An Online and Nonuniform Timeslicing Method for Network Visualisation
Jean R. Ponciano, Claudio D. G. Linhares, Elaine R. Faria, Bruno A. N. Travencolo
AA N O NLINE AND N ONUNIFORM T IMESLICING M ETHOD FOR N ETWORK V ISUALISATION
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REPRINT
J. R. Ponciano , C. D. G. Linhares , E. R. Faria , and B. A. N. Travençolo Faculty of Computing, Federal University of Uberlândia, Brazil{jeanrobertop, claudiodgl, travencolo}@gmail.com, [email protected] 25, 2020 A BSTRACT
Visual analysis of temporal networks comprises an effective way to understand the network dynamics,facilitating the identification of patterns, anomalies, and other network properties, thus resulting infast decision making. The amount of data in real-world networks, however, may result in a layout withhigh visual clutter due to edge overlapping. This is particularly relevant in the so-called streamingnetworks , in which edges are continuously arriving (online) and in non-stationary distribution. Allthree network dimensions, namely node , edge , and time , can be manipulated to reduce such clutterand improve readability. This paper presents an online and nonuniform timeslicing method, thusconsidering the underlying network structure and addressing streaming network analyses. Weconducted experiments using two real-world networks to compare our method against uniform andnonuniform timeslicing strategies. The results show that our method automatically selects timeslicesthat effectively reduce visual clutter in periods with bursts of events. As a consequence, decisionmaking based on the identification of global temporal patterns becomes faster and more reliable. K eywords Temporal network visualization · Nonuniform timeslicing · Streaming network · Network sampling.
Networks represent a useful and widely adopted structure to model systems from distinct areas, such as computerscience, biology, sociology, and others [Est15]. A network is defined in terms of nodes (instances) and edges (therelationship involving them) [AB02]. In this way, a network may be used to represent the World Wide Web (Web pagesconnected by hyperlinks), an organism cell (chemicals linked by chemical reactions), social interactions (any socialrelationship connecting individuals - e.g., friendship or collaboration), and many others [Est15]. In several situations,using only information about nodes and edges may not be enough to represent and comprehend the relations in thenetwork. In social network analysis, for example, the information of when the connections occur may be crucial todescribe such relations with less (or even without) loss of context. Networks in which the time information is considered,in addition to nodes and edges information, are studied in several disciplines and received different nomenclatures:temporal network, dynamic network, time-varying network [HS12].In temporal networks, the appearance of a new node or edge represents the occurrence of an event at the respectivetimestamp. In this type of network, all events are known and available to be used in the analysis (offline scenario) [HS12].A temporal network has a delimited observation period, its data usually fit in primary memory and unrestricted randomaccess is allowed. However, in several real-world applications, data are produced in a massive and continuous way(online scenario). These types of data are referred as data streams . Examples include credit card transactions, phonecalls, neuronal data, content sharing social networks, and others. In such applications, the volume of data may be solarge that the storage may be impossible and mining tasks become more challenging [Agg06]. Temporal networks canbe used to model such streaming data [ANK13]. Since new events (edges) are continuously arriving in non-stationarydistribution and typically at high speed, such networks are called streaming networks [ANK13, Zha10, Est13]. The a r X i v : . [ c s . S I] S e p . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation non-stationary distribution of incoming data increases complexity and makes streaming network manipulation evenmore challenging.Both temporal and streaming networks can be analysed by adopting different strategies. Statistical analysis representsa common approach and is useful to identify specific trends and patterns in the data, being used, for example, inconnection prediction [BWL ∗
19] and burst analysis [JH19]. When there is only a numeric output, however, it mayrepresent a “black-box” to the user, thus impairing pattern comprehension. Another approach involves InformationVisualisation [CMS99,War13], whose strategies assist data analysis by providing interactive and graphical computationaltools, thus including the user in the entire process of exploration and validation. An adequate Information Visualisationstrategy allows a visual analysis that is as much intuitive as possible and also helps the user in finding unexpectedpatterns, anomalies, and other behaviours in the data, thus resulting in a fast and reliable decision making. Examplesof recent visualisation methods applied to temporal and streaming networks can be found in [LPP ∗ ∗
20, GdBG12, CCM17, SCG14, DPF16, LPP ∗ ∗ Mas-sive Sequence View (MSV) layout [HCvW07, vHBv13]. Such layouts may suffer from visual clutter caused by theamount of information, impairing the analysis. To address this issue, all three network dimensions, namely node , edge (or connection ), and time , can be manipulated to reduce clutter and improve readability. Node ordering tech-niques [vdEHBvW14, LTPR17], for instance, try to reduce edge overlapping by repositioning nodes, while samplingapproaches [RMH17, ZSC ∗
