Counterexample: scale-free networked graphs with invariable diameter and density feature
CCounterexample: scale-free networked graphs with invariablediameter and density feature
Fei Ma a, , Xiaomin Wang a, and Ping Wang b,c,d, a School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China b National Engineering Research Center for Software Engineering, Peking University, Beijing, China c School of Software and Microelectronics, Peking University, Beijing 102600, China d Key Laboratory of High Confidence Software Technologies (PKU), Ministry of Education, Beijing, China
Abstract:
Here, we propose a class of scale-free networks G ( t ; m ) with some intriguingproperties, which can not be simultaneously held by all the theoretical models with power-law degree distribution in the existing literature, including (i) average degrees (cid:104) k (cid:105) of all thegenerated networks are no longer a constant in the limit of large graph size, implying thatthey are not sparse but dense, (ii) power-law parameters γ of these networks are preciselycalculated equal to 2, as well (iii) their diameters D are all an invariant in the growth process ofmodels. While our models have deterministic structure with clustering coefficients equivalentto zero, we might be able to obtain various candidates with nonzero clustering coefficient basedon original networks using some reasonable approaches, for instance, randomly adding somenew edges under the premise of keeping the three important properties above unchanged.In addition, we study trapping problem on networks G ( t ; m ) and then obtain closed-formsolution to mean hitting time (cid:104)H(cid:105) t . As opposed to other previous models, our results showan unexpected phenomenon that the analytic value for (cid:104)H(cid:105) t is approximately close to thelogarithm of vertex number of networks G ( t ; m ). From the theoretical point of view, thesenetworked models considered here can be thought of as counterexamples for most of thepublished models obeying power-law distribution in current study. Keywords:
Scale-free graphs, Small-world, Assortative mixing, Trapping problem. The author’s E-mail: [email protected]. The author’s E-mail: [email protected]. The corresponding author’s E-mail: [email protected]. a r X i v : . [ c s . S I] O c t INTRODUCTION
Complex systems, such as, friendship networks, metabolic networks, protein-protein interaction networks,predator-prey networks, can be naturally interpreted as complex network, a newborn yet useful tool thathas been widely adopted in a large variety of disciplines, particularly, in statistic physics and computerscience [1]. So, in the past two decades, complex networks have attracted considerable attention andhelped us to understand some topological properties and structural dynamics on the complex systemsmentioned above. Two significant findings of which are the small-word property [2] and scale-free feature[3]. There are in general two directions in current complex network study. The one is to generate complexnetworked models, also called synthetic networks, in order to mimic some characters prevalent in real-world networks, such as power-law degree distribution, small-world phenomena, hierarchical structure [2]-[5]. The other aims at determining the influence from topological structure of networks on dynamics takingplace on networks themselves, for instance, the mean hitting time for trapping problem, synchronizationin networks, epidemic spread [6]-[8]. In this paper, we not only propose a family of networked models G ( t ; m ) and discuss some commonly used topological measures for understanding models G ( t ; m ) inmuch detail, but also consider the trapping problem on the proposed models G ( t ; m ) and final derive theclosed-form solution to mean hitting time.The main concern in the previous research of theoretical networked models is to focus on constructingmodels which have scale-free and small-world characters as described above. However, an overwhelmingnumber of models are sparse, implying that the average degree of networks will tend to a constant inthe limit of large graph size. This is because a great deal of real-life networks are found to displaysparsity feature. By contrast, current studies in some areas turn out the existence of dense networks[9]. To describe these such networks, some available networked models have been proposed [10] andanalytically investigated in some principled manners, including mean-field theory and master equation.Yet, most of them are stochastic. To