A comparative analysis of knowledge acquisition performance in complex networks
AA comparative analysis of knowledge acquisition performance incomplex networks
Lucas Guerreiro , Filipi N. Silva and Diego R. Amancio Institute of Mathematics and Computer Science,University of S˜ao Paulo, S˜ao Carlos, Brazil Indiana University Network Science Institute,Bloomington, Indiana 47408, USA (Dated: July 24, 2020)
Abstract
Discovery processes have been an important topic in the network science field. The exploration ofnodes can be understood as the knowledge acquisition process taking place in the network, wherenodes represent concepts and edges are the semantical relationships between concepts. Whilesome studies have analyzed the performance of the knowledge acquisition process in particularnetwork topologies, here we performed a systematic performance analysis in well-known dynamicsand topologies. Several interesting results have been found. Overall, all learning curves displayedthe same learning shape, with different speed rates. We also found ambiguities in the featurespace describing the learning curves, meaning that the same knowledge acquisition curve can begenerated in different combinations of network topology and dynamics. A surprising example ofsuch patterns are the learning curves obtained from random and Waxman networks: despite thevery distinct characteristics in terms of global structure, several curves from different models turnedout to be similar. All in all, our results suggest that different learning strategies can lead to thesame learning performance. From the network reconstruction point of view, however, this meansthat learning curves of observed sequences should be combined with other sequence features if oneaims at inferring network topology from observed sequences. a r X i v : . [ c s . S I] J u l . INTRODUCTION Many real-world systems can be naturally represented by sequences corresponding tochains of events or transitions between states, including human actions [9], machine work-flow [36], scientists mobility [21] and language [14]. Communication can also be accomplishedby encoding and decoding data into sequences of symbols or continuous signals. Indeed, asignificant portion of datasets derived from real-world systems is available in this form. Fora complex system, one can understand that sequences can be generated by a process drivingthe changes among states across a certain space of allowed transitions [5].Network science has been employed to represent a great variety of complex systems [2, 7,17, 18, 20, 27]. In recent studies, complex networks have displayed the potential to representthe space of transitions between states for many types of systems [14, 24, 25]. In this context,the driving processes generating sequences are represented by stochastic walks of a varietyof heuristics. An example of this case is the knowledge acquisition process [6], in whichnodes represent knowledge that is connected according to how related they are. One ormultiple agents (such as researchers) navigate in this knowledge space, which is unknownfrom the start, and discoveries are made when the agents visit new nodes. In such a system,sequences are derived by the paths taken by the agents.While Markov chains [33] are a simple way to model and recover the inherent networkof transition probabilities, it relies on considering that the studied phenomenon is drivenby a simple stochastic process with no a priori knowledge of its space. Many real-systems,however, may present more intricate driving stochastic dynamics (which may depend on longterm memory or properties of the nodes, for instance). An example of that system is urbantransportation, where agents navigate across a system of roads with possibly predefinedorigin and destinations. The paths taken by connecting these endpoints cannot be drivensolely based on local probabilities. Also, the inherent space of state transitions can display avariety of different topologies [17] in contrast to more well-defined structures, such as regulargraphs, as a consequence, even simple stochastic dynamics can lead to intricate sequences [5].In many real-world problems, only the sequences generated by the system are observed.Thus, having a way to discriminate characteristics that are either consequence of the dy-namics or from the network can lead to a better understanding of the studied phenomenon.A simple property derived from sequences that can be differently impacted by both of these2spects is the rate of appearance of new symbols. This corresponds to the exploration cov-erage of a network under the action of a walking dynamics, which is also related to thelearning curve in a knowledge acquisition process. This property is also related to how wellan agent performs in discovering knowledge.To our knowledge, no previous study focused on a systematic analysis among the dy-namics, networks, and the sequences generated by them. Here we analyzed the coveragecurves for sequences obtained from four random walk dynamics and four network modelswith different topological structures. At first, we are interested in knowing if the coveragecurves are already good criteria for determining both the model and the dynamics used togenerate a sequence.Our analysis revealed that, among the considered stochastic walk dynamics using onlylocal network