CComplex Networks of Functions
Luciano da Fontoura Costa [email protected]˜ao Carlos Institute of Physics – DFCM/USPAv. Trab. S˜ao Carlense, 400S˜ao Carlos - SP, 13566-590, Brazil
Abstract
Functions correspond to one of the key concepts in mathematics and science, allowing the representation andmodeling of several types of signals and systems. The present work develops an approach for characterizing thecoverage and interrelationship between discrete signals that can be fitted by a set of reference functions, allowing thedefinition of transition networks between the considered discrete signals. While the adjacency between discrete signals isdefined in terms of respective Euclidean distances, the property of being adjustable by the reference functions providesan additional constraint leading to a surprisingly diversity of transition networks topologies. First, we motivate thepossibility to define transitions between parametric continuous functions, a concept that is subsequently extended todiscrete functions and signals. Given that the set of all possible discrete signals in a bound region corresponds to afinite number of cases, it becomes feasible to verify the adherence of each of these signals with respect to a referenceset of functions. Then, by taking into account also the Euclidean proximity between those discrete signals found tobe adjustable, it becomes possible to obtain a respective transition network that can be not only used to study theproperties and interrelationships of the involved discrete signals as underlain by the reference functions, but whichalso provide an interesting complex network theoretical model on itself, presenting a surprising diversity of topologicalfeatures, including modular organization coexisting with more uniform portions, tails and handles, as well as hubs.Examples of the proposed concepts and methodologies are provided respectively with respect to three case examplesinvolving power, sinusoidal and polynomial functions.‘
Qualcosa corre tra loro, uno scambiarsi di sguardi come lineeche collegano una figura all ' altra e disegnano frecce, stelle,triangoli ... ’ Italo Calvino, Le citt`a invisibili. Functions are to mathematics as sentences are to linguis-tics, constituting basic resources for develping more com-plete mathematical systems and models. The importanceof functions is reflected in their widespread applicationsnot only to the physical sciences, but to virtually everyscientific field.Traditionally, the mathematical study of functions andtheir properties has been approached in continuous vec-tor spaces, involving infinite instances of a given type offunction. While this constitutes an effective and impor-tant approach, most of the signals in practical applica- tions have discrete nature, being represented as discretesignals or vectors. This follows as a consequence of thesampling of physical signals by using acquisition systemsthat inherently implies the signals to be quantized alongtheir domain and magnitude.Though discrete functions are systematically studied inareas as digital signal processing (e.g. [1, 2]), emphasis isoften placed on aspects of quantization errors and repre-sentations in the frequency domain, employing the Fourierseries or transform (e.g. [3]). However, relatively lesserattention is typically focused on the relationship betweenthe discrete signals, or on how they can be approximatedby specific functions. Though the latter subject consti-tutes one of the main motivation of the areas of numericalmethods (e.g. [4]) and numerical analysis (e.g. [5]), thissubject is typically approached from the perspective offunction approximation, not often addressing the interre-lationship between functions.The present work develops an approach aimed at char-acterizing not only which discrete signals in a discrete1 a r X i v : . [ c s . D M ] F e b egion Ω ⊂ (cid:60) can be adjusted by a given set of refer-ence functions g i ( x ), i = 1 , , . . . , N , but also how suchadjustable discrete signals interrelate one another in thesense of being similar, or adjacent. The consideration ofa pre-specified set of function types happens frequently inscience, especially when fitting data or studying dynamicsystems. In particular, the solution of linear systems ofdifferential equations is often approached in terms of lin-ear combinations of a set of eigenfunctions (e.g. [6]), whichcould also be taken as the reference functions consideredin this work.The concepts and methods developed in the presentarticle are interesting not only theoretically while study-ing how distinct types of functions are related, but alsofrom several application perspectives, such as character-izing specific discrete spaces, discrete signal approxima-tion, morphing of functions (i.e. transforming a functioninto another through incremental changes), controllingsystems underlain by specific types of function, amongmany other possibilities. In a sense, functions can beapproached as a way to constrain, in specific respectivemanners, the adjacency between continuous signals in agiven region or space. For instance, the function sine re-stricts all possible continuous signals in a given space.In addition to the relevance of the described devel-opments respectively to the aforementioned mathemat-ical aspects, they also provide several contributions tothe area of network science (e.g. [7]). Indeed, as it willbe seen along this work, the transition networks derivedfrom discrete signal spaces with respect to sets of ref-erence functions are characterized by a noticeably richtopological structure that can involve modularity, hubs,symmetries, handles and tails [8], as well as coexistenceof regular and modular subgraphs. As such, these net-works provide valuable resources not only regarding thecharacterization of complex networks, but also constitutea model or benchmark that can be used as reference instudies aimed at investigating the classification and ro-bustness of networks, as well as investigations addressingthe