What are the Best Hierarchical Descriptors for Complex Networks?
aa r X i v : . [ c ond - m a t . d i s - nn ] M a y What are the Best Hierarchical Descriptors for Complex Networks?
Luciano da Fontoura Costa , Roberto Fernandes Silva Andrade Instituto de F´ısica de S˜ao Carlos. Universidade de S˜ao Paulo,S˜ao Carlos, SP, PO Box 369, 13560-970, Brazil, [email protected] and Instituto de F´ısica, Universidade Federal da Bahia, 40210-340, Salvador, Bahia, Brazil (Dated: November 18, 2018)This work reviews several hierarchical measurements of the topology of complex networks andthen applies feature selection concepts and methods in order to quantify the relative importance ofeach measurement with respect to the discrimination between four representative theoretical net-work models, namely Erd¨os-R´enyi, Barab´asi-Albert, Watts-Strogatz as well as a geographical typeof network. The obtained results confirmed that the four models can be well-separated by usinga combination of measurements. In addition, the relative contribution of each considered featurefor the overall discrimination of the models was quantified in terms of the respective weights inthe canonical projection into two dimensions, with the traditional clustering coefficient, hierarchicalclustering coefficient and neighborhood clustering coefficient resulting particularly effective. Inter-estingly, the average shortest path length and hierarchical node degrees contributed little for theseparation of the four network models.
PACS numbers: 89.75.Hc, 89.75.Fb, 89.75.-k
It is better to know some of the questions than all ofthe answers. (J. Thurber)
I. INTRODUCTION
A relevant analysis of several features of complex sys-tems can be achieved through the recently developedcomplex network framework (e.g. [1, 2, 3, 4, 5]). Dueto the large amount of variables normally involved insuch dynamical systems, the set up of a interaction net-work, based on functional relationship among its degreesof freedom, offers a first picture to the actual internalstructure of the system. This process requires the iden-tification of the pertinent degrees of freedom as nodes,while the edges that connect them are defined by themutual influence they are subject to. The ability of iden-tifying nodes and edges in the appropriate way is a crucialstep in this modeling.The characterization of the so obtained networks con-stitute a second important step in this kind of analysis.In this process, a small number of features is chosen in or-der to measure, in an objective way, pertinent propertiesof the sets of nodes and edges. The choice of measure-ments used in the investigation constitutes the second keydecision during the structural analysis of the networks,as it defines which information can be obtained (e.g. [5]).Nowadays, it is consensual that the set of basic measuresinclude the average number of links per node h k i , the clus-tering coefficient C , mean minimal distance among thenodes h d i , and the network diameter D . However, suchmeasurements can not provide a one-to-one characteri-zation of the networks, i.e. they yield only a degeneraterepresentation from which the original network can notbe recovered. This is because several distinct networksmay be mapped into the same set of measurement values. Therefore, new and distinct measures have been proposedin order to capture new aspects not covered by the set offour parameters listed above.One particularly important aspect regards the char-acterization of individual nodes in the network, as thisallows the identification of particularly distinct nodessuch as hubs (e.g. [6]. While both the degree and clus-tering coefficient are defined for each individual node,they provide but a limited characterization of the con-nectivity around those nodes, with several nodes result-ing with identical pairs of degree/clustering coefficientvalues even when they are placed at completely differentcontexts in the network. One interesting means to ob-tain a richer (i.e. less degenerate) set of measurementsfor each node is to consider subsequent hierarchical neigh-borhoods (e.g. [7, 8, 9, 10, 11, 12, 13, 14]) around eachnode, in addition to the immediate neighbors consideredin the traditional degree and clustering coefficient.Although new measurements provide extra informa-tion, it is important to understand how they are re-lated to those in the basic set and among themselves.If one defines a measure space whose axes are spannedby the distinct parameters, one important issue regardsthe distribution of observations along each axis. A fullanswer to such task should include also an analytical re-lationship