information, the true self-avoiding dynamics (TSAW) was found to present thebest performance in coverage rate for the considered network models. In addition to that,different patterns for the performances of coverage rate were observed. Aside from TSAW,the ranking based on performance of exploration for different sets of walk dynamics tends todepend on the network structure. For instance, when the stochastic walk is biased accordingto the node degree, better performance is attained when the network is sparse and the walksare biased towards preferring highly connected nodes. On the other hand, if the network isdenser, better performance is reached when the walk avoids highly connected nodes. We alsoencountered situations in which there exists ambiguity in the coverage property for certaincombinations of dynamics and network models. This indicates that it would be possible toswap the dynamics and the inherent structure and even so, attain similar coverage curves.These developments could shed a light on the analysis of the mechanisms leading to textgeneration, for instance, to better understand how the vocabulary grows along with the text.The following section explores the related literature to the problem of modeling real-world phenomena in terms of networks, dynamics, and sequences. Next, the methodology ispresented alongside the description of the considered network models and dynamics. Resultsare presented together with discussions, which is followed by conclusions.3
I. RELATED WORKS
Random walks (RW) have been studied in many networked applications [8, 13, 15, 32]. Inthe early studies of the emergent network science field, the properties of RW was investigatedin power-law distributed networks. In [1], the authors compared the efficiency of randomand self-avoiding walks in transferring messages through the network. Hubs were found toplay the role of centralizing and distributing information to other nodes. Most importantly,this finding revealed that the efficiency of discovering new nodes depends on the topologyof the underlying network.The process of network discovering has been approached by several recent studies [5, 6,23, 28, 39]. In [6], the authors compared the learning speed of several dynamics for particularnetwork topologies. Specifically, they analyzed how effective different dynamics are whendiscovering new nodes in the network. In addition to traditional random walks, this studyconsidered also random walks with L´evy flights [38]. Thus, the agents were allowed to visitany node in the network in the next step with a certain probability. The authors found thatmore frequent jumps favors the discovery rate, specially in Barab´asi-Albert networks. Inparticular topologies, though, jumps were found not to be as effective. This is the case ofgeographic networks. Another interesting finding is that the discovery of new nodes occurswith different speed in different network regions. The core – as identified via accessibility(entropy diversity) [4, 41] – tends to be covered faster than the network borders.In [28] the authors studied the efficiency of agents walking over the network to learnthe structure of the network. Differently from other works, the authors considered a modelwhere knowledge discovered by different agents is integrated in a specific entity of the sys-tem. This system is referred to as network brain . This type of dynamics was intended torepresent e.g. the knowledge acquisition when mapping communities of similar interests inthe Web. The most surprising result arising from this study is the fact that the learningbehavior, considering variations of the self-avoiding walk, has a very weak dependence onthe considered dynamics and network topologies.The problem of knowledge acquisition in networks has also been studied in the contextof information theory applications [5]. In [5], distinct random walks are performed overdifferent topologies. The sequence of visited nodes generates a sequence of symbols, whichis further analyzed in function of the observed compression ratio – computed via Huffman4oding. Finally, such a sequence is used to reconstruct the original network, and the error isanalyzed for distinct topologies and agent dynamics. Several interesting results have beenfound using the framework combining knowledge acquisition and information theory. Inter-estingly, the best performance in the framework constructed for representing the phenomenaof compression (during transmission) and reconstruction of networks revealed that a sim-ple knitted network model [16] yielded the best performance. This finding is compatiblewith the idea that language is optimized for transmission [11], since knitted networks arerepresentations of co-occurrence language networks [12, 31, 37, 40].The study reported in [25] aimed at identifying key Physics concepts from students’ rep-resentations of perceived similarity between distinct topics. The representation used in thiswork was a concept network, where nodes represent the concepts (in the sense of quantities,laws, models, or experiments), and edges represent similarities between these concepts, suchas actions for determining a model or the realization of a experiment using some law [39].The paper studies these concept networks using subgraph and communicability betweennesscentrality. The most relevant concept