particularly challenging relationship between networktopology and possible implemented dynamics.In order to obtain the means for quantifying how dis-crete signals in a region can be approximated by refer-ence functions, and how these signals interrelate one an-other, we develop several respective concepts and meth-ods. More specifically, after defining the problem in amore formal manner, we proceed by suggesting how todefine a system of adjacencies between continuous func-tions, in terms of the identification of respective transitionpoints. These concepts are then transferred to discretesignals, allowing the proposal of indices for quantifyingthe coverage of the discrete signals by adopted referencefunctions. Subsequently, we adapt the concept of adja- cency between functions to discrete signals, allowing thederivation of a methodology for obtaining transition net-works expressing how the discrete signals in a region canbe transformed one another while being approximatedas instances of the reference functions. Several studiesinvolving the obtained transition networks are then de-scribed, including the identification of shortest paths be-tween two adjustable discrete signals, random walks re-lated to the unfolding of dynamics on the network, as wellas the possibility of identifying discrete signals that aremore central regarding the interrelationships representedin the transition networks.The developed concepts and methods are then illus-trated with respect to three main case examples involv-ing (i) four power functions; (ii) a single complete polyno-mial of forth order; and (iii) two sets of hybrid referencefunctions involving combinations of power functions andsinusoidals. Several remarkable results are identified anddiscussed. Consider the region Ω ⊂ (cid:60) in Figure 1, which corre-sponds to the Cartesian product of the intervals x min ≤ x ≤ x max and y min ≤ y ≤ y max . Figure 1: A region Ω ⊂ Re delimited as x min ≤ x ≤ x max and y min ≤ y ≤ y max , and an example of a function y = f ( x ) com-pletely comprised in this region. Let y = f ( x ) correspond to a generic signal , whichcan be associated to a function, completely bound in Ω,in the sense of having all its points comprised within Ω.No requirement, such as continuity or smoothness, arewhatsoever imposed on these functions.In addition, consider the difference between two genericfunctions y = f ( x ) and y = h ( x ), both comprised in Ω, ascorresponding to the following root mean square distance(or error): δ ( f, h ) = (cid:115) x max − x min ˆ x max x min [ f ( x ) − h ( x )] dx (1)2 possible manner to quantify the similarity between f ( x ) and h ( x ) is as σ ( f, h ) = e − α δ ( f,h ) (2)for some chosen value of α .Let y = g i ( x ), i = 1 , , . . . , N be a finite set of specificfunctions types taken as a reference for our analysis. Forinstance, we could have g ( x ) = a x + a , g ( x ) = a x + a , and g ( x ) = a x + a and g ( x ) = a x + a , with a , a , a , a , a ⊂ (cid:60) .An interesting question regards the identification,among all the possible signals y = ˜ f ( x ) in Ω, of which ofthese signals can be expressed as g i ( x ) for i = 1 , , . . . , N ,by yielding zero difference or unit similarity between ˜ f ( x )and g i ( x ). For each of the reference functions g i ( x ), weobtain a respective set S i containing all functions ˜ f ( x )that can be exactly expressed in terms of g i ( x ).It is also interesting to allow for some tolerance by tak-ing these two functions to be related provided: δ ( ˜ f , g i ) ≤ τ d (3)or, considering their similarity, as: σ ( ˜ f , g i ) ≥ τ s (4)with: τ s = e − α τ d (5)The identification of the sets S i can provide interestinginsights regarding the relative density of each type of thereference functions in the specified region Ω, paving theway to the identification of reference functions with moregeneral fitting capability as well as the interrelationshipbetween these functions, in the sense of their proximity.It should be observed that the obtained S i will alsodepend on the specific size (or even shape) of Ω, as a con-sequence of the requirement of all functions to be com-pletely bound in that region. The alternative approach ofallowing the clipping of functions can also be considered,but this is not developed in the present work.In this work, we focus on discrete signals , which aretypically handled in scientific applications and technol-ogy. These signals are sampled along their domain andquantized in their magnitude (see Section 4). In order toidentify the adjacency between discrete signals, given anΩ and a set of reference functions g i ( x ), first we identify(by using linear least squares) the discrete functions thatcan be approximated, within a tolerance, by the referencefunctions, therefore defining the sets S i , and then linkthese functions by considering their pairwise Euclideandistance. The thus obtained network Γ or network can beverified to be undirected and to contain a total of nodesequal to the sum of the cardinality of the obtained sets S i , i = 1 , , . . . , N . In addition, each of the nodes becomes intrinsically as-sociated to the respective reference functions that werefound to provide a good respective approximation. Incase a discrete function (cid:126)f is found to be adjusted by twoor more of the reference functions, only that correspond-ing to the best fitting may be associated to (cid:126)f , thereforeavoiding replicated labeling. The reference function thusassociated to each node of Γ is henceforth called the node type .The transition network Γ provides a systematic repre-sentation of the relationships between the discrete func-tions in Ω that can be reasonably approximated by thereference functions. Several concepts and methods fromthe area of network science (e.g. [7, 9]) can then be ap-plied in order to characterize the topological properties ofthe obtained network. For instance, the average degree ofa node can provide an interesting indication about howthat function