18] select specific portions of the network to be analysed. Among these, the temporalsampling allows the selection of a subset of nodes and edges by reducing the observation time (e.g., considering onlythe first day of the network) or by timislicing the network, i.e., changing the network temporal resolution by groupingedges from subsequent timestamps in a way that each timestamp may represent, for example, all edges from 1 minuteor 1 day interval [RMH17, LTPR17]).The temporal resolution plays an important role in the layout construction and, consequently, in the visual analysis. Inseveral scenarios, as, for example, when the networks are temporally sparse, an effective timeslicing may facilitatethe analysis and highlight patterns that would be difficult to see using the original resolution [LTPR17, LMM19].Choosing the length (in terms of number of timestamps) of each timeslice, however, is not a trivial task. Since it isusually chosen before the layout construction, the user needs to be a domain specialist, in order to know a priori whichtimeslicing is the more adequate for the analysis given the expected event frequency distribution; otherwise it has tobe empirically determined. A naive and widely used timeslicing approach is to consider timeslices of equal length torepresent the network (uniform timeslicing) [RMH17, LTPR17, ZSC ∗
18, LPP ∗ ∗ ∗
18, LPP ∗ This section presents fundamental concepts and discusses strategies focused on network visualisation. A discussionconcerning network dimensions ( node , edge , and time ) manipulation is also provided.2 . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation A temporal network can be represented by G = ( V, E ) , where V = { n , n , ..., n N } is the set of nodes in thenetwork and E = { e , e , ..., e M } is the set of edges. In our context, each edge e i = ( n x , n y , t k ) connects two nodes n x , n y ∈ V at a particular and discrete timestamp t k [LPP ∗ t end as the end of the observationperiod, ≤ t k ≤ t end . In fact, an edge that occurs at t k actually occurs in the interval [ t k , t k + τ ) , where τ is thetemporal resolution [LPP ∗ ∗ streaming network [ANK13,McG09]. In a telecommunication context, phone calls form a network between the involved phone numbers [Zha10]. Inthe same way, web pages and the links between them form a web network. Methods for processing streaming networksrequire efficient and real-time processing. This means that a streaming (or online) algorithm, besides the restrictedaccess to the stream data (edges), must process the stream in a single scan, or in a small number of scans [Zha10].In temporal networks, all edges and nodes are known and available to be used in the analysis. In streaming, edgesare continuously arriving, typically at high speed, and in a way that the volume of data does not fit in the primarymemory [ANK13]. We define a streaming network S = { e , e , ..., e m , ... } as a temporal network G = ( V, E ) asfollows: e i = ( n x , n y , t k ) , e i ∈ E , represents an edge that occurs at a discrete timestamp t k , ≤ t k ≤ ∞ , between n x , n y ∈ V , | V | → ∞ . Note that it is possible to have more than one edge per timestamp. This definition is differentfrom the ones that consider each edge arriving in a different timestamp [CCM17, ADWR17, EL19]. The employment of an effective temporal network visualisation strategy helps the user in the network evolutioncomprehension and facilitates the identification of patterns, anomalies, and other network properties. In this context,several visual strategies may be adopted, such as matrix-based [Bac16, BPF14] and circular approaches [vdEHBvW14],node-link diagrams [ET94, LTPR17] and
Massive Sequence View (MSV) layouts [HCvW07, vHBv13]. Among these,node-link diagrams and MSV represent the best strategies when the task is to analyse the edge (event) distributionover time [vdEHBvW14, LPP ∗ ∗ Massive Sequence View [HCvW07, vHBv13] is a timeline-based layout [BBDW17] similar to BioFabric [Lon12].Its x -axis represents the timestamps and the y -axis represents the nodes of the network. In this layout, nodes cannotchange their positions over time. Every time there is a connection (edge) between a pair of nodes, a vertical line isdrawn linking them in the respective timestamp. The construction of the standard MSV layout using the tabular (raw)data is illustrated in Fig. 1(a,b).When applied in real-world networks with a large amount of data, node-link diagrams and MSV suffer from visualclutter caused by overlapping edges, and thus important patterns may not be perceived. The Temporal Activity Map(TAM) [LTPR17], which omits all edges of the MSV layout and changes the shape of nodes from circles to squares forbetter sense of continuity, represents an alternative layout useful to identify patterns based only in the node activity(Fig. 1(b)). However, when the edge information is still needed or the amount of visual information in the layout is stilllarge, these issues must be solved.One strategy to enhance the layout is by changing the node positioning, which affects the length of the edges and,consequently, the number of overlapping edges and visible patterns. For this purpose, several node ordering algorithmshave been proposed in the literature. For the MSV layout, examples include naive approaches, such as the onesbased on the node appearance order, degree (in/out) and lexicographic, as well as more complex approaches, such as
Optimized MSV [vHBv13],
Recurrent Neighbors [LTPR17], and
Community-based Node Ordering (CNO) [LPP ∗ ∗ ∗
18, ZCS ∗
19, SCG15]. In this case, not only edges can be sampled, but also nodes – onlythose edges connecting the sampled nodes are maintained –, as illustrated in Fig. 1(c-d).