networks were identified using an importance rankingcoefficient, which is a normalized geometric mean of the considered centrality measurements.While this study does not relies on random walks to represent the acquired network, theconcepts networks are used as examples of networks representing the knowledge acquired bystudents, according to unknown knowledge acquisition dynamics.The study conducted in [23] analyzed the properties of self-avoiding walks (SAW) inclustered scale-free networks. The study investigated how the number of SAWs changesas the desired walk length increases. The main result of the paper shows that, for scale-free networks with same average degree, there are more SAWs in clustered networks whencompared to unclustered networks. This result suggests that the modular organization inthe same topological family of networks may impact the discovery process in the network.Differently from most of the works in the literature, here we analyze the knowledgeacquisition problem in terms of a generalist point of view. We analyze whether differentnetwork topologies and dynamics can lead to the same behavior in the observed learningcurves. In other works, we analyze the behavior of learning curves by comparing, at thesame time , different configurations of network topology and agents dynamics.5 a) Networks (b) Walk dynamics (c) Coverage curves (d) PCA
FIG. 1. Methodology employed to analysis the behavior of learning curves. In (a), we selecteddifferent network topologies. In (b), dynamics based on variations of random walks were consideredto explore the networks. In (c), we obtain the learning curves describing how many nodes arediscovered as the network is explored. Finally, in (d) each curve is mapped into a 2-dimensionalspace and similarities in the behavior of learning curves for different topologies and dynamics areanalyzed.
III. METHODOLOGY
The main objective of this paper is to compare the efficiency of different walking strategyto discover new nodes in the network. We compare well known random walk strategiesin different network topologies. Most importantly, we analyze the behavior of “learningcurves” for each pair topology/dynamics in order to analyze whether different combinationsof topology and random walks can lead to the same learning curve (and vice-versa). Theadopted methodology is illustrated in Figure 1 and summarized in the following steps:1.
Network topology : we selected different network topologies. We have selected well-known network models reproducing the characteristics of real-world networks. A briefdescription of the adopted models is provided in Section III A.2.
Network dynamics : different ways to walk over the networks were considered, includingdynamics based on traditional random walks and dynamics biased towards particularneighbor properties. A brief description of the adopted network dynamics is providedin Section III B.3.
Learning curves : For each pair of topology and dynamics, we obtain the learningcurves. This learning curve describes how fast new nodes are discovered as the dy-namics unfolds (see Section III C).4.
Cluster analysis : in this phase, each learning curves are mapped into a vector. This isused to measure the similarity between two curves. Similar curves are then identified6ia cluster analysis. This step is important to show that the behavior curve A briefdescription of this process is provided in Section III D.
A. Network topology
Artificial networks were built for each set of network models. The following parameterswere used to create the networks: number of nodes ( N ) = { , , } and averagedegree ( (cid:104) k (cid:105) ) = { , , , } . We have worked with four well-known undirected networktopology models: • Erd˝os-R´enyi (ER) : this model generates small-world networks, adding the character-istic to have all the nodes with similar degrees, i.e., the probability of creating an edgeis equally distributed among the nodes. • Barab´asi-Albert (BA) : this topology implements the scale-free model, inherent to manyreal networks. BA networks are characterized by a few hubs with a very high degree,while most nodes have small degrees. • Waxman (WAX) : this a traditional geographic model, which comprehends a set ofnodes in a two dimensional space that incorporates new edges through an algorithmin which the probability decays exponentially as the distance between each pair ofnodes grows. More specifically, the probability of two nodes to be linked is given by: π ij = a exp( d ij /β ) , (1)where a is a normalization factor, d ij is the geographic distance between nodes v i and v j and β is a parameter that defines the connectivity of the network. • Modular Networks (LFR) : networks with community structure were implemented us-ing the methodology described in [26]. In this model, each community is representedas a scale-free network. In addition to the number of nodes and average degree, addi-tional parameters can be considered to generate the networks. The main parametersdescribing this model are the number of communities ( n C ), the minus exponent forthe degree sequence ( t ), the minus exponent for the community size distribution ( t ),7he maximum degree (max k ), and the the mixing parameter ( µ ), which determines thefraction of edges linking distinct network communities. Here we used n C = 5, t = 3, t = 0, µ = 0 .
