if Npq governs events from cluster p to cluster q, then Nqp\ngoverns events from cluster q to cluster p. Equation (7) shows how the rates of events in these two\nprocesses are intimately intertwined.\nSince Nqp is an atomic measure, whose atoms correspond to the times of events, we can express the\nrate of Npq given in (7), by conditioning on the events of its reciprocating processes Nqp, as:\n\n\u03bbpq(t) = \u03b3pqnpnq +\n\ngpq(t \u2212 tqp\ni )\n\n(8)\n\ni:tqp\n\ni <t\n\nwhere tqp\ni denotes the times of the ith event of process Nqp. Thus the rate of the process Npq at time\nt is some base rate at which events occur, \u03b3pq, plus an additional rate of gpq(t \u2212 tqp\ni ) for each event\nin the reciprocating process Nqp. Figure 1(top) shows an example of how \u03bbpq(t) and \u03bbqp(t) vary for\nthese pairs of processes.\nIf gpq(\u00b7) = 0 then the process is a Poisson process with rate \u03b3pqnpnq. When p = q, the process\nis self-exciting: its current rate depends solely on its own previous events. In our application, self-\nexciting processes model interactions within a social group, as they model cohesion in reciprocity:\nindividual reciprocation within a group is as if towards oneself. In the case of p (cid:54)= q, each pair of\nprocesses Npq and Nqp mutually excite one another. An event in one increases the probability of an\nevent from the other, and so on. Importantly, the type of reciprocation (parameterised by gpq and\ngqp, respectively) differs between events from group p to group q and events from group q to group\np. This difference in reciprocity is what we would like our model to leverage to learn about social\ngroups.\nHawkes processes are an example of doubly stochastic point processes. The rate of events is itself\na random variable. By integrating out the events of Nqp we can see that this process is stationary,\nas its rate does not depend upon time, and also gain further insight into the role of the functions gpq\nand gqp. For self-exciting Hawkes processes, where p = q, the marginal rate is:\n\nE[\u03bbpp(t)] = n2\n\np\n\n\u03b3pp\n\n1 \u2212 Gpp\n\nwhilst for a pair of mutually-exciting Hawkes processes the marginal rate is:\n\n(9)\n\n(10)\n\nwhere Gpq = (cid:82) t\n\nE[\u03bbpq(t)] = npnq\n\n\u03b3pq + \u03b3qpGpq\n1 \u2212 GpqGqp\n\n\u2212\u221e gpq(t \u2212 u)du which tempers the effect of the rate of events from one process\non the rate of the other. The closer Gpq is to zero, the more Poisson-like Hawkes processes behave.\nWhilst as Gpq approaches one, the rate of events in Npq are increasingly caused by those in Nqp.\n\n4 Hawkes Processes with the In\ufb01nite Relational Model\n\nWe combine Hawkes processes with the IRM as follows. We pick the form for the gpq functions as\n\u03c4pq [5, 6, 8, 9]. Examples of using this parameterisation are shown in Figure 1(top).\ngpq(\u03b4) = \u03b2pqe\n\n\u2212 \u03b4\n\n4\n\n\fDue to the memorylessness property of the exponential distribution, inference with Hawkes pro-\ncesses with this parameterisation takes time linear in the number of events [10]. Our generative\nmodel is as follows:\n\n\u03c0 \u223c CRP(\u03b1)\n\n(cid:90) t\n\n\u2212 t\u2212s\n\n\u03c4pq dNqp(s)\n\n\u03bbpq(t) = \u03b3pqnpnq + \u03b2pq\ne\nNpq(\u00b7) \u223c HawkesProcess(\u03bbpq(\u00b7))\nNuv(\u00b7) \u223c Thinning(N\u03c0(u)\u03c0(v)(\u00b7))\n\n\u2212\u221e\n\n\u2200p, q \u2208 range(\u03c0)\n\n\u2200u, v \u2208 V\n\n(11)\n\n(12)\n\n(13)\n(14)\n\nwhere, as before, \u03c0 is a partition of the individuals, drawn from a Chinese restaurant process (CRP)\nwith concentration parameter \u03b1. For each pair of clusters p and q, we associate a time-varying rate\n\u03bbpq(t) which dictates the rate of events from individuals in cluster p to individuals in cluster q, and\na Hawkes process Npq. As described in the previous section, this rate depends upon the speci\ufb01c\nevents sent in the opposite direction, from cluster q to cluster p, whose measure is also random and\nis denoted Nqp(\u00b7).