Advances in Network Complexity || Three Types of Network Complexity Pyramid

4Three Types of Network Complexity Pyramid�

Fang Jin-Qing, Li Yong, and Liu Qiang

4.1Introduction

As is well known, nature, technology, and society are full of complexity and diversityarising from intricate mechanisms of the interactions in complex systems. Now,network science [1,2] can be applied to explore various complex systems, which areinherently difficult to understand. The complexity and diversity of complex networksmay be viewed from the following different perspectives [1,2]. First, structuralcomplexity: Topological structure has been the most basic issue in network science;the study of which pervades all of science, from biology to physics, from the WWWand the Internet to the brain network. Network topological structure may vary intime and space, for example, the WWW has new web pages to be added and oldwebsites to be removed every day, and even in minutes. Second, dynamic evolvingcomplexity: From the viewpoint of nonlinear dynamics, It is well-known that how anenormous network of interacting dynamical systems works since a node in thenetwork can be a dynamical system, for example, brain neurons, traffic, powerstations, lasers, and so on, they can behave collectively, given their individualevolving dynamics and coupling architecture. They may have bifurcating andeven chaotic behaviors, for example, gene networks have dynamically evolvingnodes. Third, connection diversity: A network may have different kinds of nodes, forinstance, various subsystems of linear or nonlinear oscillator, or chaotic subsystem,and may have different weights, directions, and signs. For example, synapses in thenervous system can be strong or weak, inhibitory or excitatory, and brain cells can bestimulated or restrained. Why are topology and dynamics of networks so importantto explore and characterize? Because structure always affects dynamics or function,and vice versa; for example, the structure of social networks affects the spread ofinformation and disease, and the topology of the power grid affects the robustnessand stability of power transmission dynamics. Fourth, mutual interactions amongvarious factors: A real-world network is typically affected by many internal andexternal factors; for example, if the coupled brain cells are repeatedly excited by

� Parts of this chapter have previously been printed in [38], reprinted with kind permission.

Advances in Network Complexity, First Edition. Edited by M. Dehmer, A. Mowshowitz, and F. Emmert-Streib.� 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

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certain stimuli then their connections will be strengthened. Furthermore, the closerelations between networks or subnetworks make the already complicated behaviorsof each of themmuchmore complex and intrinsic. In fact, various couplings and thecomplexity can influence each other, and interact among networks. Fifth, supernet-works or network of networks with different levels and scales: Main features are couplinglarge-scale networks with different property, levels, and scales. For example, humanmobility networks include short-range connection (bicycle and car) and long-rangeconnection (flight, high iron, and ship). Internet of things may be comprise ofnetworks such as transportation, telecommunication, logistical, and/or financialnetworks. They may be multilevel as when they formalize the study of supply chainnetworks or may be multitiered as in the case of financial networks with interme-diation. The brain is another typical network of networks. Exploring the complexityof the brain networks has become an advancing frontier of science.Over past years, one of the most successful theoretical models. One of the most

successful theorymodels to capturing the fundamental features of the structure anddynamics of complex systems has been a significant issue of network science [1,2].Most complex systems have an inherently hierarchical organization and, corre-spondingly, their complex networks also exhibit hierarchical features. The issues ofhierarchy and complexity of complex networks have attracted the attention of a greatnumber of natural and social scientists in recent years.In this chapter, we mainly focus on three types of network complexity pyramid

(NCP) with complexity–diversity and university–simplicity [2–5], which exist exten-sively in nature, human society, living systems, and animal world. The first type ofthe NCP is the life’s complexity pyramid (LCP) in life science, which was firstproposed by Zoltvai and Barab�asi [3]. They pointed out that “the topologic propertiesof cellular networks share surprising similarities with those of natural and socialnetworks. This suggests that universal organizing principles apply to all networks,from the cell to the WWW.” The second type of the NCP is the network modelcomplexity pyramid (NMCP) proposed by Fang and his colleagues [2,4,5], which isbased on a profound analysis for network model development progress and iscombined with the unified hybrid network theory framework put forward by us.Another type of the NCP is the generalized Farey organized network pyramid(GFONP) [3] suggested by our group. These three types of the NCP are described indetail in this chapter.

4.2The First Type: The Life’s Complexity Pyramid (LCP)

Life science is perhaps the most exciting revolutionary area of scientific researchtoday. Based on the view of network science, Zolt�an and Barab�asi first proposed theLCP [3], as shown in Figure 4.1. The following introduction is based on Ref. [3]. It isseen from the bottom to the top in Figure 4.1 that the bottom of the pyramid showsthe traditional representation of the cell’s functional organization: genome, tran-scriptome, proteome, and metabolome (level-1). There is remarkable integration of

64j 4 Three Types of Network Complexity Pyramid

the various layers, both at the regulatory and at the structural level. Insights into thelogic of cellular organization can be achieved when we view the cell as a complexnetwork in which the components are connected by functional links. At the lowestlevel, these components form genetic-regulatory motifs or metabolic pathways(level-2), which in turn are the building blocks of functional modules (level-3).These modules are nested, generating a scale-free hierarchical architecture (level-4).Although the individual components are unique to a given organism, the topologicproperties of cellular networks share surprising similarities with those of naturaland social networks. This suggests that universal organizing principles apply to allnetworks, from the cell to the WWW. Cells and microorganisms have an impressivecapacity for adjusting their intracellular machinery in response to changes in theirenvironment, food availability, and developmental state. Add to this an amazingability to correct internal errors – battling the effects of mistakes such as mutationsor misfolded proteins – and we arrive at a major issue of contemporary cell biology:we need to comprehend the staggering complexity, versatility, and robustness ofliving systems. Although molecular biology offers many spectacular successes, it isclear that the detailed inventory of genes, proteins, and metabolites is not sufficientto understand the cell’s complexity. However, I. Lee et al. and R. Milo et al. havedemonstrated and offered key support for the cellular organization suggested by the

Figure 4.1 The Life’s complexity pyramid from [3], reprinted with permission from AAAS.

4.2 The First Type: The Life’s Complexity Pyramid (LCP) j65

complexity pyramid, and view the cell as a network of genes and proteins offers aviable strategy for addressing the complexity of living systems. According to thebasic dogma of molecular biology, DNA is the ultimate depository of biologicalcomplexity. Indeed, it is generally accepted that information storage, informationprocessing, and the execution of various cellular programs reside in distinct levels oforganization: the cell’s genome, transcriptome, proteome, and metabolome. Thisintegration of different organizational levels increasingly views cellular functions asdistributed among groups of heterogeneous components that all interact withinlarge networks. There is clear evidence for the existence of such cellular networks.For example, the proteome organizes itself into a protein interaction network, andmetabolites are interconverted through an intricate metabolic web. The finding thatthe structures of these networks are governed by the same principles comes as asurprise, however, offering a new perspective on cellular organization.A simple complexity pyramid composed of the various molecular components of

the cell – genes, RNAs, proteins, and metabolites – summarizes this new paradigmin Figure 4.1. These elementary building blocks organize themselves into smallrecurrent patterns, called pathways in metabolism and motifs in genetic–regulatorynetworks. In turn, motifs and pathways are seamlessly integrated to form functionalmodules – groups of nodes, for example, proteins and metabolites – that areresponsible for discrete cellular functions. These modules are nested in a hierar-chical fashion and define the cell’s large-scale functional organization.The genes to whose promoter regions these transcription factors (regulators) bind

are identified systematically. After establishing transcription factor binding atvarious confidence levels, they uncovered from 4000 to 35 000 genetic–regulatoryinteractions, generating the most complete map of the yeast regulatory network todate. The map allows one to identify six frequently appearing motifs, ranging frommulti-input motifs (in which a group of regulators binds to the same set ofpromoters) to regulatory chains (alternating regulator promoter sequences generat-ing a clear temporal succession of information transfer). A similar set of regulatorymotifs was already uncovered in the bacterium [10]. They provide evidence thatmotifs are not unique to cellular regulation but emerge in a wide range of networks,such as food webs, neural networks, computer circuits, and even the WWW. Theyidentified small subgraphs that appear more frequently in a real network than in itsrandomized version. This enabled them to distinguish coincidental motifs fromrecurring significant patterns of interconnections.An important attribute of the complexity pyramid is the gradual transition from the

particular (at the bottom level) to the universal (at the apex). Indeed, the preciserepertoire of components – genes,metabolites, proteins – is unique to each organism.For example, 43 organisms for which relatively complete metabolic information isavailable share only�4% of their metabolites. Key metabolic pathways are frequentlyshared however, and, as demonstrated in this issue, so are someof themotifs. An evenhigher degree of universality is expected at the module level. It is generally believedthat key properties of functional modules are shared across most species. Thehierarchical relationship among modules, in turn, appears to be quite universal,shared by all examinedmetabolic and protein interaction networks. Finally, the scale-


free nature of the network’s large-scale organization is known to characterize allintracellular relationships documented inmetabolic, protein interaction, genetic, andprotein domain networks. The Milo et al. study raises the possibility that thecomplexity pyramidmight not be specific only to cells. Indeed, scale-free connectivitywith embedded hierarchical modularity has been documented for a wide range ofnonbiological networks. Motifs are known to be abundant in networks as different asecosystems and the WWW. These results highlight some of the challenges systemsbiology will face in the coming years. Despite all of these recent challenges, an initialframework offering a rough roadmap appears to have been established. To seekfurther insights and to capture the system-level laws governing cell biology for thedeeper patterns common to complex systems and networks in general, cell biologists,engineers, physicists, mathematicians, and neuroscientists will need to equallycontribute to this fantastic voyage.

