Spatio-Temporal-Network Point Processes for Modeling
Crime Events with Landmarks

Our research is placed within the domain of predictive policing (Perry,, 2013), which includes four general categories: methods for predicting crimes (Chainey et al.,, 2008; Neill and Gorr,, 2007; Wang and Brown,, 2012), methods for predicting offenders (Bonta et al.,, 1998; Grann et al.,, 1999), methods for predicting perpetrators’ identities (Lev-Wiesel et al.,, 2004; Tarzia et al.,, 2018), and methods for predicting victims of crime (Gottfredson,, 1981; Russo et al.,, 2013). Our study belongs to the first category, which aims to forecast places and times with an increased crime risk. Unlike the other three categories that require the collection of extensive information about crime incidents, such as police reports, to identify certain individuals or groups that may get involved in criminal activities, the prediction of spatio-temporal occurrences of crime can be mainly achieved by leveraging historical crime data without the necessary access to sensitive information.

Many mathematical models have been used to understand the complex phenomenon of crime; a family of those includes tools that aim to detect potential hotspots based on empirical observations of spatial clusters of crime incidents (Levine and CrimeStat,, 2002; Bowers et al.,, 2004; Chainey et al.,, 2008). However, most hotspot modeling approaches do not consider the temporal dynamics of the hotspot, despite some exploring the overall evolution of hotspots rather than focusing on individual events (Short et al.,, 2008). Other models use regression-based methods (Meera and Jayakumar,, 1995; Kennedy et al.,, 2011, 2016) to quantitatively assess the effects of different factors on the total number of crimes in a specific region, and the interpretable results from the regression models can aid in more targeted and effective interventions. These methods usually run over a discretized space and require the collection of contextual information, such as demographic and socioeconomic data, to establish the regression models. Compared with them, our method focuses on modeling the spatio-temporal near-repeat effect of crime using discrete crime event data and can provide the prediction and risk evaluation on a fine-grained level over the street network.

In recent decades, there has been a substantial body of research (Kinney et al.,, 2008; Johnson and Bowers,, 2010; Groff,, 2011; Weisburd et al.,, 2012; Xu and Griffiths,, 2017) examining the relationship between urban land use and crime patterns. These studies have pinpointed environmental characteristics linked to increased crime risks in specific urban settings. Our approach differs from the aggregated-statistics-based analysis often taken in such research (Fleming et al.,, 1994; Kinney et al.,, 2008; Stucky and Ottensmann,, 2009; Browning et al.,, 2010; Xu and Griffiths,, 2017). Instead, we model the spatio-temporal crime patterns through the lens of individual crime incidents, providing a dynamic perspective for explaining the crime and integrating effective surveillance.

The application of self-exciting point processes, motivated by the modeling of earthquake occurrences in seismology (Ogata,, 1988), has been widely explored to characterize the dynamics of criminal activities (Mohler et al.,, 2011; Lewis et al.,, 2012; Mohler,, 2013; Reinhart,, 2018; Zhuang and Mateu,, 2019; Zhu and Xie,, 2022). Previous attempts demonstrate the effectiveness of point processes in modeling crime using residential burglary data in Los Angeles (Mohler et al.,, 2011), civilian death reports in Iraq (Lewis et al.,, 2012), and gunshot data in Washington, DC (Loeffler and Flaxman,, 2018). Later approaches (Mohler,, 2014; Reinhart and Greenhouse,, 2018; Zhu and Xie,, 2022) improve point process models for crime by incorporating the events’ type, location, and textual information to capture complex crime patterns in different modeling tasks. Compared with them, our approach extends the traditional modeling of crime events in Euclidean space by adopting a network distance between crime events that is more realistic to estimate the travel distance of criminals in the urban environment. A recent paper considers crime events on linear street network (D’Angelo et al.,, 2024) that focus on improving the estimation of the non-parametric influence kernel and event intensity. Our model differs from theirs by considering an influence kernel that can leverage multiple levels of information.