\nEach random measure Nuv(\u00b7) governs events between a particular pair of individuals within clusters\np and q respectively. Nuv(\u00b7) are drawn by thinning the cluster random measure Npq(\u00b7) among all\nof the edges between individuals in clusters p and q. Thinning means distributing the atoms of\nu,v Nuv(\u00b7). Constructing the edge measures by\n0 gpq(u)du < 1 for the process to be stationary.\nThis condition, under the chosen parameterisation, implies that \u03c4pq\u03b2pq < 1. When all \u03b2pq = 0,\nthis model is equivalent to the Poisson process IRM in section 2. Henceforth we will use uniform\nthinning\u2014each event in Npq(\u00b7) is assigned uniformly at random among all Nuv(\u00b7) where p = \u03c0(u)\nand q = \u03c0(v)\u2014but in principle any thinning scheme may be used.\nFor a Hawkes process Npq, the rate at which no events occurs in the interval [s, s(cid:48)) is:\n\nNpq(\u00b7) among each Nuv(\u00b7), such that Npq = (cid:80)\nthinning means it is suf\ufb01cient to ensure that(cid:82) \u221e\n\ne\u2212(cid:82) s(cid:48)\n\n(cid:82) T\n\nnuv(cid:89)\n\ni=1\n\n(15)\nSuppose we observe the times of all the events in [0, T ), {tuv\ni=1 for process Nuv (nuv being the\ntotal number of events from u to v in [0, T )). Suppose that individual u is in cluster p and that\nindividual v is in cluster q. Furthermore, assume there are no events before time 0. The likelihood\nof each edge between individuals u and v is thus:\n\ni }nuv\n\ns \u03bbpq(t)dt\n\np({tuv\n\ni }nuv\n\ni=1|\u03b8pq,{tqp\n\ni }nqp\n\ni=1) = e\n\n\u2212 1\n\nnp nq\n\n0 \u03bbpq(t)dt\n\n\u03bbpq(tuv\ni )\nnpnq\n\n(16)\n\nwhere \u03b8pq = (\u03b3pq, \u03b2pq, \u03c4pq), {tqp\nuniform priors on log \u03b1, \u03b3pq, \u03b2pq, and \u03c4pq, enforcing the constraint that \u03c4pq\u03b2pq < 1.\n\ni=1 are the times of the reciprocal events. We place proper\n\ni }nqp\n\n5\n\nInference\n\nWe perform posterior inference using Markov chain Monte Carlo. Our model is a departure from\nprevious IRM-based models as there is no conjugate prior for the likelihood. Thus we cannot simply\nintegrate out these parameters, and must sample them.\nTo infer the partition of individuals \u03c0, the concentration parameter \u03b1, and the parameters of each\nHawkes process \u03b8pq = (\u03b3pq, \u03b2pq, \u03c4pq), we use Algorithm 5 [11] adapted to the IRM and slice\nsampling [12] to draw samples from the posterior. We initialise the chain from the prior. Slice\nsampling is used for \u03b1 and each of \u03b3pq, \u03b2pq, and \u03c4pq. When setting the bounds of the slice sampler\nfor \u03b2pq (\u03c4pq) we set the upper bound to 1\n\u03c4pq\n\n) respectively, to ensure that \u03b2pq\u03c4pq \u2264 1.\n\n( 1\n\u03b2pq\n\n6 Related work\n\nSeveral authors have considered modelling occurrence events [13, 14, 15] using piecewise constant\nrate Markov point processes for known number of event types. Our work directly models interaction\n\n5\n\n\fevents (where an event is structured to have a sender and recipient) and the number of possible\nevents types is not limited. [16] describes a model of occurrence events as a discrete time-series\nusing a latent \ufb01rst-order Markov model. Our model differs in that it considers interaction events in\ncontinuous time and requires no \ufb01rst-order assumption.\nThe model in Section 2 relates the work of [17] to the IRM [1], yielding a version of their model\nthat learns the number of clusters whilst maintaining conjugacy. However our model does not use a\nPoisson process to model event times, instead using processes which have a time-varying rate.\nSimma and Jordan [10] describe a cascade of Poisson processes, forming a marked Hawkes process.\nHawkes processes are also the basis of this work, however our work does not use side-channel\ninformation to group individuals by imposing \ufb01xed marks on the process; instead we learn structure\namong several co-dependent Hawkes processes and use Bayesian inference for the parameters and\nstructure.\nPaninski et al [18, 19] describe a process similar to a Hawkes process that uses an additional link\nfunction to allow for inhibition amongst neurons. The interest is in modelling the activation and\nco-activation of neurons and as such they do not directly model cluster structure among the neurons,\nwhile our model does model this structure. Learning such structure among neurons is a potential\ninteresting future application of this model.