20. The maximum degree max k were chosen so as to obtain networkswith the desired average degree (cid:104) k (cid:105) .A visualization of the considered models for selected parameters is illustrated in Figure2. The visualizations were generated using the Networks3d software [35]. It is clear that fordifferent models the nodes with highest degrees (orangish nodes) are distributed in differentways. (C) WAX (d) LFR(a) ER (b) BA
FIG. 2. Force-directed visualizations of the considered network models. Different colors correspondto different node degrees. The visualizations were generated using the
Networks3d software [35]. . Network dynamics In order to recover the symbols from these models we have worked with the following walkdynamics: traditional random walk (RW) [29], random walk biased by degree (RWD) [10],random walked biased by the inverse of the degree (RWID) [10], and true self-avoiding walk(TSAW) [3, 24]. These walks have been widely employed to study the dynamics of learningcurves in the last few years [5, 6, 28]. The main differences among these walk dynamics aredetailed below: • Traditional random walks : the random walk dynamics is one of the most used inliterature, and a very simple one. If the walker is at node v i and Γ i is the set ofneighbors of v i , all nodes in Γ i have the same probability to be chosen as next node inthe walk. In other words, the probability of transition from v i to v j ∈ Γ i is p ij = k − i . • Degree-biased random walk : in this walking dynamics, a higher probability of transition p ij is given to those neighbors with higher degrees. Mathematically, p ij is proportionalto the degree k j of v j ∈ Γ i : p ij = k j (cid:80) l ∈ Γ i k l . (2)In other words, the RWD dynamics always tries to explore the network by prioritizingvisits to nodes with the highest number of neighbors. • Low degree-biased random walk : a different variation of the traditional random walkis the walk biased towards the inverse of the degree. In this case, the probability oftransition from v i to v j ∈ Γ i is : p ij = k − j (cid:80) l ∈ Γ i k − l . (3)Therefore, in this case, the walker tends to select nodes with low-degree in the nextstep of the random walk. • True self-avoiding walk : in a true self-avoiding walk dynamics, already visited nodesare avoided. This is achieved this by memorizing edges that have already been visited.9he transition probability is computed as p ij = e − λf ij (cid:80) l ∈ Γ i e − λf il , (4)where f ij is the frequency of visits to the edge linking nodes v i and v j . The parameter λ > λ = ln 2.The main advantage of this dynamics is that it tends to present a higher learning ratewhen many nodes have already been visited. When the walker is visiting a region withno visited nodes, this random walk behaves similarly to the RW dynamics. C. Learning Curves
The measure used to characterize each dynamics is the so-called learning rate. Thisis an important property in network science and is related to many processes on complexnetworks, including knowledge acquisition, discovery processes, diffusion and spreading [19].For each pair of network and random walk dynamics, we considered 5,000 iterations (steps).Learning curves are then obtained as the fraction of the total number of different nodesvisited after a given number of steps.The dynamics observed by visiting sequentially network nodes has an analogy with theprocess of generating written texts [5]. If we consider that, at each step, a symbol is generatedto represent that the current node has been visited, after 5,000 steps we have a sequence ofsymbols (i.e. a text) comprising 5,000 words. The learning curve can thus be seen as thevocabulary observed for a given text length. While in written texts the relationship betweenvocabulary size and text length is well described by the Heaps’ Law [30], the learning curveobserved in network discovery processes tends to follow a different pattern [6].
