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Efficient modeling of higherorder dependencies in networks: from algorithm to application for anomaly detection
EPJ Data Science volume 9, Article number: 15 (2020)
Abstract
Complex systems, represented as dynamic networks, comprise of components that influence each other via direct and/or indirect interactions. Recent research has shown the importance of using HigherOrder Networks (HONs) for modeling and analyzing such complex systems, as the typical Markovian assumption in developing the First Order Network (FON) can be limiting. This higherorder network representation not only creates a more accurate representation of the underlying complex system, but also leads to more accurate network analysis. In this paper, we first present a scalable and accurate model, BuildHON+, for higherorder network representation of data derived from a complex system with various orders of dependencies. Then, we show that this higherorder network representation modeled by BuildHON+ is significantly more accurate in identifying anomalies than FON, demonstrating a need for the higherorder network representation and modeling of complex systems for deriving meaningful conclusions.
Introduction
Networks are a popular way of representing rich and sparse interactions among the components of a complex system. It is, thus, critical for the network to truly represent the inherent phenomena in the complex system to avoid incorrect conclusions. Conventionally, edges in networks represent the pairwise interactions of the nodes, assuming the naive Markovian property for node interactions, resulting in the firstorder network representation (FON). However, the key question is—is this accurately representing the underlying phenomena in the complex systems? And if the network is not accurately representing the inherent dependencies in the complex system, can we trust the analysis and results stemming from this network? The Markovian assumption for network modeling of complex system can be limiting for network analysis tasks, including community detection [3, 4], node ranking [5], and dynamic processes [6] in timevarying complex systems.
Recent research has brought to fore challenges with the FON view, especially its limitations on capturing the sequential patterns or higher and variableorder of dependencies in a complex system and its impact on resulting network analysis. This has led to the development of network representation models that capture such higherorder dependencies, going beyond the traditional pairwise Markovian network representation [1, 2].
Our prior work [2] tackles the limitations stemming from the Markovian assumption for node interactions (as in FON), and proposes BuildHON for extracting higherorder dependencies from sequential data to build the HigherOrder Network (HON) representation. BuildHON, although accurate, faced the challenge of computational complexity as well as parameter dependency. In this work, we address these limitations by proposing a scalable and parameterfree algorithm, BuildHON+, for accurate extraction of higherorder dependencies from sequential data. Given BuildHON+, we are also interested in downstream network analysis tasks, adn we focus on the following question in this paper that has not been addressed in prior HON work: Does incorporating higherorder dependencies improve the performance of existing networkbased methods for detecting anomalous signals in the sequential data?
To answer the above question, we define anomalies (or change points) as deviations from the norm or expected behavior of a complex system. We note that the anomalies could also be important change points in the behavior of the complex system. The key here is to be able to accurately flag such deviations or events in a complex system. While there exists a wide range of anomaly detection methods on dynamic networks [7, 8], all of them use the firstorder network (FON) to represent the underlying raw data (such as clickstreams, taxi movements, or event sequences), which can lose important higherorder information [2, 3]. As FON is an oversimplification of higherorder dynamics, we hypothesize that anomaly detection algorithms that rely on FONs will miss important changes in the network, thus leaving anomalies undetected. We systematically demonstrate why existing networkbased anomaly detection methods can leave certain signals undetected, and propose a higherorder network anomaly detection framework. Consider the following example.
Example
Fig. 1 illustrates the challenge of detecting certain types of anomalies, using a minimal example of web clickstreams data (sequences of web page views produced by users) collected by a local media company. Given the web clickstreams as the input to networkbased anomaly detection methods, conventionally, a web traffic network is built for each time window (two onehour windows illustrated here), with the nodes representing web pages and the edges representing total traffic between web pages. A change in the network topology indicates an anomaly in web traffic patterns. According to the original clickstreams, in the first hour, all users coming from the soccer web page to the weather page proceed to the ticket page, and all users coming from the skating page to the weather page go to TV schedules. But the flow of users is completely flipped in the next hour, possibly the weather forecast has updated with much colder weather which is in favor of winter activities. However, despite the significant changes in user web viewing patterns, the pairwise traffic between web pages in this example remains the same, thus the FON topology shows no changes. Therefore, no matter what networkbased anomaly detection method is used, if the method relies on FON, the company will not be able to detect such type of anomalies, thus failing to respond (e.g., caching pages for visits, or targeted promotion of pages) to the changes in user behaviors.
Contributions. We make three main contributions in the paper.
We develop a scalable and parameterfree algorithm for higherorder network representation, BuildHON+, building on our prior work [2]. We demonstrate the efficiency of BuildHON+ through comprehensive complexity and performance analysis on the global ship movement data, which is known to exhibit dependencies beyond the fifth order.
We showcase the performance of BuildHON+ in the task of networkbased anomaly detection on a realworld taxi trajectory data. We explain why the parameter dependency in our prior work can be limiting for efficient network construction and as a result, anomaly detection.
Using a largescale synthetic taxi movement data with 11 billion taxi movements, we show how multiple existing anomaly detection methods that depend on FON collectively fail to capture anomalous navigation behaviors beyond firstorder, and how BuildHON+ can solve the problem.
Related work
Higherorder networks. Recent research has highlighted the limitations of the conventional network model for representing the sequential and indirect dependencies between the components of complex systems. Multilayer higherorder models [9, 10], motif and cliquebased higherorder models [4, 11, 12], and nonMarkovian higherorder models [2, 3, 6] try to embed complex patterns that are stemming from the raw data into the network representation. Specifically, nonMarkovian network models has gained a lot of attraction in many applications including social networks [13, 14], human transportation networks [2, 3, 6, 15], trade networks [16, 17], and citation networks [3]. Several research studies show how incorporating higherorder dependencies affects various network analysis tasks, including community detection [3, 4], node ranking [5], and dynamic processes [6] in the network. However, from current research studies, it is unclear what is the effect of using a higherorder network model on detecting anomalies in dynamic networks.