between the co-linear (correlated) measures.Another related issue refers to deciding, given a set ofdistinct measures, which of them are most effective inidentifying and discriminating between distinct kinds ofnetworks. The purpose of this work is to address thesequestions, by working with a set of hierarchical measuresand using sound concepts and methods of multivariatestatistics (e.g. [5, 15, 16]). We probe a large numberof networks generated according to four representativetheoretical models, Erd¨os-R´enyi (ER), Barab´asi-Albert(BA), Watts-Strogatz (WS) as well as a geographicaltype of network (GG), which can be put in connection todistinct complex network paradigmatic types, displayingfeature that are associated to the random, scale-free andsmall-world behaviors. In this way, it becomes possibleto quantify the relative importance of each measurement,with respect to the discrimination between the considereddistinct network models. Although illustrated for thesespecific four types of networks, the reported methodol-ogy is completely general and can be applied virtually toany problem involving the choice of measurements givenspecific types of theoretical or real-world networks.The hierarchical measurements we take into accounthave been discussed in a series of previous investigations(e.g. [10, 11, 12, 14]), in which the authors have inquiredhow a node sees not only its immediate neighborhood,but also successive neighborhoods up to a maximal dis-tance D from the reference node. The concepts of hierar-chies and higher order neighborhoods, that have been in-dependently introduced (e.g. [7, 8, 9, 10, 11, 12, 13, 14]),aim at providing a description of the relationship amonggiven sets of nodes which are not necessarily linked by im-mediate edges, but for which the minimal distance alongthe network is bound to a value 1 ≤ ℓ ≤ D . ℓ dependentclustering coefficients and node degrees have been investi-gated and compared for several sets of networks. In a sec-ond line of investigation, a recent contribution raises theissue of the interdependence among distinct measures,while reviewing the most relevant measures that havebeen introduced so far [5]. The results reported herein,heavily based on the ideas developed in the quoted ref-erences, are aimed at quantifying the role of the severalhierarchical measurements while discriminating betweenthe four considered theoretical network models. In or-der to quantify the influence of each measurement on theseparation between the four classes of networks, we applysound and objective concepts from multivariate statis-tics, namely standardization and canonical projections(e.g. [5, 16]).This work is organized as follows: In Section II, wepresent the basic notions of complex networks and of thetheoretical models used in our investigation. In SectionIII, we discuss the hierarchical measurements that willbe taken into account for the selection method. Thesemethods are presented and discussed in Section IV. Re-sults from our analyzes are discussed in Section V, whileSection VI closes the work with the concluding remarks. II. BASIC CONCEPTS
This section introduces the main concepts used in ouranalysis, including network representation as well as thefour theoretical models.
A. Complex Networks Basic Concepts
A non-weighted complex network Γ with N nodes and E edges can be fully specified in terms of its adjacency matrix K , so that K ( i, j ) = 1 indicates the existence ofan edge extending from node j to node i . All networksconsidered in this work are undirected, which implies K to be symmetric. They are also devoid of multiple or self-connections. Networks whose nodes have well-definedspatial positions within an embedding space are called geographical networks .The degree k i of a node i is defined as the number ofedges connected to it. In case a node j can be reachedfrom a node i , we can say that there is a path betweenthese two nodes. Two nodes can be connected throughmore than one distinct path. The shortest path d i,j be-tween two nodes i and j corresponds to the path with thesmallest number of edges connecting those nodes. The immediate neighborhood of a node i is the set of nodeswhich are directly connected to i , i.e. the nodes j forwhich d i,j = 1. The average shortest path h d i i to a node i is the mean value of d i,j over all nodes i = j , while thenetwork average shortest path h d i is obtained by tak-ing the mean value of h d i i over the whole set of networknodes.The clustering coefficient C i of node i can be calcu-lated as the ratio between the number of edges amongthe immediate neighbors of i and the maximum possiblenumber of edges between those nodes. Although mea-surements such as the node degree and clustering coeffi-cient apply to individual nodes, it is common to take theiraverage along the network, yielding the average node de-gree h k i and the average clustering coefficient C . B. Complex Networks Models