D. Principal Component Analysis
Here different learning curves are compared and similar learning curves is observed. Toquantify the similarity between curves we represent each curve as n -dimensional vector,where the i -th position of the vector represents the fraction of nodes visited after the i -th10tep. Because such a representation of curves yields several strongly correlated features, weuse Principal Component Analysis (PCA) [22] to remove possible correlations. In fact, aswe shall show, two dimensions of the PCA analysis accounts for more than 95% of the datavariation.After the learning curves are represented in a two-dimensional space, clusters can be iden-tified. Because our objective is to analyze whether similar learning curves can be obtainedwith different topology/dynamics choices, the identification of clusters was performed viavisual inspection. However, a scenario with several instances could also be analyzed by usingtraditional clustering algorithms [34]. IV. RESULTS AND DISCUSSION
Our analyses take into account the exploration coverage over time for agents discoveringknowledge in network models as they explore nodes through edges. The first step is obtainingthe learning curves for the considered pairs of dynamics (RW, TSAW, RWD, and RWID)and network models (ER, BA, WAX, and LFR models). For each network model setup, wegenerated 5 networks and recorded the coverage curves for 50 realizations of each dynamics.The starting position of each realization was drawn uniformly from the network nodes andfor each configuration we computed the average and standard deviation of the coverage(learning) curves. The resulting curves are shown in Figure 3. Each row and columncorresponds to different network models and average degree, respectively. The panels containcurves colored according to the considered dynamics.An initial observation shows that the TSAW dynamics outperformed the other dynamicsin all the experiments, corroborating previous studies in which TSAW was found to beamong the most optimal stochastic walks [6]. On the other hand, the RWD and RWIDdynamic resulted in the worst performance among the considered configurations.All curves seem to present similar shapes but different growing speeds, with faster cov-erage as (cid:104) k (cid:105) increases, a behavior that is stronger for the RWD and RWID dynamics. Inparticular, for ER, the performance among the dynamics becomes substantially similar asthe average degree increases. This indicates that the considered dynamics performs verysimilarly for denser networks. An exception to this rule is the RWD for the BA and LFR. Inthese cases, the performance of RWD gets slightly worse as network connectivity increases.11 .00.20.40.6 ERk=4RWRWDRWIDTSAW ERk=6 ERk=8 ERk=100.00.20.40.6 WAXk=4 WAXk=6 WAXk=8 WAXk=100.00.20.40.6 BAk=4 BAk=6 BAk=8 BAk=100 2000 40000.00.20.40.6 LFRk=4 0 2000 4000LFRk=6 0 2000 4000LFRk=8 0 2000 4000LFRk=10iterations a cc u m u l a t e d l e a r n i n g r a t e FIG. 3. Learning curves for N = 5 ,
000 nodes and the models ER, BA, WAX and LFR. Each rowand column correspond to different network topologies and average degrees, respectively.
This is probably related to the fact that a scale-free network (such as BA or LFR) allows theexistence of extremely connected nodes in which a walker could get stuck given its preferenceto move to nodes with high degrees.Another important aspect of the analysis is how the ranking of dynamics performancechange amongs the experiments. In general, TSAW is followed by RW, except for the LFRand BA networks with high connectivity. In this case, RWID attains a second place. This12eveals that, in these networks, avoiding hubs can be a good strategy to explore them morequickly. When the degree is lower, however, RWD performs better than RWID, indicatingthat, in this case, it is preferable to reach the hubs than avoiding them to attain betterperformance. l e a r n i n g r a t e g r o w t h FIG. 4. Learning rates for the considered models and N = 5 , In addition to the previous analyses, we observe two distinct patterns for the behavior ofthe curves among the network models, one for ER and WAX, and another for BA and LFR.13hile these pairs do not necessarily display exactly the same behavior, the performancerankings of the dynamics within these pairs of models do not change much. We also analyzedthe differences (or rates of growth) of the cumulative discovery curves. Figure 4 shows theobtained rate curves for all the considered