Abstract
Many evolving complex realworld systems can be modeled via dynamic networks. An important problem in dynamic network research is community detection, which finds groups of topologically related nodes. Typically, this problem is approached by assuming either that each time point has a distinct community organization or that all time points share a single community organization. The reality likely lies between these two extremes. To find the compromise, we consider community detection in the context of the problem of segment detection, which identifies contiguous time periods with consistent network structure. Consequently, we formulate a combined problem of segment community detection (SCD), which simultaneously partitions the network into contiguous time segments with consistent community organization and finds this community organization for each segment. To solve SCD, we introduce SCOUT, an optimization framework that explicitly considers both segmentation quality and partition quality. SCOUT addresses limitations of existing methods that can be adapted to solve SCD, which consider only one of segmentation quality or partition quality. In a thorough evaluation, SCOUT outperforms the existing methods in terms of both accuracy and computational complexity. We apply SCOUT to biological network data to study human aging.
Introduction
Networks (or graphs) are elegant yet powerful abstractions for studying complex systems in various domains, from biological entities to social organizations. Realworld systems evolve over time. Until relatively recently, dynamic measurements about their functioning have been unavailable, owing mostly to limitations of technologies for data collection. Hence, an evolving system has traditionally been analyzed by studying its static network representation, which discards the system’s time dimension by combining all of its interacting elements and their connections across multiple times into a single aggregate network. For example, dynamic cellular functioning has traditionally been modeled as a static proteinprotein interaction (PPI) network that combines biomolecular interactions across different time points and other contexts^{1}. Such an aggregate approach loses important temporal information about the functioning of evolving systems^{2}. Analyzing dynamic network representations of evolving systems is crucial for understanding mechanisms behind various dynamic phenomena such as human aging in the computational biology domain^{3} or opinion formation in the social network domain^{4}, especially with the increasing recent availability of temporal realworld data in these and other domains. The dynamic network representation of an evolving system models its temporal measurement data as a series of network snapshots, each of which encompasses the temporal data observed during the corresponding time interval. We refer to this snapshotbased representation as a dynamic network.
Approaches for studying dynamic networks can be categorized into: 1) those that extend wellestablished static network problem formulations and solutions to their dynamic counterparts, and 2) those that consider novel network problems and solutions that arise specifically from the time dimension and are thus native only to the dynamic setting. A popular problem from category 1 above that is of our interest is community detection. A popular problem from category 2 above that is of our interest is time segmentation, or segment detection (also known as change detection). We next discuss these two problems.
Community detection studies network structure (or topology) from mesoscopic (intermediate or groupsofnodes level) perspective, in contrast to doing so from macroscopic (global or network level) or microscopic (local or node level) perspective^{5}. Its goal is to identify groups of topologically related (e.g., densely interconnected^{6,7} or topologically similar^{8,9,10}) nodes called communities (or clusters), which are likely to indicate important functional units within the network. Examples of communities are groups of proteins with similar functions in a biological network or groups of friends in a social network^{5,11,12}. We focus on the mathematical notion of a partition, which divides a network into nonoverlapping communities. Yet, our work can be extended to handle overlapping communities as well. For an evolving realworld system, community detection in its dynamic network representation is likely to yield additional insights compared to community detection in its static representation^{13,14}. Two extremes of community detection in a dynamic network are: 1) snapshot clustering and 2) consensus clustering. On the one hand, snapshot clustering finds a separate partition for each temporal snapshot^{15,16,17,18,19}. Given the snapshotlevel partitions, one can then track their evolution by matching individual clusters in adjacent snapshots^{20,21,22,23}. On the other hand, consensus clustering finds a single partition that fits well all snapshots^{24,25,26,27}. In the real life, community organization most often lies between these two extremes. Finding this real life community organization is one of key goals of our study.
Segment detection aims to divide a dynamic network into continuous segments (groups of snapshots), such that the “border” between each pair of adjacent segments marks a prominent shift in the network structure^{28}. As a result, all snapshots within a given segment have similar network structure, while every two adjacent segments have snapshots with dissimilar structure. The set of all segments covering the whole dynamic network is called the segmentation of the network. Time points that separate the segments are called change points. Since change points correspond to shifts in the network structure, they likely indicate functionally important events in the life of the underlying system^{28}. For example, change points can correspond to transitions between different functional states in brain networks or to stock market changes in financial networks^{29}. Finding change points indicating important structural shifts in the dynamic network is the other key goal of our study.
Community detection partitions a dynamic network along the node dimension (by grouping nodes into communities), while segment detection does this along the time dimension (by grouping snapshots into segments). Their combination, which is our focus and which we refer to as segment community detection (SCD), can be seen as twodimensional clustering: simultaneously grouping snapshots of the dynamic network into segments based on community organization of the snapshots, and grouping nodes of the snapshots into communities based on the segments these snapshots belong to (Fig. 1(a)).