4.3The Second Type: Network Model Complexity Pyramid

On the basis of network model development progress, NMCP [1,2,4–6] is proposedby Fang’s group, as shown in Figure 4.2.The top three levels of the NMCP are the Euler graph (EG, level-7), the Erd€os and

R�enyi random graph (ERRG, level-6), the Watts–Strogatz (WS) small world model

Figure 4.2 Network model complexity pyramid.

4.3 The Second Type: Network Model Complexity Pyramid j67

and Barab�asi-Alb ert (BA) scale-free networks (level-5), respectively. These networkmodels mark the three milestones in network science development history.The level-4 of the NMCP is the weighted evolution network models (WENMs)

[27,28]. The top four levels have grabbed the main intrinsic quality of complexnetwork respectively. In further study of network science, however, how exactlydepict and fully mirror all characteristics of most real-world networks is still achallenging subject because the real world is one harmonious and unified worldwith both determinacy and randomness. Therefore, we have put forward the unifiedhybrid network theoretical framework with three unified hybrid network models[15–26], which can be constructed as the following three levels of the NMCP. Thelevel-3 is the harmonious unification hybrid preferential network model(HUHPNM), the level-2 is the large unified hybrid network model (LUHNM).After comparison with RWNs, it is obvious that, even so, the level-2 still does notfully reflect the actual network growth situation, because actual networks usuallydisplay variable speed growing process, such as high-tech network, the Internet, theWWW, human social networks, communication networks, and so on. Therefore, itis necessary to introduce the unified hybrid network model with various speedgrowing (UHNM-VSG) so-called as the level-1 of the NMCP. The UHNM-VSG has aflexible and a rich set of fresh features, which includes most current importantproperties of network models.Various levels of the NMCP include the top three levels, which are briefly

described in the following subsection. Retrospective network in the footsteps ofscientific development, network theoretical model research has been one of themostsignificant issues in the network sciences. So far, the history of this area has gonethrough three milestones, which are all breakthroughs from the theoretical model.

4.3.1The Level-7: Euler (Regular) Graphs

A graph that has an Euler circuit is called an Eulerian graph [7]. The first milestonein graph theory was the Euler graphs, conceptualized in the 1736, which attributedto Euler’s pioneering work. He first solved the famous Konigsberg Seven Bridgeproblem and the many facets of the Euler theorem. The Euler’s theorem states that(a) If a graph has more than two vertices of odd degree then it cannot have an Eulerpath. (b) If a graph is connected and has just two vertices of odd degree, then it has atleast one Euler path. Any such path must start at one of the odd vertices and end atthe other odd vertex. The Euler graphs have been studied for the longest period sincethen. The regular EG theory has laid the foundation of the regular graph theorydevelopment and should be at the top level-7 of the pyramid.

4.3.2The Level-6: Erd€os–R�enyi Random Graph

In graph theory, the Erd€os–R�enyi (ER) model, the so-called ER random graphtheory due to Paul ErdÅs and Alfr�ed R�enyi, is either of two models, G(n, p) and


G(n, M), for generating random graphs, including one that sets an edge betweeneach pair of nodes with equal probability, independently of the other edges [8]. Itcan be used in the probabilistic method to prove the existence of graphs satisfyingvarious properties, or to provide a rigorous definition of what it means for aproperty to hold for almost all graphs. The G(n, p) model was first introducedby Edgar Gilbert in a 1959 paper, which studied the connectivity threshold. TheG(n, M) model was introduced by Erd€os and R�enyi in their 1959 paper. As withGilbert, their first investigations were related to the connectivity of G(n, M), withmore detailed analysis following in 1960. The ER theory impacted graph theory for40 long years. Erd€os is known as the 20th-century Euler and received the WolfAward in 1984. The ER random graph obeys the Poisson degree distribution, andhas a smaller average path length and smaller cluster coefficient. After the ERmodel, from the late 1950s to late 1990s, large-scale networks with no clear designprinciples primarily used this simple and easy random graph topology, which isaccepted by the majority of people. Many mathematicians give random graphtheory strict mathematical proof, and obtain many similar and accurate results. Sofar, the ER random graph has succeeded in revealing the emergence of certainstructural properties and multithreshold function and so on. Thus, it should be atthe level-6 of the pyramid.

4.3.3The Level-5: Small-World Network and Scale-Free Models

In the level-6, both the major assumptions of the G(n, p) model (that edges areindependent and that each edge is equally likely) may be unrealistic in modeling realsituations. In particular, an Erd€os–R�enyi graph will likely not be scale-free like manyreal networks. Therefore, the Watts and Strogatz model attempts to correct thislimitation. In 1998, Watts and Strogatz proposed small-world (SW) network mode[9–12]. They revealed that the SW effect of the complex network is a kind of hybridresults of determinacy and randomness. Soon Newman and Watts made someimprovements to the SWmodels. The degree distribution of ER randommodel andthe WS model are not completely in line with many networks in reality and havecertain limitations. Many empirical graphs are well modeled by small-world net-works. Social networks, the connectivity of the Internet, and gene networks allexhibit small-world network characteristics. A certain category of small-world net-works was identified as a class of random graphs by Watts and Strogatz. They notedthat graphs could be classified according to two independent structural features,namely the clustering coefficient and average node-to-node distance, the latter alsoknown as the average shortest path length. Purely random graphs, built according tothe ER model, exhibit a small average shortest path length (varying typically as thelogarithm of the number of nodes) along with a small clustering coefficient. Wattsand Strogatz measured that, in fact, many RWNs have not only a small averageshortest path length, but also a clustering coefficient significantly higher thanexpected by random chance. Watts and Strogatz then proposed a novel graphmodel, now currently named the WS model, with (i) a small average shortest


path length and (ii) a large clustering coefficient. The first description of thecrossover in the WS model between a “large world” (such as a lattice) and a smallworld was described by Barthelemy and Amaral in 1999. This work was followed by alarge number of studies including exact results.In 1999, BA proposed a scale-free (SF) network model [12] and found the power-

law nature of the complex networks, that is, degree distribution follows p(k)� k�Y.Two discoveries of the SW and the SF networks mark the third milestone ofnetwork development, and network sciences were born [1,2]. The formationmechanism of the SF network is based on two rules: growth and preferentialattachment in accordance with the degree of nodes. The BA model is the firstmodel of a random network with the SF property. Further, networks with complextopology describe as diverse as the cell, the WWW, or society. One of the mostsurprising finding is that despite their apparent differences and although theyshare the same large-scale topology, each have an SF structure. Subsequently, itwas found that the formation mechanisms of the SF are also as diverse asreplication, nearest neighbor connections, hybrid preferential linking, and localconnective information.In summary, the main feature of the evolution of complex network is driven by

self-organizing processes that are governed by simple but generic scaling laws.Many subsequent empirical research of RWNs has demonstrated that the RWNs areneither regular nor random, but they belong to a large class of hybrid network bothdeterminacy and randomness, and commonly possess both the SW and the SFproperties, as well as the statistical property, which is completely different from thelevel-7 regular graph and the level-6 random graph.