configurations. Both the ranks and other overallobservations drawn from the cumulative curves can also be drawn for the rate curves.To summarize the main characteristics of the obtained learning curves, we applied PCAas a way to reduce their dimension. For each experiment, we derive a set of 50 featurescorresponding to the values of the learning rate curves (i.e., the derivatives shown in Figure 4)at epochs 100 iterations apart (see Section III D). −10 −5 0 5 10 15 20PCA1 (95.14%)−2024 P C A ( . % ) ER RW 4ER RW 6ER RW 8ER RW 10 ER RWD 4ER RWD 6ER RWD 8ER RWD 10 ER RWID 4ER RWID 6ER RWID 8ER RWID 10ER TSAW 4ER TSAW 6ER TSAW 8ER TSAW 10 BA RW 4BA RW 6BA RW 8BA RW 10 BA RWD 4BA RWD 6BA RWD 8BA RWD 10BA RWID 4BA RWID 6BA RWID 8BA RWID 10 BA TSAW 4BA TSAW 6BA TSAW 8BA TSAW 10 WAX RW 4WAX RW 6WAX RW 8WAX RW 10 WAX RWD 4WAX RWD 6WAX RWD 8WAX RWD 10 WAX RWID 4WAX RWID 6WAX RWID 8WAX RWID 10 WAX TSAW 4WAX TSAW 6WAX TSAW 8WAX TSAW 10 LFR RW 4LFR RW 6LFR RW 8LFR RW 10 LFR RWD 4 LFR RWD 6LFR RWD 8LFR RWD 10LFR RWID 4LFR RWID 6LFR RWID 8LFR RWID 10LFR TSAW 4LFR TSAW 6LFR TSAW 8LFR TSAW 10
RWRWDRWIDTSAW ERBAWAXLFR
FIG. 5. PCA results for ER, BA, WAX, and LFR for N = 5 ,
000 nodes. Each instance represents alearning curve obtained for a specific pair of network topology and agent dynamics. Interestingly,in some cases, different combinations of topology/dynamics can lead to similar learning curves.
The obtained data projection, shown in Figure 5, reveals that almost 100% of the variancein the curves can be explained by only two components. In particular, the first componentcovers about 95 .
1% of the variance. This outcome indicates a high correlation among thecurves. At the positive extreme of the first principal component, we find a separated groupcorresponding to the curves obtained for RWD dynamics simulated on the BA and LFRnetworks. These correspond to the curves with worst performance among the considered ex-periments. The RWID curves spread across the PCA1 axis, revealing its diversified behaviorwith each curve depending on the network model and connectivity.Along the negative segment of the first principal component, we observe a substantialoverlap among the curves for different experiment configurations. This region correspondsto configurations of high node degree or simulated through the TSAW dynamics. Amongthe notable overlapping configurations are ER and WAX. This is a surprising result, since14hey present very distinct characteristics in terms of global structure. At least three otherregions are shared by different combinations of networks and dynamics. This includes thoseobtained from ER, WAX and LFR models when the dynamics are TSAW for LFR, andRW for the others. Another example are the RW curves for the BA, WAX, and ER. Theseresults indicate that just by looking at the coverage performance curves it is not trivial todistinguish between network models and dynamics.The profile of the PCA axes in the original space, shown in Figure 6, reveals that thefirst principal component (PCA1) is almost flat along the iterations. This indicates that allepochs are equally important for the principal component. Conversely, PCA2 seems to cap-ture the difference of rates at the beginning and end of the curves. To further explore theseaspects we plotted together all the averaged cumulative learning curves of the consideredconfigurations colored by PCA1 and PCA2. This result is shown in Figure 6. We note thatPCA1 (a) indeed correspond to the inverse of total learning coverage, which is somewhatindependent from the shape of the curves. A second order effect seems to be captured byPCA2 (b), corresponding to how fast the rates of the learning curves are increasing acrossthe epochs. This becomes more clear when all the curves are aligned so that the starts andends match, as shown in (c). Curves with low values of PCA2 tends to be more concave(presenting high curvature) and vice-versa. All in all, PCA1 corresponds to the averagelearning speed, while PCA2 seems to be related to the acceleration of the curves. PCA1 P r o j e c t i o n w e i g h t Iterations PCA2
FIG. 6. Projection profiles of PCA1 and PCA2 axes along the original space. Iterations T o t a l c o v e r a g e −2−10123 P C A Iterations (b)(a) (c) T o t a l c o v e r a g e −5051015 P C A FIG. 7. Averaged learning curves for all the considered configurations. The color of each curveindicates the PCA1 (a) or PCA2 (b). The insight (c) shows all the curves normalized by theirrespective maximum value.