SCD naturally allows for compromising between the extremes of snapshot clustering and consensus clustering to identify the real life community organization. While snapshot clustering “zooms in” to each snapshot and consensus clustering “zooms out” to the whole network, SCD can automatically choose an appropriate “zoom level” by focusing on segments, each spanning coherent snapshots while still capturing important changes in the community organization (Fig. 1(b)). Consider studying how protein modules evolve with age: it may be more desirable to focus on different stages of aging (infancy, childhood, adolescence, adulthood, etc.^{30}) via SCD than on each day/month/year of the lifespan via snapshot clustering or on the entire lifespan via consensus clustering. Similar holds when studying e.g., evolution of protein modules with disease progression.
Existing approaches that can be adapted to be able to deal with the SCD problem are GraphScope^{31}, MultiStep^{25}, and GHRG^{32} (Supplementary Section S1). These methods can produce both segments and the corresponding partitions, which is a solution that SCD aims to find. However, they have drawbacks. 1) They generally cannot produce a highquality solution with respect to both SCD aspects (segmentation quality and partition quality), as we show in Results. 2) For each method, the number of segments can only be either: a) determined automatically but not set by the user, or b) set by the user but not determined automatically. In applications where domain expert knowledge on the desired number of segments is available, the user should be able to feed this knowledge into the method by setting the number of segments, but the methods of type “a” above (GraphScope and GHRG) cannot handle this. In applications where such knowledge is unavailable, the method should be able to determine an appropriate number of segments automatically, but the methods of type “b” above (MultiStep) cannot handle this. For a method to be generalizable to both application types, it should be able to handle both automatic and userdefined determination of the appropriate number of segments. 3) Each of the existing methods has a single builtin intuition about what a good segment or partition is. Hence, each approach could be biased towards the particular parameters that it implements. Thus, a generalizable approach that would offer flexibility in terms of parameter choices is desirable. To address these three drawbacks, we introduce SCOUT, a new general framework for segment community detection, as follows.
We propose a novel SCD formulation as an optimization process that integrates the two aspects (segment detection and community detection) more explicitly than the existing methods. Also, we propose SCOUT, a general framework for solving the new formulation, which addresses the above drawbacks: 1) it can produce a highquality solution with respect to both SCD aspects; 2) it can handle both automatic and userdefined determination of the appropriate number of segments; 3) it offers high level of flexibility when it comes to the choice of segmentation or partition quality parameters.
SCOUT’s key algorithmic components (Supplementary Section S2.1 and Figures S1–S3) are: objective function (a measure of what a good SCD solution is), consensus clustering (given a set of change points, how to find a good partition for each segment), and search strategy (how to search through the space of possible change point sets). We vary choices for these components to trade off between different goals, such as segmentation quality and partition quality, or accuracy and speed.
We comprehensively evaluate SCOUT against the existing methods on both synthetic and realworld networks of varying sizes (Supplementary Section S2.2 and Figures S4–S5). In particular, to illustrate generalizability of our approach, we first perform evaluation on synthetic dynamic network data. For this purpose, we introduce an intuitive model for automatic generation of a synthetic dynamic network of an arbitrary size with known ground truth segmentation as well as community organization; we analyze 20 different synthetic ground truth configurations. In addition, we analyze six realworld dynamic networks from domains that do offer such data and that offer such data with some ground truth knowledge embedded into them; these networks span studies of human proximity, communication, and political relationships. To evaluate how well each method can reconstruct the ground truth knowledge, we rely on established partition quality and similarity measures as a basis for developing new SCD accuracy measures that can simultaneously account for both segmentation quality and partition quality. We show that SCOUT outperforms the existing methods with respect to both segmentation quality and partition quality, while also being more computationally efficient. In a case study, we show that SCOUT, when applied to dynamic, agespecific human PPI network data^{3,33}, correctly identifies different stages of the aging process. We make our SCOUT implementation publicly available (http://nd.edu/~cone/SCOUT).
Results
For each method (GraphScope, MultiStep, GHRG, and SCOUT), we first evaluate the effect of parameter choices on its performance (Supplementary Section S3 and Figures S6–S12). Then, we compare the methods under their best parameter values on synthetic and realworld networks, via network structurebased and ground truth knowledgebased measures (see Methods). As a measure of the former type, we use average snapshot partition quality Q_{P} based on modularity. As a measure of the latter type, we use a) similarity of a method’s output to the ground truth and b) change point classification. For case “a” above, we compute segmentation similarity Sim_{T}, partition similarity Sim_{P}, and overall similarity Sim_{B}; for all three measures, we rely on Normalized Mutual Information (NMI). For case “b” above, we use area under the precisionrecall curve (AUPR).
When we have the complete ground truth SCD information (on both the segmentation aspect and the partition aspect) available, which is the case for our synthetic networks, we use all of the above measures, but we trust Sim_{B} the most, since it captures similarity between a given method’s solution and the ground truth solution with respect to both SCD aspects. When we do not have the complete ground truth information, which is the case for our realworld networks, we cannot use the twoaspect Sim_{B}. Instead, we use the structurebased measure (Q_{P} based on modularity) and whichever ground truth knowledgebased measure we can compute based on the partial ground truth information. Since in our case the available ground truth information is the list of change points (see Methods), for the latter, we can use any measure that captures the segmentation aspect of the solution quality. We have two such measures: Sim_{T} and change point classification. Since we show below that the two measures yield consistent results on synthetic networks, and since per our discussion in Methods change point classification is theoretically more meaningful than Sim_{T} (as it accounts for ranking of all time points rather than only for the identified change points), for brevity, we focus on change point classification for realworld networks.