our knowledge, almost no deterministic models with both densitystructure and scale-free feature are built in the past. Although there exist some disadvantages inheritedby the latter in comparison with stochastic models, the deterministic structure of model allows us toprecisely derive the solutions to some quantities of great interest, such as clustering coefficient, degreedistribution, average path length. To some extent, determining some invariants on networked modelswith deterministic structure has the theoretical flavor. Motivated by this, we present a class of novelnetworked models with hierarchical structure that are precisely proved to not only show scale-free and2mall-world characters but also be dense. Throughout this paper, all graphs (models) addressed aresimple and the terms graph and network are used indistinctly.The rest of this paper can be organized by the following several sections. Section II aims at intro-ducing networked models and discussing some widely studied structural parameters, including averagedegree (cid:104) k (cid:105) , diameter D . Among of them, while our models are analytically proved to show scale-freefeature, the power-law exponents γ is equal to an unexpected constant 2. This suggests that the densityfeature can be found on our models. Surprisingly, all the networked models have an identical diameterthat is always invariable in the growth process ( t ≥ p , to switch the original graphs with deterministic structure into stochastic ones.Last but not least, we take account into a type of random walks, called trapping problem, and thenanalytically determine the mean hitting time (cid:104)H(cid:105) t . The result shows an interesting phenomenon that theclosed-form solution to quantity (cid:104)H(cid:105) t is asymptotically close to the logarithm of vertex number of ourmodels under consideration, which is not covered by almost all theoretical models characterizing complexnetworks with scale-free feature. Final, we close this paper with a concise conclusion and some futuredirections in Section III. The goal of this section is to build up our networked models, denoted by graphs G ( t ; m ), with hierarchicalstructure and study some topological structural parameters on these proposed models both analyticallyand experimentally. In addition, we also consider the trapping problem on the proposed models G ( t ; m )where the trap is allocated on the largest degree vertex. Here we will introduce the graphs G ( t ; m ) ( m ≥ G (0; m ), is a star with m leaves as shown in the top-left panel of Fig.1. The next graph G (1; m ) can be obtained from G (0; m )in the following manner, ( ) generating m duplications of seed G (0; m ) labelled as G i (0; m ), ( ) takingan active vertex, ( ) connecting that active vertex to each leaf in star G i (0; m ). Obviously, the resultinggraph G (1; m ) has hierarchical structure as plotted in the bottom-left of Fig.1. For convenience, we divideall vertices of G (1; m ) into three classes, i.e., that active vertex allocated at the level 0, denoted by L = 0,3igure 1: The diagram of first three examples of graph G ( t ; 2).all the central vertices of stars G i (0; m ) at the level L = 1 and the remaining vertices of graph G (1; m )at the level L = 2. Henceforth, for time step t ≥
2, the young graph G ( t ; m ) can be built based on m duplications of the preceding graph G ( t − m ) by connecting an active vertex to all the vertices at thelevel L = t of graphs G i ( t − m ). As an illustrative example, the graph G (2; 2) is shown in the rightmostpanel of Fig.1.In view of the growth manner of graph G ( t ; m ), it is not hard to obtain a couple of equationssatisfied by vertex number | V ( t ; m ) | and edge number | E ( t ; m ) | as follows | V ( t ; m ) | = m | V ( t − m ) | + 1 | E ( t ; m ) | = m | E ( t − m ) | + m t +1 . (1)With the initial conditions | V (0; m ) | = m + 1 and | E (0; m ) | = m , we can solve for | V ( t ; m ) | and | E ( t ; m ) | from Eq.(1) to obtain | V ( t ; m ) | = m t +2 − m − , | E ( t ; m ) | = ( t + 1) m t +1 . (2)So far, we already accomplish the construction of our networked models exhibiting hierarchy phe-nomena. Note that the related work to construction of hierarchical networks has been widely reported inthe literature, such as Refs.