V. CONCLUSION
With many real-world phenomena being modeled and represented as sequences, one wayto characterize their respective complex system is by separating the dynamics encoding thesequences from their underlying state space. In this context, a certain stochastic walk dy-namics acts as the encoder while a complex network can be used to represent the state space.While this framework has been used to model several real-world problems, no systematicanalysis of the relationships among these three aspects of the systems exists in the literature.In this paper, we performed a systematic analysis of the behavior of different dynamics inwell-known network topologies. Whenever a dynamics (or exploration strategy) is performedon a network, one obtains a sequence of visited nodes. We aimed at studying how bothtopology and network dynamics affects the observed sequence of visited nodes. Here wefocused in one property of the sequences, the total number of different visited nodes. Thisproperty has many applications in network science, and is oftentimes related to the processof knowledge acquisition [5, 6]. In a semantic network, for example, each visited node canbe considered as a new learned concept.We adopted a framework to study the behavior of learning curves. For each combinationof network topology and dynamics, we obtained the corresponding learning curves. Then,each learning curve was mapped into a two-dimensional space via Principal ComponentAnalysis. This allowed us to compare curves in a more systematic way, with the advantage16f removing correlations while keeping the variability of the original learning curves space.Several interesting results have been found with our approach. Overall we found that trueself avoiding walks outperformed all other dynamics, while the variations of random walksbiased towards high or low degree displayed the worst learning curve performances. Despitesuch differences in performance, we found that all learning curves presented similar shapes.A further investigation of growth rates (i.e. the derivatives) of learning curves revealedthat no additional information can be obtained from such an analysis. This means that thelearning curves are sufficient to discriminate different network topologies and dynamics.The Principal Component Analysis confirmed that, despite distinct performances, allcurves shapes are similar. This could be confirmed by the fact that curves could be mappedinto a two-dimensional space virtually without any lost in the original data variation. Sur-prisingly, the first component accounted for 95% of the original variation. The visualizationprovided by PCA allowed us to observe some interesting patterns. Some regions were foundto share different combinations of topologies and dynamics. For example, similar learningcurves were found in ER and WAX, showing that the same behavior can be obtained evenin very distinct network topologies. The PCA visualization also revealed the variability oflearning curves with different topologies. While RWD and RWID were found to be verydependent upon topology, learning curves obtained with TSAW dynamics were found to bemuch less sensitive to distinct network topologies.The ambiguity of the behavior of learning curves observed in the PCA space can beuseful in practical scenarios. For example, in a knowledge acquisition scenario, the networktopology can represent how concepts are linked to each other, while the chosen dynamicscan be interpreted as the methodology used to cover the concepts being taught. In sucheducational scenario, our results suggest that one can be able to deliver the same learningexperience by adopting completely different knowledge organization (i.e. network topology)and teaching sequence (i.e. network dynamics).Our results show that when one uses learning curves to describe sequences of visitednodes ambiguous behaviors may arise. In other words, sequences with similar behaviorcan be observed from distinct pairs of topology/dynamics. This result suggests that thereconstruction of the processes underlying network construction and topology cannot relyonly on learning curves as descriptive features of sequences. For this reason, in future works,we intend to study additional sequence features to identify a minimum set of sequence17escriptors that are able to discriminate both the topology and dynamics generating theobserved sequence. Because sequences are used to construct embeddings, further studiescan analyze if similar embeddings can be obtained from distinct topologies and walks. [1] L. A. Adamic, R. M. Lukose, A. R. Puniyani, and B. A. Huberman. Search in power-lawnetworks.
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