Along with node positioning and sampling strategies, the temporal dimension can also be manipulated to improvelayout readability. In this case, one possibility is to choose an observation time of interest (e.g., only the first or thesecond day of the network), as adopted by Zhao et al. [ZSC ∗
18] (Fig. 1(e)). Another possibility is to change thenetwork temporal resolution scale through timeslicing strategies. Timeslicing in our context means that events from3 . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation (f)EDGE DIMENSION
ABBCD ABBCD edge sampling
MSV
ABBCD ABBCD
ABBCD ABBCD standard MSVtemporal activity map
NODE DIMENSION
ACBD ACBD
ABBC D ABBC D node orderingnode sampling TEMPORAL DIMENSION
ABBCD ABBCD observation time sampling resolution 2
TABULAR DATA source target time
A BB CA BC D 027273536A B 1A BB DB CA C 2A CC DA D 4
ABBCD ABBCD (a) (b)(d) (e) (c)
Figure 1: MSV layout and possible types of manipulation. (a) Tabular (raw) data; (b) MSV layout: standard MSV,showing nodes and edges, and temporal activity map (TAM). In real-world networks that contain a large amount of data,MSV suffers from visual clutter. In this case, it is possible to improve the layout by manipulating the three networkdimensions. (c) Node dimension manipulation – ordering and sampling; (d) Edge dimension manipulation – sampling;(e) Temporal dimension manipulation – observation time sampling; (f) Temporal dimension manipulation – timeslicing.subsequent timestamps will be grouped in a single timestamp [RMH17]. The higher the temporal resolution scale, themore subsequent timestamps will be grouped into one and, as a consequence, the longer the time interval representedby the resulting timestamp.In the simpler timeslicing approach (known as uniform timeslicing ), the temporal resolution scale is a global and staticvalue, so all timestamps of the network represent the same length of time without considering cases in which thenetwork has non-stationary event distribution. As an example, if each timestamp in resolution 1 represents a 20-secondinterval, then each timestamp in resolution 2 will represent a 40-second interval. Linhares et al. [LTPR17] changes thetimestamp in which each event occurs by following a uniform timeslicing as follows (Eq. 1). t new = (cid:22) t ori − t s τ (cid:23) τ + t s (1)where t new is the new timestamp of the event, t ori is the timestamp of the event in the original temporal resolution, t s isthe first timestamp of the network and τ is the desired resolution scale. Repeated events (edges) are considered as asingle one if their timestamps are merged. As a result, one may identify temporal patterns that would be difficult to seein the original resolution, especially in temporally sparse networks [LTPR17]. This timeslicing process is exemplifiedin Fig. 1(f), where in resolution 2 (new resolution defined by τ = 2 ) each pair of adjacent timestamps from the originalnetwork (Fig. 1(b)) are merged into one, thus the events represented in (A,B,0) and (A,B,1) become a single event inresolution 2 (A,B,0) and so on.For convenience, several studies that visualise temporal networks employ uniform timeslicing in the networks underanalysis [LTPR17, ZSC ∗
18, LPP ∗ ∗ ∗ . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation network temporal resolution is a 20-second interval per timestamp. In the same way, the Enron network [SA04] wasanalysed using a specific temporal resolution in [ZCS ∗
19] and a different one in [LTPR17]. Such global and statictemporal resolution scale is empirically chosen through initial exploratory analyses or by a domain specialist thatknows a priori which one is adequate for the analysis given the event distribution. This scale does not consider theunderlying network structure and thus may not faithfully represent the non-stationary distribution of events over time.Notwithstanding, in temporal networks all events are available in the visual analysis, so different scales may be testeduntil an adequate one is found.In streaming scenarios, although one can adopt uniform timeslices as well, the employment of uniform approaches iseven more difficult due to the non-stationary distribution of future events. Exploratory analysis may not be possiblebecause usually there are no a priori data to support the decision. Since the event distribution can change, consideringonly an initial set of events in the stream to support the choice may be inefficient as well. This is often ignored when itis assumed a uniform event distribution, as if the events came in consecutive timestamps [CCM17, ADWR17, EL19].Wang et al. [WAH ∗
19] proposed a nonuniform timeslicing method for temporal network visualisation that createstimeslices with a balanced number of events (equal visual complexity) by using an approach similar to the histogramequalisation, well-established in the discipline of digital image processing. Their strategy (hereafter named
BalancedVisual Complexity – BVC ) uses more timestamps to represent high-activity periods (with bursts of events) and lesstimestamps otherwise. The method we propose in this paper goes in the opposite direction, i.e, we also consider thathigh-activity periods contain too much visual information, but we propose to represent them with higher resolutionscales (consequently reducing the number of timestamps) instead of redistributing them in more timestamps. In theproduced layout, the identification of global temporal patterns (e.g., birth and death of highly-active groups of nodes,bursts of events) is facilitated. Moreover, contrary to BVC, our method runs online and thus is suitable for streamingnetwork analysis. To the best of our knowledge, no other study has proposed online and nonuniform timeslicingmethods for network visualisation.