[11] and [12]. For instance, Ravasz et al have generated a class of hierarchicalnetworked models in [12] and then discussed some structural parameters on them. Whereas, it is worthynoticing that the generative method adopted in this paper is slightly different from that used in [12].While the both are manipulated in an iterative manner, our method is based on an idea that the nextnetworked model G ( t ; m ) is created by connecting a portion of vertices of each duplication G i ( t − m )of the proceeding model G ( t − m ) to a new external vertex instead of a designated vertex in model4 ( t − m ). In addition, as will become clear in the coming sections, some properties of our models arenot found on those models in [11] and [12].The next tasks are to in-detail discuss several common topological properties on graphs G ( t ; m ),for instance, average degree (cid:104) k (cid:105) and clustering coefficient (cid:104) C (cid:105) , as tried in the existing literature [13],[14]. As described above, the deterministic nature of the proposed graphs allows us to evaluate some well-studied topological parameters associated with the underlying structure. This section aims to calculatethe closed-form solutions for several structural indices, including average degree (cid:104) k (cid:105) . As the simplest yet most important structural parameter, average degree (cid:104) k (cid:105) , defined as the ratio of 2times edge number and vertex number, can be adopted to determine whether a given network is sparseor not. In general, almost all published networked models with scale-free feature (discussed later), bothstochastic and deterministic, are by definition sparse, that is, the value for (cid:104) k (cid:105) being finite in the largegraph size limit. By contrast, our networked models G ( t ; m ) turn out to be of density due to (cid:104) k ( t ; m ) (cid:105) = 2 | E ( t ; m ) || V ( t ; m ) | ∼ t = O (ln | V ( t ; m ) | ) . (3)From Eq.(3), it is clear to see that average degree (cid:104) k ( t ; m ) (cid:105) is linearly correlated with time step t and no longer a constant compared with those models in [11]-[13], such as, Apollonian networks [15].Meantime, as recently reported in [9], many real-world network examples have been proven to have nosparsity topological structure. Therefore, Our networked models G ( t ; m ) may be able to be selected aspotential models to unveil some unseen properties behind those dense networks in real life. In the past years, there are two significant findings in complex network study. One of which is scale-freefeature due to Barabasi and Albert [3] using statistical method for depicting vertex degree distribution ofmany real-world networks. Such types of networks show a fact that a small fraction of vertices possess agreat number of connections and however the rest of vertices have a small number of connections. Sincethen, given a complex networked model, one always estimate using degree distribution whether or not it is5cale-free. Taking into account deterministic structure of graphs G ( t ; m ), we make use of the cumulativedegree distribution in discrete form P cum ( k ≥ k i ) = (cid:80) k ≥ k i N k | V | , (4)where symbol N k represents the total number of vertices with degree exactly equal to k in graph G ( V, E ).As said in Eq.(4), we need to classify all vertices of graphs G ( t ; m ) according to vertex degree in orderto determine whether the presented models show scale-free feature.For a graph G ( t ; m ) with t + 1 levels, it is evident to see that the greatest degree vertex is thatactive vertex added at time step t and has degree m t +1 , the second greatest degree vertices are all at thelevel L = 1, and so on. With such a classification, there is in fact an unexpected case where all verticesat level L = t + 1 have degree t + 1. This degree value must be in some range between m t i and m t i +1 .Therefore, we will adjust the initial ranks of vertices with respect to vertex degree alone for simplicity.The end list is as follows L ... t i − ... t − t t + 1 k L,t ; m m t +1 m t ... m t i ... m m t + 1 N L,t ; m m ... m t − t i +1 ... m t − m t m t +1 In practice, the above list is usually called degree sequence of graph G ( t ; m ) in the jargon of graph theory.Based on such a list and Eq.(4), we can calculate the degree distribution of networked model G ( t ; m ) inthe following P cum ( k ≥ k i ) = k − γ α i , k i > t + 1 k − γ α i + , k i ≤ t + 1 (5)where power-law parameter γ α = 1. Performing the derivative of both sides in Eq.(5) with respect to k yields P ( k ) ∼ k − γ , γ = γ α + 1 = 2 . (6)This disapproves a statement in [16] that no network with an unbounded power-law degree distributionwith 0 ≤ γ ≤ G ( t ; m ) are all denseas in Eq.