Several theoretical models of complex networks havebeen proposed (e.g. [1, 2, 3, 4, 5]). As announced inSection I, the current work considers four of such models(ER, BA, WS and GG), the most important features ofwhich we briefly describe below. All networks used forthe comparison of the hierarchical measurements in thiswork have the same number of nodes N and average nodedegrees h k i as similar as possible.The ER model (e.g. [1]) is characterized by havingconstant probability ρ of connection between any pos-sible pair of nodes. Its average degree is given as h k i =2 E/N = 2( N − ρ . The BA model can be obtainedby starting with randomly interconnected m m nodesin the current network such that each connection is pref-erential to the degree of the previous nodes. The averagedegree of a BA model is given as 2 m . Therefore, in orderto have ER and BA networks with the same node degree,we need to enforce that m = ( N − ρ . WS networks canbe produced by starting with the N nodes distributedalong a ring and connecting each node to its h k i / h k i being an even number. Then,a small percentage of edges are randomly rewired. Fi-nally, the geographical model considered in this work isobtained by considering a Poisson spatial distribution ofpoints with density γ in a two-dimensional embeddingspace with uniform connecting all pair of nodes whichare at Euclidean distance smaller than p h k i / ( γπ ). III. HIERARCHICAL MEASUREMENTS
Two of the most ubiquitously accepted network mea-sures, namely the average number of links per node h k i and the clustering coefficient C , reflect the immediatelandscape of the nodes, as they just consider, respec-tively, the number of neighbors each node is connectedto by a direct edge, and how the neighbors of a nodeare connected among themselves. The hierarchical mea-surements introduced in [10, 11, 12, 13, 14] first requirethe identification of the sets named the hierarchical shells H i ( ℓ ) or, alternatively, the neighborhoods N i ( ℓ ), of order ℓ of a node i as the nodes that lie at a minimal distance ℓ along the edges of the network of a given node i . Forthe sake of uniqueness, from now on we call these setsas H i ( ℓ ). The hierarchical measurements result from theextension of the two basic concepts to the sets H i ( ℓ ).For the feature selection analysis we consider, respec-tively, two and three distinct types of node degrees andclustering coefficients, which are so defined. The averagedegree h k ( ℓ ) i = n X i =1 k i ( ℓ ) , (1)where k i ( ℓ ) counts the number of neighbors which are ata minimal distance ℓ of node i , indicates how the higherorder neighborhoods of each node are populated. The average hierarchical degree h k H ( ℓ ) i = n X i =1 k Hi ( ℓ ) , (2)has a different meaning, as k Hi ( ℓ ) counts the number oflinks between elements of the two sets H i ( ℓ ) and H i ( ℓ +1).It expresses how deep connected are the nodes that lie intwo successive hierarchical shells, namely ℓ and ℓ + 1, ofnode i . Observe that we have h k ( ℓ = 1) i = h k H ( ℓ = 0) i = h k i , if we consider that the 0-th order neighborhood of anode is the node itself.The three distinct ℓ dependent clustering coefficientscoincide with the usual C when ℓ = 1. The hierarchi-cal clustering coefficient C H ( ℓ ) counts how many of the k i ( ℓ )( k i ( ℓ ) − / H i ( ℓ ) are directly linked by one edge. On theother hand, the neighborhood clustering coefficient C N ( ℓ )takes into account those pairs of the same set that areneighbors of order ℓ . The original clustering coefficient C is a direct measure of the presence of nearby trianglesin a network, and it indirectly hints to the presence ofconnected structures as cliques. The higher order C H ( ℓ )and C N ( ℓ ) give information on the how the nodes on more distinct hierarchical shells are related among them-selves.Finally, the hierarchical clustering coefficient by balls C B ( ℓ ), which was also previously introduced, constitutesthe third measurement that takes into account the hi-erarchical structure of neighbors of a node. This is acumulative measure in the sense that, instead of consid-ering the nodes in a single set H i ( ℓ ), it considers all nodesin the set H i ( ℓ ) = S ℓi =1 H i ( ℓ ).To evaluate all the above hierarchical measures weprofited from the formalism introduced in [14], whichamounts to first identifying all the higher order neigh-borhoods of the networks and storing the information ina single matrix c M = D X ℓ =0 ℓM ( ℓ ) . (3)All distinct hierarchical measures can be easily definedin terms of the elements of c M . IV. FEATURE SELECTION METHODS