Synthetic networks
We evaluate the methods on 20 synthetic network configurations: five options for the number of segments times four options for the number of nodes per snapshot; each configuration has 16 snapshots. These configurations span the whole “spectrum” between the extreme cases of snapshot clustering (where the number of ground truth segments corresponds to the number of snapshots) and consensus clustering (where there is only one ground truth segment – the whole dynamic network). For each configuration, we generate multiple random network instances and report results averaged over the multiple instances.
The idea of our synthetic network generator (snapshots within the same segment having the same community organization) aligns well with the intuition of each considered method. Thus, each method has a fair chance for recovering the ground truth knowledge, except MultiStep, which has an unfair advantage. Namely, to be able to evaluate MultiStep, we need to provide the ground truth number of segments as input to it (Supplementary Section S3). We denote this modification as MultiStep*. This a priori knowledge gives an unfair advantage to MultiStep for all configurations and especially for the extreme configurations with the minimum and maximum possible numbers of ground truth segments (i.e., with one and 16 segments, respectively). This is because for these two types of configurations, the knowledge of the ground truth number of segments guarantees that MultiStep’s solution will have the correct segmentation: given all 16 snapshots, there is only one way to group the 16 snapshots into one segment (the resulting segment will encompass all 16 snapshots) and only one way to group the 16 snapshots into 16 segments (each segment will encompass exactly one snapshot). For the other nonextreme configurations, while knowing the ground truth number of segments still gives an advantage to MultiStep (as it will produce the correct ground truth number of segments, or equivalently, the correct number of change points), it does not necessarily guarantee that MultiStep will obtain the correct segmentation (i.e., that the identified change points will be correct). This is because for the nonextreme configurations, there are multiple ways to group snapshots into the given number of segments.
Below, we discuss results for the segmentation aspect (Sim_{T} and change point classification), partition aspect (Q_{P} and Sim_{P}), and both aspects (Sim_{B}) of SCD. For Sim_{B}, since we trust it the most, we test the statistical significance of SCOUT’s improvement over the existing methods (Supplementary Section S2.2). Finally, we compare the methods’ running times.
Segmentation aspect of the solution quality
For Sim_{T}, SCOUT is superior to all other methods, as it achieves the highest scores for 90% of all synthetic network configurations, while the other methods are comparable to each other (Supplementary S13). The remaining 10% (i.e., two) of all configurations in which an existing method (in this case, GraphScope) achieves higher scores are configurations with the two largest numbers of nodes per snapshot and with the maximum possible number of segments (Supplementary S14). The fact that GraphScope has higher Sim_{T} for these configurations is not surprising. Namely, GraphScope produces solutions with more segments than the other methods do, often overestimating the ground truth number of segments (Supplementary S15). So, for the configurations with the maximum possible number of segments, the most that GraphScope can overestimate is the maximum number of segments itself, i.e., the correct solution. Note that when measuring Sim_{T} for the extreme configurations with the minimum and maximum possible numbers of segments, we exclude MultiStep from comparison. This is because of MultiStep’s unfair advantage (see above), which for these extreme configurations means a priori knowing the correct segmentation and thus achieving the perfect Sim_{T} (Supplementary S14). For the remaining nonextreme configurations, MultiStep is always outperformed by SCOUT and at least one of the existing methods (Supplementary S13). Thus, MultiStep, which knows the ground truth number of segments a priori typically does not yield a high quality segmentation with respect to Sim_{T}, whereas SCOUT does so (and typically better than the other methods) despite not having this prior knowledge (Fig. 2(a) and Supplementary S14). This is further confirmed by SCOUT being able to automatically determine the ground truth number of segments more accurately than the existing methods (Supplementary S15).
For change point classification, the trends are consistent to those for Sim_{T}: SCOUT is again superior in over 90% of all synthetic network configurations (Fig. 2(b) and Supplementary Figures S13 and S16). The consistency is expected, since the two measures focus on the same aspect of SCD. Note that unlike for Sim_{T} above, for change point classification, MultiStep does not have the unfair advantage over the other methods (Supplementary Section S2.2).
Partition aspect of the solution quality
For Q_{P}, SCOUT is superior to all other methods, achieving the highest Q_{P} for 70% of all synthetic network configurations (Supplementary S17). Of the existing methods, MultiStep is the best, followed by GraphScope and GHRG that are comparable to each other (Fig. 2(c) and Supplementary S16). Importantly, SCOUT overall outperforms MultiStep in terms of Q_{P} despite the fact that MultiStep explicitly maximizes modularity (which is the basis of Q_{P}), while the version of SCOUT under consideration does not rely on Q_{P} (Supplementary Section S3). The configurations on which MultiStep outperforms SCOUT are mostly those with the maximum possible number of ground truth segments (Fig. 2(c) and Supplementary S16). This is not surprising, since for these 16segment configurations, SCOUT can produce a solution with at most 16 segments, while MultiStep is guaranteed to produce the solution with exactly 16 segments. That is, intuitively, MultiStep’s solution has a separate segment partition for each snapshot, and each of the partitions aims to maximize modularity and consequently Q_{P}. For the configurations where MultiStep outperforms SCOUT, MultiStep’s Q_{P}based superiority is not necessarily an advantage. This is because MultiStep achieves higher Q_{P} scores even compared to Q_{P} scores of the ground truth solution (Fig. 2(c) and Supplementary S16). Thus, even if MultiStep obtains the highest Q_{P}, its partitions might not necessarily be closer to the ground truth than SCOUT’s partitions, as we justify next.