(3) because of the feature of Riemann ζ (1) function ζ (1) = (cid:88) i =1 i − . (7)6n the limit of large i , the right-hand side of Eq.(7) is divergence. Thus, average degree (cid:104) k ( t ; m ) (cid:105) ofnetworked models G ( t ; m ) must be infinite due to (cid:104) k ( t ; m ) (cid:105) = (cid:82) k max k min kP ( k ) dk .As showed in [17], Caldarelli et al have pointed out the widespread occurrence of the inverse squaredistribution in social sciences and taxonomy and also provided some detailed discussions about mech-anisms causing such a phenomenon. Meanwhile, they showed that the treelike classification method iscompetent to lead to this behavior mentioned above. At the same time, several stochastic scale-freegraphs with degree parameter γ = 2 have been constructed in [18]. Nonetheless, there are no determin-istic models following power-law degree distribution with γ = 2 found in the current theoretical modelsstudy. Roughly speaking, our models, graphs G ( t ; m ), can be regarded as the first attempt in context ofconstructing deterministic models. In addition, graphs G ( t ; m ) have many other interesting topologicalstructural properties as we will show shortly. Some of which are not displayed by those stochastic models.In particular, the finding of ultrasmall diameter is a surprising result in this sense. Following the previous subsection, the other intriguing finding in the complex network studies is small-world property attributed to Watts and Strogatz [2] by empirically capturing the diameter of somereal-world and synthetic network models. Mathematically, diameter of a graph, denoted by D , is themaximum over distances of all possible vertex pairs. For a pair of vertices u and v , distance betweenthem, denoted by d uv , is the edge number of any shorted path joining vertex u and v . Most generally,diameter can be viewed as a coarse-granularity index for measuring the information delay on a networkin question.With the help of concrete construction of networked models G ( t ; m ), it is easy to find the diameters D ( t ; m ) to obey D ( t ; m ) = 4 = O (1) . (8)This is because ( ) all vertices at level L = t + 1 are connected to that active vertex at the highestlevel, namely, L = 0; ( ) each vertex at the intermediate levels, L = 1 , ..., L = t , always connects to avertex at the most bottom level L = t + 1. In fact, diameters D ( t ; m ) are equal to the distance betweenthat two vertices that are both at the intermediate levels and in different branches of graphs G ( t ; m ).Taking into account a trivial character in a connected graph G ( V, E ) that average path length, definedas (cid:104) d (cid:105) = (cid:80) u,v ∈ E,u (cid:54) = v d uv / [ | V | ( | V | − G ( t ; m ). Using the exact value of diameter in Eq.(8), onecan be convinced that our graphs have captured an ingredient of small-world property. The other will bediscussed in the subsequent section.It is worthy noting that in [19], the authors have reported some results about diameter of scale-freegraphs G ( V, E ), for instance, those with 2 < γ < D ∼ ln ln | V | . However, there are few discussions about diameter of thescale-free graphs with γ = 2 in published papers. Perhaps one of important reasons for this is that theprevious researches focus mainly on sparse models with scale-free feature. Here, our networked models G ( t ; m ) are proved to have an invariable diameter in the evolution process. As a result, graphs G ( t ; m )can serve as stronger evidences for illustrating that our ability to understand the fundamental structuralproperties of graphs including all scale-free ones is always limited to some specific models and hence someearly demonstrations may be not complete.Compared to numerous pre-existing networked models with scale-free feature, such as those in [19],the power-law parameter γ of our graphs G ( t ; m ) is not in the range 2 < γ <
3. Hence, one has mostlylikely to conjecture whether the density feature of networked models G ( t ; m ) leads to such an ultrasmalldiameter directly. In principle, the dense graphs should show smaller diameters in comparison with thosesparse ones. In practice, we would like to note that density feature planted on graphs G ( t ; m ) indeedshrinks the distance between any pair of vertices and thus has vastly effect on emergence of smallerdiameter, but it is not a sufficient condition. As described in our recent work [20], the diameter of densegraphs G ( V, E ) obeying power-law distribution may also be quite large compared with the widely-usedvalue ln | V | . Again, this strongly