can be transformed (or ‘morphed’), by aminimal perturbation, into other functions in Ω.The definition of a system of adjacencies between thefunctions of Ω as proposed above also paves the way forperforming respective random walks (e.g. [10]). Start-ing at a given node, adjacent nodes are subsequently vis-ited according to a given criterion (e.g. uniform proba-bility), therefore defining sequences of incremental trans-formations of the original function. These trajectories offunctions can provide insights about how a function canbe progressively transformed into another (morphing), todefine minimal distances between any of the adjustablefunctions in Ω or, when associated to energy landscapes,to investigate the properties of respectively associated dy-namical systems (e.g. [11]), including possible oscillations(cycles) and chaotic behavior. A mathematical function often involves parameters, cor-responding to values determining its respective instantia-tion. For instance, the function: g ( x ) = a x + a (6)corresponds to a straight line function whose inclinationand translation is specified by the parameters a and a ,respectively.Given two generic functions in the region Ω, a partic-ularly interesting question is whether one of them can bemade identical to the other, which will be henceforth beexpressed as these functions being mutually adjacent , inthe sense of providing an interface between these two func-tions, which can be therefore transitioned. More specifi-cally, let the two following functions g i ( x ) and g j ( x ), with3espective parameters a i , a i , . . . , a iN i and a j , a j , . . . , a jN j : g i ( x ; a i , a i , . . . , a iN i ) g j ( x ; a j , a j , . . . , a jN j )It should be kept in mind that, throughout this work,the superscript value j in the terms a j corresponds to anindex associated to the respective reference function, notcorresponding to the j -power of a .The functions g i () and g j () can be said to be adjacent provided it is possible to find respective configurations ofparameters ˜ a i , ˜ a i , . . . , ˜ a iN i and ˜ a j , ˜ a j , . . . , ˜ a jN j so that: g i ( x ; ˜ a i , ˜ a i , . . . , ˜ a iN i ) = g j ( x ; ˜ a j , ˜ a j , . . . , ˜ a jN j ) (7)for every value of x in Ω.The set of parameters ˜ a i , ˜ a i , . . . , ˜ a iN i and˜ a j , ˜ a j , . . . , ˜ a jN j are henceforth understood to rep-resent a transition point in the parameter space[˜ a i , ˜ a i , . . . , ˜ a iN i , ˜ a j , ˜ a j , . . . , ˜ a jN j ], namely: P g i ↔ g j : [˜ a i , ˜ a i , . . . , ˜ a iN i , ˜ a j , ˜ a j , . . . , ˜ a jN j ] (8)Observe that each transition point defines a respectiveinstantiation of both involved functions, therefore alsocorresponding to a specific instantiated function in Ω.As an example, let’s consider the following four para-metric power functions: g ( x ) = a x + a g ( x ) = a x + a g ( x ) = a x + a g ( x ) = a x + a (9)with a , a , a , a , a , a , a , a ⊂ (cid:60) .All pairwise combinations of these functions g i () and g j () have respective transition points corresponding to˜ a i = ˜ a j = 0 for any values of a i and a j for which thefunctions remain completely comprised within Ω. Thoughthe four reference functions above have an infinite num-ber of pairwise transitions points, each of them defines arespective transition network Γ as presented in Figure 2.It is interesting to observe that, in this particular ex-ample, each of the transition points corresponds to theconstant functions g ( x ) = a = a = a = a , whichtherefore acts as a quadruple transition point for each a ∈ [ y min , y max ] with ˜ a = ˜ a = ˜ a = ˜ a = 0: P : [˜ a = ˜ a = ˜ a = ˜ a = a , ˜ a = ˜ a = ˜ a = ˜ a = 0](10)Observe that other sets of reference functions canpresent many other types of transition points, which canbe of types other than the null function. Actually, anyshared term between two parametric functions potentiallycorresponds to a transition point. Figure 2: The four reference functions in Eq. 9 share the transitionpoint P given as a ∈ [ y min , y max ] with ˜ a = ˜ a = ˜ a = ˜ a = 0.For one of the reference functions g i to transition to another function g j , it is necessary that g i be instantiated to the function correspond-ing to P through a respective parameter configuration, from whichit can then follow to g j . Observe that, though this diagram involvesonly five basic nodes (functions), there is actually an infinite num-ber of respectively defined situations in Ω as a consequence of itscontinuous nature. In addition to transitions between types of referencefunctions as developed above, it is also possible to havetransitions between incrementally different instances of asame type of function. This can be achieved by adoptinga tolerance τ regarding the similarity of two instances ofthe same type of function g ( x ), i.e.: ˆ x max x min [ g ( x ; a , a , . . . , a N − ) −− g ( x ; a + δ , a + δ , . . . , a N i + δ N − )] dx ≤ τ In this manner, it is possible to obtain long sequencesof transitions between instances of a same function as therespective parameters are incrementally variated ( (cid:126)δ ), typ-ically giving rise to handles and tails [8] in respectivelyobtained network representations.Given that the approach reported in this work is respec-tive to discrete signals and functions given a tolerance τ ,both types of function transitions identified in this sectionare expected to be taken into account and incorporatedinto the respectively derived transition networks. Though interesting in itself, the above described probleminvolves infinite and non-countable sets S i . Though thiscould be approached by using specific mathematical re-sources, in the present work we focus on regions Ω thatare discrete in both x and y , taken with respective reso-lutions: ∆ x = x max − x min N x − y = y max − y min N y − N x and N y correspond to the number of discretevalues taken for representing x and y , respectively.The so-obtained discretized region Ω is depicted in Fig-ure 3. Figure 3: A discretized region Ω ⊂ Re , with N x values along the x -axis and N y values along the y -axis. More specifically, we now have that: X j = ( j −