A number of studies have considered the problem of modeling discrete events observed within network structures using self-exciting point processes in addition to modeling crime incidents. Most of them (Fang et al.,, 2023; Cai et al.,, 2024; Sanna Passino et al.,, 2024) only model the temporal occurrences of the events that come from the nodes in the networks. Besides, a previous work (Cho et al.,, 2013) studies the missing data problem in spatio-temporal social networks, where the network nodes are geographically distributed entities. The missing data problem also exists in criminology, such as detecting unreported crimes, which is not the focus of our study. Another work of network Hawkes (Linderman and Adams,, 2014) adopts a similar decomposed representation for the mark interactions as ours. However, their approach includes the estimation of a binary random matrix using Gibbs sampling, which is fundamentally different and more complicated, and the events’ location information is processed on an aggregated level.

Last but not least, incorporating neural networks in point process models has recently been a popular research topic (Shchur et al.,, 2021). Various neural point processes focus on leveraging recurrent neural networks (RNNs) (Du et al.,, 2016; Mei and Eisner,, 2017) or Transformer structure (Zuo et al.,, 2020) to encode the historical information. Compared with our method, they did not consider the statistical framework of the self-exciting point process and often lacked model interpretability. Another line of work (Dong et al., 2023b, ; Dong et al.,, 2022; Zhu et al., 2021a, ; Zhu et al., 2021b, ; Zhu et al., 2021c, ) focuses on representing the influence kernel using neural networks, allowing for the modeling of a wider range of event dynamics such as non-stationary and inhibiting effects. However, they do not consider contextual information such as the latent network structure or mark features. Graph neural networks have been extensively developed within the machine learning community. Nevertheless, their application in point processes has received scant investigation. Two recent works use message-passing GNNs in point processes (Xia et al.,, 2022; Wu et al.,, 2020) for the task of temporal link prediction rather than modeling discrete marked events. Another concurrent study of graph point processes (Dong et al., 2023a, ) shares similarities with ours by approximating influence kernels using graph neural networks. However, they do not consider the spatial aspect of the data.

Urban areas with different facilities and functionalities, known as urban functional zones (Yuan et al.,, 2014), can have different crime patterns based on the citizen activities exhibited in those areas (Kinney et al.,, 2008). For instance, commercial or public places attract more human activities and, potentially, more crime and disorder events (Andresen,, 2007; Wuschke and Kinney,, 2018). The identification of the urban functional zones is critical for implementing targeted crime prevention strategies and mitigating potential hotspots.

In our study, we partition the entire city area of Valencia into various urban functional zones based on the 1,975 city landmarks. This approach aligns with the point-of-interest (POI) method commonly referenced in the literature (Gao et al.,, 2017; Hu and Han,, 2019; Long et al.,, 2015; Yuan et al.,, 2014), which involves geographic entities that can be abstracted as points for zone identification, such as schools, banks, companies, restaurants, and supermarkets (Jiang et al.,, 2015). Specifically, we use the $k$-nearest neighbors algorithm to identify different functional zones based on their proximity to the city landmarks. For a given location $s\in\mathcal{S}$, we find its $k$ nearest landmarks and assign to it a landmark category $l\coloneqq\ell(s)$ as the most common landmark category among the $k$ landmarks. Thus, the function

serves a labeling mechanism that maps each location within the city $\mathcal{S}$ to a corresponding landmark category in the set $\mathscr{L}$. Locations sharing the same landmark category (e.g., $l$) are grouped to form the functional zone $\mathcal{S}_{l}$, and we have $\mathcal{S}=\cup_{l\in\mathscr{L}}\mathcal{S}_{l}$. The left panel in Fig 2 visualizes the partition of urban functional zones in Valencia.