4.3.4The Level-4: Weighted Evolving Network Models

Up to now three milestones from the level-7 to the level-5 are all unweightednetworks. They reflect most of topological properties and dynamical behaviorbetween network nodes and connectivity, but they could not describe the differentroles of nodes and all characteristics of the RWNs completely because almost RWNsbelong to weighted networks. Only weighted evolving networks can carefully portraythe nodes connection and mutual interaction [13–15]. Thus, it is a natural boost thatfrom the unweighted network models above toward WENM, which has became thelevel-4 of the NMCP. Along with more and more empirical studies on weightednetworks, and fresh properties related link weight are obtained by some typicalWENMs. In the level-4 there are several preferential driving mechanisms: (1) nodestrength; (2) edged weight; (3) both strength and edged weight; (4) both weight andfitness; (5) both topological growth and strength driving; (6) geographical link ofposition neighborhood; (7) local information or both local world and weight driving;(8) topological growth with strengths’ driving, and so on. In the level-4, the WENMshave revealed some common characteristics: the SWs as well as the three SFs fornode degree, strength and weight distributions, that is, all obey the power-lawproperty with different exponents.


4.3.5The Bottom Three Levels of the NMCP

It is noted that all WENMs in the level-4 belong to generalized random networks,which always ignored deterministic linking. It is useful for theoretical analysis easilyand reproduces main topological properties for the RWNs. However, based on thefoundational observation fact for a unifying world in natural and social networks,one cannot ignore anyone of order and random since their interactions in the realworld are neither completely regular nor completely random and lie between theextremes of order and randomness.To overcome the weakness of the level-4, the unified hybrid network model

framework (UHNMF) [6] with trilogy was proposed as the following three levels ofthe NMCP [16–33]. The diagram of UHNMF is shown in Figure 4.3. The main ideais that four hybrid ratios (dr, fd, gr, vg) are introduced by

dr ¼ DPA=RPA

f d ¼ HPA= HPAþ DPAð Þgr ¼ GRA= GRAþ RPAð Þvg ¼ DVG=RVG

8>>>>><>>>>>:

ð4:1Þ

where dr is the total hybrid ratio, which is the first level ratio; DPA is time interval(step) for the deterministic preferential attachment; RPA is time interval for therandom preferential attachment; the second level hybrid ratio includes two: one is fd,the so-called helping poverty ratio, the other is general randomhybrid ration gr;HPA

Figure 4.3 Diagrams of the unified hybrid network theory framework (UHNTF).


is time interval for the helping poverty attachment (e.g., choosing nodes whosedegree is smallest to be linked), GRA is time interval for the general randomattachment (e.g., ER random attachment); vg is the so-called acceleration ratio,DVGis time interval for deterministic variable growth, RVG is time interval for randomvariable growth. The unified hybrid theory framework consists of three level models(HUHPNM, LUHNM, LUHNM-VSG), they can be constructed according to differ-ent hybrid ratios. A hybrid of merit form has a wide range of practical basis in natureand human society, in line with the natural, social, physical, and technical as well asthe lives of the majority to seek an answer to this question and the correspondingsolutions and means.

4.3.5.1 The Level-3: The HUHPNMThe level-3 is the harmonious unifying hybrid preferential network model(HUHPNM), in which one total hybrid ratio is introduced by dr [16–23]. It wasfound in the level-5 that some universal topological properties, including theexponents of the three power laws (degree, node strength, and edged weight),are highly sensitive to the total hybrid ratio dr. A threshold of the exponent is atdr¼ d/r¼ 1/1.Through theoretical analysis for the HUHPNM, we have obtained [15] a relation-

ship between power exponent c and dr for some weighted HUHPNM, which quitecoincide with the numerical curves. Moreover, for all well-known models (such asunweighted BA and weighted model BBV), where c has quite a complicated relationwith the weighted parameters and the total hybrid ratio dr for theHUHPNM-BA andHUHPNM-BBV, we obtained their relation as follows:

cHUHPNMBA ¼ 1

bþ 1 ¼ A1

expdrA2

� �A3" #þ A4 ð4:2Þ

cHUHPNMBBV ¼

4dþ A1

expdrA2

� �A3" #þ A4

2dþ 1ð4:3Þ

where A1;A2;A3;A4 are relevant parameters.The results reflect both mutual competition and harmonious unification. The

level-3 has both the SF and the SW properties. For example, it was found that theHUHPNM-BA model is of the shortest average path length (APL) and the largestaverage clustering coefficient (ACC).

4.3.5.2 The Level-2: The LUHNMTo describe diverse complex networks and improve the HUHPNM, we haveextended the HUHPNM toward a large unifying hybrid network model (LUHNM),which become the level-2 of the NMCP [24–29]. Two new hybrid ratios: deterministhybrid ratio fd and random hybrid ratio gr are introduced in (4.1). Here areDA¼HPAþDPA; RA¼GRAþRPA. It is found in the level-2 that much more


complex relation of topological properties depends on three hybrid negative 1 topositive 1 in both the unweighted and weighted LUHNM. First, only if the fd� 0.9/1,whatever the gr value is, the rt curves appear multiple peaks phenomena as (dr, fd, gr)change. As dr increases, the rt increases and can reach the largest positive 1. Thefd¼ 0.9/1 plays a key role for the transition features of the r depending on theunderstanding the degree–degree correlation r change in different hybrid rations.Obviously, the results in the level-2 are closer to the RWNs and can give a reasonableanswer to the concerned question: Why social networks are mostly positive degree–degree correlation but biological and technological networks tend to be negativedegree–degree correlation? The LUHNM can further increase additional hybridratio according to actual need, andmakes it a more flexible and potential application.

4.3.5.3 The Level-1: The LUHNM-VSGFurther comparison to the RWNs and in-depth analysis, it is obvious that, even so, inthe level-2 still doesnot fully reflect the actual network growth situation, because actualnetworks usually display variable speed growing process, such as high-tech network,the Internet, theWWW,human social networks, communicationnetworks, and so on.Therefore, it is necessary to introduce a variable growth hybrid ratio, vg, which isdefined in (4.1). Thus we propose and construct the large unified hybrid networkmodelwith various speedgrowingLUHNM-VSGas the level-1 of theNMCP [2,30–33].The level-1may have two possible variable growth pictures: deterministic and randomgrowth, for example, one may take a growing format as follows:

mðtÞ ¼ pðnðtÞÞa ð4:4Þwherem(t) is the number of connecting edges at t time, n(t) is the number of nodes inthe network at t time,a is growth index, pmay be a constant for deterministic growth;but for random growth the linking probability may be 0< p(t)< 1. According to thevalue of the variable speed index a, there may be four cases: normal (a¼ 0),deceleration (a< 0), acceleration (0<a< 1), and superaccelerated situation(a> 1). It is seen from the level-1 the LUHNM-VSG can provide much moreinformation about topological properties.It is well-known in the level-1 that there exist several kinds of cumulative degree

distributionsP(k), mainly, including: the first is single SF, that is, power law: pðkÞ � k�c

The second is the stretched exponential distribution (SED), single SED is definedby [24]

pðkÞ ¼ exp � kk0

� �cð4:5Þ

where k0 is a parameter, the SED is characterized by a stretched exponent c smallerthan 1. When c¼ 1, Eq. (4.5) corresponds the usual exponential distribution. Forc< 1, Eq. (4.5) presents a clear curvature in a log–log plot while exhibiting largerapparent linear behavior, which belongs to a power-law distribution. If 0< c< 1,Eq. (4.5) falls between the stretched exponential and the power-law distributions, thesmaller the c value is, the closer to the scale-free case the curve is. Equation (4.5) canthus be used to account for a limit scaling regime and a cross-over to nonscaling.


The third is the normal Gaussian distribution (GD) defined by

PðxÞ ¼ y0 þA

wffiffiffiffiffiffiffiffip=2

p e�2ðx�xc=wÞ2 ð4:6Þ

where y0, xc, w, and A are the Gaussian parameters.The fourth distribution is the delayed exponential distribution (DED) defined by

pðkÞ ¼ Ae�kt1 þ y0 ð4:7Þ

where A, y0, and t1 are three parameters.The above interconversion distributions depend on matching condition of four

hybrid ratios (vg, dr, fd, gr). Some of fresh features/results of the LUHNM -VSG arebriefly extracted below.

P(k) Transition from Single-Scale to Multiscale as vg Changes Table 4.1 and thecorresponding Figure 4.4 show four kinds of the cumulative degree distributions

Table 4.1 A list of two kinds of distribution parameters under three work modes.