Anomaly detection in dynamic networks. Unlike the task of detecting anomalous nodes and edges in a single static network (such as [18]), anomaly detection in dynamic networks [7, 19] uses multiple snapshots of networks to represent the interactions of interest (such as interacting molecules [20], elements in frames of videos [21], flow of invasive species [22], etc.), then identifies the time when the network topology shows significant changes, using network distance metrics [23–25], probability methods [26], subgraph methods like [27] and more. There are many advantages of using networkbased methods for the task of anomaly detection in sequential data. Aside from the availability of several different networks, a graph structure represents the relational nature of the data, which is essential for addressing the anomaly detection problem [7]. Furthermore, the interdependencies of the raw data can be captured more efficiently with graph representation. This feature can be further enhanced in the higherorder representation of the network, as done in this work. The importance of higherorder patterns in different network analysis tasks has gained a lot of attention recently [1, 28]. However, one of the major challenges is that the graph search space is very large, requiring the anomaly detection methods to be scalable and efficient for large data sets [7].
Moreover, using snapshots of networks may cause the finegrained timestamps to be lost. Therefore, the optimal timestamp is often datadependent and should be identified empirically through sufficient experiments.
Nevertheless, existing methods on anomaly detection rely on conventional FON; as we will show, certain types of anomalies cannot be detected with any networkbased anomaly detection methods if FON is used. Rather than proposing another approach to identify the anomalous network from a series of networks, our innovation lies in the network construction step, which ensures anomalous signals are preserved in the network in the first place.
Methods
We first present a scalable and parameterfree approach for constructing HON, namely BuildHON+. We then show how this new approach enables more accurate anomaly detection (compared to using FON) by incorporating several different network distance measures. Our previous algorithm, BuildHON required two parameters that had to be specified experimentally, depending on the data set. Furthermore, it uses an exhaustive search for extracting the dependency rules and constructing the network, which becomes impractical for various network analysis tasks, including anomaly detection. It needs two parameters in addition to the detection threshold: a MaxOrder parameter which governs how many orders of dependencies the algorithm will consider in HON, and a MinSupport parameter that discards infrequent observations. These limitations mitigate its applicability to Big Data.
BuildHON+: building HON from big data
Here we introduce BuildHON+, a parameterfree algorithm that constructs HON from big data sets. BuildHON+ is a practical approach that preserves higherorder signals in the network representation step (\(S_{i} \rightarrow G_{i}\)) which is essential for anomaly detection. The difference between BuildHON and BuildHON+ is similar to the difference between pruning and early stopping in decision trees. BuildHON first builds a HON of all orders from firstorder to MaxOrder and then selects branches showing significant higherorder dependencies. BuildHON+ reduces the search space beforehand by checking in each step if increasing the order may produce significant dependencies. Furthermore, BuildHON can only discover dependencies up to MaxOrder. BuildHON+ however, finds the appropriate dependency order hidden in the raw data and is not limited by MaxOrder. Therefore, the output network resulting from BuildHON+ is a more reliable and accurate representation of the raw data, which is essential for the task of anomaly detection.
The core of BuildHON is the dependency rule extraction step, which answers whether higherorder dependencies exist in the raw sequential data, and how high the orders are. The dependency rules extracted are then converted to higherorder nodes and edges as the building blocks of HON. Rather than deriving a fixed order of dependency for the whole network, the method allows for variable orders of dependencies for more compact representation. Figure 2 illustrates the dependency rule extraction step. BuildHON first counts the observed ngrams in the raw data (step ), then compute probability distributions for the next steps given the current and previous steps (step ). Finally test if knowing one more previous step significantly changes the distribution for the next step—if so, higherorder dependency exists for the path (step ); this procedure (“rule growing”) is iterated recursively until a predefined MaxOrder (shown here \(\mathit{MaxOrder}=3\)). In this example, the probability distribution of the next steps from C changes significantly if the previous step (coming to C from A or B) is known (step ), but knowing more previous steps (coming to C from \(E \rightarrow A\) or \(D\rightarrow B\)) does not make a difference (step ); therefore, paths \(CA \rightarrow D\) and \(CA \rightarrow E\) demonstrate secondorder dependencies.
Formally, the “rule growing” process works as follows: for each path (ngram) \(\mathcal{S} = [S_{tk}, S_{t(k1)}, \dots , S_{t}]\) of order k, starting from the firstorder \(k=1\), assume k is the true order of dependency, which \(\mathcal{S}\) has the distribution D for the next step. Then extend \(\mathcal{S}\) to \(\mathcal{S}_{\mathrm{ext}} = [S_{t(k+1)}, S_{tk}, S_{t(k1)}, \dots , S_{t}]\) by adding one more previous step; \(\mathcal{S}_{\mathrm{ext}}\) has order \(k_{\mathrm{ext}}=k+1\) and distribution \(D_{\mathrm{ext}}\). Next, test if \(D_{\mathrm{ext}}\) is significantly different than that of D using Kullback–Leibler divergence [29] as \(\mathcal{D}_{\mathrm{KL}}(D_{\mathrm{ext}}D)\), and compare with a dynamic threshold δ—if the divergence is larger than δ, order \(k+1\) is assumed instead of k for the path \(\mathcal{S}_{\mathrm{ext}}\). The dynamic threshold δ is defined as \(\delta = \frac{k_{\mathrm{ext}}}{\log _{2} (1+ \mathit{Support}_{\mathcal{S}_{\mathrm{ext}}})}\), so that lower orders are preferred than higherorders, unless higherorder paths have sufficient support (number of observations). The whole process is iterated recursively until MaxOrder.