Given L classes of networks (in the case of the cur-rent article the four theoretical models ER, BA, WS andGG) and Q respective measurements of their topology,an important question is: which subset of measurementsis more effective for discriminating between such classes?Such a problem provides a good example of feature se-lection .Two main approaches have been considered for fea-ture selection: filter and wrapper . The difference be-tween these two families of methods is that the latterevaluates the features by considering the results obtainedafter feeding them into a classifier, while the formermethods investigate the intrinsic relationship betweenthe measurements between and/or within the classes(e.g. [16, 17, 18])). For instance, the canonical projec-tion method used in this work provides an example of afilter approach to feature selection.It should be observed that none of the feature ap-proaches are absolutely optimal. While wrapper meth-ods will select features which are most effective for givenclassifiers, filter approaches will depend on the definitionof some optimality criterion. For instance, the canonicalprojection method adopted in this work quantifies theseparation between the classes by maximizing the dis-tance between the classes and minimizing the dispersioninside each class (see Section IV C). Because our interestin the current work is to characterize the discriminationpower of the several hierarchical measurements, we limitour attention to filter feature selection methods.The following subsections present the basic conceptsfrom multivariate statistics as well as the principal com-ponent analysis and the canonical projection methodolo-gies. A. Basic Concepts in Multivariate Statistics
Let each of the Q objects of interest (e.g. networks)be characterized in terms of R measurements x ( i ), i =1 , , . . . , R . It is convenient to organize the set of mea-surements obtained for each object p = 1 , , . . . , Q intothe respective feature vector ~v p = [ x p (1) , x p (2) , . . . , x p ( R )] T . (4)The mean feature vector ~µ can be calculated as µ ( i ) = 1 Q Q X p =1 x p ( i ) . (5)The elements C ( i, j ) of the covariance matrix C of themeasurements of the objects can be estimated as C ( i, j ) = 1 Q − Q X p =1 ( x p ( i ) − µ ( i ))( x p ( j ) − µ ( j )) (6)The standardized feature vector can be obtained as ~s p = (cid:20) x p (1) − µ (1) σ (1) , x p (2) − µ (2) σ (2) , . . . , x p ( R ) − µ ( R ) σ ( R ) (cid:21) T , (7)where σ ( i ) is the standard deviation of measurement x ( i ).Note that each normalized measurement s ( i ) has zeromean and unity standard deviation.The Pearson Correlation Coefficient between two mea-surements x ( i ) and x ( j ) can be given by the covariancebetween the standardized measurements s ( i ) and s ( j ). B. Principal Component Analysis — PCA
The multivariate statistical method known as principalcomponent analysis (PCA) allows dimensionality reduc-tion while maximizing the data variance along the firstprojected axes (e.g. [5, 13, 15]). Because the class of eachpoint is not taken into account in this method, it cor-responds neither to filter nor wrapper feature selection.This method is considered in this work for two reasons.First, it can be used to obtain preliminary visualizationsof the distribution of points and classes. Second, it pro-vides an introduction and a comparison standard to themore sophisticated canonical projections methodology, towhich it is related.Given the set of Q objects, characterized by R mea-surements, it is possible to project such measurementsinto a reduced space with W < Q dimensions. In orderto do so, the covariance matrix C of the measurementsis estimated as described in Section IV A and its eigen-values and respective eigenvectors are calculated. The eigenvectors corresponding to the W largest eigenvalues(in decreasing order of absolute values) are organized intoa matrix A such that each line corresponds to an eigen-vector. The matrix A defines the statistical linear trans-formation of the original set of data that maximizes vari-ances along the first new axes. Provided W is equal to 2or 3, the so-transformed data can now be visualized as a2D or 3D distribution of points. The original classes ofeach point can be visualized with different marks. C. Canonical Projections