Hybridratio

I. First work mode: dr¼ 1/49 fd¼ 0/1 gr¼ 0/1

First-half curved shapedelayed exponential distribution

Second-half curved shapepower-law distribution

vg t1 g

1/49 3.97� 106 3.621/4 2.25� 106 3.631/1 232.18 3.534/1 108.62987 3.6649/1 4.42 3.68

II. Second work mode: dr¼ 1/1 fd¼ 0/1 gr¼ 0/1

First-half curved shapeGaussian distribution

Second-half curved shapestretched exponential c

vg y0 xc W A y0 xc W A

1/49 0.98 36.67 11.43 �13.72 0.04 2198.40 1945.43 �290.421/4 0.94 38.27 16.12 �19.03 0.05 1993.96 2238.00 �267.41/1 0.93 45.69 30.66 �35.49 c¼ 0.894/1 0.94 38.76 16.88 �20.13 c¼ 2.1349/1 0.98 37.03 11.98 �14.55 c¼ 2.51

III. Third work mode: dr¼ 49/1 fd¼ 0/1 gr¼ 0/1

First-half curved shapedelayed exponential distribution

Second-half curved shapestretched exponential c

vg t1 C

1/49 � 16.57 3.311/4 4.03 E6 2.651/1 2.31 E6 1.944/1 � 25.58 2.8349/1 � 15.93 3.53


Figure 4.4 P(k) versus vg under three work modes of dr for fixed fd¼ 0/1and gr¼ 0/1.(a) dr¼ 49/1: random prevailing; (b) dr¼ 1/1: divide equally (between the two); (c) dr¼ 1/49:determininacy prevailing.

p(k) versus vg, including SF, SED (4.5), GD (4.6), and DED (4.7), and theircorresponding parameters under the dr three work modes. It is seen from Table 4.1that for random prevailing (dr¼ 1/49) work mode first-half curve p(k) follows DGDbut second-half curve p(k) is the SF distribution. For dr¼ 1/1 work mode, the first-half curve is the GD but the second-half curve p(k) is the SED. For determinacyprevailing (dr¼ 49/1) work mode, the first-half curve p(k) follows the DED butsecond-half curve p(k) is the SED.

P(k) Transition from Single-Scale to Multiscale as a Changes The variable growthindex a in Eq. (4.4) is also a key control parameter for topological property in theLUHNM-VSGand has great effects on it. Let us see what are the effects of different aand the hybrid ratios on topological property. First of all, under the fixed fd¼ 0/1 andgr¼ 0/1 for both the DPA and RPA cases, Figure 4.5 and Table 4.2 give thecumulative degree distribution P(k), in comparison of the normal case (a ¼ 0)with the acceleration case (a ¼ 0:3; 0:6).


Figure 4.5 Cumulative degree distribution p(k) versus k for different dr work modes with fd¼ 0/1and gr¼ 0/1: (a) a¼ 0; (b) a¼ 0.3; (c) a¼ 0.6.

Table 4.2 A list of two kinds of exponent in Figure 4.5 under different dr work modes, where g ispower-law exponent, c1 and c2 are the fist stretched exponent and second stretched exponent inthe double SED.

dr a¼ 0Power exponent g

a¼ 0.3Double stretchedexponent first c1, second c2

a¼ 0.6,Double stretchedexponent first c1,second c2

1/49 1.79 1.39157 0.31581 1.83 —

1/4 1.91 1.5475 0.97936 1.86 0.541/1 2.88 1.66723 — 1.66 0.894/1 4.11 1.69526 — 1.38 0.9849/1 6.00 1.57938 — 1.83 0.87


For Figure 4.5a with a¼ 0, P(k) always follows the single SF, that is, the power-law distribution (4.5), the power exponent c increases (see insert curves) as totalhybrid ratio dr increases, also seen in Table 4.2. However, Figure 4.5b and cshows that for a¼ 0.3, 0.6 and under the different dr work modes, P(k) emergesas double stretched exponential distribution. Table 4.2 gives a list of two kinds ofexponents in Figure 4.5, the stretched exponent c in Figure 4.5a increases withthe dr, but in Figure 4.5b and c have a maximum and a minimum value,respectively.It can be seen from Figure 4.5 and Table 4.2 that cumulative degree distribution is

changed from a single scale-free (power-law) to double SED as the growth indexa> 0.3 under different dr work models. The topological properties and inter-conversion between seven levels of the NMCP are summarized in Table 4.3 andFigure 4.2.In addition, the average clustering coefficient can be changed from 0 to 1 in a

nonlinear fashion. The degree–degree corrective coefficient rt can be changed fromþ1 to �1, whose value only depends on various matching of the four hybrid ratios.Compared with the level-3 (the HUHPNM) and the level-2 (the LUHNM), the level-1(the LUHNM-VSG) can include almost current network models and may approachto RWNs. The SED often may provide a better description for economical networksand high technology networks.

Table 4.3 Comparison of the pyramid levels under the different hybrid ratios in the NMCP.

Hybrid ratiosModel

dr gr fd vg Properties Pyramidlevel

EG 1/0 0/0 0/0 0/0 Simplicity 7ER 0/1 1/0 0/0 0/0 Simplicity, emerge 6WS 1/0.1 (a few) 1/0 0/1 0/0 Small world

simplicity5

BA 0/1 0/1 0/0 0/0 Scale freeuniversality

BB, BBP, BBV, etc 0/1 0/1 0/0 0/0 Scale free3 power lawssmall word

4

HUHPNM Tunable 0/1 0/1 0/0 Complexityscale freediversitysmall word

3

LUHNM Tunable Tunable Tunable 0/0 Complexity"diversity"

2

LUHNM-VSG Tunable Tunable Tunable Tunable SF$ SEDComplexity"diversity"simplicity#SF$ SED

1


In short, we construct and summarize the network model complexity pyramidwith seven levels. It is seen from Figures 4.2–4.5 and Tables 4.1–4.3 that all modelsof the pyramid levels can be well studied in the unification hybrid network theoryframework depending on the four hybrid ratios (dr, fd, gr, vg). It is found thatuniversality-simplicity is increasing but complexity-diversity is decreasing from thebottom level-1 to the top level-7 of the pyramid.

4.4The Third Type: Generalized Farey Organized Network Pyramid

Before we describe GFONP [34–40], it is necessary to introduce the generalized Fareytree and its pyramid, which have been applied to describe the complexity of nonlinearcomplex systems [41–43] since 1980s. In an earlier study, Kim and Ostlund [41]constructed a Farey triangle to obtain rational approximants of a pair of irrationalnumber motivated by the interest in nonlinear systems. They have shown thatfrequency locking in maps can be organized by the Farey arithmetic that providesrational approximants of irrational numbers, and showed that frequency locking on athree torus can be organized by the generalized Farey arithmetic that gives rationalapproximants for pairs of mutually irrational numbers. Then Maselko and Swinney[42] found that the Farey triangle can provide a natural compact description ofsequences of periodic state observed in their experiments on the Belousov–Zhabo-tinskii reaction. Fang [43] have also found that in nonlinear dynamical system modelocking and complicated multipeaked periodic oscillations can be described by thegeneralized Farey tree, triangle, and pyramid, which demonstrated that this classifi-cation provides a natural, compact, and elegant means of organizing complexity andreveals the self-similar structure of the nonlinear dynamics. Calvo and his colleagues[44] have also investigated the hierarchical structure of three-frequency resonances innonlinear dynamical systems with three interacting frequencies, in which theyhypothesized an ordering of these resonances based on a generalization of the Fareytree organization from two frequencies to three, and the experiments and numericalsimulations demonstrated that their organization may describe the hierarchies ofthree-frequency resonances in representative dynamical systems, and may be univer-sal across a large class of three-frequency systems.In this section, we introduce another type of the GFONP, which also depicts and

reveals the complexity and universality of complex network systems. The maintheoretical results of topological properties in the GFONP, including degreedistribution, clustering coefficient, and degree–degree correlation coefficient(assortative coefficient), are deduced theoretically.

4.4.1Construction Method of the Generalized Farey Tree Network (GFTN)

In the ordinary Farey analysis, an infinite tree of rational numbers can be con-structed from a pair of rational numbers: The Farey sum of the pair a/b and a0=b0 is


ðaþ a0Þ=ðbþ b0Þ, which is the rational mediant between a/b and a0=b0 with thelargest denominator. This Farey addition can be continued infinitely, yielding theFarey tree. Constructional rule of the GFTN is as follows:

1) It is similar to ordinary Farey tree, but in the GFTN three nodes of network startfrom the adjacent three values of (0/1, 1/1, 1/0), which represent the firstgeneration, also the so-called first level.

2) First level (generation) is (0/1, 1/1), (1/1, 1/0), and (1/0, 0/1), respectively. Thus,they format three families (branches).

3) Second level of the GFTN has (0/1þ 1/1¼ 1/2), (1/1þ 1/0¼ 2/1), and (1/0þ0/1¼ 1/1), the GFTN can be constructed in a recursive way. Figure 4.6 gives theGFTN with the 6 levels, which consists of the three branches (families).