The idea behind our method is that intervals with the same length but that have different numbers of events must berepresented by different resolution scales. Having more events leads to higher resolutions and thus in fewer timestamps,and so the amount of visual information is reduced to an appropriate level in an attempt to optimise the identification ofpatterns.Our method considers the number of events and their distribution on a fixed size window of timestamps (window ofsize w size ) to decide the temporal resolution scale that will be applied to the next window (i.e., next timeslice). Inside awindow, the event distribution is considered by reducing the importance of older events using the forgetting mechanismfading sum, highly used in stream scenarios [Gam10, GSR13].Initially, we adopt the original resolution scale in the first window (cold start). From there, the resolution value for eachsubsequent non-overlapping window is calculated according to Eq. 2: σ n = (cid:98) δ.σ c + (1 − δ ) .f s ( w size ) (cid:99) (2)where σ n is the resolution value for the next window, δ (0 ≤ δ ≤ is a constant that determines the importance of thecurrent resolution value ( σ c ) in the computation of the new resolution, and f s ( w size ) is the fading sum of all events inthe current window, which is calculated according to a recursive formula (Eq. 3): f s ( i ) = x i | T wc | + α.f s ( i − (3)where x i is the number of events in position i of the window, | T wc | is the number of timestamps considered by thewindow that presents at least one event, f s (1) = x | T wc | is the initial term, and α (0 (cid:28) α ≤ is the fading factor. Thehigher the fading factor, the more importance is given to old events and, consequently, the higher the adopted resolutionscale inside this window. If σ n = 0 , then σ n is set as the average value of all past resolutions, so large inactivity periods(i.e. without events) may be represented by a resolution scale different from the original.With the new resolution scale computed, it is possible to change the timestamp attribute of incoming events. Inspired byEq. 1 [LTPR17], we define the new timestamp t new of an event e as: t new ( e ) = (cid:22) t orig ( e ) − t ini σ n (cid:23) + t ref (4)5 . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation where t orig ( e ) is the timestamp of e in the original resolution, t ini is the first timestamp considered by the currentwindow, σ n is the new resolution value (Eq. 2), and t ref is the timestamp that acts as a reference in order to applythe resolution scale in inactive timestamps. The value of t ref is computed when dealing with the first event of a newresolution (new timeslice) and is defined according to Eq. 5: t ref = (cid:22) t ini − t orig ( e (cid:48) ) σ p (cid:23) + t new ( e (cid:48) ) (5)where t orig ( e (cid:48) ) is the original timestamp of the last event from the previous window (event e (cid:48) ), σ p is the previousresolution (the resolution applied on e (cid:48) ), and t new ( e (cid:48) ) is the timestamp of e (cid:48) in σ p .Figure 2 shows an example of how the timestamp of an event is changed according to Eq. 4. In this figure, t orig ( e (cid:48) ) = t new ( e (cid:48) ) because, up to this point, the original resolution (value 1) was maintained in the network. In t = 100 , thenew resolution value was computed (value 2) and so it was necessary to change the original timestamp of e . Eachtimestamp in resolution 2 is twice the time interval represented by a timestamp from resolution 1, thus t orig ( e ) = 130 and t new ( e ) = 115 . Assuming an event x with t orig ( x ) = 131 , then t new ( x ) would be equal to 115 as well, and so on.As stated, Eq. 4 takes into account inactivity periods, respecting their occurrence in the converted timestamps. ABC t (e’) = t (e’) = 70 orig new
RESOLUTION 1 t = 100(change resolution) e’ RESOLUTION 2 t (e) = 115 new t (e) = 130 orig e e
Figure 2: Example of timeslicing using our proposal. In resolution 2, the timestamp of e is changed from 130 to 115. In this section, we present visual analyses of two real-world temporal networks manipulated timestamp-by-timestamp tosimulate streaming scenarios. Our goal is to compare our nonuniform timeslicing method against the original networkresolution, uniform timeslicing approaches, and BVC [WAH ∗ δ = 0 . as the importance of the current resolution in the computationof the new one (see Eq. 2). To validate our method and illustrate its application, we rely on MSV [vHBv13] andTAM [LTPR17]. All experiments were performed using the software DyNetVis [LTPR17], which implements ourmethod and all layouts and features presented in this section. DyNetVis is freely available at . The first network,
Primary School [GBC14, SVB ∗ st - nd of 2009. This network contains 242 nodes and 125,773 edges distributed in 5,846timestamps. The original temporal resolution is 20 seconds, which means that each timestamp in Res. 1 comprises a20-second interval. The data represent contacts from the first to fifth grade, each of them having two classes (A andB). In this network, the majority of contacts occurs between students of the same class and each class has an assignedteacher [SVB ∗ . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation first daysecond day Tch
Figure 3: TAM layout showing four classes and all teachers of the Primary School network using resolution 1 (original).The interval between both days (from 5.21pm to 8.29am) does not present any edge and was omitted due to its sizein the layout. Nodes are grouped according to the classes and grades. The “Tch” profile refers to the teachers of theschool. The layout is horizontally large, dense and has few visible patterns, as, for example, the absence of classes 4Aand 4B near the end of the second day.Moreover, the layout is dense (a lot of events over time) and only a few patterns are easily identified, as, for example,the absence of classes 4A and 4B students near the end of the second day. The network does not register contacts duringsports activities [SVB ∗ F F ) and window size ( w size ) were evaluated and their impact in the adoptedresolution scales is shown in Fig. 4. In the figure, the whole network is considered (activity in day 1, interval, activity inday 2). By comparing the plots in which the F F value is the same (