To accurately depict patterns of criminal activity across the city, it is crucial to consider contextual information about crime events, such as the type of crime and the environment setting in which it occurs. Currently, crimes are grouped by their crime types, for instance, Assault (or Agresión, in the original name). This categorization, however, may overlook important contextual differences. For instance, an assault near a restaurant and another near a bank are both categorized under Assault, despite the distinct human activity patterns typical of dining and financial areas. By refining our crime categorization to account for these specific environment settings, we can enhance our understanding of crime dynamics.

We design a novel mark associated with each event (Reinhart,, 2018) to categorize the crime events. The mark is designed to combine the event’s crime category $c$ and the landmark category $\ell(s)$ of its location $s$, thus considering the urban functional zone that the event falls in. We denote the event mark as $c\times\ell(s)$. For instance, as illustrated in the right panel of Fig 2, the Assault occurring in a restaurant zone on July 9th, 2016, is assigned the label $1\times 6$ (representing Assault$\times$ restaurant), while another Subtraction on October 12th, 2018, in a financial zone receives the label $2\times 1$ (representing Subtraction$\times$ financial). The value space of the mark is a finite set $\mathscr{C}\times\mathscr{L}$ with a size of 21 (three crime categories and seven landmark categories). We refer to the mark “crime-landmark label” of the event in the later discussion to reveal its practical meaning. As we can see, this new mark derives a comprehensive categorization of crime events by including contexts of the observed event. Meanwhile, it allows for a detailed examination of crime patterns across different urban functional zones by analyzing incidents through the lens of their specific crime-landmark labels.

Crime events are ordered in time, and historical events will impact the probability, timing, or characteristics of future events (Mohler et al.,, 2011; Loeffler and Flaxman,, 2018). Such an impact is referred to as event dependence. To model the dependence among temporal events, we consider their relations over the geographic space with an underlying street network structure and the mark space (crime-landmark labels) characterized by an interaction network.

To model the spatial relationship between crime events on the urban streets of Valencia, we overlay a street network structure $G$ on top of the continuous geographic space $\mathcal{S}$. This street network is constructed using the data from OpenStreetMap database (OpenStreetMap contributors,, 2017). The streets in Valencia are represented as linear segments linked at their endpoints. Note that the endpoints of these segments do not necessarily align with actual street intersections; they can be located in the middle of a street, such as on a curved street divided into multiple segments. These endpoints are treated as the nodes of the street network, while the street segments become the edges connecting these nodes. Each network edge is associated with an attribute, known as the edge weight, indicating the length of the corresponding street segment measured in kilometers. The network is processed to be undirected to reflect the mobility patterns in street crimes in Valencia, where perpetrators commonly travel on foot or by bike in either direction along the streets (Bounce,, 2024). The street network consists of 8,043 nodes and 12,309 weighted, undirected edges, covering the entire city area of Valencia. It is worth noting that crime events are integrated into this network by being mapped to random locations on the network edges based on their geographic coordinates rather than being assigned to specific nodes. Two layers at the bottom in Fig 3 illustrate such an overlay of the street network $G$ and the mapping of the crime events to the network edges.

Understanding the relation between event marks also provides valuable insights into characterizing the dependencies of crime events. By analyzing the sequence of observed marks, we can determine if certain events tend to be triggered by others in a specific pattern. In this study, such dependencies are represented through a mark network, denoted as $A$. Each node of the mark network represents a distinct crime-landmark label (total of 21 nodes), and the events are assigned to the corresponding nodes based on their crime-landmark labels. The edges between these nodes indicate the potential relation between the crime-landmark labels they connect with. Such a relation can be directional, i.e., an observed crime with label $c\times\ell(s)$ may influence the occurrence of a future event with label $c^{\prime}\times\ell(s^{\prime})$ but not vice versa. Hence, the edges of the mark network are directional. Unlike the street network, which is derived directly from available geographic data, the mark network is established by learning a point process model detailed in the next section from the crime data. This model learns from the crime data to establish the directed and weighted edges of the mark network, indicating both the direction and strength of the dependencies between different event marks. An example of the mark network is presented at the top of Fig 3, highlighting directional relations among various event marks (crime-landmark labels).