Topological properties of the GFTN, including four important characteristics,degree distribution, clustering coefficient, diameter, and degree–degree correla-tions, are deduced and calculated numerically, and both of their results are nodifferent.

Figure 4.6 The GFTN with the 6 levels, which consists of huge three families, t¼ 5. Reprintedfrom [36] with kind permission from Springer Science and Business Media.

4.4 The Third Type: Generalized Farey Organized Network Pyramid j79

In the following subsections, we denote the network after t steps by Gt, t� 1, andthe total numbers of nodes or edges in Gt are nt or et, respectively. For t¼ 0, G0 is atriangle composed by first level’s three nodes (0/1, 1/1, 1/0), which link to eachother.When t� 1, the number of next level’s nodes added to the networkDnt is threetimes the number of new nodes appearing in a single Farey sequence. Thus, we have

Dnt ¼ 3 � 2t�1. Each new node has two edges that link the new node to two nodes.The p/q of each new node is taken in increasing level from the GFTN. Thus, at timeinterval t� 1, according to the evolving rules, the numbers of total nodes and totaledges in the GFTN are nt ¼ nt�1 þ Dnt; n0 ¼ 3; et ¼ et�1 þ 2ðnt � nt�1Þ; e0 ¼ 3:

Their solutions are nt ¼ 3 � 2t�1; et ¼ 3 � ð2tþ1 � 1Þ:Topology properties are of fundamental significance to understand the complex

dynamics of real-world systems. Here, we have reduced solutions of four importantcharacteristics (degree distribution, clustering coefficient, diameter, and degree–degree correlations).

4.4.2Main Results of the GFTN

4.4.2.1 Degree DistributionDegree ki of a node i is the number of edges connected to it. In general, degree k isone of the most important statistical characteristics of a network. Let ki,t is thedegree of node i at step t, and tj is the current step j at which node i is added to thenetwork. Then, by construction, initial degree of each new node at tj is 2, and theincrement of old node at tj� 1 is 2 for j� 1. It is not difficult to find ki;j ¼2ðj þ 1Þ; j ¼ 0; 1; . . . ; t: Therefore, the degree in the GFTN at step t is discrete, andthe nodes having maximum degrees are the first three nodes in G0. We can givethe relationship between k and Nk, which is the number of total nodes whose

degrees are all k in Gt of the GFTN, we have Nk ¼ N2j ¼ 3 � 2t�j; j ¼ 1; 2; . . . ; t: andNk¼ 3 if for j¼ tþ 1.Since the number of all nodes in network at step t is nt, we can deduce the degree

distribution P2,j according to the knowledge of classical probability:

P2j ¼N2j

nt¼

3 � 2t�j

nt¼ 3 � 2t�j

3 � 2t ¼ 1

2j; j ¼ 1; 2; . . . ; t

3nt

¼ 33 � 2t ¼

12t; j ¼ tþ 1

8>>><>>>:

ð4:8Þ

Let k¼ 2j, we have j¼ k/2, and 2� j¼ 2� k/2. Through substitution ofvariable,

Pk ¼ P2j ¼2�

k2; k ¼ 2; 4; . . . ; 2t

2�t; k ¼ 2ðtþ 1Þ

(/ 2�

k2; k ¼ 2; 4; . . . ; 2t; 2ðtþ 1Þ ð4:9Þ


From Eq. (4.9), degree distribution of the GFTN follows the form of an expo-nential. Theoretic curve is taken as Eq. (4.10):

Pk ¼ 2�k2 ð4:10Þ

The theoretical result of degree distribution is in agreement with numericalsimulation.

4.4.2.2 Clustering CoefficientWhile clustering coefficient (C) is considered another characteristic of a network,we also derive the analytical expression of the C. In a network, the local clusteringcoefficient Ci of a node i is defined as Eq. (4.11) and C is the algebraic average ofall Ci.

Ci ¼ 2Ei

kiðki � 1Þ

C ¼ 1nt

Xi2Gt

Ci

8>>><>>>:

: ð4:11Þ

where Ei is the number of links presenting among its neighbors of the given node iin a network and ki is the degree of node i, and nt is the total number of nodes innetwork. By construction, at step t¼ 0, it is straightforward to calculate exactly the Ci

and C¼ 1. When a node i joins the network at step j, ki and Ei are 2 and 1,respectively. After that, if the degree ki of node i increases by 1, its new neighbormust connect to one of its presenting neighbors. So Ei increases by 1 at the sametime. All the edges among neighbors of node i can almost be a closed polygon exceptthat an edge is missing. However, for a node i whose degree is ki, there is only ki� 1links among its neighbors:

Ei ¼ k� 1 ð4:12ÞSo, combining with (4.12), Eq. (4.11) can be rewritten as

Ci ¼ 2ðki � 1Þkiðki � 1Þ ¼

2ki

ð4:13Þ

And for t� 1,

C ¼ 1nt

XkiNkiCi ¼ 1

3 � 2t 3 � 2t�1 22þ 3 � 2t�2 2

4þ 3 � 2t�3 2

6þ � � � þ 3 � 2t�t 2

2tþ 3 � 2

2ðtþ 1Þ� �

¼ 12t

2t�1 11þ 2t�2 1

2þ 2t�3 1

3þ � � � þ 2t�t 1

tþ 1tþ 1

� �¼ 1

2tXt

i¼1

2t�i

iþ 1tþ 1

� �

ð4:14ÞIn the limit for large t, we get

limt!1C ¼ lim

t!112t

Xt

i¼1

2t�i

iþ 1tþ 1

� �¼ lim

t!1

Xt

i¼1

1

i2ið4:15Þ


According to the D’Alembert discriminance of convergence criteria of positiveterm series, we have

limi!1

1

ðiþ 1Þ2iþ1

1

i2i

¼ 12< 1 ð4:16Þ

So Eq. (4.12) is constringent. Expanding the formula using the logarithmic serieswe get

lnð1þ xÞ ¼ x � 12x2 þ 1

3x3 � 1

4x4 þ � � � ; ð�1 < x � 1Þ

lnð1� xÞ ¼ �x � 12x2 � 1

3x3 � 1

4x4 � � � � ; ð�1 < x � 1Þ

ð4:17Þ

We obtain Eq. (4.18) given by

½lnð1þ xÞ � lnð1� xÞ � ½lnð1þ xÞ þ lnð1� xÞ2

¼ x þ 12x2 þ 1

3x3 þ 1

4x4 þ � � � ¼ lim

t!1

Xt

i¼1

1

i2i

ð4:18Þ

Taking the value x¼ 2�1, Eq. (4.15) can be rewritten as

limt!1C

12

ln 1þ 12

� �� ln 1� 1

2

� �� ln 1þ 1

2

� �þ ln 1� 1

2

� �� ¼ ln 2

ð4:19Þ

It is demonstrated that the relationship between C and t is consistent betweentheoretical and numerical results.

4.4.2.3 Diameter and Small WorldThe length of the geodesic, lij, is defined as the minimum number of edges fromnode i to node j in a network. The maximum value of lij between any pair of its nodesis called the diameter of the network at step t, which is denoted byDt. Diameter playsan important role in the transport and communication within a network and it canmeasure maximum delay of them. By construction, when t¼ 0 there are the firstthree nodes in the network and D0¼ 1. As increasing of step t, Dt is associated withthe new nodes joined to the network at step t, and we can observe the relationshipbetween Dt and t as

Dt ¼ tþ 1 ð4:20ÞFigure 4.7 shows the relationship betweenDt and t from theoretical and numerical

results, which are consistent.We observe fromEq. (4.20) thatDt¼ tþ 1 for theGFTN.Since at time t the network diameter is also equal to tþ 1, the APL of the network isless than tþ 1. In themean time, the logarithmvalue of the total number of nodes nt is


ln(3 � 2t)¼ ln 3þ t ln 2, which equals approximately to t ln 2 for large t. It shows thatDt

grows logarithmically with the network size nt, so its APL grows slower than ln(nt). Itmeans obviously that the GFTN has the small-world characteristic.