The method of canonical projections, also calledcanonical analysis or canonical variables (e.g.[15, 16]),also performs a projection of the original measurementspace, but now considering explicitly the original classesof each object. The projection is performed not in orderto maximize the variances along the first new axes, butso as to obtain maximum separation of the classes, quan-tified by an optimality index ξ reflecting the distributionof the data both inside and among classes. More specif-ically, ξ will favor well-separated classes, with small dis-persions of the respective objects. The inter- and intra-class dispersion matrix, respectively D e and D a , can becalculated as described in [5, 15, 19]. The eigenstructureof the matrix ( D a ) − D e provide the basis for the soughtlinear transformation (as in the PCA, the eigenvectorsassociated to the largest absolute eigenvalues are stackedas lines in the transformation matrix) maximizing theseparation between the classes. For instance, in the caseof canonical projections into two-dimensional spaces, theeigenvectors v v importance of each measurement i as the sum of theabsolute values of the respective weights in v v I ( i ) = | v i ) | + | v i ) | (8)In order to avoid intrinsic biases implied by the relativeamplitude of each measurement, it is interesting to per-form the canonical projections on standardized versionsof the measurements. V. RESULTS AND DISCUSSION
In order to investigate, in a comparative fashion, therelative contributions of each measurement for the char-acterization and discrimination between the four con-sidered complex network models, 30 realizations of eachmodel, all with mean degree equal to 6 and sizes N of100, 200 and 300 nodes, were first obtained. Two types ofWS networks were obtained, considering 0 . N and 0 . E connection rewirings, where E is the overall number ofconnections. These two types of WS networks are hence-forth abbreviated as WS-R and WS-S. Observe that theformer type corresponds to almost regular networks (i.e.similar node degrees throughout), while the latter typepresents the small world property. All considered modelshad their average traditional and hierarchical measure-ments (for ℓ = 1 , , and 3) calculated and used as featurevectors. Table I lists the considered measurements aswell as their respective symbols and abbreviations. Measurement Symbol Abbrev-iationHierarchical clustering coefficient by balls C B ( ℓ ) cb Hierarchical clustering coefficient C H ( ℓ ) cl Neighborhood clustering coefficient C N ( ℓ ) cn Average number of nodes h k ( ℓ ) i n Average shortest path h d i sp Average hierarchical degree h k H ( ℓ ) i hd TABLE I: The symbols and abbreviations of the consideredhierarchical measurements defined in Sections II and III. Therespective hierarchical level ( ℓ ) is henceforth represented infront of each abbreviation, e.g. the hierarchical node degreeat level ℓ = 3 is abbreviated as hd Figure 1 shows the two-dimensional phase spaces ob-tained by PCA projection of the original 13-dimensionalphase spaces for N = 100 and 300. Each point in thisphase space corresponds to a specific network realization.The ER and BA clusters resulted near one another, whichwas also obtained for the WS/GG pair of clusters. TheGG networks resulted in the most dispersed cluster.The phase spaces obtained while considering all 13measurements were also projected into two dimensionsby using the canonical methodology described in Sec-tion IV C. Figure 2 shows the projected phase spacesobtained for networks with size of N = 100 (a) and 300(b) nodes, respectively. It is clear that the separationbetween the four networks modes is much better thanthat obtained by using PCA (Fig. 1). It is also clearfrom the two dimensional spaces in Fig. 2 that the fourmodels could be very well separated as a consequence ofusing such a comprehensive set of features. Interestingly,the ER/BA and WS/GG models again tended to clus-ter together. Observe that the dispersion of the pointsfor all distinct classes decreased for larger N . The rel-ative separation between the ER and BA models alongthe v direction also decreased for this case. Also, theWS-R and WS-S families of networks resulted near oneanother. (a)(b)FIG. 1: The distribution, in the PCA projected phase space,of the four theoretical complex network models obtained fornetworks with 100 (a) and 300 (b) nodes. The axes p p The seven most important measurements consideringall the three network sizes, in decreasing order, were: cb ch cn C , cb cn cn cb cb hd h k i and h d i , as well as of thehigher order hierarchical node degrees, for the purposeof identifying the distinct