For Sim_{P}, SCOUT is superior to all other methods, achieving the highest Sim_{P} score for all configurations (Supplementary S17). The other methods are relatively comparable to each other, with slight superiority of MultiStep (Fig. 2(a) and Supplementary S14). Trends for Sim_{P} are not always consistent with those for Q_{P}, although the two capture the same SCD aspect. For example, for the configuration with 100 nodes per snapshot and 16 ground truth segments, although MultiStep achieves the highest Q_{P} score (Fig. 2(c)), it has the lowest Sim_{P} score (Supplementary S14). The inconsistency between Q_{P} and Sim_{P} is not be surprising, since modularity is not always able to capture well the ground truth communities^{7}.
Overall solution quality
For Sim_{B}, SCOUT again outperforms the other methods for all configurations (Fig. 3). The other methods are comparable to each other (Fig. 2(d) and Supplementary S16). Importantly, in most cases, SCOUT not only improves upon the existing methods but also its improvement is statistically significant. Namely, SCOUT statistically significantly improves upon the best existing method in 75%, 65%, and 55% of all cases at pvalue threshold of 0.05, 0.01, and 0.001, respectively (Supplementary Table S1). Note that the above percentages could not be perfect, since for 20% of all configurations (namely, the four configurations with the minimum number of ground truth segments), in addition to SCOUT that achieves the perfect Sim_{B}, MultiStep also (unfairly, per our above discussion) achieves the perfect Sim_{B} and is thus comparable to SCOUT.
Running time
Theoretic computational complexities of all existing methods and SCOUT are discussed in Supplementary Sections S1 and S2.1, respectively. Here, we focus on comparing the methods’ empirical running times. SCOUT is the fastest of all methods, over all configurations (Supplementary S18). Specifically, across all network sizes, SCOUT is at least 2.6 times faster than the next fastest method. It is followed by MultiStep, GraphScope, and GHRG, respectively. GHRG, even when parallelized, cannot be run for the larger networks due to its high computational complexity.
Realworld networks
We evaluate all methods on six realworld networks. Since the complete ground truth knowledge (both change points and segment partitions) is unavailable for any of the networks, we perform evaluation based on change point classification and Q_{P}.
Segmentation aspect of the solution quality
For change point classification, SCOUT is superior to all other methods, achieving the highest accuracy for all considered realworld networks, and it is followed by GHRG, GraphScope, and MultiStep, respectively (Fig. 4(a)). This method ranking is consistent with that for synthetic networks. We have formal lists of change points only for four of the six networks, and thus the above change point classification is performed only on those four networks. Yet, we can still intuitively (informally) analyze segmentation results of the methods for the remaining two networks, High School and Senate, as follows.
Regarding High School network, this network captures proximity of students in a high school. Intuitively, we do not expect largescale changes in the students’ interaction patterns over time (meaning that we expect very few change points, if any, i.e., very few segments, possibly only one), since students typically interact with other students from the same classes^{34}. Consistent with this intuition, SCOUT (as well as GraphScope and MultiStep) detects only one segment for High School network (Supplementary S19). Moreover, SCOUT (as well as MultiStep) produces the partition for this single segment that perfectly matches the (static) partition of students according to their classes^{34}. Hence, it is encouraging that SCOUT (as well as MultiStep) captures the intuition about the expected dynamics and structure of High School network.
Regarding Senate network, for a given method, we identify its 10 topranked “change point”like time points. The lists of top ranked time points produced by the different methods have little overlap (Supplementary S20). Specifically, the four methods combined identify 33 out of all 4 × 10 = 40 possible distinct time points. Also, there is only one time point (the 83^{rd} Congress in 1953) that is identified by more than two methods (SCOUT, GraphScope, and GHRG). Clearly, the results of the different methods are quite complementary. We aim to empirically evaluate whether the top ranked points correspond to important historical events. If so, this would further validate the given method. This evaluation needs to be performed qualitatively (rather than quantitatively, as done so far), since it is hard to determine the ranking of all historical events in terms of their importance and consequently to correlate this ranking with the methods’ ranking of the time points. Because of this, and because the resulting qualitative evaluation is time consuming, while we illustrate the top 10 ranked change points for each method (Supplementary S20), here we do not compare the different methods. Instead, we discuss SCOUT’s results only, to at least intuitively assess their meaningfulness. SCOUT’s top four time points (1953, 1879, 2003, and 1979, respectively) correspond to Congresses with shifts in the structure of the Senate’s majority between the Democratic and Republican parties. Of SCOUT’s next three time points (the 86^{th}, 88^{th}, and 67^{th} Congress, respectively), the first two brought major civil rights acts (Civil Rights Act of 1960 and Civil Rights Act of 1964, respectively), and during the third one, “Teapot Dome” Scandal occurred, which is considered one of the most significant investigations in the history of the Senate^{35}. SCOUT’s remaining three time points correspond to divided Congresses: the 112^{th} Congress that almost lead to government shutdown^{36}, plus the 80^{th} Congress and the 109^{th} Congress, both of which were nicknamed as “donothing”^{37}. It is encouraging that SCOUT identifies as likely change points such important historical events.