Self-exciting spatio-temporal point processes (Moller and Waagepetersen,, 2003; Reinhart,, 2018) are widely used in crime modeling to capture the contagious nature of crime events (Mohler et al.,, 2011). Let $\mathcal{H}_{t}=\{(t_{i},s_{i},c_{i}\times l_{i})\in\mathcal{H}_{T}|t_{i}<t\}$ denote the observed crime events happened before time $t$; we adopt a conditional intensity function for each event category $c\times l$ to suggest the possibility of observing a new event with label $c\times l$ conditioning on the history. Specifically, the conditional intensity function at time $t$ and location $s$ is defined as

where $B(s,\Delta s)$ is a ball centered at location $s$ with radius $\Delta s$. The $\mathbb{N}_{cl}$ is the counting measure for events with label $c\times l$, i.e., $\mathbb{N}_{cl}(A)$ is defined as the number of events with label $c\times l$ occurring within any subset $A\subseteq[0,T]\times\mathcal{S}$. This function essentially measures the rate at which events are expected to occur at a specific time and place based on historical data, with $\lambda_{cl}\left(t,s|\mathcal{H}_{t}\right)\geq 0$ for any arbitrary $c$, $l$, $t$ and $s$. To simplify the notation, we omit the $\times$ between $c$ and $l$ in the subscript.

Hawkes processes proposed in (Hawkes,, 1971) provide the self-exciting model formulation for capturing the triggering effects among events. It assumes that the occurrences of future events are positively influenced by the observed history, and the influence of past events is linearly additive. In this study, we model the conditional intensity function as follows:

Here, $\mu_{cl}$ is a constant representing the base intensity of events with label $c\times l$. The $k$ function is the so-called influence kernel that captures the influence of a past incident $(t^{\prime},s^{\prime},c^{\prime}\times l^{\prime})$ on a current event $(t,s,c\times l)$. This formulation allows for characterizing the influence of historical events on the likelihood of future events within the framework of the Hawkes process.

A separable form of the influence kernel has been commonly assumed in previous literature (Dong et al., 2023b, ; Mohler,, 2014; Reinhart,, 2018; Reinhart and Greenhouse,, 2018; Zhu and Xie,, 2022). The influence kernel $k$ can be expressed by the product of three individual kernel functions as

The kernel functions $f,g,h$ characterize the event influence over the space of times, locations, and event marks, respectively. We note that the separable form of the influence kernel enables a computationally efficient procedure for model fitting, given the large size of the data set. Meanwhile, the separable influence kernel can also provide us with interpretable results, as illustrated in Section 6. In the following, we introduce the construction of these kernel functions in our context of modeling the street crime events within an urban environment.

The learning of the coefficients $\{\alpha_{cl,c^{\prime}l^{\prime}}\}_{c,c^{\prime}\in\mathscr{C},l,l^{\prime}\in\mathscr{L}}$ plays an essential role in understanding the mark interactions and the characterization of the event dynamics. Traditionally, these coefficients have been directly estimated from data, as outlined in various studies (Mohler,, 2014; Reinhart and Greenhouse,, 2018; Zhu and Xie,, 2022). However, recent advancements in point process models have showcased the value of incorporating prior knowledge of event marks, known as features, into the modeling of these coefficients and the mark interactions. For example, when modeling the interactions between different social media users (marks), the work of Group Network Hawkes Process (Fang et al.,, 2023) treats these users as network nodes and leverages their characteristics (the features of the marks) to effectively identify the group interactions and influential users in social networks. In our case, the event marks are defined by the combinations of multiple crime and landmark categories. It is reasonable to believe that marks sharing the same crime or landmark category tend to exhibit stronger interactions than those with differing categories. Therefore, these crime or landmark categories that compose the mark can be regarded as the mark features that we can leverage in the coefficient modeling.