4.4.2.4 Degree–Degree CorrelationsBesides above three network characteristics, degree–degree correlations (or a so-called assortative coefficient) are usually used to measure the remaining degree –the number of edges leaving the node other than the one we arrived along. Awidely accepted formula is the one of assortative coefficient rt proposed byNewman [12] as

rt ¼e�1t

Pipiqi � e�1

t

Pi12ðpiþqiÞ

� �2e�1tP

i12 ðp2i þ q2i Þ � e�1

tP

i12ðpiþqiÞ

� �2 ð4:21Þ

where pi and qi are the degrees of the vertices at the ends of the ith edge, with i¼ 1,2, . . . , et. And rt lies in the range �1� rt� 1. A network is said to show assortativemixing when rt> 0, and it indicates that the nodes in the network that have manyconnections tend to be connected to other nodes with many connections. A networkis said to show disassortative mixing when rt< 0, and it indicates that the nodes inthe network that have many connections tend to be connected to other nodes withfew connections. The value of rt is zero for no assortative mixing or disassortativemixing.By the above construction, the GFTN is composed of three Farey trees, and each

of them has similar topology structure. An arbitrary edge ep,q in a single Farey treecan be analyzed through the two nodes it connects, p and q. According to thecurrent steps tp and tq, at which the two nodes joined to the network, we can easilyobtain their characteristics as filled in the Table 4.4, where Nkp,kq is the number ofedges ep,q.

Figure 4.7 Comparison of the diameter of networks between theoretical one (line) and numericalone (circle) with the increasing of step t. Reprinted from [36] with kind permission from SpringerScience and Business Media.


Combining with Table 4.3, Eq. (4.21) can be rewritten as

M ¼ 1

3 �ð2tþ1�1ÞA ¼ 3 �Pipiqi

¼Pti¼1½2 �2ðtþ1Þ�2ðtþ1� iÞþ½2ðtþ1Þ2þPt�1

j¼1

Pti¼jþ1½2j �2ðt� jþ1Þ�2ðtþ1� iÞ

B ¼ 3 �Pi12ðpiþqiÞ

¼ 3 �2Pt

i¼1½2ðtþ1Þþ2ðtþ1� iÞþ2 �2ðtþ1Þ2

þ3 �Pt�1

j¼1

Pti¼jþ1f2j½2ðt� jþ1Þþ2ðt� iþ1Þg

2

C ¼ 3 �Pi12ðp2i þq2i Þ

¼ 3 �2Pt

i¼1½22ðtþ1Þ2þ22ðtþ1� iÞ2þ2 �22ðtþ1Þ22

þ 3 �Pt�1

j¼1

Pti¼jþ1f2j½22ðt� jþ1Þ2þ22ðt� iþ1Þ2g

2

rt ¼MA�ðMBÞ2MC�ðMBÞ2

¼ 6þ23tþ3þ2t2�24tþ4þ9tþ28ð4tÞ�38t2t�26ð2tÞ�23tþ3t�8t24tþ44t4tþ16t28t�4t22t

½6t22tþ t2þ6t2tþ5t�16ð4tÞþ16ð2tÞð2tþ1�1Þ2

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð4:22Þ

Table 4.4 Degrees kp and kq of an arbitrary edge ep,q.

tp tq Nkp,kq kp kq

0 0 1�3 2(tþ 1) 2(tþ 1)1 2�3 2(tþ 1) 2t2 2�3 2(tþ 1) 2(t� 1)— — — —

t 2�3 2(tþ 1) 2[t� (t� 1)]1 2 2�3 2t 2(t� 1)

3 2�3 2t 2(t� 1)— — — —

t 2�3 2t 2[t� (t� 1)]— — — — —

i (i� 1) iþ 1 2�2i�1 �3 2[t� (i� 1)] 2(t� i)— — — —

iþ j 2�2i�1�3 2[t� (i� 1)] 2[t� (iþ j� 1)]— — — —

t 2�2i�1 �3 2[t� (i� 1)] 2[t� (t� 1)]— — — — —

t� 1 t 2�2t�2 �3 4 2

1� i� t� 1 and 1� j� t� 1.


From the expression of rt in Eq. (4.22), we can easily calculate its value with t toinfinity,

limt!þ1 rt ¼ 1

4¼ 0:25 ð4:23Þ

Figure 4.8 gives the relationship of rt with t and compares the theoretical and thenumerical results, which are consistent to each other. While t< 4, rt< 0 and networkshow disassortative mixing. While t� 4, rt> 0, and rt tends to a constant, thenetwork show assortative mixing.

4.4.3Weighted Property of GFTN

A complex topology is often associated with a large heterogeneity in the capacityand intensity of the connections. Since we generated GFTN combining Fareynumbers as described in above subsection, it is thus natural to study itsweighted property in which each node is treated on the numerical value of itsFarey number.The second reason is that we have proposed several network models, including

harmonious unifying hybrid preferential model, large harmonious unifyingnetwork model, and unified hybrid variable speed growth network model. Themain theory framework is based on the hybrids between determinist andrandomness, for example, multiplicity of attachment patterns among helping-poor attachment, preference attachment, random attachment, and other generalselecting patterns for other model. The numerical and theoretical analysis hasrevealed some evolution features and universal characteristics depending on fourhybrid ratios (dr, fd, gr, vg).

Figure 4.8 The rt versus t and comparison of theoretical (line) with numerical (circle) results.Reprinted from [36] with kind permission from Springer Science and Business Media.


So the unified hybrid theory framework consists of the above three-levelmodels, which can be constructed according to different hybrid ratios. A hybridof merit form has a wide range of practical basis in nature and human society, inline with the natural, social, physical, and technical as well as the lives of themajority to seek an answer to this question and the corresponding solutions andmeans.It is pointed out that we set up a bridge between the Farey organized generalized

pyramid and the unified hybrid network theory framework by using 3d Fareysequence (three binary number) as determinate, random, and hybrid weight ofnetwork nodes, respectively. We calculated and fit strength and weight of networkpyramids, again revealed the complexity of the characteristics of GFTN, andobserved the values at different levels and at different sequence (or hybrid ratio)changes. Cumulative degrees distribution, strength distribution, and weight distri-bution can be widely transformed among extend exponent, exponent distribution,and other distribution.Since Farey number and the hybrid ratios have similar numerical expression

values, we want to know its characteristic under the form of weighted network. Itmay be helpful to apply to most current unweighted and weighted complex networkevolution models. Figure 4.9a shows the cumulative distributions of numericalFarey value v, (Pc(v)), of GFTN at different t and they seem to have the sameform. When t is larger, the whole curve has a slow excursion to the right direction.In log–log plot, it follows the form of SED shown as formula (4.5). Correspondingwith the fitting result in Figure 4.9b, c¼ 1.064 0.005. However, Farey number ofthe network follows an exponential distribution.Given a pair of nodes i, j with different Farey number vi and vj, the weight of the

link connecting them, wi,j, can be defined accordingly,

wi;j ¼ vivj ð4:25Þ

Figure 4.9 (Pc(v)) of GFTN and curve is the fitting result, (a) different t, (b) t¼ 10. Reprintedfrom [36] with kind permission from Springer Science and Business Media.


And the strength of a node in GFTN is the sum of weight of links connectedwith it.

si; ¼Xj

wi;j ð4:26Þ

Figure 4.10 shows the cumulative distributions of w and s, denoting by (Pc(w)) and(Pc(s)), respectively, at different t. They all fit well with the sigmoidal (logistic) formas

PcðxÞ ¼ A2 þ A1 � A2

1þ ðx=A3Þq ð4:27Þ

where x represents w or s, and Ai and q are all parameters, i¼ 1, 2, 3.Using formula (4.27), we numerically get all the values of exponential parameter q

for different t, and find that for (Pc(w)), q and t have a linear relationship:

q ¼ A4tþ A5 ð4:28Þ

where the coefficients A4¼ 0.08 and A5¼ 0.41.But for (Pc(s)), q and t have not directed expression and their relationship is more

complex.

4.4.4Generalized Farey Organized Network Pyramid (GFONP)

4.4.4.1 MethodsOn the basis of the aforementioned research, the ordinary Farey pyramid can beconstructed from a pair of rational numbers: the Farey pyramid sum of the pair abcand a0b0c0 is ðaþ a0Þðbþ b0Þðc þ c0Þ, which is similar to the Farey sum. This Fareyaddition can also be continued infinitely, yielding the Farey pyramid. Format of the

Figure 4.10 Pc(w)(a) and Pc(s)(b) in different t. Curves are corresponding fitting results.Reprinted from [36] with kind permission from Springer Science and Business Media.


GFOPN may be diverse and rich [34–40]. As one of typical GFOPN, the construc-tional rule of the GFOPN with hexagon is suggested as follows:

1) Typical structure of a pyramid consists of its top and each level. Here we treat anode as top of GFOPN, and the basic shape of its level is hexagon. Each node inthe pyramid has a number with three digits represented by the positive integersgreater than or equal to 0. They are potentially used to characterize differenthybrid ratios in our model.