Eliminating all parameters
The reason for having the MaxOrder and MinSupport parameters in BuildHON is to set a hard stop for the rule growing process, otherwise, it will iterate indefinitely and keep extending \(\mathcal{S}\). However, we show that we can predetermine if extending \(\mathcal{S}\) will not produce significantly different distributions, which forms an important basis for BuildHON+.
Lemma 1
The significance threshold\(\delta = \frac{k_{\mathrm{ext}}}{\log _{2} (1+ \mathit{Support}_{\mathcal{S}_{\mathrm{ext}}})}\)increases monotonically in rule growing when expanding\(\mathcal{S}\)to\(\mathcal{S}_{\mathrm{ext}}\).
Proof
On the numerator, the order \(k_{\mathrm{ext}}\) of the extended sequence \(\mathcal{S}_{\mathrm{ext}}\) increases monotonically with the inclusion of more previous steps. Meanwhile, every observations of \([S_{t(k+1)}, S_{tk},\dots , S_{t1}, S_{t}]\) in the raw data can find a corresponding observation of \([S_{tk}, \dots , S_{t1}, S_{t}]\), but not the other way around. Therefore, the support of \(S_{\mathrm{ext}}\), \(\mathit{Support}_{S_{\mathrm{ext}}} \leq \mathit{Support}_{S}\) of the lower order \(k = k_{\mathrm{ext}} 1\). As a result, the denominator decreases monotonically with the rule growing process. □
Given the next step distribution \(D=[P_{1}, P_{2}, \dots , P_{N}]\) of sequence \(\mathcal{S}\), we can derive an upperbound of possible divergence:
The equal sign (maximum possible divergence) is taken iff the least likely option for the next step \(P(i)\) in \(\mathcal{S}\) becomes the most likely option \(P_{\mathrm{ext}}(i)=1\) in \(\mathcal{S}_{\mathrm{ext}}\), and all other options have \(P=0\). Therefore, we can test if \(\log _{2}(\min (P_{\mathrm{Distr}}(i))) < \delta \) holds during the rule growing process; if it holds, then further increasing the order (adding more previous steps) will not produce significantly different distributions, so we can stop the rule growing process and take the last known k (which passed the actual divergence test, not the order which passes the maximum divergence test) as the true order of dependency. Note that, the dynamic threshold is chosen heuristically in its current form. This threshold meets our design requirements: (1) enforce higher support for higherorders, and (2) fast to compute, as it is a frequently used module in the innermost loop.
Furthermore, BuildHON+ no longer requires a MinSupport parameter. Recall that using \(\mathit{MinSupport} >1\) in BuildHON helps reduce the search space as a crude form of early stopping, with the risk of losing valid higherorder patterns. In BuildHON+, the dynamic threshold takes care of early stopping without requiring any extra parameter (MinSupport) to limit the search space. This parameter is left in the algorithm only for backward compatibility and is set to 1 by default, but does not serve any initial seeding purpose. In other words, MinSupport is not used in BuildHON+.
An advantage of this proposed parameterfree approach is that rather than terminating the rule growing process prematurely by the MaxOrder threshold, the algorithm can now extract arbitrarily orders of dependency.
Scalability for higherorders
BuildHON builds all observations and distributions up to MaxOrder ahead of the rule growing process (Fig. 2 left). This procedure becomes prohibitively expensive for big data with high orders of dependencies: to extract sparse tenth order dependencies, BuildHON needs to enumerate ngrams from firstorder to tenth order and compare probability distributions, which already exceeds a personal computer’s capacity using a typical realworld data set (see Sect. 4).
BuildHON+, on the other hand, uses a lazy construction of observations and distributions that has a much smaller search space, and can easily scale to arbitrarily high order of dependency. Specifically, BuildHON+ does not require the counting of the occurrences of ngrams or calculating the distribution of the next steps, until the rule growing step explicitly asks for such information.
Example
BuildHON+ first builds all firstorder observations and distributions (Fig. 2 right step –). Given that \(A\rightarrow C\), \(B\rightarrow C\), \(D\rightarrow B\), \(E\rightarrow A\) all have single deterministic options for the next step with \(P=1\), according to \(\log _{2}(\min (P_{\mathrm{Distr}}(i))) = 0 < \delta \), BuildHON+ knows no higherorder dependencies can possibly exist by extending these bigrams (step ). Only the two paths \(C\rightarrow D\) and \(C\rightarrow E\) will be extended; since the corresponding secondorder observations and distributions are not known yet, BuildHON+selectively derives the necessary information from the raw data (Fig. 2 right step –), and finds that the secondorder distributions show significant changes. At this point, both \(CA\rightarrow D\) and \(CB\rightarrow E\) have single deterministic options for the next step, so again, BuildHON+ determines no dependencies beyond secondorder can exist (step ), so the rule growing procedure stops, without the need for further generation and comparison of distributions.
The challenge is how to count the ngram of interest on demand—seemingly every ondemand construction requires a traversal of the raw sequential data with the complexity of \(\varTheta (L)\). However, given the following knowledge:
Lemma 2
All observations of the sequence\([S_{tk1}, S_{tk}, \dots , S_{t1}, S_{t}]\)can be found exactly at the current and one preceding locations of all observations of sequence\([S_{tk}, \dots , S_{t1}, S_{t}]\)in the raw data.
Proof
Instead of traversing the raw data, we use an indexing cache to store the locations of known observations, then use that to narrow down higherorder ngram lookups. As illustrated in Fig. 2, if we cache the locations of \(C\rightarrow D\) and \(C\rightarrow E\) in the raw sequential data, then \(CA\rightarrow D\) and \(CB\rightarrow E\) can be found at the same locations.
During the rule growing process, if \(\mathcal{S}_{\mathrm{ext}}\) has not been observed, recursively check if the lowerorder observation is in the indexing cache, and use those cached indexes to perform a fast lookup in the raw data. New observations from \(\mathcal{S}_{\mathrm{ext}}\) are then added to the indexing cache. This procedure guarantees the identification of observations of the previously unseen \(\mathcal{S}_{\mathrm{ext}}\), and the lookup time for each observation is \(\varTheta (1)\) when the indexing cache is implemented with hash tables. □
Complexity analysis. We formally analyze and compare the computational complexity of BuildHON and BuildHON+.
