{"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