2) For clarity, Figure 4.11 illustrates the diagram of the first step of the GFOPN.As shown in Figure 4.11a, initial network is starting with seven nodes. Sixnodes of them, which are represented by (010, 001, 110, 011, 101, 111) andcalled loop 1, link to a hexagon and all of them link to the top node, which isrepresented by 100. Seven values are right combinations of 0 and 1 except 000.Then for the six pairs nodes of bottom hexagon, which is loop 1; six middlenodes are born, respectively, which sequential link to a new hexagon seen asloop 2. According to the rule of Farey pyramid sum, each node is representedby (011, 111, 121, 112, 212, 121). New hexagon seen as loop 3 can be generatedfrom loop 2, and so on m loops can be generated in a recursive way. For eachloop, its six nodes all link with the top node of pyramid and we called a part ofpyramid. When m¼ 2, it is shown in Figure 4.12b. So pyramid of t¼ 0 iscomposed of m parts, which have a common top node and each loop has thesame edges except for loop 1.

3) When time interval t> 0, we can only show the attachment rule of new nodesand edges for an arbitrary part connecting with part else. For example, we take

Figure 4.11 Illustration of GFOPN when t¼ 0,m¼ 2. (a) Part 1, including top node 100 andloop 1(010, 001, 110, 011, 101, 111). (b) GFOPN, including Part 1 and loop 2(011, 111, 121, 112,212, 121). Reprinted from [38] with kind permission.


Part 1 of Figure 4.11a as shown in Figure 4.12. In the middle of the neighbornodes, which are in a side face and not in a hexagon, a new node is added to thenetwork. Its number is the corresponding value for the sum of numbers of theneighbor nodes. For example, they are 111, 210, 101, 110, 211, 201 at t¼ 1 andcan be called loop 1(1). Each new node is linked to two nodes of its neighbornodes and the linking between them is alive. At the same time, six nodes in ahorizontal level are linked end to end. The aforementioned rule is repeatedm� 1 times for else part of the GFOPN. Figure 4.12b shows the result form¼ 2. New nodes, joined to the Part 2 of the solution of Eq. (4.30) is network,are 221, 211, 111, 221, 312, 212, which can be called loop 2(2). It is obviouslythat each node of loop 2(2) is the corresponding middle node of loop 1(1).

4) Sequentially, the new hexagonal structure can generate an unlimited number ofhexagons in a generated Farey mode, and the network is an organized pyramidwith m parts that each has high levels of exponential quantity. Since the GFOPNcan be constructed in a recursive way.

According to the above recursive rule, the numbers of total nodes nt and totaledges et in the GFOPN with m parts at time interval t are

nt ¼ nt�1 þ Dnt; n0 ¼ 6m þ1

et ¼ et � 1þ4mnt; e0 ¼ 12m þ6ðm � 1Þ ¼ 18m � 6

Dnt ¼ nt � nt�1 ¼ 6m � 2t�1

8>><>>:

ð4:29Þ

nt ¼ 6m � 2t þ 1

et ¼ ð24m � 6Þ � 2t � 6m

(ð4:30Þ

Along the steps above subsection, we can deduce the expression of properties ofGFOPN, such as degree distribution, clustering coefficient, and degree–degreecorrelations.

Figure 4.12 Illustration of GFOPN when t¼ 1,m¼ 2. (a) Part 1. (b) GFOPN. Reprinted from [38]with kind permission.


4.4.4.2 Main Results of GFONP

Degree Distribution From Eq. (4.30), we have the average number of degree ofnetwork at time interval t,

�k ¼ 2etnt

�!t!18� 2

m�!m!1

8 ð4:31Þ

For a node belonging to type i and joining the network at step tj, ki, tj, t is itsdegree at step t. Then by construction, degree of the each node in GFOPN can becategorized into five types. First is the uppermost node, which has the largestnumber of the edges, and increment of it is 6m. Second are six nodes of theoutermost hexagon base (loop 1) at t¼ 0 and the third are the nodes from loop 2to loop m in the underside at t¼ 0. Fourth is each outermost hexagonal nodejoined to the network at t> 0 and the fifth type are the nodes except for the onesfrom the first to the fourth. We have the following relation:

k1;0;t ¼ k1;0;t�1 þ 6m; k1;0;0 ¼ 6m; nodes 2 type 1

k1;0;t ¼ k1;0;t�1 þ 6m; k1;0;0 ¼ 6m;nodes 2 type 1

k2;0;t ¼ k2;0;t�1 þ 1; k2;0;0 ¼ 3;nodes 2 type 2

k3;0;t ¼ k3;0;t�1 þ 1; k3;0;0 ¼ 5;nodes 2 type 3

k4;tj ;t ¼ k4;tj ;t�1 þ 2; k4;tj ;tj ¼ 4;nodes 2 type 4

k5;tj ;t ¼ k5;tj ;t�1 þ 2; k1;tj ;tj ¼ 6;nodes 2 type 5

8>>>>>>>><>>>>>>>>:

ð4:32Þ

where 1� tj� t.And corresponding Nk, which is the number of total nodes whose degree are all k

in Gt of the GFOPN, is shown as

Nk1;0;t ¼ 1

Nk2;0;t ¼ 6

Nk3;0;t ¼ 6ðm � 1ÞNk4;t�j;t ¼ 6� 2tj�1; 1 � tj � t

Nk5;t�j;t ¼ ðm � 1Þ � 2tj�1; 1 � tj � t

8>>>>>>>><>>>>>>>>:

ð4:33Þ

Solutions of Eq. (4.32) are

k1;0;t ¼ 6m tþ 1ð Þk2;0;t ¼ tþ 3

k3;0;t ¼ tþ 5

k4;tj;t ¼ 2ðt� tjÞ þ 4; 1 � tj � t

k5;tj ;t ¼ 2ðt� tjÞ þ 6; 1 � tj � t

8>>>>>>>><>>>>>>>>:

ð4:34Þ


Therefore, the degree of GFOPN is discrete as 4, 6, 8, 10, . . . , 2tþ 4, and threecase: tþ 3, tþ 5, and 6m(tþ 1). Combining Eq. (4.33) with Eq. (4.34), we have

Nk¼6mðtþ1Þ ¼ 1

Nk¼tþ3 ¼ 6

Nk¼tþ5 ¼ 6ðm � 1ÞNk¼4 ¼ 6 � 2t�1

Nk¼2tlþ4 ¼ 6ð2m � 1Þ2t�tl�1; 1 � tl � t� 1

Nk¼2tþ4 ¼ 6ðm � 1Þ � 2t�t

8>>>>>>>>>>><>>>>>>>>>>>:

ð4:35Þ

Since the number of all nodes in the network at step t is nt, we can deduce thedegree distribution Pk,t according to the knowledge of classical probability:

Pk;t ¼ Nk

ntð4:36Þ

When t is large, we approximately have

P2tlþ4 ¼ 6ð2m � 1Þ2t�tl�1

6m2t þ 1; 1 � tl � t� 1 ð4:37Þ

Let k¼ 2tlþ 4, we have tl¼ (k� 4)/2. Through substitution of variable, we have

Pk � 2m � 1m

2�tl�1 ¼ 2m � 1m

2�k2�1ð Þ; k ¼ 6; 8; . . . ; 2tþ2 ð4:38Þ

From Eq. (4.38), degree distribution of the GFOPN follows the form of anexponential. Figure 4.13 shows the curve of degree distribution at step t¼ 10and t¼ 100 and the theoretical results fit well with numerical ones. Theoreticalcurve is taken from Eq. (4.37). It is seen from Figure 4.13 that the theoretical result ofdegree distribution is in agreement with numerical simulation.

Figure 4.13 Comparison of the degree distribution of network form¼ 5 between theoretic result(line) and numerical one (circle). (a) t¼ 10, (b) t¼ 100. Reprinted from [36] with kind permissionfrom Springer Science and Business Media.


Clustering Coefficient By the definition of C shown in Eq. (4.11), when step tincreasing, we can classify the nodes in the network into four types according to thestep in which they joined to the network: First is a top node, second are polygonalnodes in the base of the network, third are polygonal nodes nearest to the top node,and fourth are the nodes elsewhere.Except for type 1, each type has a different initial value of Ei of node i according to

the changing of loop m. For an arbitrary polygon of the network, loop 1 is theoutermost hexagon, and inner hexagon is gradually accord with the increasing ofm.By construction, in Table 4.5 we give different expressions of incremental change ofEi at step t, which is shown as DE. 1� tl1� tl2� t, Et¼ 0 is the initial value, ki isdegree of node i and Nki is its number.Combining Eq. (4.11) with Table 4.5, we can calculate the value of Cm,t for

different m.For m¼ 5, Figure 4.14 shows the curve between C5,t and t. The corresponding

theoretical and numerical results are consistent as well.