network classes. The fact thatall networks considered in this work had nearly the sameaverage degree explains why h k i had little contributionfor the discrimination. However, the fact that the distri-bution of average hierarchical degrees have been foundto vary between different network models [12] should, atleast in principle, imply better discrimination potentialfor those measurements. The relatively minor contribu-tion provided by the average shortest path is also sur-prising.The traditional clustering coefficient, C resulted themost important measurement in all cases, but the rele-vance of considering higher hierarchies for the character- (a)(b)FIG. 2: The distribution, in the projected phase space, of thefour theoretical complex network models obtained for net-works with 100 (a) and 300 (b) nodes. The axes v v ization of the considered networks was corroborated byrelatively high importance obtained for the other mea-surements. Interestingly, the importance of the cluster-ing coefficient by balls for hierarchy 3 tended to increasewith N , while the neighborhood clustering coefficient forhierarchy 3 decreased with that parameter. The formereffect is a consequence of the fact that the hierarchicaldegree becomes more relevant in larger networks, becausesuch networks have larger diameter and therefore allowmore elaborated and unfolded neighborhoods.Figure 4 illustrates several scatterplots obtained con-sidering pairs of the adopted measurements. Fig-ures 4(a,b), corresponding to C N (2) × C N (2) and k H (2) × k H (2), show that these two measurements provided gooddiscrimination between the four network models. Forother measurements (not shown), we can observe the ex-istence of fewer distinct disjoint regions. The other scat-terplots in Figure 4 correspond to pairwise associationsbetween distinct measurements. For instance, the panel C B (2) × C B (3) (Fig. 4c) has most of the points alignedalong the diagonal, indicating that these measures pro-vide almost the same kind of information (statistically, FIG. 3: The importance of the most relevant measurementsfor each of the considered three network sizes, i.e. 100, 200and 300 nodes. the two measurements are said to be correlated). A simi-lar tendency was observed for the same hierarchical mea-surements with distinct values of ℓ , as in the panels for C N (2) × C N (3) (Fig. 4d) and h k H (2) i×h k H (3) i (Fig. 4e).Note, however, that the points in these cases are alignedin a less clear way in comparison to C B (2) × C B (3)(Fig. 4c). The panels which combine hierarchical mea-sures of distinct classes ( C H , h k H i , C N ) have the pointsaway from the diagonal (e.g. Fig. 4e-h). This is a indica-tion that these measurements are uncorrelated, tendingto provide non-redundant information and, consequently,enhanced discrimination power. It is however importantto stress that the overall discrimination can not be fullypredicted simply from pairwise relationships between themeasurements, such as those illustrated above. Observealso that none of the two-dimensional scatterplots in Fig-ure 4 provide separation between the four models as goodas that shown in Figure 2b. That is because the latterscatterplot was obtained from the much higher dimen-sional phase space by projecting into the plane allow-ing the best separation between the four models. Sucha result clearly corroborates the increased separabilityallowed by the consideration of a comprehensive set ofdistinct hierarchical measurements.Finally, we proceeded with a further test to investigatethe ability of the used feature section methods to uncovernetwork specificities, by adding one extra network to twoof the previous groups of 120 specimens. We consideredthe Apollonian network (AN) [20, 21], a geometrical as-sembly of nodes and links, which is defined on the basisof the classical problem of finding the optimal covering ofa plane by circles. As they are defined in a recursive way,we considered two successively AN generations, respec-tively with N = 124 and 367 nodes, and included theminto the groups consisting of networks with N = 100 and300. It is important to recall that AN shows several fea-tures that are typical both of small world (small D and h d i ), and scale free scenarios ( p ( k ) ∼ k − γ ). The fea-ture selection method can help to identify whether AN FIG. 4: Scatterplots respective to pairwise combinations of the adopted measurements for N = 300. Measurements cn hd lies closer to the WS or BA clusters and, hence, to in-dicate which of the quoted scenarios it stays closer to.The results ( v , v
2) = ( − . , .
53) for N = 124, and( − . , .