BuildHON. Suppose the size of raw sequential data is L, and there are \(D_{i}\) distinct ngrams of order of i. All firstorder observations (bigrams) take \(\varTheta (2D_{2})\) space, second order observations (trigrams) take \(\varTheta (3D_{3})\) space, and so on; building observations and distributions up to \(k^{\mathrm{th}}\) order takes \(\varTheta (2D_{2}+3D_{3}+\cdots +kD_{k})\) storage, with k being the maximum order allowed, because BuildHON always keeps raising order until k is reached, while keeping all the breadthfirst search results for lower orders. with \(D_{3}\geq D_{2}\), \(D_{4} \geq D_{3}\), resulting in a complexity of \(\varTheta (k^{2}D_{2})\).
If N is the number of unique entities in the raw data, then the time complexity of the algorithm is \(\varTheta (Nk^{2}D_{2})\). All observations will be traversed at least once, and evaluating if adding a previous step significantly changes the probability distribution of the next step takes up to \(\varTheta (N)\) time (assuming Kullback–Leibler divergence [29] is used).
BuildHON+. Assume there are \(R_{i}\) distinct ngrams that are exactly of order i. By definition, we have \(R_{i} \leq D_{i}\). Therefore, BuildHON+’s space complexity is \(\varTheta (2R_{2}+3R_{3}+\cdots +tR_{t})\) (including observations, distributions, and the indexing cache) where \(R_{k}\) is the exact number of higherorder dependency rules for order k. Note that, \(R_{k}\leq L\), but it is not necessarily \(R_{(i+1)}\geq R_{(i)}\). Also \(t\leq k\).
In practice, what makes BuildHON+ different from BuildHON is its sensitivity to the underlying data. If the dataset contains very few nonsignificant ngrams up to maximum specified order, the space complexity of BuildHON+ would not be very different from BuildHON. However, for very noisy data \((D_{i} \gg R_{i})\) or data with an actual order much smaller than the specified maximum order \((t \ll k)\), the space complexity of BuildHON+ would be significantly smaller than BuildHON. The same applies to time complexity: while BuildHON has \(\varTheta (Nk^{2}D_{2})\), BuildHON+ has \(\varTheta (N(2R_{1}+3R_{2}+\cdots ))\). A sidebyside comparison between BuildHON and BuildHON+ in running time and memory consumption on a realworld data set is provided in Sect. 4.
Higherorder anomaly detection
Definition. The procedure of a networkbased anomaly detection method takes the sequential data, \(\mathcal{S} = [S_{1}, S_{2}, \dots , S_{T}]\) which is divided into T time windows \(t \in [1, T]\) as the input. In each time window, the sequential data is represented as a network, i.e., \(S_{i} \rightarrow G_{i}\), yielding a dynamic network \(\mathcal{G} = [G_{1}, G_{2}, \dots , G_{T}]\) composed of the sequence of networks. The dynamic network \(\mathcal{G}\) is then used to find the change point(s) \(t \in [1, T]\) when \(G_{t}\) is significantly different from \(G_{t1}\). The difference between networks in neighboring time intervals, i.e., \(d_{t}=\mathcal{D}(G_{t1}, G_{t})\), can be quantified by network distance metrics \(\mathcal{D}\) (e.g., [23–25]). Then the problem of anomaly detection in networks reduces to anomaly detection in the time series of \([d_{2}, d_{3}, \dots , d_{T}]\). Next, to determine if the network difference \(d_{t}\) is significantly high, straightforwardly, if \(d_{t}\) is larger than a fixed threshold Δ, \(G_{t}\) is anomalously different than \(G_{t1}\). Another more robust way is to establish the norm of network differences by computing the running average and standard deviation of network differences in the last k time intervals, the null hypothesis being \(d_{t}\) not significantly large; if \(d_{t}\) deviates from the running average by two standard deviations, the null hypothesis is rejected and time t is considered a change point.
Existing networkbased anomaly detection methods mostly differ at the network distance calculation step. However, for the \(S_{i} \rightarrow G_{i}\) step, i.e., where raw sequential data is represented as networks, existing methods all use FON as G to represent the underlying sequential data S, by counting the occurrences of pairs (bigrams) as edge weights in the network. Here, we propose to use the higherorder network (HON) that selectively embeds ngrams for the \(S_{i} \rightarrow G_{i}\) step. HON, using BuildHON+, keeps all structures of FON, and when higherorder dependencies exist in the raw sequential data, it splits a node into multiple nodes representing previous steps. We show that certain types of anomalies will remain undetected for all existing networkbased anomaly detection methods using FON, but can be revealed by using HON.
Example
Fig. 3 illustrates a sidebyside comparison of FON and HON in the network representation step. Suppose there are four taxi trajectories in the raw data. In time window I, taxis in location c randomly navigate to d or e, regardless if the taxis came to location c from a or b. In this time window, HON is identical to FON and there are no higherorder dependencies. In time window II, the traffic patterns are randomly shuffled, and the pairwise traffic between pages a, b, c, d, e remains the same as time window I. Neither FON nor HON shows changes.
In time window III, secondorder patterns emerge: all taxis that had navigated from a to c go to d, and all taxis from b to c go to e. Since the aggregated traffic from c to d and e remains the same, the FON remains exactly the same, missing this newly emerged pattern. In contrast, HON uses additional higherorder nodes and edges to capture higherorder dependencies: node c is now splitted into a new node \(ca\) (representing c given the last step being a) and node \(cb\) (representing c given the last step being b). The path \(a\rightarrow c \rightarrow d\) now becomes \(a \rightarrow ca \rightarrow d\); the edge \(c \rightarrow e\) rewired similarly. Therefore, the emergence of the secondorder pattern in the raw data is reflected by the nontrivial changes in the topology of HON. If we use the weight distance [23] defined as
with w being the edge weights and \(E\) being the total number of edges, due to the complete changes in four out of the nine edges on HON, the network distance \(\mathcal{D}(G_{2}, G_{3}) = 0.44 > 0\), successfully captures this higherorder anomaly (a significant change in higherorder navigation patterns).