Partition aspect of the solution quality
For Q_{P}, except for Hypertext and AMD Hope networks, SCOUT and MultiStep are comparable, and they outperform GraphScope and GHRG (Fig. 4(b)); this is the same trend as for synthetic networks. For Hypertext network, SCOUT is outperformed by GHRG and MultiStep, respectively (Fig. 4(b)). For AMD Hope network, SCOUT is outperformed by MultiStep and GraphScope, respectively (Fig. 4(b)). These results for Hypertext and AMD Hope networks are not necessarily surprising. Namely, different methods can produce solutions with different numbers of segments. In particular, for these two networks, GHRG and GraphScope produce more segments than SCOUT and MultiStep (Supplementary S19). The more segments exist in a solution, the easier it is for this solution to obtain a high partition quality score (i.e., Q_{P}). Hence, a direct comparison of Q_{P} scores of the solutions with different numbers of segments may be inappropriate. Consider this: if method 1 has a slightly higher Q_{P} score than method 2, but it also achieves its score with ten times as many segments as method 2, does it mean that method 1 has a better partition accuracy than method 2? Probably not. Thus, ideally, we want to compare Q_{P} scores of the solutions with equal numbers of segments. Since SCOUT can produce a solution with not only an automatically determined but also userprovided number of segments, we can compare Q_{P} score of each existing method and Q_{P} score of SCOUT when SCOUT is asked to produce a solution with the same number of segments as the solution of the given existing method. This way, we avoid the bias arising from the fact that the two compared methods might have different numbers of segments. According to this evaluation, SCOUT outperforms all methods (Fig. 4(c) and Supplementary S21).
The shape of the Q_{P}curve as a function of the number of segments l from e.g., 4(c) could provide insights into the dynamics of the given network. Although the xaxis of the curve does not correspond to time, and thus it cannot say when changes in community organization (if any) occur, the fact that the xaxis corresponds to l can say something about the number of such changes and their scale. Namely, if Q_{P} increases slowly (or does not increase) as l increases, this could mean that the community organization does not change a lot with increase in the number of segments, and thus, the increase in the number of segments in unnecessary. This is the case for e.g., High school network (Supplementary S21), which agrees with our discussion above. On the other hand, if Q_{P} increases drastically as l increases, this could mean that the community organization indeed changes a lot with increase in the number of segments, and thus, the increase in the number of segments is justified. This is the case for e.g., Senate network (Fig. 4(c)), which also agrees with our discussion above.
Running time
When we compare the methods’ empirical running times, just as for synthetic networks, SCOUT is the fastest of all methods, over all realworld networks (Fig. 4(d)). Again, GHRG is the slowest method, which means that it cannot be run for the larger networks due to its high computational complexity (Supplementary Section S1).
Case study: using SCOUT on biological network data in the context of studying human aging
Studying human aging is important, because the risk of disease increases with age. Studying human aging experimentally is hard due to long lifespan and ethical constraints. Thus, human aging has typically been studied computationally, via genomic sequence or gene expression analyses^{3}. Recently, we used network analysis to deepen our knowledge about human aging. Specifically, we inferred dynamic, agespecific PPI network data of human, capturing which proteins interact at which age, and analyzed the resulting network data to identify key players in aging, i.e., proteins whose PPI network positions significantly change with age^{3}. Here, we analyze the agespecific PPI network data with a different goal: to test whether SCOUT’s top ranked time points will correspond to ground truth ages that reflect shifts from one stage of human lifespan to another.
Specifically, in the agespecific PPI network data, each network corresponds to a certain age, and there are 37 networks (i.e., shapshots) for ages between 20 and 99^{3}. Since we do not have available the ground truth knowledge for ages above 80 (see below), we focus only on ages between 20 and 80. We apply SCOUT, the best of all considered methods, on the resulting 23 agespecific snapshots (xaxis of Fig. 5). We study SCOUT’s top ranked time points (i.e., ages) to evaluate their meaningfulness. In particular, we test whether SCOUT’s top ranked ages correspond to ground truth borders between known aging stages. In our considered age range between 20 and 80 years, there are two such ground truth borders: 35 years, the change from early adulthood to midlife, and 50 years, the change from midlife to mature adulthood (vertical red lines in Fig. 5)^{30}. Since the data does not contain snapshots corresponding to these two ages, we consider the next closest ages that are present in the data. For age of 50, the closest ages present in the data are 48 and 52 (Fig. 5). For age 35, we also consider ages two years before and after, i.e., ages 33 and 37 (both of which are present in the data). Moreover, it is known that an important agingrelated change happens between 60 and 80 years, though it is not known exactly when^{38}.