F F = 0 . in (a,c) and F F = 0 . in (b,d)), it ispossible to see that a large window makes that the perception of changes in the number of events be late, delaying theresolution change. As a consequence, patterns related to these changes may be lost or identified only many timestampslater. This is especially relevant in streaming network analysis, in which the past data may have already been discarded.By comparing the plots in which the w size is the same ( w size = 50 in (a,b) and w size = 200 in (c,d)), one can noticehigher resolution values when adopting higher F F . This is expected since high
F F values increases the importance ofold events. Finally, the plots show the resolution adopted in the interval between both days of the network, in whichthere is no event. This value is computed based on the average value of the past resolutions. This decision is related tothe space of the layout required to represent such interval, which would be many times greater in the original resolution. (c) (d)(a) (b)
Figure 4: Our nonuniform timeslicing and the relation between the adopted resolution scales and the event distributionfor the Primary School network. (a) w size = 50 and F F = 0 . (1,443 timestamps). (b) w size = 50 and F F = 0 . (353 timestamps). (c) w size = 200 and F F = 0 . (4,880 timestamps). (d) w size = 200 and F F = 0 . (541timestamps). The choice of the Fading Factor ( F F ) and the window size ( w size ) affects the resolution scale and,consequently, the layout and visible patterns. 7 . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation Figure 5 shows TAM layouts for different timeslicing scales considering the same four classes and teachers from Fig. 3.Figure 5(a-e) shows TAM layouts for uniform timeslices adopting different resolutions (Res. 10, 25, 39, 100, and200, respectively). Figure 5(f) shows the TAM layout generated by our nonuniform timeslicing method ( w size = 100 and F F = 0 . , chosen empirically). Resolutions 10, 25, and 39 (Fig. 5(a-c)) were chosen because they representthe lower, the average and the higher resolution values adopted by our method for this network. Resolutions 100and 200 (Fig. 5(d-e)) are arbitrary values. As expected, higher resolution values generate denser and (horizontally)smaller layouts, which impairs the visual analysis and the identification of patterns. Our method (Fig. 5(f)), however,automatically defines resolution scales that represent appropriate levels of visual density. (b)(d) (e)(c) (a) (f) Tch
Figure 5: TAM layouts showing four classes and all teachers of the Primary School network according to our nonuniformtimeslicing method and five uniform resolution scales. (a) Res. 10. (b) Res. 25. (c) Res. 39. (d) Res. 100. (e) Res. 200.(f) Our method ( w size = 100 and F F = 0 . ).The adopted timeslicing strategy highly affects pattern identification. Figure 6 presents visual analyses over the TAMlayouts generated by our method (adopting w size = 100 and F F = 0 . , Fig. 6(a)) and by uniform timeslices usingresolutions 25 and 200 (Fig. 6(b,d), respectively). These are the same layouts from Fig. 5(f,b,e). The layout generatedby BVC is also considered (Fig. 6(c)). In the best-case scenario, at least seven patterns can be identified: (1) all studentsfrom class 2B joined the network after the other classes and the group of teachers; (2) lunch break – several studentsgo home for lunch, which reduces the number of interactions in such interval [SVB ∗ ∗ . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation resolution 1 (a) (b)
393 timestamps (d)
72 456
30 timestamps234 timestamps
41 32
Tch (c)
Figure 6: Visible patterns in TAM layouts generated by different timeslicing approaches for the Primary School network.(a) Our method ( w size = 100 and F F = 0 . ). (b) Res. 25. (c) BVC. (d) Res. 200. A maximum of seven patterns canbe identified: (1) class 2B students joined the network after the others; (2) lunch break; (3) no interaction involvingclass 2B students near the end of the st day; (4) absence of classes 4A and 4B near the end of the nd day; (5) twoteachers left the network after lunch in the nd day; (6) some students did not join the network in the st day; and (7)inactivity period. Continuous rectangles represent patterns considered as easy to identify. Dotted rectangles representpatterns with difficult perception.for this network spent 393 timestamps (against 5,845 from BVC) while preserving all seven analysed patterns. Lesstimestamps leads to less screen space and decreases the need of (horizontal) scrolling, which tends to facilitate theperception of temporal changes in the network (better mental map preservation). Considering uniform timeslicing,one should test different resolution scales until the better one is found. This approach, however, is only possible whendealing with (non-streaming) temporal networks (see Section 2.3). Our method not only provides adequate timeslices,but is suitable for streaming scenarios in which events are continuously arriving in non-stationary distribution.Figure 7 shows the spread of events over time according to different timeslicing approaches: the original resolution,BVC, our method ( w size = 100 and F F = 0 . ), and uniform Res. 25. The absence of events in the middle of plots(a,c,d) corresponds to the inactivity period between both days of the network. While BVC (Fig. 7(b)) changes theevent distribution because of its histogram equalisation – which impacts pattern identification –, our method (Fig. 7(c))provides a distribution similar to those from uniform approaches (Fig. 7(a,d)). Since our timeslicing adopts the originalresolution in the first window (cold start), one may see a “shift” in the time dimension at the plot (Fig. 7(c)).Figure 8 shows the empirical cumulative distribution function (ECDF) considering the events from our method’slayout ( w size = 100 and F F = 0 . , Fig. 8(a)) and from resolution 1’s layout (Fig. 8(b)). Our method produces lesstimestamps without events when compared with the original resolution (36.6% vs 47% – blue dotted lines), which isjustified by the resolution scale used in inactivity periods, that is different from the original one. Furthermore, 25.4% ofthe time contains very few events in our layout (cold start window). By observing the third quartile (red dotted lines),after 75% of the time our layout contains a maximum of 213 events per timestamp (26% of the maximum numberof events per timestamp), while in resolution 1 the number of events per timestamp is almost 43% of the maximumnumber of events per timestamp (40 out of 94 events). 