\nOur model may also be seen as a probabilistic interpretation of the interaction rank of [20], which\nwe leverage to discover global clustering structure. An interesting future direction would be to learn\na per-person (i.e., ego-centric) clustering structure.\n\n7 Experiments\n\nIn our experiments we compared our model to the Poisson process IRM (Section 2), a single Hawkes\nprocess and a single Poisson process. These latter two models are equivalent to the \ufb01rst two models\nwhere just one cluster is used.\nWe compared these models quantitatively by comparing their log predictive densities (with respect\nto the space of ordered sequences of events) on events falling in the \ufb01nal 10% of the total time of the\ndata (Table 2). We normalised the times of all events such that the \ufb01rst 90% of total time lay in the\ninterval [0, 1]. We ran our inference algorithm for 5000 iterations and discarded the \ufb01rst 500 burn-in\nsamples, repeating each experiment 10 times from different initialisations from the prior.\n\nSynthetic data We generated synthetic data to highlight differences between our model and the al-\nternatives. The data involves three individuals and is plotted in Figure 1. Table 1 shows details of the\n\ufb01t of the model to the data, and Table 2 shows the predictive results. The Poisson IRM is uncertain\nhow to cluster individuals as it cannot model the temporal dependence between individuals, while\nthe Hawkes IRM can and so performs better at prediction as well. A single Hawkes process does\nnot model the structure among individuals and so performs worse than the Hawkes IRM, although\nit is able to model dependence among events.\n\nEnron email threads We took the \ufb01ve longest threads from the Enron 2009 data set [21]. We\nidenti\ufb01ed threads by the set of senders and receivers, and their subject line (after removing common\nsubject line pre\ufb01xes such as \u201cRe:\u201d, \u201cFwd:\u201d and so on, removing punctuation and making all letters\nlower case). All of these threads involve two different people so there is little scope for learning\nmuch group structure in these data: either both people are in the same cluster, or they are in two\nseparate clusters. However as can be seen in Table 2 these data suggest a predictive advantage\nto using mutually-exciting Hawkes processes, as automatically determined by our model, instead\nof a single self-exciting Hawkes process and of both of these approaches over their corresponding\nPoisson processes model. A self-exciting Hawkes process is unable to mark the sender and receiver\nof events as differing, whilst Poisson process-based models are unable to model the causal structure\nof events.\n\nSanta Barbara Conversation Corpus We took \ufb01ve conversations from the Santa Barbara Con-\nversation Corpus [22] involving the largest number of people. These results are labelled \u201cSB conv\u201d\nfollowed by the conversation identi\ufb01er in this corpus, in the results in Tables 1 and 2. These con-\n\n6\n\n\fFigure 2: Graphs of clusters of individuals inferred by our model. Edge width and temperature (how\nred the colour is) denotes the expected rate of events between pairs of clusters (using equations (9)\nand (10); edges whose marginal rate is below 1 are not included). On the left is the graph inferred\non the \u201cSB conv 26\u201d data set. On the right is the graph inferred on the \u201cSmall MID\u201d data set.\n\nversations cover a variety of social situations: questions during a university lecture (12), a book\ndiscussion group (23), a meeting among city of\ufb01cials (26), a family argument/discussion (33), and a\nconversation at a family birthday party (49). We modelled the turn-taking behaviour of these groups\nby taking the times of when one speaker switched to the next. In Figure 2(left) we show the cluster\ngraph found by our model for conversation 26, involving city of\ufb01cials discussing a grant application.