1) ∆ x − x min Y k = ( k −
1) ∆ y − y min (12)for j = 1 , , . . . , N x and k = 1 , , . . . , N y .Now, the possible functions in Ω can be expressed asthe finite set of vectors or discrete signals: (cid:126)f = [ f f . . . f N x ] T (13)with f j taking values in the set { Y k } respectively to theabscissae X j .It is assumed henceforth, typically with little loss ofgenerality, that x min = − x max = 1, y min = − y max = 1.The total number of possible vectors in the discretizedregion Ω can now be calculated as being given as corre-sponding to the number of permutations: N T = N N y x (14)Henceforth, we identify each of the N T possible dis-crete signals (or functions) in Ω in terms of a respectivelabel n = 1 , , . . . , N T . In case N y is relatively small,it is possible to implement this association by derivingthe discrete signal from its respective label n by firstrepresenting this value in radix N y , yielding the number[ p N x − . . . p p ] N y , and then making: Y i +1 = p i ∆ y + y min (15)for i = 0 , , . . . , N x − (cid:126)f and (cid:126)h can now be expressed in terms of the following root mean square error: δ ( (cid:126)f ,(cid:126)h ) = (cid:118)(cid:117)(cid:117)(cid:116) x max − x min N x (cid:88) j =1 [ f [ X j ] − h [ X j ]] (16)while the similarity between those functions can still begauged by using Equation 2.In order to verify if a given function (cid:126)f can be approx-imated by a reference function g i ( x ), we apply the lin-ear least squares methodology (e.g. [12]). This approachprovides the set of fit parameters (e.g. the coefficients ofa polynomial) so as to minimize the error of the fittingas expressed by the sum of the square of the differencesbetween (cid:126)f and g i ( x ) (taken at the abscissae X f ). For in-stance, if g i ( x ) is a third degree polynomial and N x = 5,we first obtain the matrix: A = X X X X X X X X X X X X X X X and then express the respective coefficients in terms ofthe vector: (cid:126)p = (cid:2) a a a a (cid:3) T So that the fitting can be represented in terms of thefollowing overdetermined system: (cid:126)f = A (cid:126)p (17)The respective solution can be obtained in terms of the pseudo-inverse of A as: (cid:126)p = ( A T A ) − A T (cid:126)f (18) Figure 4: Example of the linear least squares methodology for fittinga discrete signal Y i = f ( X i ), with N x = 7 and N y = 5, by areference function of the type a x + a . The respectively obtainedroot mean square error was τ d = 0 . τ s = e − τ d = 0 .
256 (for α = 10). Discrete Signals Coverage
The discretization of Ω implies that not all signals in (cid:126)f can be expressed with full accuracy in terms of referencefunctions (cid:126)g i , so that it becomes important to adopt somedifference tolerance τ d , or respectively associated similar-ity tolerance τ s . Henceforth, every discrete signal (cid:126)f thatcan be approximated by a reference function g i within agiven tolerance τ will be said to be adjustable by thatreference function.It is important to keep in mind that, when a toler-ance is allowed, more than one of the reference functionscan be verified to provide a good enough (i.e. with er-ror smaller than the specified tolerance) approximation,in which case a same function (cid:126)f will be identified as be-ing adjustable by more than one reference function, whichis reasonable given that this actually happens in discretedomains. However, in case the mapping is required to bemade unique, it is possible to keep only one of the fit-tings for each possible (cid:126)f , such as that corresponding tothe smallest approximation error. In this work, however,multiple adjustments will be considered.The sets S i ( τ d ), which are defined by τ d , will now con-tain a finite number of discrete functions. Thus, givena discrete signal (cid:126)f and a set of reference functions g i , i = 1 , , . . . , N , the total number of adjustable signals N a can be expressed as: N a = N (cid:88) k =1 S k ( τ d ) (19)We can now take the relative frequency of each refer-ence function g i with respect to the whole of adjustablefunctions as: r ( g i , τ d ) = { S i ( τ d ) } N a (20)where { S i ( τ d ) } corresponds to the cardinality of theset S i ( τ d ).This measurement, which is henceforth referred to as relative coverage , can be used to compare the fitting po-tential of each of the considered reference functions.It is also possible to consider the following densitiesrelative to the total number of functions in Ω as: q ( g i , τ d ) = { S i ( τ d ) } N T (21)In case only one fitting is associated to each possiblediscrete signal (cid:126)f in Ω, we will have that 0 ≤ q ( g i , τ d ) ≤ (cid:80) Nk =1 q ( g i , τ d ) = 1. This can be achieved byconsidering the sets ˜ S i = S i − (cid:83) Nk =1 S k (cid:54) = i instead of S i in Equation 21. Otherwise, this measurement maytake values larger than 1 and we will also have that (cid:80) Nk =1 q ( g i , τ d ) ≥
1, indicating that the possible discretefunctions in Ω is being covered in excess. The relative density q ( g i , τ d ), henceforth called the cov-erage index of g i provides a means to quantify of how wellthe reference function g i covers the discrete signals in thegiven region Ω and resolution τ d . Larger values of q ( g i , τ d )will typically be observed when τ d is increased (or τ s isdecreased).Also, observe that the above relative densities also de-pend on the choice of the discretization resolutions ∆ x and ∆ y , with { S i ( τ d ) } increasing substantially with N x and N y . While the relative densities r ( g i , τ d ) can provide interest-ing insights about the generality of each considered ref-erence function g i , these measurements can provide noinformation about the proximity or interrelationship be-tween the discrete functions (cid:126)f as fitted by a set of refer-ence functions g i , i = 1 , , . . . , N . However, it is possibleto quantify the proximity between all the possible discretefunctions in Ω in terms of some distance between the re-spective vectors and then define links between the pairsof functions that have respective distances smaller than agiven threshold L .Consider the Euclidean distance between two dis-cretized functions (cid:126)f [ i ] and (cid:126)f [ j ] in the region Ω as: ω (cid:16) (cid:126)f [ i ] , (cid:126)f [ j ] (cid:17) = (cid:118)(cid:117)(cid:117)(cid:116) N x (cid:88) k =1 (cid:104) f [ i ] k − f [ j ] k (cid:105) (22)The whole set of Euclidean distances between everypossible pair of functions in a given Ω can then be repre-sented in terms of the following distance matrix: W i,j = w (cid:16) (cid:126)f [ i ] , (cid:126)f [ j ] (cid:17) (23)The symmetric matrix W can be immediately under-stood as providing the strength of the links between thenodes of a graph, each of these nodes being associated toone of the possible N T discrete functions in a given Ω.However, such a graph would express the distances be-tween functions, not their proximity . Though these dis-tances could be transformed into similarity measurementsby adopting an expression analogous to Equation 2, there-fore yielding a weighted respective graph, in this work weadopt the alternative approach of understanding two dis-crete functions as being adjacent provided the respectiveEuclidian distance as defined above is smaller or equal toa given threshold L .Overall, obtaining the transition network for a set ofreference functions g i ( x ) and a respective discrete regionΩ, with N x and N y , involves the following 3 main pro-cessing stages:6 Assign a label n to each of the N T = N N y x possiblediscrete signals in Ω; • For each value n = 1 , , . . . , N T , obtain the respec-tive function (cid:126)f n = [ Y N x − , . . . , Y , Y ] by using Equa-tion 15 and apply least square approximation re-spectively to each of the reference functions g i ( x ), i = 1 , , . . . , N . In case the similarity between (cid:126)f n and g i as obtained by applying Equations 16 and then 2,is larger or equal to τ s , assign a respective node withlabel n , also incorporating the type i of the respectiveapproximating function g i ( x ); • Interconnect all pairs of nodes obtained in the previ-ous step which have Euclidean distance smaller than L , therefore yielding the transition network Γ.It is also important to keep in mind that one so obtainedtransition network can be understood as constraining theoverall adjacency network between all possible functions (cid:126)f in Ω so that only the nodes associated to cases thatcan be adjusted with good accuracy by a respective ref-erence function g i are maintained. In brief, the transitionnetwork therefore provides a representation of the adja-cency between the possible discrete functions that can beadjusted by the reference functions. The derivation of the transition network Γ respective toa set of reference functions and a discrete region Ω pavesthe way to several interesting analysis and simulations,some of which are discussed in this section.One first interesting possibility is, given two functions (cid:126)f i and (cid:126)f j in Γ, to identify the shortest paths between therespective nodes. We mean paths in the plural because itmay happen that more than one shortest path exist be-tween any two nodes of a network. Each of these obtainedshortest paths indicate the smallest number of successivetransitions from (cid:126)f i to (cid:126)f j that are necessary to take one ofthose functions into the other (or vice-versa) while usingonly instances of the considered reference functions. Thisresult is potentially interesting for several applications,including implementing optimal controlling dynamics un-derlain by the reference functions, or optimal morphingbetween two or more signals underlain by the respectivelyconsidered reference functions.Given a transition network Γ and all the shortest pathsbetween its pairs of nodes, it also becomes interesting toconsider statistics of the length of those paths, such astheir average and standard deviation, which can provideinteresting information about the overall potential of the reference functions for implementing optimal transitionsand morphings as mentioned above.Another interesting approach considering a transitionnetwork consists in performing random walks (e.g. [10])along its nodes. Several types of random walks canbe adopted, including uniformly random and preferen-tial choice of nodes according to several local topologi-cal properties of the network nodes, such as degree andclustering coefficient. These random walks can be un-derstood as implementing respective types of dynamicsin the network. For instance, a random walk with uni-form transition probabilities is intrinsically associated todiffusion in the network. In this manner, random walkson transition networks provide means for simulating andcharacterizing properties related to dynamics involvingtransition between the discrete signals in Γ.Yet another interesting perspective allowed by thederivation of the transition matrices Γ concerns studiesinvolving betweenness centrality (e.g. [9]) or accessibility(e.g. [13]) of edges and nodes in Γ, which can comple-ment the two aforementioned analyses. For instance, itcould be interesting to use the accessibility to identify thediscrete signals in Ω, as underlain by the reference func-tions g i ( x ), leading to the largest and smallest number ofnodes, therefore providing information about the role ofthose nodes regarding influencing or being influenced byother nodes. The accessibility measurement can also beapplied in order to identify the center and periphery ofthe obtained transition networks [14]. This section presents a case example of the proposedmethodology assuming the four power functions in Equa-tion 9. First, we consider the region Ω as being sampledby N x = 5 abscissae values and N y = 7 coordinate sam-ples, assuming τ s = 0 . α = 10, and L = 0 .