In time window IV, the secondorder navigation pattern changes: all taxis that navigated from location a to c now visit e instead of d, and all from b to c now visit d instead of e. Since the pairwise traffic from c to d and e remains the same, FON remains the same. However, HON captures the changes with two edge rewirings: now \(ca \rightarrow e\) and \(cb \rightarrow d\), resulting in \(\mathcal{D}(G_{3}, G_{4}) = 0.22 > 0\).
In brief, FON is an oversimplification of sequential data produced by complex systems, and conventional networkbased anomaly detection methods that use FON may fail to capture the emergence and changes of higherorder navigation patterns. If HON is used instead, without changes to distance metrics, existing methods can capture these previously ignored anomalies.
Distance metrics
After successful construction of HON (using BuildHON+) we apply five network distance measures to detect anomalies.
 1.
Weight distance. This metric was introduced earlier (Equation (2)).
 2.
Maximum common subgraph (MCS). The MCS distance is defined similarly to the weight distance in Equation (2) but operates on MCS [23]:
$$ \mathcal{D}(G, H) = \vert E_{G}\cap E_{H} \vert ^{1}\sum_{u,v\in V} \frac{ \vert w^{G}(u,v)w^{H}(u,v) \vert }{\max \{w^{G}(u,v),w^{H}(u,v)\}} $$(3)  3.
Modality. This distance function can be defined as follows [24]:
$$ \mathcal{D}(G, H)= \bigl\Vert \pi (G)\pi (H) \bigr\Vert $$(4)where \(\pi (G)\) and \(\pi (H)\) are the Perron vectors of graphs G and H, respectively.
 4.
Entropy graph distance. This can be defined as follows [25]:
$$ \mathcal{D}(G, H)= E(G)E(H) $$(5)where \(E(*)\) is the entropy measure of the edges:
$$ E(*)=\sum_{e\in E_{*}}\widetilde{W}_{*}^{e} \sum_{e \in E_{*}}\ln \widetilde{W}_{*}^{e} $$(6)and:
$$ \widetilde{W}_{*}^{e}= \frac{W_{*}^{e}}{\sum_{e\in E_{*}}W_{*}^{e}} $$(7)is the normalized weight for edge e.
 5.
Finally, we also use the spectral distance, which is defined as [25]:
$$ \mathcal{D}(G, H)=\sqrt{ \frac{\sum_{i=1} ^{k}(\lambda _{i}\mu _{i})^{2}}{\min (\sum_{i=1} ^{k}\lambda _{i}^{2},\sum_{i=1} ^{k}\mu _{i}^{2})}} $$(8)where \(\lambda _{i}\) and \(\mu _{i}\) represent the eigenvalues of the Laplacian matrix for graph G and G, respectively.
Note that, in order to calculate network distance in HON, all higherorder nodes are treated as firstorder ones. That is, a change from \(DB,C \rightarrow E\) to \(DB,C,A \rightarrow E\) results in total removal of \(DB,C\) and new addition of node \(DB,C,A\). The reason is that in many cases, anomalous patterns result in a change of higherorder patterns. It is desirable that the anomaly detection method detects the “emergence”, “change” and “dissipation” of higherorder patterns. We leave the task of classifying different higherorder anomalies for future work.
Results
In this section, we first compare BuildHON+ with BuildHON in terms of running time and memory consumption on realworld data of various sizes and multiple orders of dependency. Next, we present the anomaly detection results.
For the anomaly detection experiments, we first construct a largescale synthetic taxi movement data with 11 billion movements and variable orders of dependencies, and show that five existing anomaly detection methods based on FON collectively fail to capture anomalous navigation behaviors beyond firstorder, while using our framework, all methods show significant improvements.
We also demonstrate HON on realworld taxi trajectory data, showing its ability in capturing the higherorder anomaly signals and revealing the exact location of anomalies.
Scalability analysis: performance improvement of BuildHON+ over BuildHON
To highlight the scalability advantage of BuildHON+, instead of the taxi data or the synthetic data (which demonstrates up to third order of dependency), we use the same shipping trajectories data as used in the HON paper [2]. This data was shown to demonstrate dependencies of more than the fifthorder due to ships’ cyclic movement patterns. It consists of up to three years of shipping data (between May \({1^{\mathrm{st}}}\), 1997 and April \({30^{\mathrm{th}}}\), 2003), aggregated over 3months intervals. The smallest and largest data contains 372,500 and 4,721,936 voyages, respectively.
For a fair comparison, we use the Python implementation for both BuildHON+ and BuildHON. Both implementations run singlethreaded on the same Linux machine (Intel Quad 16core @ 2.10 GHz, 128 GB RAM). BuildHON+ is parameterfree (no limit to the maximum order, optional \(\mathit{MinSupport} = 1\)) and does not require further configuration. We set \(\mathit{MinSupport} = 1\) and \(\mathit{MaxOrder}= 15\) for BuildHON. We start with the first 3months of the data and aggregate the trajectories over the next 6 months, 9 months, and so on. Figure 4 illustrate the time and memory usage of both algorithms as the size of the data increases. We observe that BuildHON is highly sensitive to the size of the data. For the maximum data size, BuildHON requires approximately 7.2 times more memory than BuildHON+ and takes 4.5 times longer to run.