Degree–Degree Correlations By construction above, the GFOPN contains manypolygons composed of hexagon, and inerratic links lies among hexagons. Anarbitrary edge ei linked node p and node q in the network can be decompoundedinto two parts, inside of a polygon and among polygons at different sides in trenddirections of the network. Especially, considering links with the top node and linkswith the polygonal nodes in the base polygon, we can easily obtain their character-istics as listed in Table 4.6, whereNkp,kq is the number of edges ei and the parameters

Table 4.5 Node partitioning of the GFOPN.

Type Loop DE Et¼0 Ei ki Nki

1 \ 18m� 6 12m� 6 12m� 6þ (18� 6)t 6m(tþ 1) 12 Loop 1 1 2 tþ 2 tþ 3 6

Loop 2 toLoop m� 2

1 4 tþ 4 tþ 5 6(m� 3)

Loop m� 1 1 5 tþ 5 tþ 5 6Loop m 1 6 tþ 2 tþ 6 6

3 Loop 1 2 3 2ðt� tl1 Þ þ 3 2ðt� tl1 Þ þ 4 6Loop 2 toLoop m� 2

2 5 2ðt� tl1 Þ þ 5 2ðt� tl1 Þ þ 6 6(m� 3)

Loop m� 1 2 6 2ðt� tl1 Þ þ 6 2ðt� tl1 Þ þ 6 6Loop m 2 7 2ðt� tl1 Þ þ 7 2ðt� tl1 Þ þ 6 6

4 Loop 1 2 1 2ðt� tl2 Þ þ 1 2ðt� tl2 Þ þ 4 6ð2tl2�1 � 1ÞLoop 2 toLoop m� 2

2 1 2ðt� tl2 Þ þ 3 2ðt� tl2 Þ þ 6 6ðm� 3Þð2tl2�1 � 1Þ

Loop m� 1 2 2 2ðt� tl2 Þ þ 2 2ðt� tl2 Þ þ 6 6ð2tl2�1 � 1ÞLoop m 2 1 2ðt� tl2 Þ þ 1 2ðt� tl2 Þ þ 6 6ð2tl2�1 � 1Þ


corresponding to the current step in which node p or q joined to the network.Cumulative value of all the Nkp,kq is equal to et.Combining with Table 4.6, form� 2, Eq. (4.21) can be rewritten as Eq. (4.39). The

result of m¼ 1 has been deduced in another paper [37].

rm;t ¼A� B

ð24m�6Þ2t�6m

C � Bð24m�6Þ2t�6m

ð4:39Þ

Figure 4.14 Comparison of the cluster coefficient of the GFOPN between theoretical result (line)and numerical one (circle) with the increasing of step t,m¼ 5. Reprinted from [36] with kindpermission from Springer Science and Business Media.

Table 4.6 Edge partitioning of the GFOPN.

partitioning Nkp,kq kp kq Parameters

Inside of a polygon 6 tþ 3 tþ 3 m¼ 112 tþ 3 tþ 3 m� 212m� 18 tþ 5 tþ 5 m� 26�2s�1 4þ 2(t� s) 4þ 2(t� s) s¼ 1,2, . . . , t;m¼ 112�2s�1 4þ 2(t� s) 6þ 2(t� s) s¼ 1,2, . . . , t;m� 2(12m�18)2s�1 6þ 2(t� s) 6þ 2(t� s) s¼ 1,2, . . . , t;m� 2

Among polygons 6 tþ 3 6m(tþ 1)6m� 6 tþ 5 6m(tþ 1)6 4þ 2(t� s) 6m(tþ 1) s¼ 1,2, . . . , t6m� 6 6þ 2(t� s) 6m(tþ 1) s¼ 1,2, . . . , t6 4þ 2(t� s) tþ 3 s¼ 1,2, . . . , t6m� 6 6þ 2(t� s) tþ 5 s¼ 1,2, . . . , t6�2w 4þ 2(t� s) 4þ 2(t�w) w¼ 1,2, . . . , s� 1; s¼ 2,3, . . . , t(6m�6)�2w 6þ 2(t� s) 6þ 2(t�w) w¼ 1, 2, . . . , s� 1; s¼ 2, 3, . . . , t


where

A ¼ 606þ 420t� 1884m � 1056 � 2t � 1278mt� 324mt2 � 18mt3

þ 2112m2t þ 180m2 þ 396m2tþ 252m2t2 þ 78t2 þ 36m2t3

B ¼ ð36þ 18t� 84 � 2t � 141m � 72mt� 9mt2 þ 216m2t

þ 18m2 þ 36m2tþ 18m2t2Þ2

C ¼ 738þ 2256m2t þ408t� 1881m þ 108m3t3 � 1119mt� 255mt2

� 21mt3 � 1032 � 2t þ66t2 þ 108m3 þ 324m3t2 þ 324m3t

From the expression of rm,t in Eq. (4.39), we can easily calculate its value with t toinfinity:

limt!þ1 rm;t ¼ 28m2 � 12m � 5

52m2 � 14m � 6ð4:40Þ

Figure 4.15 gives the relationship of rm,t with t and compares the theoreticalresults with the numerical results for m¼ 5, which are consistent to eachother. While t< 6, rm,t< 0 and network show disassortative mixing. Whilet� 6, rm,t> 0, and rm,t tends to a constant, 635/1224, the network showsassortative mixing.

The Other Types It should be pointed out that we can construct much morecomplex multiarchitecture types of deterministic weighted GFONP, such as dia-gram of three kinds (GFONP-1, GFONP-2, GFONP-3) is shown in Figure 4.16,which are studied in more detail [38–40], their topological characteristics (degree

Figure 4.15 The rm,t versus t and comparison of theoretical (line)with numerical (circle) results,m¼ 5. Reprinted from [36] with kind permission from Springer Science and Business Media.


distribution, average path length, clustering coefficient, assortativety coefficient andso on) are also obtained by theoretical analysis and numerical simulation.

4.4.4.3 Brief SummaryWe have described another type family of complex network: GFTN and itsGFONP. The analytical expressions of characteristic quantities of the GFTN andGFONP are given, and the computed corresponding numerical results are ingood accordance with each other. These characteristic quantities include degreedistribution, average clustering coefficient, diameter, and the degree–degreecorrelations.The main features of the GFTN have been studied, such as the degree distribu-

tions is exponential form with discrete. The average clustering coefficient of thenetworks decreases as step t (or level number) increases and tends to a constant, forexample, ln 2 for the GFTN. The degree–degree correlations are independent when tis large and the networks almost show assortative mixing.So far, we have introduced several kinds of weighted GFONPs [34–40], the

topological characteristics of the network complexity pyramids are studied byboth theoretical analysis and numerical simulations.

Figure 4.16 Diagram of three kinds of theWGFONP architecture [39]. (a) WGFONP-1only has a signal cycle in each level; (b)WGFONP-2 has in-connection multicycle in

each level; (c) WGFONP-3 has multicycle forouter joins between two levels. a) and b)reprinted from [38] and c) from [39] with kindpermission.


4.5Main Conclusions

So far, we have described three large types of the NCP and discuss main results inmore detail in this chapter. The first type of the NCP is the LCP, which is heuristicfor us. We suggest that the aforementioned universal organizing principles canapply to the NMCP and multikind of the GFONP. It is worth noting that we havebuilt the bridges between the Farey organized generalized pyramid and the unifiedhybrid network theory framework using 3d Farey three sequence (dr, fd, gr) asdeterminate, random, and hybrid weight of network nodes. We have revealed thecomplexity characteristics of three large types of the network complexity pyramids.One of the highlights of the network complexity pyramids is that from the top level

to the bottom level complexity-diversity of the pyramids is increased but universality-simplicity is decreased, and vice versa. Another feature is that topological propertiesinterconversion may occur between different kinds of degree distributions for thelevels of the network pyramid. It is found in the NMCP that topological properties,including cumulative degrees distribution, strength distribution, and weight distri-bution, may be widely transformed among power law, stretched exponentialdistribution, extend exponential distribution, and others, if matching of four hybridratios is changed suitably. The results can provide a newway to study the complexity-diversity and universality-simplicity in the real-world networks, and may help tounderstanding mutual transition between simplicity-complexity and universality-diversity in some types of the network pyramids. Therefore, it has a certain potentialfor applications in real-world networks. Moreover, exploring different types of theCNP are still an open and interesting issue.

Acknowledgment

This work was supported by Nature Science Foundation of China: Nos. 70431002,61174151 and 60874087; also by Science Foundation of China Institute of AtomicEnergy: YZ2011-20.

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