9 . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation (c) no r m ( e ) time (d) no r m ( e ) time (a) no r m ( e ) time (b) no r m ( e ) time Figure 7: Spread of events according to different timeslicing approaches for the Primary School network. (a) Res. 1. (b)BVC. (c) Our method ( w size = 100 and F F = 0 . ). (d) Res. 25. “norm(e)” refers to the normalisation of the numberof events to values between 0 and 1. (a) (b) % t i m e s t a m p s s t a m p s t i m e Figure 8: Empirical cumulative distribution function (ECDF) and event distribution (ED) considering the events fromthe Primary School network. (a) ECDF our method ( w size = 100 and F F = 0 . ). (b) ECDF Res. 1. (c) ED ourmethod (393 timestamps, w size = 100 and F F = 0 . ). (d) ED Res. 1 (5,846 timestamps).The visual analysis can be performed from a different perspective by showing only edges, as illustrated in Fig. 9, thatshows the interactions involving classes 2A, 2B, and 4A over a MSV layout generated by our nonuniform timeslicing( w size = 100 and F F = 0 . ). This layout reaffirms: (i) students from class 2B joined the network after the others; (ii)students from class 4A left the network earlier than the others in the second day; (iii) the absence of the majority of2A students, as well as 2B students, during a period after the lunch break in the first day. Besides, this layout revealsnew patterns, such as the perception that the only two students from class 2A that stayed in the network during thetime interval after lunch in the first day connected to one another. Moreover, the layout shows that students from oneclass have few interactions with students from other classes, with the majority of these interactions occurring duringlunch. Not least, students from the nd grade interact more between themselves than with class 4A. This behaviour isalso observed in the rest of the network (a lot of interactions among students of the same grade and few interactionsinvolving different grades). These situations are expected in the network [SVB ∗
11] and easily perceived in this layout.10 . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation pattern 2pattern 1 resolution 1 lunch inactivity lunch
Figure 9: MSV layout with our nonuniform timeslicing ( w size = 100 and F F = 0 . ) showing the interactions amongclasses 2A, 2B and 4A. Pattern 1: many interactions between 2A and 2B during lunch. Pattern 2: few interactionsbetween 2B and 4A in the network. The second network,
Enron [KS05, SA04], contains email communications from Enron Inc., a former energy companyinvolved in the biggest American accounting fraud [SFF ∗ ∗ ∗ w size and F F for the Enron network. Comparingthe plots (a,c), it is possible to see the impact of the fading factor in the resolution computation. As can be seen, the highnumber of events near the end of the network is reflected in the timeslicing for the two
F F values tested. Comparingthe plots (b,c,d), one can see how frequent the timeslicing occurs according to the window size. As discussed, largewindows make the change in the resolution scale less frequent and, as a consequence, each resolution may not faithfullyrepresent the different number of events and their distribution. One can see such situation occurring in the Enronnetwork by analysing the resolution evolution under w size = 200 and F F = 0 . (Fig. 10(d)): at the end of thenetwork, the number of events decreases abruptly, but the resolution scale remains high. (c) (d)(a) (b) Figure 10: Our nonuniform timeslicing and the relation between the adopted resolution scales and the event distributionfor the Enron network. (a) w size = 100 and F F = 0 . (921 timestamps). (b) w size = 50 and F F = 0 . (357timestamps). (c) w size = 100 and F F = 0 . (448 timestamps). (d) w size = 200 and F F = 0 . (579 timestamps).Figure 11 shows an approximation of the same time interval (near Dec. th , 1999 to near May th , 2000) and thesame group of nodes in three distinct layouts obtained by adopting w size = 100 and three different Fading Factor values Since the resolution scale may aggregate different days in a single timestamp, the first timestamp may also represent few daysbefore the first day of the interval depending on the adopted resolution. In the same way, the last timestamp may also represent fewdays after the last day of the interval. . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation ( F F = 0 . , F F = 0 . and F F = 0 . ). The first layout, with F F = 0 . , maintained the original resolutionscale during the whole interval. By doing so, each timestamp refers to a 1-day interval and so it was possible to identifydays without events. Such days are usually weekends and holidays, such as the highlighted weekend May th − th ,2000 and holiday May th , 2000 (Memorial Day). By adopting F F = 0 . , one can see that the weekend/holidaypattern is lost due to the aggregation of days in a single timestamp. Another pattern, however, is revealed: it is easier toidentify a node without interactions, i.e., a person in the company that did not receive or send any emails in this period.By analysing the layout with F F = 0 . , it is possible to see that the node without interactions from the previouslayout appears in the network in the last timestamp. Moreover, one can notice that the first and the last nodes of thelayout had interactions only in the first timestamps. These last two patterns are visible in all three layouts, but they aremore easily perceived in the layouts with higher F F values.