Given the three resulting ground truth age intervals (33–37, 48–52, and 60–80; gray rectangles in Fig. 5), since we expect one major change within each interval, we consider SCOUT’s top three ranked ages, hoping that each would fall within one of the intervals. SCOUT’s output are ages 37, 52, and 69 (Fig. 5). Indeed, they cover the three intervals (pvalue of 0.03162; Supplementary Section S2.2). It is encouraging that SCOUT can perfectly recover the expected agingrelated shifts.
Discussion
We combine community detection with segment detection in dynamic networks to formulate a new problem of SCD. To address the drawbacks of the existing methods that can be employed to solve the SCD problem, we introduce a new and superior approach called SCOUT. We show that the SCD problem and SCOUT in particular is a useful framework for studying community organization of dynamic networks, as it can identify both when communities evolve by identifying change points and how communities look like at each stage of their evolution by identifying segment partitions. The solution of the SCD problem provides a concise yet informative description of the dynamic network from the perspective of its community organization.
Our work has potential future extensions. Methodologically, SCOUT could be extended to different problem settings, such as dealing with weighted networks or overlapping communities. Applicationwise, an important problem in dynamic network analysis is to choose a meaningful time scale for defining network snapshots. Usually, the time scale is chosen so that each snapshot is assumed to have the same duration, and the duration is determined empirically to fit the given application. Instead, the output of SCD could provide a systematic way for defining snapshots. Namely, the smallest meaningful traditional empirical equallength snapshots would be used to define the initial dynamic network. Then, this network would be given as input to SCOUT to group the small snapshots with consistent community organization into larger segments. Finally, the time interval of each segment would correspond to a new, more meaningful snapshot, and collection of all such new snapshots would form a new, more meaningful dynamic network. This way, each snapshot of the new network would capture the period during which community organization is consistent. Also, the duration of different snapshots could be different. These newly constructed snapshots could then be used as input to various methods for dynamic network analysis, which could improve the quality of results compared to using the same methods on the traditionally determined empirical samelength snapshots.
This paper focuses on the problem of SCD in dynamic networks. Other important network science problems exist, such as network comparison/alignment^{39,40,41}, link prediction^{42}, information spreading^{43}, graphlet (subgraph) mining^{10}, etc. Most of these problems have already been extended from the static to dynamic network context. For example, while in a static network, link prediction is used to denoise the network by identifying missing and spurious links^{44}, in a dynamic network, its goal is to predict interactions at time t based on network information up to time t − 1^{42}. Also, while graphlets have traditionally been used to characterize the structure of a static network, they have been extended into dynamic graphlets to allow for analyzing temporal data^{10}. Similarly, traditional static network alignment has been extended to its dynamic counterpart^{45}. Clearly, the field of dynamic network analysis, and thus our SCOUT approach that contributes to the SCD problem within this field, will only continue to gain importance with the increase in availability of temporal realworld network data.
Methods
Problem formulation
A dynamic network D is a sequence of k snapshots {G_{0}, G_{1}, …, G_{k−1}}, where each snapshot G_{i} = (V_{i}, E_{i}) is a static graph capturing network structure during time interval i. A sequence of consecutive snapshots can be grouped into a segment. Formally, a segment s is a sequence of consecutive snapshots {G_{i}, G_{i+1}, …, G_{j}}, i ≤ j, with i being its start time, j being its end time, and j − i + 1 being its length. A sequence of nonoverlapping segments (meaning that each segment in the sequence starts right after the previous one ends) that covers the whole dynamic network (meaning that the first segment in the sequence starts at time 0 and the last segment in the sequence ends at time k − 1) forms a segmentation of this network. Formally, a segmentation S is a sequence of l adjacent segments {s_{0}, s_{1}, …, s_{l−1}} such that s_{0} starts at time 0 and s_{l−1} ends at time k − 1. We can specify such a segmentation via a set T = {t_{1}, t_{2}, …, t_{l−1}} of l − 1 time points called change points, such that t_{i} is the start time of segment s_{i}, i ∈ [0, l − 1] (by convention, we always assume that t_{0} = 0).
Given a dynamic network D, SCD aims to simultaneously find a segmentation (or equivalently a change point set ) and a sequence of partitions such that identifies important shifts in the community organization of D and each (called segment partition) reflects well the community organization of each snapshot within segment (Fig. 1(a)). The output (i.e., solution or answer) of SCD can be represented as . Intuitively, in a good output, should be parsimonious (meaning that it should capture all important shifts in the network with as few change points as possible), while should be accurate (meaning that segment partitions should correctly capture community organization of all snapshots within the corresponding segment). That is, output should aim to simultaneously satisfy two objectives: segmentation parsimony and partition accuracy. We can now state the problem:
Problem 1 (SCD)
Given a dynamic network D = {G_{1}, G_{2}, …, G_{k}}, find a number of segments l, a sequence of l − 1 change points , and a sequence of l segment partitions such that the output forms a parsimonious segmentation with accurate segment partitions.