We introduce a novel approach via GNNs to model the coefficients, leveraging their ability to integrate nodal features in learning node similarity. We first decompose the coefficients into two components as follows:

where $a_{cl,c^{\prime}l^{\prime}}>0$ and $0\leq p_{cl,c^{\prime}l^{\prime}}\leq 1$ are both scalars. Together, these two components can be viewed as the strength and the chance of the interaction between two marks. The term $p_{cl,c^{\prime}l^{\prime}}$, modeled by a GNN, will incorporate the mark features and capture the graph topology by providing the likelihood for any $c\times l$ to have a connection to the rest of $c^{\prime}\times l^{\prime}$; meanwhile, the $a_{cl,c^{\prime}l^{\prime}}$ captures the weights on the edges in the mark network, indicating the strength of the connection.

We model these two components separately. For the strength $a_{cl,c^{\prime}l^{\prime}}$, we treat it as a trainable scalar that is learned from data. For the modeling of the chance, we use Graph Attention Networks (GAT) (Veličković et al.,, 2018) to take the mark features into account. The feature of mark $c\times l$ can be denoted by a column vector $X_{cl}\in\mathbb{R}^{D}$, which is the concatenation of the one-hot vectors of the crime category $c$ and the landmark category $l$. Based on the feature vectors, GAT computes the chance of the influence from mark $c^{\prime}\times l^{\prime}$ to mark $c\times l$ through a so-called multi-head self-attention mechanism over graphs. The attention mechanism in the $r$-th attention head is carried out as

Here, the weight matrix $\bm{W}^{r}\in\mathbb{R}^{D^{\prime}\times D}$ defines a shared linear transformation of each mark feature, and $\gamma\colon\mathbb{R}^{D^{\prime}}\times\mathbb{R}^{D^{\prime}}\to\mathbb{R}$ takes a pair of transformed features as input and computes the attention weight $e^{r}_{cl,c^{\prime}l^{\prime}}$ (defined in the next equation). The attention weights are normalized through the softmax function, which leads to the output of the $r$-th head

Here, $\|$ stands for the concatenation operation, and the Leaky ReLU (Maas et al.,, 2013) is an element-wise non-linear function defined as ${\rm LeakyReLU}(x)=\max(0,x)+b\min(0,x)$. In our numerical experiments, we set $b=0.2$ as used in the original GAT (Veličković et al.,, 2018). The results from different attention heads are averaged to get the final chance $p_{cl,c^{\prime}l^{\prime}}$ as

Note that the GAT introduces a constraint of $\sum_{c^{\prime}\in\mathscr{C},l^{\prime}\in\mathscr{L}}p_{cl,c^{\prime}l^{\prime}}=1$ to ensure that the $p_{cl,c^{\prime}l^{\prime}}$ collectively form a probability distribution over potential connections. The hyper-parameter to be determined in advance is the number of attention heads $R$ to achieve the balance between model flexibility and generability. The learnable parameters are $\{\bm{b}^{r},\bm{W}^{r}\}_{r=1}^{R}$ in GAT and the interaction strength $\{a_{cl,c^{\prime}l^{\prime}}\}_{c,c^{\prime}\in\mathscr{C},l,l^{\prime}\in\mathscr{L}}$.

We first validate our model from two aspects: the determination of the hyper-parameter $R$ and the goodness-of-fit of the chosen model on the crime data.

An appropriate choice of the number of attention heads $R$ in GAT needs to be determined in advance, which can be achieved using cross-validation. We first divide the training data from 2015 to 2018 into 12 subsequences with the same time window length of 120 days. Then, we adopt 4-fold cross-validation on the training data to determine the value of $R$. Given a choice of $R$, all the 12 subsequences are shuffled randomly and split into four groups. One round of cross-validation involves taking one group as the hold-out data, training the model with the remaining groups, and evaluating the trained model on the hold-out data. The final model performance is obtained by averaging the metrics over four independent rounds. We compare the performance of the model with $R=2,4,6,8,12$, and $16$ attention heads in terms of the model log-likelihood on the hold-out data, which evaluates the model generalization ability to the unseen data. Better model performance is indicated by a higher hold-out log-likelihood. Fig 5 reports the averaged hold-out log-likelihood with different attention heads. According to the results, we choose $R=8$ as an optimal choice in the remaining experiments in this study.