6. Theresulting transition network is depicted in Figure 5, as vi-sualized by the Fruchterman-Reingold methodology [15].Several remarkable features can be identified in the ob-tained transition network. First, we find the nodes or-ganized according to a well-defined bilateral symmetry,which can be verified to correspond to the sign of thecoefficients a i , i = 1 , , ,
4. In addition, the nodes cor-responding to approximations by the power functions g and g , both of which presenting odd parity, tend to beadjacent one another, with a similar tendency being ob-served for the nodes respective to the evenly symmetricpower functions g and g . Five main clusters of nodescan also be identified along the diagonal of the figure run-ning from bottom-left to top-right, each of which with a7 igure 5: Visualization, by using the Fruchterman-Reingoldmethod, of the transition network obtained for the reference powerfunctions in Eq. 9, N x = 5 and N y = 5, assuming τ s = 0 . L = 0 .
6. The colors indicate, according to the legend, the respec-tive type of power function approximating the discrete signals. Thetwo semiplanes of the bilateral symmetry corresponds to the signof the coefficients a i . The five main clusters of nodes correspondto the constant functions a i = − , − . , , . ,
1. Observe the hubsat the center of each of the 5 clusters of nodes. See text for moreinformation. respective central hub. These hubs correspond to the con-stant functions a i = − , − . , , . , reduced version of theabove transition network. Basically, all nodes associatedto each of the 4 categories of nodes (i.e. the adopted 4reference functions) are subsumed by a respective node,while the interconnections between all the original nodesare also collected into the links between the agglomer-ated nodes. Figure 6 illustrates the reduced version ofthe transition network in Figure 5.The obtained reduced graph corroborates the predomi-nance of transitions between odd ( g and g ) and even ( g and g ) power functions. In addition, we also have thatthe largest number of transitions is observed between in-stances of g , and that the smallest number of transitionstakes place between instances of g and g . The largestnumber of transitions between odd and even functions Figure 6: The reduced version of the transition network in Fig. 5.Each of the 4 nodes correspond to a respectively indicated type ofpower function, while the links between each pair of nodes accumu-late all connections between the original subsumed nodes. take place between g and g .One interesting question regards to what an extent thetransition networks may vary with respect to distinct val-ues of the tolerance τ d or τ s . Figure 7 depicts 9 additionaltransition networks obtained for the same configurationadopted in the previous example, with respect to severaldifferent values of τ s respectively indicated above eachnetwork.As illustrated in Figure 7, the size and connectivity ofthe transition network decreases steadily with τ s , and sev-eral markedly distinct types of networks, most of whichpresenting bilateral symmetry, are respectively observed.Given that the more generalized ability of the power func-tions to adjust the discrete signals when larger tolerancevalues are allow (i.e. small values of τ s ), the initial net-works tend to present a more widespread and uniforminterconnectivity. Observe also that the networks splitinto two or more connected components for values of τ s larger than approximately 0 . τ s =0 . , . , . . . , . τ s thattend to increase from left to right in Figure 8(a), up toa point, near τ s = 0 .
35, where the relative coverages be-come nearly constant and markedly distinct between the4 considered types of reference functions.As expected, the coverage index decreased steadily with τ s for all the four considered reference functions, also pre-senting values similar. Observe that only a small per-8 igure 7: Additional examples of transition networks obtained for the same configuration used in the previous example, but with respectto several other values of τ s , as respectively indicated in each network. The colors follow the same convention as in Fig. 5. Networkvisualizations obtained by using the Fruchterman-Reingold method. centage of the possible discrete signals are adjustable at τ s = 0 . n = 53.Self avoiding operation was adopted in not to repeatnodes. Observe the relatively smooth transition, involvingminimal modifications of the discrete signals, along each9 igure 8: The relative coverage (a) and coverage index (b) forthe case example 1 with N x = 5 and considering the referencefunctions in Equation 9, as obtained for similarity tolerance values τ s = 0 . , . , . . . , . N x = 5 and N y = 5, leading fromsignal n = 53 to signal n = 105. The numbers within parenthesisindicate the type of respectively fitted power function (1 = g , 2 = g , 3 = g , and 4 = g ). of the implemented transitions.Figure 11 depicts the transition network obtained forthe same situation above, but now with N x = 7 instead of N x = 5.The resulting transition network again presents severalinteresting features. As before, we have the bilateral sym-metry corresponding to the sign of the coefficients asso-ciated to the x -term. In addition, clusters and respectivecentral hubs have again be obtained, corresponding to theconstant (null) transition functions as observed before.However, unlike the network obtained for N x = 5, nowwe most of the nodes separated also along the up-downorientation, corresponding to interactions between blue-yellow (up) and red-green (down). These two portionsof the transition network can therefore be understood asbeing directly associated to the odd/even parity of theinvolved reference functions. Of particular interest is thefact that the discrete signals associated to the blue nodes,associated to the reference function g ( x ) = a x + a , de-fine a relatively regular pattern of interconnection that ismarkedly distinct to the more sequential pattern of in-terconnections observed for the 3 other reference func-tions. Observe that this transition network also incor-porates several handles , corresponding to relatively longsequences of links [8]. Such sequences are associated toincrementally distinct instances of the same type of refer-ence function, as discussed in Section 3. While the previous case example assumed power func-tions containing only two terms, we now address the moregeneral situation where only one complete polynomial oforder P is adopted as reference function, i.e.: g = a P x P + . . . + a x + a x + a (24)Figure 12 illustrates the transition network obtained forthe above polynomial reference function assuming P = 4, N x = 7, N y = 5, τ s = 0 . α = 10, and L = 0 . igure 10: One of the many possible random walks with 23 steps in the transition network shown in Figure 5, considering self-avoidinguniform transition probabilities. Observe the incremental change implemented in the involved discrete signals at each successive step. Thenumbers within parenthesis indicate the type of respectively fitted power function (1 = g , 2 = g , 3 = g , and 4 = g ).Figure 11: Visualization, by using the Fruchterman-Reingoldmethod, of the transition network obtained for the reference powerfunctions in Eq. 9, N x = 7 and N y = 5, assuming τ s = 0 . L = 0 .