means that there are a great number of structural properties of scale-freegraphs incompletely uncovered until now and so more efforts should be paid to better understand thiskinds of fascinating graphs in the future. The other ingredient of small-world property is clustering coefficient that plays an important role inevaluating the level of clusters in a networked model under consideration. For instance, it in essencedescribes a phenomenon that in a friendship network two arbitrary friends of one person will have ahigher likely to be friends with each other, usually called the triadic in social analysis. For the theoreticalpoint of view, such a connection trend among neighbors of one vertex v with degree k can be abstractlydepicted in the following form c v = n v / [ k ( k − n v represents the actually existing edges between8igure 2: The diagram of clustering coefficients (cid:104) C (cid:105) of graphs G ( t ; m ). (a) (b) (c) (d) Figure 3: The diagram of clustering coefficients (cid:104) C (cid:105) of graphs G ( t ; m ) where parameters m are equalto 2 , , v . For the whole graph G ( V, E ), the clustering coefficient (cid:104) C (cid:105) can be defined as theaveraged value over clustering coefficients of all vertices, as follows (cid:104) C (cid:105) = (cid:80) v ∈ V n v | V | . (9)By definition, we can without difficulty obtain that the clustering coefficients (cid:104) C ( t ; m ) (cid:105) of networkedmodels G ( t ; m ) are equivalent to 0 because there are no triangles contained in the evolution of graphs G ( t ; m ). This suggests that small-world property can not be found in graphs G ( t ; m ). Nevertheless, thereare still plenty of other potential characters behind networked models G ( t ; m ). To be more concrete, allthe generated graphs G ( t ; m ) are planar, meaning that for an arbitrary parameter m , graph G ( t ; m ) canbe embedded in the plane so that its edges intersect only at their ends [21].9o make our models G ( t ; m ) small-world, i.e., achieving the transformation from zero clusteringcoefficient to nonzero, we have to take some effective measures. There are in fact a great deal of methodsfor addressing problems of this kind in present research. The simplest one is to add new edges betweenunconnected vertex pairs for generating enough triangles. Here, our goal is not only to obtain nonzeroclustering coefficient but also to remain the properties discussed above unchanged. To this end, we makeuse of a simple replacement of the seed from star G (0; m ) to wheel G (0; m ) [22]. Besides that, all thegrowth mechanisms keep the same as previously. This leads to a new graph G ( t ; m ). By definition, it isstraightforward to get (cid:104) C (cid:105) = (cid:80) ti =0 2 m t − i m i +1 − + t +1) m t +1 ( t +3)( t +2) | V ( t ; m ) | , (10)where the vertex number | V ( t ; m ) | of graph G ( t ; m ) is equal to | V ( t ; m ) | . In the large graph size limit,the clustering coefficient (cid:104) C (cid:105) of Eq.(10) tends to a nonzero constant for small parameters m as shown inFig.2, suggesting that deterministic graph G ( t ; m ) will have scale-free feature and small-world propertysimultaneously.To make further progress, we may delete with probability p each edge between vertices at the level L = t + 1 in graph G ( t ; m ), leading to another graph G ( t ; m ). The introduction of randomly deletingedges will switch deterministic graphs into the opposite case, namely, stochastic ones. As before, theclustering coefficient (cid:104) C (cid:105) of stochastic graphs G ( t ; m ) can be calculated as follows (cid:104) C (cid:105) = (1 − p ) (cid:80) ti =0 2 m t − i m i +1 − + (cid:104) (1 − p ) t +1)( t +3)( t +2) + p (1 − p ) t +2 (cid:105) m t +1 | V ( t ; m ) | , (11)here | V ( t ; m ) | is the vertex number of graphs G ( t ; m ). To determine the tendency of (cid:104) C (cid:105) in thelimit of large size, we feed graphs G ( t ; m ) into computer. As plotted in Fig.3, for distinct parameters m ,the values (cid:104) C (cid:105) are different from each other initially but all show similar tendency in the large- t limit.In a word, the three types of networked models, graphs G ( t ; m ), G ( t ; m ) and G ( t ; m ), can beselected as counterexamples for disproving some previous statements about scale-free graphs, such as thescale-free graphs with small-world property may have invariable diameter. Meantime, the lights shed bythem may be helpful to construct new networked models in the future. Many real-world networks [23] have been observed to show assortative