Another model assumption – the stationarity of the influence kernel needs to be validated by investigating the model’s fit with the training data. Stationarity means that the model parameters do not vary over time, indicating that the pattern of event influence remains consistent. Previous research on crime modeling with point processes has frequently made this assumption, but often without adequately verifying its validity. The work of the non-stationary ETAS model (Kumazawa and Ogata,, 2014) presents a method to test the goodness-of-fit of a stationary point process model to the data by comparing the expected cumulative number of events computed from the learned model and empirical cumulative number of events. In our context, given the learned model $\hat{\lambda}_{cl}$, the expected cumulative number of events in the time interval $[0,t]$ is computed as $\Lambda(t)=\int^{t}_{0}\int_{\mathcal{S}}\sum_{c\in\mathscr{C},l\in\mathscr{L}}\hat{\lambda}_{cl}(t,s)dsdt$. If the model represents a good approximation of the real data, we expect that $\Lambda(t)$ and the empirical cumulative event counts $\mathbb{N}(t)=\sum_{c\in\mathscr{C},l\in\mathscr{L}}\mathbb{N}_{cl}(t)$ are close. We fit the model using the entire training set, and plot the $\mathbb{N}(t)$ and $\Lambda(t)$ from 2015 to 2018 in Fig 6. The consistent match between the empirical and expected cumulative event numbers suggests that the underlying data dynamics are stationary, and the assumption of kernel stationarity is reliable when fitting this data set.

We then fit the model on the entire training data from 2015 to 2018 and analyze the results. To quantitatively demonstrate the effectiveness of our model, we compare our model with different baselines in terms of the fitted log-likelihood on the training data, the Akaike Information Criterion (AIC) (Akaike,, 1974, 1998) of the model, and the mean absolute error (MAE) of the in-sample estimation of the number of the crime events. The log-likelihood, computed by (5) using training data, measures the model goodness-of-fit to the training data. The AIC considers both the model fit to the data and the model complexity. It is described as $AIC=-2\max_{\theta}\log L(\theta)+2k$, where $\log L(\theta)$ is the model log-likelihood and $k$ is the number of model parameters to be estimated. The in-sample estimation of the event number over a given time interval can be performed as follows: we fit the model using the entire training data, feed the same data into the fitted model, and calculate the integral of the conditional intensity function over the time interval as the estimated number of events. In practice, the in-sample estimation of number of events with mark $c\times l$ over $[t_{1},t_{2}]$ can be calculated by $\int_{t_{1}}^{t_{2}}\int_{\mathcal{S}}\hat{\lambda}_{cl}(t,s)dsdt$. We evaluate the in-sample estimation of event numbers during each week using our model STNPP, and compare its performance with four baselines, including two predictive time series models, one point process model, and an ablated variant of our model: (1) The persistence forecast (Persistent) that uses the event number in the previous week as the estimation; (2) Vector autoregression (VAR), which is a statistical model for analyzing and predicting multivariate time series data; (3) Epidemic-type aftershock sequence (ETAS) model (Ogata,, 1998) with a diffusion-type kernel using Euclidean distance; (4) The STNPP without GAT (STNPP-GAT). We slightly modify the ETAS model by incorporating a set of coefficients to account for the interactions between different event marks, since the original ETAS model cannot deal with multiple event types (see Appendix B for details). Fig 7 visualizes the in-sample estimations by different models on the number of each event type and the total events from 2015 to 2018, alongside the actual observed values. Our model effectively recovers both the overall temporal trend in total event numbers and the specific temporal patterns for event types that occurred either frequently or infrequently during the training period. Note that such heterogeneity in the event dynamics is simultaneously captured by a holistic model instead of fitting independent models for each type of event.