The two objectives, segmentation parsimony and partition accuracy, are competing with each other, as optimizing one does not necessarily lead to optimizing the other. For example, at the extreme of snapshot community detection (bottom of Fig. 1(b)), each snapshot is a separate segment that has its own wellfitting partition, which yields high partition accuracy. However, such a finegrained output with the maximum possible number of segments might contain redundancies, because some adjacent snapshots might have similar community organizations. In this case, segmentation parsimony will be low. To optimize (increase) segmentation parsimony, adjacent snapshots with similar community organizations should be grouped together. At the other extreme of consensus community detection (top of Fig. 1(b)), all snapshots form one segment with a single segment partition for the whole network, which yields high segmentation parsimony. However, the single segment partition will have to “compromise” between many possibly quite distinct snapshots. In this case, the segment partition will not be able to fit well all of the distinct snapshots, and consequently, partition accuracy will be low. In real life, the SCD solution typically lies between these two extremes, and finding it requires balancing between the two somewhat contradicting goals of optimizing both segment parsimony and partition accuracy. We formalize the ways of finding such a solution below.
Recall from Introduction the need of being able to find a solution with a userspecified number of segments l, in addition to being able to determine this parameter l automatically. Our current SCD problem formulation (Problem 1) can handle the latter scenario, but we can extend it to handle the former scenario as well. Specifically, when finding an SCD solution, in addition to allowing for simultaneously optimizing both aspects of SCD quality (segmentation parsimony and partition accuracy), we can allow for optimizing only one aspect (partition accuracy) while setting the other one (segmentation parsimony, expressed as the number of segments l) as a constraint. So, we extend the problem formulation by adding to the existing SCD objective from Problem 1 the following new objective: given a dynamic network D and the desired number of segments l as input by the user, find an output with l segments that achieves the highest partition accuracy. We refer to this new objective as the constrained SCD problem (CSCD). We propose SCOUT to solve any of the SCD and CSCD problems, to allow for handling both of the above scenarios (automatic vs. userdefined selection of the number of segments l, respectively).
Our SCOUT approach
Given a dynamic network D, we aim to find an output by directly optimizing an objective function that measures both segmentation parsimony and partition accuracy (see below for details on how we deal with SCD versus CSCD). Supplementary Algorithm S1 provides a highlevel overview of SCOUT and Supplementary Section S2.1 provides further details. SCOUT has the following five steps. 1) Select the initial change point set as the current change point set T (line 2 in Supplementary Algorithm S1). For example, the initial change point set could correspond to a set of all possible snapshotlevel segments (bottomup search) or just one large networklevel segment (topdown search). Given T, the method iteratively performs the following steps. 2) Perform consensus clustering within each segment s_{i} to get its corresponding partition p_{i} (line 7). In general, the consensus clustering method should aim to obtain the partition set P that maximizes the objective function for T. Step 2 results in A = (T, P) (line 9). 3) Use a search strategy to search for the next change point set that will become the new current change point set T (line 11). Clearly, the search strategy guides how we explore the space of possible change point sets. For example, in bottomup search, the next change point set is obtained by merging two adjacent segments, while in topdown search, the next change point set is obtained by splitting a segment into two. 4) Repeat steps 2 and 3 above until the exploration of the space is finished (corresponding to T = ∅ in line 3), e.g., until one largest possible networklevel segment is reached in bottomup search or until all possible snapshotlevel segments are reached in topdown search. 5) Choose the best output out of all outputs computed in step 2 as the final output (line 13). When solving the SCD problem, the best output is the one maximizing the objective function. When solving the CSCD problem, the best output is the one maximizing the objective function while satisfying the constraint (the solution consisting of l segments).
Experimental setup
Methods for comparison
We compare SCOUT with GraphScope, MultiStep, and GHRG (Supplementary Section S2.2).
Datasets
We evaluate the methods on two types of networks: synthetic networks and realworld networks.
Synthetic networks
To generate a synthetic dynamic network D with the embedded ground truth A^{(gt)} = (T^{(gt)}, P^{(gt)}), we introduce a new dynamic random graph model, called segment community generator (SCG). Intuitively, we first select change points T^{(gt)} and generate segment partitions P^{(gt)}. Then, we use these to generate D, assuming that each of its snapshots is generated via a stochastic blockmodel based on the corresponding segment partition. For details, see Supplementary Section S2.2 and Algorithm S2. For our experiments, we generate synthetic dynamic networks with 16 snapshots and 1, 2, 4, 8, and 16 ground truth segments, while considering various network sizes: 50, 100, 500, and 1000 nodes in each snapshot. This results in 5 × 4 = 20 synthetic network configurations. For each configuration, we generate 10 instances in order to account for the randomness in the synthetic network generator. This totals to 20 × 10 = 200 synthetic networks.