We further analyze the run time and memory usage of both algorithms on the same shipping dataset to analyze the effect of setting different values for MaxOrder. For this experiment, we use one year of data which consists of 3,415,577 voyages between May \({1^{\mathrm{st}}}\), 2012 and April \({30^{\mathrm{th}}}\), 2013.
We set \(\mathit{MinSupport} = 1\) for BuildHON, and gradually increase MaxOrder from the firstorder. Same as above, BuildHON+ does not require further configuration. BuildHON+ was able to find up to \(11^{\mathrm{th}}\) order within 2 minutes, with a peak memory usage less than 5 GB, as the reference lines displayed in Fig. 5. In comparison, BuildHON already exceeds the running time and memory consumption of BuildHON+ at \(6^{\mathrm{th}}\) order, reaches the physical memory limit at \(8^{\mathrm{th}}\) order, and would need about 22 GB memory and 6 minutes (3× time and 5× memory than BuildHON+) to achieve the same results as BuildHON+ can. Both implementations run singlethreaded on the same laptop (Intel i76600U @ 2.60 GHz, 16 GB RAM, SSD).
Anomaly detection: largescale synthetic taxi movements
We first use the synthetic data with known higherorder anomalies to test the effectiveness of the HONbased anomaly detection framework. With synthetic data, we know exactly when, where, and what types of anomalies exist. To begin with, we assume 100,000 taxis are navigating through a \(10\times10\) grid with cells numbered from 00 to 99. At each timestamp, every taxi moves 100 steps, resulting in 10,000,000 movements.
Our goal is to synthesize input sequences with variable orders of taxi navigating patterns. We start from the basic case where all taxis navigate randomly, then gradually add or change firstorder and higherorder navigation rules, and see if the proposed method can successfully identify these anomalies.
For each of the following 11 cases, we maintain the taxi navigation behavior for 100 time windows. In total, we generate 11,000,000,000 taxi movements for the subsequent anomaly detection task. The full process to synthesize the input trajectories is illustrated in Fig. 6.
Initial random movement case. At \(t=[0,99]\), each taxi has a 50% chance of navigating to the cell on the right and 50% chance of navigating down in each move.
Emergence of the firstorder dependency. At \(t=[100,199]\), we impose the following firstorder rule of movement: all taxis coming to cell 00, 03 and 06 will have a 90% chance of moving to the right and 10% chance of moving down in the next step. This new rule incurs a significant change of firstorder traffic at \(t=100\) between pairs of cells 00–01, 00–10, 03–04, 03–13, 06–07 and 06–16. The locations of these dependency rules are highlighted on the right of Fig. 6.
Change of the firstorder dependency. At \(t = [200,299]\), we change the existing firstorder movement rules: all taxis coming to cell 00, 03 and 06 will now have a 90% chance of moving down in the next step, and a 10% chance of moving right. This change at \(t = 200\) should also be reflected in both FON and HON.
Emergence of secondorder dependency. At \(t=[300,399]\), we keep the previous firstorder rules and impose a new secondorder rule: all taxis coming from cell 27 to 28 will have a 90% chance of moving to the right in the next step, and a 10% chance of moving down. This change at \(t=300\) not only introduces new higherorder dependencies, but also slightly influences firstorder traffic (traffic of \(27\rightarrow 28 \rightarrow 29/38\) changes from 1:1 to 7:3).
Emergence of complementary secondorder dependencies. At \(t=[400,499]\), we impose a pair of new secondorder rules: (1) all taxis coming from cell 30 to 31 (and 34 to 35) will have a 90% chance of moving to the right in the next step, and a 10% chance of moving down; (2) all taxis coming from page 21 to 31 (and 25 to 35) will have a 90% chance of moving down, and a 10% chance of moving right. The combined effect of these two new complementary secondorder dependencies at \(t=400\) is that the firstorder taxi traffic from cell 31 and 35 remains unchanged.
Change of complementary secondorder dependencies. At \(t=[500,599]\), we flip the rules for the complementary secondorder dependencies: (1) all taxis coming from cell 30 to 31 (and 34 to 35) will have a 90% chance of moving down, and a 10% chance of moving right; (2) all taxis coming from page 21 to 31 (and 25 to 35) will have a 90% chance of moving right, and a 10% chance of moving down. At \(t=500\) the firstorder taxi traffic from cell 31 and 35 still remains unchanged.
Emergence of thirdorder dependency. At \(t=[600,699]\), we impose a new thirdorder rule: all taxis coming from cell 61 through 71 to 81 will have a 90% chance of moving to the right in the next step, and a 10% chance of moving down. This introduction of thirdorder dependencies at \(t=600\) also slightly influences the firstorder traffic (from 1:1 to 3:2).
Emergence of complementary thirdorder dependencies. At \(t=[700,799]\), we impose a pair of new thirdorder rules: (1) all taxis coming from cell 64 through 74 to 84 (and 67 through 77 to 87) will have a 90% chance of moving to the right in the next step, and a 10% chance of moving down; (2) all taxis coming from 73 through 74 to 84 (and 76 through 77 to 87) will have a 90% chance of moving down, and a 10% chance of moving right. Here at \(t=700\) firstorder traffic does not change when these two complementary thirdorder dependencies are introduced together.
Change of complementary thirdorder dependencies. At \(t=[800,899]\), we flip the rules for the complementary thirdorder dependencies. Firstorder traffic at \(t=800\) again remains unchanged.
Emergence of complementary mixedorder dependency. At \(t=[900,999]\), we impose a new thirdorder rule and a firstorder rule: (1) all taxis coming from cell 39 through 49 to 59 will have a 90% chance of moving to the right in the next step, and a 10% chance of moving down; (2) all taxis at cell 59 will have \(11/30\) chance of moving right and \(19/30\) chance of moving down. At \(t=900\) firstorder traffic does not change, because the influence of the new thirdorder rule on pairwise traffic is canceled by the new firstorder rule.