More quantitative results about the in-sample estimations are summarized in Table 1. To showcase our model’s versatility in handling different types of events with distinct underlying mechanisms, we present separate in-sample estimation MAE assessments for events characterized by frequent or rare occurrences of the marks. The frequent event marks include those with landmark categories of “financial,” “industrial,” and “restaurant,” corresponding to the crime-landmark categories with the top nine total observations. The remaining crime-landmark categories are treated as rare event marks. We report MAE (rare), MAE (frequent), and MAE (total) as the final metrics, representing the estimation MAEs averaged over rare, frequent, and all event marks. The results in Table 1 demonstrate the comparable or superior predictive performance of STNPP against baselines. Note that although VAR has performance metrics that are close to our method, it is a time series model for predicting the event numbers, and it is not designed for dealing with discrete spatio-temporal events or providing any insights on the underlying event dynamics.

Table 1 also reports the training log-likelihood and the model AIC for three spatio-temporal point processes (we omit the comparison with AIC of VAR, which is not meaningful). The highest training log-likelihood and the lowest model AIC show that STNPP enjoys the best goodness-of-fit to the data. Besides, the improved performance from ETAS to STNPP-GAT highlights the advantages of using street-network distance, and the performance gain of STNPP against STNPP-GAT emphasizes the benefits of incorporating nodal (mark) features in capturing complex event dynamics.

The model’s predictive power can be assessed by the out-of-sample prediction task on the testing data set. We perform a one-week-ahead prediction of the number of events over the time window of 2019. Specifically, at a given time $t^{*}$ in 2019, we feed the data before $t^{*}$ into the learned model and evaluate the conditional intensity function over the next week. The predicted number of events with mark $c\times l$ in the following week can be estimated by the integral of the evaluated intensity function $\hat{\lambda}_{cl}$ over time and space, similarly as those in the in-sample predictions. The MAE between the predicted event numbers and the number of true observations are computed to indicate the model’s predictive performance. We perform the out-of-sample prediction on a weekly basis over the year of 2019 and report the average prediction MAE. Table 2 presents the average MAEs of the predictions for the number of rare events, frequent events, and total events by our model STNPP and four baselines, indicating the superior performance of our model against other baselines on predicting the future. Besides the MAE, we also compare the fitted log-likelihood of the testing data using different point process models and report them in the table. The highest log-likelihood of STNPP showcases the best generalization ability of our model to the unseen data.

We visualize the predictive power of STNPP in Fig 8 by showing the predicted conditional intensity function for three types of crimes over the street network at different times in 2019. Each panel compares the predicted conditional intensity of one type of crime over space given the observed history with the true distribution of that type of crime in the next two days. As we can observe, the predicted event intensity by our model is consistent with the true distribution of future events, showing a higher intensity in those areas with a higher likelihood of observing crime events. Meanwhile, our model discerns the spatial patterns of different crimes by learning from the historical data, such as the risk for Assault victims in major busy areas (e.g., the financial zone in the north part of the city) and a more regional, concentrated pattern for Subtraction and Others.

The coefficients $\{\alpha_{cl,c^{\prime}l^{\prime}}\}_{c,c^{\prime}\in\mathscr{C},l,l^{\prime}\in\mathscr{L}}$ learned by GAT capture the direction and magnitude of the influence between different event marks and is crucial in interpreting the model in practice. We visualize the learned coefficients by our model STNPP in Fig 9(a) by stacking them together into a matrix. Each matrix entry represents the coefficient that models the triggering effect from the event mark at the corresponding column to the mark at the corresponding row. As the matrix can be regarded as the weighted adjacency matrix of the mark network we established in Section 3.3, we adopt the Louvain algorithm (Blondel et al.,, 2008) to perform community detection on the event marks. The detected communities tell us the groups of marks that are more closely connected, which are indicated by the square frames with red dashed lines in the visualized matrix. Five communities are detected based on the coefficients, suggesting different types of human daily activities. For instance, the largest community with six marks, including Assault and Subtraction in industrial, market, and nightclub zones, showcases the clustering patterns of certain crime events related to citizen activities after hours, such as grocery shopping or night amusement. Other communities also reveal criminal activities that are relevant to specific urban facilities, such as restaurants (the first community) and industrial zones (the last community).