Fading Factor 0.99 Fading Factor 0.9
Weekend and Memorial DayNode without connections
Fading Factor 0.99999
Connectionsonly in the firsttimestamps resolution 3 resolution 6 resolution 5 resolution 10resolution 1 resolution 1
Figure 11: Impact of different Fading Factor (
F F ) values on the layout ( w size = 100 ). Different FF values lead todifferent visual patterns. The change in the node colour represents a change in the resolution scale (new timeslice).Node ordering defined by Recurrent Neighbors [LTPR17] using resolution 1. (a) (c) (e)(d)(b)(f) (h) (i)(g) Figure 12: TAM layouts for the Enron network considering different resolution scales. (a) Res. 1. (b) Res. 2. (c) Res. 7.(d) Res. 15. (e) Res. 25. (f) Res. 50. (g) Res. 75. (h) Res. 100. (i) Our method ( w size = 100 and F F = 0 . ). Nodeordering defined by Recurrent Neighbors [LTPR17] using Res 1.12 . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation Figure 12 shows TAM layouts for the Enron network considering different timeslicing approaches. Figure 12(a-h) showsthe TAM layouts for uniform timeslicing using resolutions 1, 2, 7, 15, 25, 50, 75, and 100, respectively. Figure 12(i)shows the TAM layout generated by our method ( w size = 100 and F F = 0 . ). Resolutions 1, 2, and 7 (Fig. 12(a-c))are shown because they represent the lower (and original), the average and the higher resolution scales adopted by ourmethod for this network. The other resolutions (Fig. 12(c-h)) are arbitrary values.As illustrated in Fig. 13, depending on the timeslicing being used, more or less patterns can be identified. The layoutgenerated by our method allows the identification of at least 5 patterns (Fig. 13(a)): (1) weekdays, in which there areinteractions among nodes, and weekends (without interactions); (2) perception of the growing number of events overtime; (3) identification of highly active groups of nodes; (4) a time interval with a burst of events near the end of thenetwork; and (5) abrupt decrease in the number of events followed by the end of the network. The uniform timeslicingusing resolution 2 (Fig. 13(b)) also allows the perception of all five patterns. However, recall that this resolutionrepresents the average value adopted by our method, which supports the claim that it chooses resolution scales that areindeed suitable for the network analysis. As expected, BVC redistributed the events along the timestamps, and so thesetemporal patterns (all but pattern 3) are lost (Fig. 13(c)). By using resolution 7 in a uniform timeslicing (Fig. 13(d)),patterns 1 and 2 are lost. Each timestamp in this resolution represents 7 days and so there is no separation of weekdaysand weekends or the perception of growing node activity. One can note that layouts with temporal resolutions above 7(see Fig. 12(d-h)) are even worse for the Enron network visual analysis. (d) visible patterns: 3-5 (c) (b)(a)
41 2 53 visible patterns: 1-5visible pattern: 3visible patterns: 1-5
Figure 13: TAM layouts generated by different timeslicing approaches and their visible patterns in the Enron network.(a) Our method (921 timestamps, w size = 100 and F F = 0 . ). (b) Resolution 2 (673 timestamps). (c) BVC (1,345timestamps). (d) Resolution 7 (193 timestamps). Depending on the layout, a maximum of five visual patterns can beidentified: (1) weekdays, in which there are interactions among nodes, and weekends, that are days without interactions;(2) perception of the growing number of events over time; (3) identification of highly active groups of nodes; (4) a timeinterval with a burst of events near the end of the network; and (5) abrupt decrease in the number of events followed bythe end of the network. Node ordering defined by Recurrent Neighbors [LTPR17] using resolution 1.The ideal timeslicing depends on the network being analysed. For the Primary School network, the uniform timeslicingusing resolution 25 allowed the identification of several patterns (see Fig. 6(b)). On the other hand, resolution 25 is nota good choice for the Enron network (Fig. 12(d)). In the same way, a uniform timeslicing using resolution 2 would notimprove Primary School analysis. Our method is capable of adapting the resolution scale according to the number anddistribution of events, thus enhancing the network visual analysis.Figure 14 presents two other patterns observed in the layout generated by our method when zooming in the time intervalwith a burst of events showed in Fig. 13(a). According to Sun et al. [SFF ∗ “Rove divests his stocks in energy” . In the layout, it is possibleto see a decrease in the number of events (emails) in the majority of the days in June and July involving the majorityof the nodes. Such pattern may be related to this important episode. The layout also shows the moment in whichthere is an abrupt decrease in the number of events followed by the end of the network. Such decrease is related toanother company episode, “Lay [Enron CEO] implicated in plot to inflate profits and hide losses” [SFF ∗ . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation June, 2001 July, 2001Nodesmaintaining level ofactivity Nodeswith activitydecreasingover time CEO implicated in fraud