mixing on their degrees, that is, apreference for high-degree vertices to attach to other vertices like them, while others show disassortative10igure 4: The diagram of assortativity coefficient r ( t ; m ) of graphs G ( t ; m ).mixing, i.e., high-degree vertices attach to ones unlike them. Particularly, this is a popular phenomenonin social networks. For instance, it is mostly willing of people to establish friendships with those at thesame level as them rather than to get in touch with others. For the purposes of quantifying this featureof networks G ( V, E ), Newman defined the following measure r , usually called assortativity coefficient, r = | E | − (cid:80) e ij ∈ E k i k j − (cid:34) | E | − (cid:80) e ij ∈ E ( k i + k j ) (cid:35) | E | − (cid:80) e ij ∈ E ( k i + k j ) − (cid:34) | E | − (cid:80) e ij ∈ E ( k i + k j ) (cid:35) , (12)in which k i is the degree of vertex v and e ij denotes an edge connecting vertex i to j . With such an index,most social networks turn out to have significant assortative mixing, while technological and biologicalnetworks seem to be disassortatively constructed.As explained above, the scalar measure r in fact figures the degree of similarity between two end-points of any edge on an observed network by means of vertex degree. Empirically, our networked models G ( t ; m ) should be disassorative. To show this, we can write based on Eq.(12) r ( t ; m ) = m t +2 − mm − − (cid:104) t +12 + m t +2 − m (2 m − t +1) (cid:105) t +1) + m t +4 − m (2 m − t +1) − (cid:104) t +12 + m t +2 − m (2 m − t +1) (cid:105) . (13)As t → ∞ , for a given parameter m , r ( t ; m ) tends to zero, seeing Fig.4 for a lot. By analogy with Eq.(13),we can evaluate assortativity coefficient r ( t ; m ) of stochastic graphs G ( t ; m ) by plugging the following11quations into Eq.(12) | E | = | E ( t ; m ) | = m t +1 ( t + 2 − p ) , (14a) (cid:88) e ij ∈ E k i k j = m t +1 t (cid:88) i =0 [ p ( t + 1) + 2 p (1 − p )( t + 2)]+ m t +1 t (cid:88) i =0 (1 − p ) ( t + 3) + (1 − p ) ( t + 3) m t +1 + 4 p (1 − p ) [ p ( t + 2) + (1 − p )( t + 2)( t + 3)] m t +1 , (14b) (cid:88) e ij ∈ E ( k i + k j ) = m t +1 ( t + 1) + t (cid:88) i =0 (1 − p ) ( t + 3) m i + t (cid:88) i =0 m i [ p ( t + 1) + 2 p (1 − p )( t + 2)]+ 4 p (1 − p ) [ p (2 t + 4) + (1 − p )(2 t + 5)] m t +1 + (1 − p ) (2 t + 6) m t +1 , (14c) (cid:88) e ij ∈ E ( k i + k j ) = m t +1 t (cid:88) i =0 m i +1 + t (cid:88) i =0 (1 − p ) ( t + 3) m i + t (cid:88) i =0 m i [ p ( t + 1) + 2 p (1 − p )( t + 2) ]+ 2(1 − p ) [4 p ( t + 2) + (1 − p ) ( t + 3) ] m t +1 + 4 p (1 − p ) [( t + 2) + ( t + 3) ] m t +1 . (14d)In order to evaluate whether the random deletion of edges has influence on assortativity coefficient r ( t ; m ), we conduct extensive simulations in terms of Eq.(12) and Eqs.(14a)-(14d) and experimentalresults are shown in Fig.5.Interestingly, from the panels in Fig.5, it can be easy to see a phenomenon that all theoretical valuesfor assortativity coefficients r ( t ; m ) are bounded from above the critical condition 0 while approachingto zero in the large graph size limit. This is sharply different from those previously reported results in[23] where all assortativity coefficients associated with most of studied networked models are negativewhile also tending to zero. 12 a) (b) (c) (d) Figure 5: The diagram of assortativity coefficients r ( t ; m ) of graphs G ( t ; m ) where parameters m areequal to 2 , , In this section, we formulate the trapping problem on our graphs G ( t ; m ). In practice, this is a simpleunbiased Markovian random walk with a tap, say a perfect absorber, allocated on a designated vertexon a graph in question. As shown above, that active vertex at the level L = 0 of graphs G ( t ; m ) has thelargest degree and hence is called the hub vertex, denoted by h t . In order to further probe its importanceunder trapping problem, we put an absorber on the hub vertex. And then, a particle located on vertex v but for the hub will hop to one of its neighbor N v with the transition probability 1 /d v ( t ; m ) beforearriving at that absorber where d v ( t ; m ) is the degree of vertex v in graphs G ( t ; m ).Consider that a particle starts from vertex v at initial time, the jumping probability P vu of startingout from v to u satisfies the following master equation P vu ( l + 1) = (cid:88) i ∈ V ( t ; m ) a iu d i ( t ; m ) P vi ( l ) , (15)where a iu is the element of adjacency matrix of graph G ( t ; m ), a iu = 1 if this pair of vertices i and u areconnected by an edge and a iu = 0 otherwise.According to the rule above, we are particularly interested in the quantity, called hitting time H ,for measuring the expected time for a particle, which starts from an arbitrary vertex, to first visit at thetrap in the trapping problem. For graph G ( t ; m ) as a whole, we denote the hitting time for a particleplaced on vertex v by H v and then let P ( H v = l ) be the probability for that particle to first hit the trap,i.e., hub vertex h t , after l steps. By analogy with Eq.