We also visualize the mark network $A$ established from the learned coefficients in Figure 9(c), with nodes representing the event marks and edges indicating their interactions. The colors of the nodes suggest the detected communities of different marks. To demonstrate the benefits of adopting GAT to learn the coefficient and their community structure, we compare the learned coefficients by the ablated model STNPP-GAT in Fig 9(b)(d) with detected communities. Although we have no ground truth to validate the community detection results, we here report the modularity (Newman,, 2010) of the mark networks. Networks with higher modularity have stronger intra-community connections and fewer inter-community connections. The modularity of the learned mark network by STNPP is much higher than the one learned by STNPP-GAT, as reported in Fig 9(c)(d). These visualizations also reveal a more distinct community structure in the network learned by STNPP, in contrast to the one of STNPP-GAT, which has more blurred community divisions. This high modularity of the mark network is beneficial for the decision-making of the local police department. For example, the detected communities highlight those closely connected marks and help identify influential crime events within specific communities. These insights can lead to more targeted and effective police patrolling against criminal activities.

To identify the most influential event marks, we plot the expected number of events that are triggered by an observed event with each mark in Fig 10. The number of triggered events by one event with mark $c^{\prime}\times l^{\prime}$ is calculated by aggregating the coefficient $\alpha_{cl,c^{\prime}l^{\prime}}$ over the index $c$ and $l$, i.e., $\sum_{c\in\mathscr{C},l\in\mathscr{L}}\alpha_{cl,c^{\prime}l^{\prime}}$, and a larger number indicates a stronger influence by an observed event with mark $c^{\prime}\times l^{\prime}$. As we can see, crime events with marks of Assault$\times$taxi have the strongest influence on the future by triggering the most number of events. Although this event mark is barely observed during the five-year period, its impact on subsequent event occurrences is not negligible. Other influential event marks include those related to police zones or assault activities, indicating the heterogeneity of the event dynamics across different crime types and urban areas.

Another important task for local practitioners in implementing effective prevention strategies is to identify particular types of crime events that can lead to an obvious risk increase in the community’s exposure to the crimes. These events not only include those that trigger the subsequent event occurrences to a large extent, such as Assault$\times$taxi, but also include those event types that have a smaller influence magnitude on future events but are frequently reported, such as Assault$\times$restaurant.

To this end, we investigate the contribution of each event mark $c^{\prime}\times l^{\prime}$ to the underlying event generation mechanisms by neutralizing its influence (i.e., set $\alpha_{cl,c^{\prime}l^{\prime}}=0,\forall c\in\mathscr{C},l\in\mathscr{L}$) and then evaluating the performance gap between the reduced model and the original model. A larger performance gap indicates a higher importance of that event mark in the effectiveness of the model. We evaluate the performance of each reduced model with influence from one type of event mark neutralized (a total of 21 reduced models) in terms of three metrics: AIC on training data, negative log-likelihood on testing data, and out-of-sample prediction MAE, and compare them with the metrics obtained by the full model. The corresponding results are presented in Fig 11. For the top two marks that have the most expected number of triggered events shown in Fig 10, the neutralization of the influence of Assault$\times$taxi leads to obvious performance degradation, while the other one of Others$\times$police have a much smaller impact, due to the scarce observation of this event mark during the investigation period. Other event marks, to which the neutralization of the influence can significantly decline the model performance, include those of Assault$\times$financial, Assault$\times$industrial, Assault$\times$restaurant, Subtraction$\times$restaurant, and so on. Crime events with these marks relate to those places and the daily activities of citizens who are more vulnerable to criminals. These events are more frequently observed than others, and their cumulative effects on attracting future criminal activities need to be paid attention to.