6. The colors indicate, according to the legend, the respec-tive type of power function approximating the discrete signals. Seetext for more information. ibility of complete polynomials, as they cater for manymore transition points corresponding to the larger num-ber P of involved terms and parameters.
10 Case Example 3: Hybrid Func-tions
As with power functions and polynomials, also sinusoidalfunctions are extensively applied in mathematics, physics,and science in general, constituting the basic componentsof the flexible Fourier series. The third case example con-sidered in the present work adopts a set of reference func-tions containing two power functions and two sinusoidals,more specifically: g ( x ) = a x + a g ( x ) = a x + a g ( x ) = a sin (3 x ) + a g ( x ) = a sin (5 x ) + a (25)Figure 13 illustrates the transition network obtainedfor the above hybrid reference functions assuming P = 4, N x = 5, N y = 7, τ s = 0 . α = 10, and L = 0 . igure 12: Visualization, by using the Fruchterman-Reingoldmethod, of the transition network obtained for a complete poly-nomial of order P = 4 as single reference function, and N x = 7, N y = 5, τ s = 0 . α = 10, and L = 0 .
6. The colors are assigned sothat increasing values are represented from cyan to magenta colortones.Figure 13: Visualization, by using the Fruchterman-Reingoldmethod, of the transition network obtained for the reference hy-brid (two power and two sinusoidal) functions in Eq. 25, for N x = 7and N y = 5, τ s = 0 . L = 0 .
6. The colors indicate, according tothe legend, the respective type of power function approximating theadjustable discrete signals. In addition to the 5 clusters observedin the previous examples involving power functions, now also tailsand handles are obtained. See text for additional discussion.
A particularly interesting structure is observed for thisexample. First, the five main clusters, corresponding to respective constant transition points, are again observedin analogous manner with the other examples involvingpower functions. bilateral symmetry is again observed,being related to the sign of the coefficients a i , i = 1 , , , g (blue) and the lower frequency sinusoidal g (yellow), both of which have odd parity. These nodesalso tend to form handles at the border of the obtainedtransition network.The nodes associated to the second order power func-tion g (red) results mostly distributed along the six pro-jecting tails at the periphery of the network, which cor-respond to incremental instantiations of the same type offunction. Contrariwise, the nodes corresponding to dis-crete signals adjustable by the high frequency sinusoidal g (green) are found concentrated in the three most cen-tral clusters of nodes of the network, despite the fact thatboth g and g share even parity.In order to study the effect of extending a set of refer-ence functions on the topology of the respectively definedtransition network, we incorporate two additional powerfunctions, respective to third and forth orders, into theset of reference functions adopted in the previous example(Eq. 25), yielding the following extended set of referencefunctions: g ( x ) = a x + a g ( x ) = a x + a g ( x ) = a x + a g ( x ) = a x + a g ( x ) = a sin (3 x ) + a g ( x ) = a sin (5 x ) + a (26)Figure 14 illustrates the transition network obtainedfor the above hybrid reference functions assuming P = 4, N x = 5, N y = 7, τ s = 0 . α = 10, and L = 0 . g in pink and g in cyan) have been incorpo-12 igure 14: Visualization, by using the Fruchterman-Reingoldmethod, of the transition network obtained for the second case ofreference hybrid (four power and two sinusoidal) functions as inEq. 26, for N x = 7 and N y = 5, τ s = 0 . L = 0 .
6. The col-ors indicate, according to the legend, the respective type of powerfunction approximating the adjustable discrete signals. In additionto the 5 clusters observed in the previous examples involving powerfunctions, now also tails and handles are obtained. See text foradditional information. rated, being characterized by several respective handles.Also of particular interest is the fact of the tails in Fig-ure 14 being assimilated into the inner structure of thenetwork.