Change of complementary mixedorder dependency. At \(t=[1000,1099]\), we flip the rules for the mixedorder dependencies. Firstorder traffic at \(t=1000\) remains unchanged.
Results
For all five distance metrics, we present a sidebyside comparison between anomaly detection results using FON and HON in Fig. 7. The Yaxis shows the graph distances between neighboring time windows; given that we have injected 10 anomalous movement patterns at \(t=[100, 200, \dots , 1000]\), we should expect to see 10 “spikes” in graph distances.
Methods using FON can detect at most 4 out of the 10 anomalies: the addition and changes in firstorder movement patterns (\(t=100\), \(t=200\)), the addition of secondorder (\(t=300\)), and the addition of thirdorder (\(t=600\)) movement patterns. Because the latter two cases also slightly change the firstorder traffic, FON does reflects the changes, but the spikes incurred are not as significant as when changes are made directly to firstorder rules. For the other six cases, all five distance metrics appear as if there are no anomalies, as long as they rely on FON topology.
In contrast, methods using HON (1) capture all firstorder anomalies (\(t=100\), \(t=200\)); (2) show stronger signals for the addition of secondorder and thirdorder rules (\(t=300\), \(t=600\)) because not only the firstorder traffic is changed but BuildHON+ also creates additional higherorder nodes and edges for higherorder dependencies; (3) capture the six additional cases where higherorder movement patterns are changed but firstorder traffic remains the same. Here the topological changes of HON are best reflected with weight distance and spectrum distance (detecting \(10/10\) anomalies); MCS weight method misses the addition of higherorder nodes and edges (\(t=400, 700, 900\)) because those topological changes are excluded from common subgraphs; entropy method misses changes in edge weights (\(t=200, 500, 800, 1000\)), also because by definition a swap in edge weights do not change a graph’s entropy. Nevertheless, all these distance metrics are able to identify more types of anomalous signals simply by using HON instead of FON, with no changes to these distance metrics. In other words, BuildHON+ can be plugged into existing networkbased anomaly detection methods directly, and extend their ability in detecting higherorder anomalies.
Anomaly detection: real world taxi data
We use the ECML/PKDD 2015 challenge data,^{Footnote 1} which contains one year (Jul. 1, 2013 to Jun. 30, 2014) of all the 442 taxi GPS trajectories in Porto, Portugal. The coordinates of each taxi were collected every 15 seconds. To discretize the geolocation data into points of interest that are representative of population density, we map all coordinates to the nearest 41 police stations (Fig. 8). As a preprocessing step and to void introduction of bias / noise, we removed the taxis that have been idle for more than 5 days because that can arise due to data collection errors (on average 5.29% of the trajectories were removed). The highlighted box in Fig. 8 indicates the detection of anomalies. Figure 9 shows the week to week difference in higherorder dependencies, yielding in 52 time windows and 442 trajectories of points of interest. We consider both FON and BuildHON+ with a fixed maximum order of 2 and BuildHON+ with a variable higherorder (discovered to be 3 by the algorithm). We show that BuildHON+ when allowed to discover the maximum order, results in the highest indication of potential anomalies.
Note that the choice of timewindow is quite datadependent. We initially attempted daily timewindows but noticed that the weekly fluctuation patterns (weekday commute traffic vs weekend recreational traffic) dominate any other signals. Besides, daily time windows have sparser observations, resulting in a very sparse network for each time step. On the other hand, because anomalous traffic patterns usually last for no more than a few days, using monthly aggregation will dilute the signal and result in too coarse a granularity.
Graph distance analysis
We compare the 52 networks for FON, MaxOrder of 2 as a constraint for BuildHON+ (indicated as HON2), and no MaxOrder constraint on BuildHON+ (indicated as HON+) in Fig. 10. Our goal is to see the improvements afforded by allowing BuildHON+ to automatically discover the requisite higherorder for a given data, versus specifying the maximum order of 2 using BuildHON and the FON representation. We compute the graph distances (using weight distance) for neighboring time windows. The histograms of graph distances for each network is shown in Fig. 10 (a), (b), and (c). We also compute the running average and standard deviation using the graph distances in the previous 10 weeks, with the null hypothesis as “the network is not significantly different if the graph distance does not deviate more than 2σ from the mean”. Variation of graph distances for FON, HON2 and HON+ is shown in Fig. 10 (d), (e), and (f) respectively. While the trend of HON+ resembles that of HON2 and FON, the graph distances in weeks 43 and 44 are particularly more significant in HON+ than HON2 and FON (HON2 offers more significance over FON as well). Such differences are also indicated in the histograms of graph distances in Fig. 10 (a), (b), and (c), where the red circles highlight the correct anomalous signals, which is observable in HON+, while it is not as significant in FON and even HON2.
We focus on the case of week 43 and 44 to understand why HON+ produces a stronger signal than HON2 and FON in this time window. We notice that Porto’s second most important festival, “Burning of the Ribbons”, lasts from May 2 to May 9 in 2014 and falls within the end of week 43 and the entire week 44 of our study. The festival involves parades, road closures, and is popular among tourists, which could be the underlying reason for the changes in taxis’ movement patterns. After plotting the traffic HON+ of week 43 and week 44 in Fig. 8 (b) and (c), we notice that multiple higherorder nodes and edges emerge in these weeks, indicating the emergence of higherorder traffic patterns. The newly emerged higherorder patterns correspond to police stations labeled from 9 to 14, which is where the event’s main venue (Queimódromo in the City Park) and participating universities are located.^{Footnote 2} We further compared the fluctuations in the number of higherorder nodes (obtained from HON+) in Fig. 9. We notice that the number of firstorder nodes does not change significantly, while the number of second and thirdorder nodes shows a sharp change in week 44 of the data. FON (although showing deviation from average at week 44) does not capture the change in additional higherorder nodes, and HON2 does not capture the change in thirdorder nodes. HON+ is more effective in deciphering the anomalous signal. This analysis shows the importance of including variable and higherorder dependencies for anomaly detection, and the applicability of BuildHON+ in discovering the appropriate orders given the data. Depending on the data, the MaxOrder value required for accurate detection of anomalies can be different. BuildHON+ removes this dependency, ensuring accurate detection of changes in the network.