Realworld networks
We consider six publicly available realworld dynamic networks (Supplementary Section S2.2 and Table S2). 1) Hypertext^{46} network captures facetoface proximity of attendees of the Hypertext 2009 conference. This network has T^{(gt)} that corresponds to the list of events from the conference program^{46}. 2) AMD Hope^{47} network captures colocation of attendees of The Last HOPE 2008 conference. This network has T^{(gt)} that corresponds to featured/keynote talks and social events^{47}. 3) High School^{34} network captures proximity of students in a high school during a work week in 2013. This network does not have T^{(gt)}. 4) Reality Mining^{48} network captures social interactions of university students and faculty during 2004–2005 academic year. This network has T^{(gt)} that corresponds to the list of events from the academic calendar^{32}. 5) Enron^{49} network captures email communication of Enron employees during the 2000–2002 period. This network has T^{(gt)} that corresponds to the list of companyrelated events from the news^{32}. 6) Senate^{18} network captures voting similarities of US senators during the 1789–2015 period (i.e., for 113 Congresses). This network does not have T^{(gt)}.
Evaluation measures
We evaluate the methods via network structurebased and ground truth knowledgebased measures.
Network structurebased measures
Here, we evaluate a given method’s solution with respect to the structure of the input dynamic network D, without relying on any ground truth knowledge. For this, we can use any objective function from Supplementary Section S2.1. This includes four Q_{P} measures of partition quality and two Q_{B} measures accounting for both segmentation quality and partition quality. Regarding the four Q_{P} measures (modularity, conductance, normalized cut, and averageODF), in our experiments, all four measures show statistically significantly correlated results with respect to both Pearson and Spearman correlations (with all pairwise pvalues < 10^{−49}). So, in case of Q_{P}, for brevity, we report results only for modularity. Regarding the two Q_{B} measures (AIC and BIC), we do not evaluate the results with respect to them, since these are the objective functions that SCOUT explicitly optimizes, and we want to avoid circular reasoning.
Ground truth knowledgebased measures
Here, we explicitly rely on the ground truth knowledge to evaluate a given method’s solution , by: I) measuring similarity of to the known ground truth solution A^{(gt)} = (T^{(gt)}, P^{(gt)}) and II) evaluating the method’s ability to rank time points according to how “change pointlike” they are.
I) We introduce three measures of similarity between and A^{(gt)}: a) segmentation similarity Sim_{T}, capturing the segmentation aspect of and A^{(gt)}, b) partition similarity Sim_{P}, capturing the partition aspect of and A^{(gt)}, and c) overall similarity Sim_{B}, capturing simultaneously both aspects of and A^{(gt)} (Supplementary Section S2.2). All of Sim_{T}, Sim_{P}, and Sim_{B} rely on a measure H of similarity between two partitions. We test four such measures H: 1) Normalized Mutual Information (NMI)^{50}, 2) Adjusted Mutual Information (AMI)^{50}, 3) Adjusted Rand Index (ARI)^{50}, and 4) VMeasure (VM)^{51} (Supplementary Section S2.2). In our experiments, all four measures H show statistically significantly correlated results with respect to both Pearson and Spearman correlations (with all pairwise pvalues < 10^{−239}). For brevity, we report results only for NMI.
II) One way to assess a method’s ability to detect ground truth change points T^{(gt)} is via Sim_{T} from above, which directly compares the method’s change point set against T^{(gt)}. Sim_{T} only accounts for time points that were chosen as change points, i.e., it does not consider time points that were not chosen as change points, even though some of these time points may have still been good change point candidates. Namely, when determining which time points should be change points, a method assigns to each time point a score (rank) according to how “change pointlike” the time point is. Instead of using “binary” information for each time point t as Sim_{T} does (i.e., either t ∈ T^{(gt)} or ), we can make use of the more complete information on ranking of all time points. This is useful because even if some ground truth change point t ∈ T^{(gt)} is not (mistakenly) included into , we still want the method to rank t higher than some other . Sim_{T} would fail to capture this, so we use an alternative evaluation metric, change point classification, as follows. Having a ranked list of all time points (for details on how we obtain this list for each method, see Supplementary Section S2.2), we measure a given method’s performance with respect to change point classification via three measures: 1) the area under the precisionrecall curve (AUPR), 2) the maximum Fscore, and 3) the area under the receiver operator characteristic curve (AUROC) (Supplementary Section S2.2). In our experiments, all three measures show statistically significantly correlated results with respect to both Pearson and Spearman correlations (with all pairwise pvalues < 10^{−64}). So, for brevity, we report results only for AUPR.
Additional Information
How to cite this article: Hulovatyy, Y. and Milenković, T. SCOUT: simultaneous time segmentation and community detection in dynamic networks. Sci. Rep. 6, 37557; doi: 10.1038/srep37557 (2016).
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Acknowledgements
This work was supported by the National Science Foundation (CAREER CCF1452795 and CCF1319469).
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Y.H. and T.M. conceived the study. Y.H. designed the methodology. Y.H. and T.M. designed the experiments. Y.H. performed the experiments. Y.H. and T.M. analyzed the results. Y.H. and T.M. wrote the paper. T.M. supervised all aspects of the study.
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Hulovatyy, Y., Milenković, T. SCOUT: simultaneous time segmentation and community detection in dynamic networks. Sci Rep 6, 37557 (2016). https://doi.org/10.1038/srep37557
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