Figure 14: TAM layout generated by our method ( w size = 100 and F F = 0 . ) showing a portion of the network.Two patterns are visible: (i) a decrease in the number of events in June and July 2001; and (ii) an abrupt decrease inthe number of events followed by the end of the network. The change in the node colour represents a change in theresolution scale (new timeslice). Node ordering defined by Recurrent Neighbors [LTPR17] using resolution 1.happened in Feb th , 2002. After the decrease of events, our method changed the resolution scale from 7 to 3, reflectingthe new number of events. Temporal patterns such as these are probably lost when using BVC because of its eventredistribution (Fig. 15(b)). Our method (Fig. 15(c)), on the other hand, provides a distribution similar to those fromuniform approaches (Fig. 15(a,d)), and thus is capable of highlighting such temporal patterns. (c) no r m ( e ) time (d) no r m ( e ) time (a) (b) no r m ( e ) time no r m ( e ) time Figure 15: Spread of events over time according to different timeslicing approaches for the Enron network. (a) Originalresolution (1,346 timestamps). (b) BVC (1,345 timestamps). (c) Our method (921 timestamps, w size = 100 and F F = 0 . ). (d) Res. 25 (193 timestamps). While BVC changes the event distribution because of its histogramequalisation procedure, our method provides a distribution similar to those from uniform approaches. “norm(e)” refersto the normalisation of the number of events to values between 0 and 1. When timeslicing, one should be aware that events may be lost to improve network comprehension (due to theconsecutive timestamp grouping), and so relevant information may be lost in the process. Such characteristic exists inany other sampling strategy. Our method, however, considers the number of events and maintains their non-stationarydistribution in an attempt to reduce such impairment.Although our method improves the layout by manipulating the network temporal dimension, the node positioningrepresents another important aspect that has to be considered, since the ordering quality may impact layout readability.14 . R. Ponciano et al. / An Online and Nonuniform Timeslicing Method for Network Visualisation
We thus recommend the adoption of a high-quality node ordering method. Eventually, the joint employment of samplingstrategies may be required.Since two timestamps in the layout may represent completely different time intervals, one should pay attention in theresolution scale adopted in each of them when the task depends on this information (e.g., to decide which node hasbeen active for the longest time in the network). Changes in the node colour, as used in the experiments, attenuatethis limitation, but other visual encoding can be used. In such cases, where the nonuniform timeslicing impairs theanalysis, our method remains useful as the average resolution scale computed by it represents a good choice for auniform timeslicing (as occurred with resolution 25 in our Primary School analysis).Finally, we have demonstrated our method’s quality using TAM and MSV. Although our method runs online, thesevisual representations draw the network elements (nodes and/or edges) in an offline manner. This is a characteristic ofthese layouts and not a limitation of our method. Although they could be adapted to handle online scenarios by plottingconsecutive windows over time, this adaptation is out of the scope of this paper. Furthermore, our method does not relyon particular layouts’ characteristics (e.g., length/positioning of edges or animated vs timeline layouts) and thus couldbe applied in different layouts as well. In animated layouts, however, the visual analysis would probably be impaired insome cases, since the number of frames devoted to high-activity periods would reduce, potentially breaking the user’smental map.
We proposed in this paper an online and nonuniform timeslicing method for network visualisation that highlightstemporal patterns such as bursts of events, highly-active groups of nodes, and others. Without it, when handlingtemporal networks one should test different uniform timeslices until the “less worst” is found. Besides the effort ofsuch preliminary tests, analyses of different networks require different temporal resolutions. For streaming scenarios,exploratory analysis to support the timeslicing may not be possible because events are arriving online and in non-stationary distribution. For the same reason, considering an initial set of events to support the choice of the resolutionscale may be inefficient as well.Our method considers the number of events and their distribution to adapt the layout. This is possible because thechoice of each new resolution scale uses only events from a sliding window, with old information being discounted bya forgetting mechanism. The method has low time and spatial computational complexity, since there is no need forvarious scans in the data and edges can be discarded once they are processed. In our experiments, we have analysed tworeal-world networks with different characteristics and the results show that the resolution scales automatically adoptedare indeed suitable for each network analysis.As future work, we intend to perform user experiments to validate our method considering the quality of the producedlayout. Besides, the choice of both window size and fading factor value directly affects the layout. These are currentlyuser-dependent parameters and we will try to automate them. We also intend to apply our method in other networkvisualisation strategies.
Acknowledgments
This research was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq, Coordenaçãode Aperfeiçoamento de Pessoal de Nível Superior (CAPES PrInt - Grant number 88881.311513/2018-01), and Fundaçãode Amparo a Pesquisa do Estado de Minas Gerais - FAPEMIG.
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