\nThe identities of participants in all of these data are anonymised, preventing an exact interpretation.\nHowever, the model captures the discussive to-and-fro of the meeting, where PATT appears to be the\nchair of the meeting, and DAN, ROSE and GRAN are the main discussors, all of whom discuss with\nthe chair, initially in question and an answer format, and among themselves, with other members of\nthe audience chipping in sporadically.\n\nCorrelates of war We use version 3.0 of the Militarized Interstate Disputes (MIDs) data set [23] to\nmodel correlates of war. This data set spans the years 1993 to 2001, and consists of MID incidents,\nalong with the countries involved in the incidents. Incidents vary from diplomatic threats of military\nforce to the actual deployment of military force against another state. A detailed description of each\nincident is available in [24].\nThe results of all models on the correlates of war data are given in Table 2 with details of the \ufb01ts\nin Table 1 in the rows entitles \u201cSmall MID\u201d and \u201cFull MID\u201d. The full MID data set consists of 82\ncountries\u2014yielding a large graph. For exposition purposes, we show the graph (in Figure 2(right))\non part of the MID data set, by restricting to events among the USA, Kuwait, Afghanistan, Taiwan,\nRussia, China, and Iraq. Thicker and redder lines between clusters (computed from equations 9 and\n10) re\ufb02ect a higher rate of incidents directed between the countries along the edge.\nThe results of the clustering given by our model are in keeping with that discussed in [24]. There\nwere three main con\ufb02icts involving the countries we modelled during the time period this data\ncovers. These con\ufb02icts involve 1) Russia and Afghanistan, 2) Taiwan (sometimes with support from\nthe USA) and China (sometimes with support from Russia), and 3) Iraq, Kuwait, and the USA. 1)\nRevolved mostly around border disputes coming out of the Soviet war in Afghanistan, and incidents\nsometimes involved using former Soviet countries as proxies. 2) Re\ufb02ects con\ufb02ict between Taiwan\nand China over potential Taiwanese independence. Lastly, 3) deals with con\ufb02icts between Iraq and\neither Kuwait or the USA coming out of the Persian Gulf war. It is interesting to note that groups\ninvolving smaller countries were found to be more likely to initiate incidents with larger countries\nin a dispute (e.g. Iraq was almost always the instigator of disputes in their con\ufb02ict with Kuwait and\nthe USA). Since the data ends in 2001, relatively few disputes with Afghanistan involve the USA.\n\n7\n\nDan$Gran$Rose$Pa,$Bern$X$Cher$Paul$$Many$Dere$Kare$IRQ$KUW$USA$RUS$CHN$AFG$TAW$\fHawkes IRM\n\nPoisson IRM\n\nSynthetic\nSmall MID\n\nN\n3\n7\nFull MID 82\n2\n2\n2\n2\n2\n18\n11\n12\n11\n10\n\nEnron 0\nEnron 1\nEnron 2\nEnron 3\nEnron 4\nSB conv 23\nSB conv 26\nSB conv 12\nSB conv 49\nSB conv 33\n\nT\n239\n57\n412\n896\n204\n122\n117\n85\n832\n95\n133\n620\n499\n\nE[K]\n2.00\n4.30\n13.67\n2.00\n2.00\n2.00\n2.00\n2.00\n11.87\n4.26\n4.11\n8.85\n8.44\n\nlog probability\n594.04\u00b10.01\n33.59\u00b10.02\n-638.25\u00b11.16\n6724.76\u00b10.01\n1202.99\u00b10.02\n616.37\u00b10.02\n497.53\u00b10.02\n252.60\u00b10.02\n1581.72\u00b10.12\n170.34\u00b10.03\n233.41\u00b10.03\n1728.13\u00b10.07\n803.22\u00b10.16\n\nE[K]\n1.36\n1.02\n3.93\n2.00\n2.00\n2.00\n2.00\n2.00\n3.01\n2.00\n2.53\n3.40\n2.03\n\nlog probability\n533.65\u00b10.00\n-63.99\u00b10.03\n-1412.49\u00b15.38\n4516.77\u00b10.00\n692.32\u00b10.00\n336.02\u00b10.00\n318.38\u00b10.00\n192.74\u00b10.00\n599.29\u00b10.42\n-51.92\u00b10.14\n-59.12\u00b10.15\n990.75\u00b10.15\n431.59\u00b10.12\n\nTable 1: Details of data sets and \ufb01ts of the structured models. N denotes the number of individuals in the data\nset. T denotes the total number of events in the data set. E[K] is the average number of clusters found in the\nposterior. Log probability is the average log probability of the training data.