96) for N = 367, indicate that, for both gen-erations, the AN is mapped away from all four clusterscorresponding to each of the considered theoretical mod-els. This is, indeed, a very interesting result, as it showsthat our method is able to identify that this type of net-work presents quite distinct topological features, albeitit shares some common properties with those that fitinto those the small world and scale free scenarios. Inother words, the very fact that several features of a givennetwork coincide with those typical for a large networkclass, does not automatically implies that it belongs tothe same set. VI. CONCLUDING REMARKS
Much of the advances in science have only been al-lowed by ability of researchers to focus attention on themost important features and variables in each problem.Because human beings have a rather limited ability tocope with a large numbers of measurements, it becomescritical to devise and apply methods which can possiblyidentify the most relevant features. Fortunately, soundand objective concepts and methods — defining the re-search area called feature selection — have been devel-oped which can help us in such tasks. Perhaps for histor-ical reasons, such methods are not so widely known andused by the Physics community.Because highly structured complex networks can onlybe comprehensively characterized by considering severalmeasurements, the application of feature selection meth-ods presents great potential for helping researchers inthat area. In a recent work [5], canonical variablesprojections and Bayesian decision theory were appliedin order to classify complex networks and to investigatemeasurements. The current work has unfolded such apossibility with respect to the discriminative potential ofa comprehensive set of hierarchical measurements.While the traditional node degree, clustering coeffi-cient and shortest path provide quantifications of im-portant features of the networks under analysis, theyare degenerated in the sense that several networks maymap into the same measurement values. The exten-sion of such concepts to reflect also the progressiveneighborhoods around each node has been proposed(e.g. [10, 11, 12, 13, 14]) in order to obtain enhanced,less degenerated, characterizations of complex networks.Each of such measurements are defined for a series ofhierarchical levels ℓ , yielding a high dimensional mea-surement space. Actually, the values of such measure-ments taken at each level can be understood as a mea-surement in itself. Given such a large number of fea-tures, it becomes important to identify which measure-ments are potentially more effective in providing discrim- inative descriptions of specific types of networks underanalysis. In this work, we applied standardization andcanonical projections in order to identify the most im-portant measurements in Table I with respect to fourrepresentative complex networks models, namely Erd¨os-R´enyi, Barab´asi-Albert, Watts-Strogatz and a geograph-ical type of network. Each hierarchical measurementwas calculated along three successive neighborhood lev-els. The traditional average degree, clustering coefficientand shortest path lengths were also considered. A totalof 13 measurements were considered in our investigation.Several interesting findings have been obtained bythe applied methodology. First, four types of networkswere well-separated even in the two-dimensional canon-ical projected phase space (considerably worse separa-tions were obtained by PCA), with the pairs of modelsER/BA and WS/GG forming superclusters. By takinginto account the respective weights of each measurementin the canonical projections, it was possible to associatean overall importance value to each measurement. Suchvalues were calculated for three network sizes ( N = 100,200 and 300). While the traditional clustering coeffi-cient was identified as contributing more intensely forthe separation between the network types, several hierar-chical measurements resulted in relatively high comple-mentary contributions, with the hierarchical clusteringcoefficient by balls ( C B ( ℓ )) and neighborhood clusteringcoefficient ( C N ( ℓ )) providing particularly relevant con-tributions. The node degree (traditional and hierarchi-cal), as well as the average shortest path length, did notcontribute significantly to the separation of the networkmodels. It is important to recall that such results are, inprinciple, specific to the separation of the four consideredtypes of networks, in the sense that different results maybe obtained when considering other networks models.In addition to providing an objective means for se-lecting measurements for characterization and discrimi-nation of complex networks models, the multivariate ap-proach considered in this work can also provide valuableinsights about the structural differences between distincttypes of networks. For instance, the fact that the clus-tering coefficient resulted more relevant than the short-est path length suggests that the four considered mod-els present local connectivity (expressed in the clusteringcoefficient) even more distinct than shortest path lengthdistribution. It would be interesting to apply such mul-tivariate methods to the characterization of other typesof networks, especially those involving community struc-ture. Acknowledgment:
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