11 Concluding Remarks
Functions can be understood as essential mathematicalconcepts, being widely used both from the theoreticaland applied points of view in science and technology. Asa consequence of their great importance, whole areas ofmathematics and other major areas have been dedicatedto their study and applications, including calculus, math-ematical physics, linear algebra, functional analysis, nu-merical methods, numerical analysis, dynamic systems,and signal processing, to name but a few examples.The present work situates at the interface between sev-eral of these areas, also encompassing other areas, includ-ing network science, computer graphics, and shape anal-ysis. More specifically, we aimed at developing the issueof how well all possible signals in a given region Ω corre-spond to instances of a given set of reference functions.Given that infinite sets of adjustable functions would beobtained when working with continuous functions, we fo-cused instead on addressing the aforementioned problem in discrete regions, leading to finite sets of adjustable func-tions to be obtained. In particular, if the region Ω issampled by N x × N y values, the total number of possiblediscrete signals in that region is necessarily equal to N N y x .The adoption of discrete signals also paves the way to ver-ify if each of them can be adjusted, given a pre-specifiedtolerance, as instances of the reference parametric func-tions by using the least linear squares methodology.Having identified the sets of adjustable discrete signalsrespectively to each of the adopted reference functions,it becomes possible not only to study their relative den-sity, but to approach the particularly interesting issue oftransitions between adjacent functions, yielding respec-tive transition networks. The adjacency between twofunctions, as understood in this work, was first charac-terized with respect to continuous parametric functionsas corresponding to respective instances leading to theidentity between the two functions, being subsequentlyadapted to discrete signals and functions by taking intoaccount the Euclidian distances smaller than a specifiedthreshold L .A number of interesting possible investigations can thenbe performed with basis on these obtained networks, in-cluding studies of optimal sequence of transitions, randomwalks potentially associated to dynamical systems, as wellthe identification of particularly central signals in termsof betweenness centrality and accessibility.The potential of the reported concepts and methodswere then illustrated with respect to three case examplesrespective to: (i) four power functions; (ii) a single com-plete polynomial of forth order; and (iii) two sets of hy-brid reference functions involving combinations of powerfunctions and sinusoidals.As expected, the coverage index decreased steadily as τ s increased, while the four power functions presented sim-ilar potential for adjusting the discrete functions in theassumed region Ω.In addition, the obtained transition networks gave riseto a surprising diversity of topologies, including combina-tions o modularity and regularity, as well as hubs, handlesand tails. Several of the networks also were characterizedby symmetries which have been found to be related to thesign of the reference function coefficients, as well as theirparity. The power functions and sinusoidals were foundto lead to quite distinct patterns of interconnectivity inthe resulting transition networks, wth the latter leadingto peripheral handles.The intricate and diverse patterns of topological struc-ture obtained for the transition networks are also influ-enced by the discrete aspects of the lattice underlying Ω.For instance, most of the case examples involving powerand sinusoidals for N x = 5 were found to incorporate fiveclusters of nodes associated to the null discrete transi-13ion. Other topological heterogeneities of the obtainednetworks are also related to specific anisotropies of thelattice, as well as to the nature of the respective referencefunctions.One particularly distinguishing aspect of the proposedapproaches concerns the complete, exhaustive represen-tation of every possible discrete signal in the region Ω.As such, these approaches provide the basis for system-atic studies in virtually every theoretical or applied areasinvolving discrete signals or functions. In particular, itwould be interesting to revisit dynamic systems from theperspective of the described concepts and methods, as-sociating each admissible signal to a respective node inthe transition networks, and studying or modeling spe-cific dynamics by considering these networks.The generality of the concepts and methods developedalong this work paves the way to many related further de-velopments. For instance, it would be interest to extendthe approach from 1D signals to higher dimensional scalarand vector fields, as well as to other types of regions pos-sibly including non regular borders or even disconnectedparts. It would also be interesting to study other types offunctions such as exponential, logarithm, Fourier series,as well as several types of statistical distributions. Inaddition, the several types of obtainable transition net-works can be applied as benchmark in approaches aimedad characterizing classifying complex networks, as well asfor studies aimed at investigating the robustness of net-works to attacks, and also from the particularly importantperspective of relating topology and dynamics in networkscience. Another interesting possibility consists in apply-ing the developed methodologies to the analysis of realdata, such as time series, shapes and images.Though the present work focused on undirected net-works, it is possible to adapt the proposed concepts andmethods for handling directed transition networks, there-fore extending even further the possibly modeled patternsand dynamics. this can be done, for instance, by defin-ing the concept of adjacency in an asymmetric manner,such as when one of the reference functions approaches,through incremental parameter variations, approaches an-other parameterless reference function, in which case thedirection would extend from the former to the latter re-spectively associated nodes. Another possibility would beto establish the directions in terms of an external field,which could be possibly associated to a dynamical sys-tem.Last but not least, the networks generated by the pro-posed methodology yield remarkable patterns when vi-sualized into a geometric space, presenting shapes withdiverse types of coexisting regularity, heterogeneity andsymmetries. It has been verified that an even wider andricher repertoire of shapes can be obtained by the sug- gested method by varying the involved parameters. Forinstance, symmetries of types other than bilateral can beobtained by using reference functions containing 3 or moreterms instead of the 2 terms as adopted in most of the ex-amples in this work. One particularly interesting aspectof generating shapes in the described manner is that veryfew parameters are involved while determining structureswith high levels of spatial and morphologic diversity andcomplexity. Actually, the only involved parameters speci-fying each of the possibly obtained shapes are N x , N y , thereference functions, α and L . This potential for produc-ing such flexible shapes paves the way to several studiesnot only in shape and pattern generation and recogni-tion, but also for development biology, in the sense thatthe obtained structures could represent a model of mor-phogenesis through gene expression control by the refer-ence functions, while the spatial organization of the cellswould be defined in a manner similar to the Fruchterman-Reingold method, i.e. nodes that are connected attractone another, while disconnected nodes tend to repel oneanother. These interactions could be associated to mor-phic fields (e.g. biochemical concentrations, electric fields,etc.) taking place during development. Acknowledgments.
Luciano da F. Costa thanks CNPq (grantno. 307085/2018-0) for sponsorship. This work hasbenefited from FAPESP grant 15/22308-2 .
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