\n\nHawkes IRM Poisson IRM\n-6.76\u00b10.02\n43.00\u00b10.00\nSynthetic\n-50.88\u00b10.02\n12.69\u00b10.04\nSmall MID\nFull MID -134.97\u00b12.98\n-355.29\u00b15.61\n39.33\u00b10.00\n259.20\u00b10.01\nEnron 0\n436.66\u00b10.01\n133.29\u00b10.00\nEnron 1\n139.40\u00b10.01\n24.14\u00b10.00\nEnron 2\n124.22\u00b10.01\n21.06\u00b10.00\nEnron 3\n28.38\u00b10.00\n127.82\u00b10.02\nEnron 4\n-198.34\u00b10.23\n132.57\u00b10.27\nSB conv 23\n-5.85\u00b10.02\n-16.83\u00b10.09\nSB conv 26\n96.07\u00b10.03\n-97.89\u00b10.10\nSB conv 12\n220.85\u00b10.09\n-116.62\u00b10.12\nSB conv 49\n-100.83\u00b10.04\n46.19\u00b10.06\nSB conv 33\n\nHawkes\n39.88\u00b10.01\n6.37\u00b10.01\n-188.08\u00b10.00\n233.44\u00b10.00\n380.27\u00b10.01\n118.86\u00b10.00\n101.71\u00b10.01\n109.62\u00b10.00\n30.93\u00b10.00\n-6.05\u00b10.00\n33.18\u00b10.00\n126.94\u00b10.00\n21.71\u00b10.00\n\nPoisson\n-3.88\u00b10.00\n-50.86\u00b10.00\n-302.65\u00b10.00\n40.11\u00b10.00\n105.71\u00b10.00\n22.88\u00b10.00\n21.03\u00b10.00\n22.08\u00b10.00\n-213.18\u00b10.00\n-14.54\u00b10.00\n-128.53\u00b10.00\n-83.62\u00b10.00\n-83.79\u00b10.00\n\nTable 2: Average log predictive results for each model with standard errors\n\n8 Discussion\n\nWe have presented a Bayesian nonparametric approach to learning the structure among collections\nof co-dependent Hawkes processes, which on several interaction data sets consistently outperforms\nboth unstructured and Poisson-based models in terms of predictive likelihoods. The intuition behind\nwhy our model works well is that it captures part of the reciprocal nature of interactions among\nindividuals in social situations, which in turn requires modelling some of the causal relationship of\nevents. By learning this structure, our model is able to make better predictions.\nThere are several future directions. For example, individuals might contribute to groups differently\nto one another. There may be different kinds of events between individuals and other side-channel\ninformation. Both of these artefacts may be modelled by replacing the uniform thinning scheme\nproposed above, with a detailed model of these effects. It would be interesting to consider other\nparameterisations of gpq(\u00b7) that, for example, include periods of delay between reciprocation; the\nexponential parameterisation lends itself to ef\ufb01cient computation [10] whilst other parameterisations\ndo not necessarily have this property. But different choices of gpq(\u00b7) may yield better statistical\nmodels. Another interesting avenue is to explore other structure amongst interaction events using\nHawkes processes, beyond reciprocity.\n\nAcknowledgements The authors are grateful for helpful comments from the anonymous review-\ners, and the support of Josh Tenenbaum, the Gatsby Charitable Foundation, PASCAL2 NoE, NIH\naward P30 DA028803, and an NSF postdoctoral fellowship,\n\n8\n\n\fReferences\n[1] Charles Kemp, Joshua B. Tenenbaum, Thomas L. Grif\ufb01ths, Takeshi Yamada, and Naonori\n\nUeda. Learning systems of concepts with an in\ufb01nite relational model. AAAI, 2006.\n\n[2] Zhao Xu, Volker Tresp, Kai Yu, and Hans-Peter Kriegel. In\ufb01nite hidden relational models.\n\nUncertainty in Arti\ufb01cial Intelligence (UAI), 2006.\n\n[3] Edoardo M. Airoldi, David M. Blei, Stephen E. Fienberg, and Eric P. Xing. Mixed membership\n\nstochastic blockmodel. 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Suggesting friends using the implicit social graph.\nIn Proceedings of the 16th ACM SIGKDD Conference on Knowledge Discovery and Data\nMining, 2010.\n\n[21] Enron 2009 Data set. http://www.cs.cmu.edu/ enron/.\n[22] John W. DuBois, Wallace L. Chafe, Charles Meyer, and Sandra A. Thompson. Santa Barbara\n\ncorpus of spoken American English. Linguistic Data Consortium, 2000.\n\n[23] Faten Ghosn, Glenn Palmer, and Stuart Bremer. The mid3 data set, 19932001: Procedures,\n\ncoding rules, and description. Con\ufb02ict Management and Peace Science, 21:133\u2013154, 2004.\n\n[24] Dispute Narratives. http://www.correlatesofwar.org/cow2%20data/mids/mid v3.0.narratives.pdf.\n\n9\n\n\f", "award": [], "sourceid": 4834, "authors": [{"given_name": "Charles", "family_name": "Blundell", "institution": null}, {"given_name": "Jeff", "family_name": "Beck", "institution": null}, {"given_name": "Katherine", "family_name": "Heller", "institution": null}]}