(15), we can obtain13 ( H v = l ) = (cid:88) i ∈ V ( t ; m ) ,i (cid:54) = h t a iu d i ( t ; m ) P ( H v = l − . (16)A commonly used approach for the preceding equation is generating function. Without loss ofgenerality, we may define the corresponding generating function of quantity P ( H v = l ) in the followingform P v ( x ) = ∞ (cid:88) t =0 P ( H v = l ) x l . (17)As we will show later, a trial yet useful fact related to P v ( x ), that is, the expected time H v is exactlyequal to the value P (cid:48) v (1), helps us to consolidate all the results in the subsequent section.Before beginning to derive our calculations, for convenience, we need to introduce two notations P t ( s ) and Q L ( s ). The former represents the probability for a particle on an arbitrary vertex at the level L = t + 1 of graphs G ( t ; m ) to first arrive at the hub h t after s steps, and the latter is defined as theprobability that a particle originating from an arbitrary vertex w at the level L ( L = 1 , . . . , t ) hits one atrandom chosen vertex at the level L = t + 1 of graphs G ( t ; m ), which connects to vertex w , after s jumps.Based on the structure of graphs G ( t ; m ) and the statements above, we can write the following equation P t ( s ) = δ s, k L = t +1 + 1 k L = t +1 t (cid:88) L =1 s − (cid:88) i =1 Q L ( i ) P t ( s − − i ) , (18)here δ s, is the Kronecker delta function in which δ s, = 1 as s = 1 and δ s, = 0 otherwise, k L = t +1 is thedegree of vertex at the level L = t + 1 and equals t + 1 as above.Using the lights shed by Eq.(17), the generating function P t ( x ) corresponding to quantity P t ( s ) canbe expressed as follows P t ( x ) = xk L = t +1 + tx k L = t +1 P t ( x ) , (19)in which we make use of an evident result Q L ( i ) = 1 only for both L = 1 , . . . , t and i = 1, as well Q L ( i ) = 0 otherwise.At the same time, we let H tt +1 stand for the hitting time for a particle initially set on any vertex atthe level L = t + 1 which is by definition written H tt +1 = ddx P t ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x =1 . (20)14aking into consideration Eq.(20), performing the derivative of both sides of Eq.(19) produces theexact solution of H tt +1 as follows H tt +1 = 2 t + 1 . (21)For each vertex at the level L = 1 , . . . , t , combining the definition of Q L ( s ) and the hierarchicalstructure of graphs G ( t ; m ), the hitting time H tL for a particle originally allocated on any vertex at thelevel L may be obtained in terms of H tt +1 H tL = H tt +1 + 1 . (22)By far, the hitting times H tL for a particle at the level L = 1 , . . . , t + 1 are all precisely calculatedin a rigorous manner. The next task is to derive the mean hitting time (cid:104)H(cid:105) t , which characterizes thetrapping process on average, in the following fashion (cid:104)H(cid:105) t = 1 | V ( t ; m ) | − t +1 (cid:88) L =1 H tL | N t ( L ) | , (23)here we again utilize the hierarchy of graphs G ( t ; m ) and | N t ( L ) | denotes the total number of vertices atthe level L ( L = 1 , . . . , t + 1). As pointed out before, | N t ( L ) | is in essence equal to N L,t ; m .Substituting Eqs.(21)-(22) and the value of | V ( t ; m ) | in Eq.(1) into Eq.(23) yields (cid:104)H(cid:105) t = O (2 t + 1 m ) , (24)where we take useful advantage of some simple arithmetics.To make process further, we consider the logarithm of vertex number of graphs G ( t ; m ), namely,ln | V ( t ; m ) | ∼ t ln m . It is clear to see that for the whole graphs G ( t ; m ), the mean hitting time (cid:104)H(cid:105) t hasa close relationship with the vertex number of graphs G ( t ; m ) as shown below (cid:104)H(cid:105) t ∼ ln | V ( t ; m ) | . (25)This is completely different from some previous results in the existing literature, such as, thecomplete graph on N vertices having the mean hitting time exactly equal to N −