Robustness to noise
We notice that FON graph distance in week 43 and 44 falls slightly outside the 2σ threshold as well. However, HON+ deviation from the 2σ threshold is 3.25 times bigger that of FON in week 44 and 5.2 times bigger in week 43. This becomes important in the presence of noise where anomalies by FON may not be detected. Furthermore, based on FON graph distances, weeks 24, 26, 41, and 42 are also anomalous events (Table 1 and Fig. 10 (d)). However, no significant event happened during these weeks. Thus, without any noise, FON can detect anomaly but with a higher false positive rate (4), while HON+ can also correctly detect the anomalies but with only 1 false positive for week 24, with a very small value above the 2σ threshold, as indicated in Fig. 10 (f) and Table 1.
To illustrate the above point, we designed an experiment to show the robustness of HON+ and FON against noise. We randomly assigned 10% of all taxis to the next closest police station and constructed the corresponding HON+ and FON. The graph distances for FON and HON+ after adding the noise is shown in Fig. 11 (a) and Fig. 11 (b), respectively. The detected anomalies are presented in Table 1 where the values represent the difference between the graph distance and the 2σ threshold. Before adding the noise, FON detects the anomalies at week 44 and week 43 with a small margin from the 2σ threshold. Furthermore, it has a higher false positive rate (Fig. 10 (d)). After adding the noise, FON shows false positives in weeks 17, 18, 26, 41, 42 and one correct anomaly at week 44 which is very close to the 2σ threshold (Fig. 11 (a) and Table 1). Furthermore, FON does not detect anything in week 43. HON+, on the other hand, detected the anomalous event (in both weeks 43 and 44) before the noise (Fig. 10 (f)) and after the noise (Fig. 11 (b)) with only one false positive (Table 1). It is important to note that false positives can be very costly and often require manual correction by human labor.
Conclusion
This paper presented a scalable and parameterfree algorithm for extracting higherorder dependencies from the sequential data, and demonstrates the success of higherorder network modeling for anomaly detection in dynamic networks. We show that BuildHON+ is scalable and parameterfree and automates the process of discovering the appropriate variable and higherorder dependencies for each of the nodes in a network. The complexity analysis of BuildHON+, as well as running time and memory consumption benchmarking results, demonstrates the scalability of BuildHON+ to largescale networks.
We further demonstrate that FONs are weak detectors of higherorder anomalies, especially in the noisy data. This emerges because FONs do not adequately capture the sequential orders or indirect pathways in a complex system, thereby providing a limiting view of the behavior of a complex system in their network representation. BuildHON+ can accurately capture such anomalies and also work seamlessly with existing anomaly detection methods to enable more accurate detection of anomalies in comparison to using FON.
The higherorder network representation results in a more accurate representation of the underlying trends and patterns in the behavior of a complex system and is the correct way of constructing the network to not miss any important dependencies or signals. This is especially relevant when the data is noisy and has sequential dependencies within indirect pathways. This has numerous applications, ranging from information flow to human interaction activity on a website to transportation to invasive species management to drug and human tracking.
We note that changes in the HON structure can be more complex than changes in the FON structure, such as the emergence and dissipation of higherorder patterns. In order to use the graph distances that are defined for FONs, our current approach treats any changes in the node orders as total removal/addition of that node. This approach may result in more fluctuations in the graph distance and can cause HONs to become less overlapping over time. Regardless, measures defined for FONs can still be used for anomaly detection in HONs, since the 2σ criteria captures the HON fluctuations. One possible improvement can be designing a distance measure for capturing the unique fluctuations in the HON structure.
Another direction for future work is to classify different types of anomalies given different types of node changes in HON, like the emergence and dissipation of higherorder patterns. In addition to the graph distance metrics, one may also consider structurebased metrics [7] that factor in changes of clustering or ranking results and local properties such as motifs on the network. This could be considered as a supervised learning problem, where different categories of anomalies are labeled as classes in the training data and the task is to predict whether those categories of anomalies appear in the testing data. All of these extensions are directly compatible with BuildHON+, as the resulting HON representation does not impose a change in the network analysis method.
Supplementary materials are available as Additional file 1.
Abbreviations
 HON:

Higherorder network
 FON:

Firstorder network
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Acknowledgements
This research was supported in part by NSF Grants EF1427157, IIS1447795 and CRI1629914, and by the Army Research Laboratory under Cooperative Agreement no. W911NF0920053 (the ARL Network Science CTA). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation here on.
Availability of data and materials
The Portugal taxi data is available through: http://www.geolink.pt/ecmlpkdd2015challenge/data%20set.html. The Code for generating higherorder network and synthetic data is available at: https://github.com/msaebi1993/HONANOMALY.
Funding
This work is supported in part by NSF awards EF1427157, IIS1447795, CRI1629914 and in part by the Army Research Laboratory under Cooperative Agreement no. W911NF0920053 [the ARL Network Science CTA (Collaborative Technology Alliance)].
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All the authors participated in the conception of the research study. JX and MS are equal contributors. JX and MS implemented the algorithms and performed the experiments. All the authors participated in analysis and writing of the paper. All authors read and approved the final manuscript.
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Mandana Saebi and Jian Xu contributed equally to this work.
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Saebi, M., Xu, J., Kaplan, L.M. et al. Efficient modeling of higherorder dependencies in networks: from algorithm to application for anomaly detection. EPJ Data Sci. 9, 15 (2020). https://doi.org/10.1140/epjds/s1368802000233y
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Keywords
 Higherorder network
 Dynamic network
 Anomaly detection
 Sequential data