1, which is quiteapproximately close to its vertex number, the hierarchical models considered in [24] with the meanhitting time also being the same order of magnitude as their own vertex number. Compared to ourmodels, the former, i.e., complete graph, has no both scale-free feature and hierarchical structure while15ith the smallest diameter. On the other hand, the models in [24] have a larger enough diameter thanour graphs while showing scale-free feature and the hierarchy of structure. Following the discussionsabove, it has been shown that when the trap is allocated on the greatest degree vertex, those hierarchicalnetworks presented in [11] and [12] have mean hitting time approximately close to the power of numberof vertices for some exponent α in form, a demonstration that is distinct with the consequence obtainedin this paper. On average, this suggests that the proposed networked models G ( t ; m ) outperform thosemodels [11] and [12] with respect to mean hitting time in the trapping problem considered here.As a consequence, our models can be capable of serving as counterexamples for in-depth understand-ing many other fundamental properties on theoretical models, in particular, with respect to scale-freegraphs. One of the most important reasons is that scale-free feature is ubiquitously observed in a largeamount of complex networks, both synthetic and real-world. In summary, we present a family of scale-free networked models of significant interest. Based on boththeoretical arguments and experimental simulations, we derive some striking results unseen in pre-existingtheoretical models. They shows that (1) our graphs G ( t ; m ) follow power-law degree distribution withexponent 2 and thus are dense, (2) an invariable diameter can be found on our graphs G ( t ; m ) compared toalmost all previously proposed scale-free models, (3) using random method leads the end graphs G ( t ; m )to display a nonnegative assortativity coefficient, and (4) when the trap is allocated on the hub in graphs G ( t ; m ), the mean trapping time is approximately related to the logarithm of vertex number of graphs. Tothe best of our knowledge, this work seems the first to probe novel scale-free models particularly becausein the last, a significant amount of attention have been paid to discuss sparse graphs with scale-freefeature. From the respect of theoretical research, our models can be used as counterexamples to disprovesome previous demonstrations corresponding to the scale-free graph family in current study and so enableresearchers to well understand the fundamental structure properties planted on scale-free models. Acknowledgments
The research was supported by the National Key Research and Development Plan under grant 2017YFB1200704and the National Natural Science Foundation of China under grant No. 61662066.16 eferences [1] M.E.J. Newman. Networks. Oxford university press. 2018[2] D.J. Watts, S.H Strogatz. Nature. 393 (1998): 440-442[3] A.-L. Barab´asi, R. Albert. Science. 5439 (1999): 509-512[4] E. Ravasz, A.L. Somera, D.A. Mongru, Z.N. Oltvai, A.-L. Barab´asi. Science. 5586 (2002): 1551-1555[5] M, Zamani, L.C. Forero, T. Vicsek. New J. Phys. 20 (2018)023025[6] F. Coghi, J. Morand, H. Touchette. Phys. Rev. E. 99 (2019)022137[7] A. Otto, G. Radons, D. Bachrathy, G. Orosz. Phys. Rev. E. 97 (2018)012311[8] R.P. Satorras, C. Castellano. Journal of Statistical Physics. 173 (2018): 1110-1123[9] N. Blagus, L. Subelj, M. Bajec. Physica A. 391 (2012): 2794-2802[10] R. Lambiotte, P.L. Krapivsky, U. Bhat, S. Redner. Phys. Rev. Lett. 117 (2016) 218301[11] A.-L. Barab´asi, E. Ravasz, T. Vicsek. Physica A. 299 (2001): 559-564[12] E. Ravasz, A.-L. Barab´asi. Phys. Rev. E. 67 (2003)026112[13] F. Ma, P. Wang, B. Yao. Physica A. 527 (2019)121295[14] Y. Zou, R.V. Donner, N. Marwan, J.F. Dongesc, J. Kurths. Physics Reports. 787 (2019): 1-97[15] P.P. Zhang. arXiv:1901.07073[16] C.I.D. Genio, T. Gross, K.E. Bassler. Phys. Rev. Lett. 107 (2011)178701[17] G. Caldarelli, C.C. Cartozo, P.D.L. Rios, V.D.P. Servedio. Phys. Rev. E. 69 (2004) 035101(R)[18] O.T. Courtney, G. Bianconi. Phys. Rev. E. 97 (2018)052303[19] R. Cohen, S. Havlin. Phys. Rev. Lett. 90 (2003)058701[20] F. Ma, X.M. Wang, P. Wang, X.D. Luo. arXiv:1912.08923.[21] J.A. Bondy, U.S.R. Murty. Graph theory. Springer. 2008[22] The wheel graph G (0; m ) can be obtained from star G (0; m ) by adding new edges to connect all leaves in a mannerthat generates a cycle C m .[23] M.E.J. Newman. Phys. Rev. Lett. 89 (2003)208701[24] Z.Z. Zhang, Y. Lin, S.Y. Gao, S.G. Zhou, J.H. Guan, M. Li. Phys. Rev. E. 80 (2009)051120.[23] M.E.J. Newman. Phys. Rev. Lett. 89 (2003)208701[24] Z.Z. Zhang, Y. Lin, S.Y. Gao, S.G. Zhou, J.H. Guan, M. Li. Phys. Rev. E. 80 (2009)051120