Evolving Network Models Under a Dynamic Growth Rule

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S)prop- (SF) eredis- degree static (TPLDD) Dtd ac 7 2018) 17, March (Dated: ,2 1, with net- eHu, Ke P a and e.[7,w osdrdasml case: simple a In considered respectively. we studied, [17], net- are PA growing ref. without of and properties with topological works step, proba- time a each with Based node at a nodes. adding by of bility rule, (AD) growth AD com- deletion we such which and [17], on rule addition work growth network recent the of a prises type In node new of a [22–25]. deletion proposed account the been into when taken already affected is strongly has be networks The to growing shown [19–21]. of nodes) behavior of permanent (deletion scaling by elements ad- accompanied old constant but of are removal elements, matter there new networks, a of growing As dition real system. in real fact, of of evolution the to description exceptions diver- recent 18]. the some 17, of with [8, origin FTDDs, observed the the on of studies sity attention systematical some to lesser in paid relatively EDD is However, the or regimes. TPLDD specialized mecha- parameter more the the These created as indeed [9]. well models filtering as the information [8], of [13–16], cost adding nism and nodes of ageing existing mechanism of between the mechanism edges as rewiring such and models, added two were this mechanisms into other many degree strict [8–10], nor observed power-law exponential those strict neither explain are to which distributions order In to obtained. growth connect was nodes the added by chosen newly (REN grow the randomly networks while model BA-type, which of network in rule evolving 12], [11, random also were model) the PA without in is networks fact studied growing PLDD the addition, the In model, yield PA BA [5]. with the networks growing In that revealed FTDDs. observed these of rae scntns ( constants. as treated Dcnla otedvriyo TDi elsystems. real in FTDD of diversity of the mechanism to the lead that can indicating were AD systems model, real the of in FTDDs observed obtained the All model.) the 2 h Atp rwhrl ie oehtsimplified somewhat a gives rule growth BA-type The origin the explain to been has challenge theoretical One h Dgot ueitoue n[7 ie oede- more a gives [17] in introduced rule growth AD The n iTang Yi and P d n,cagn rmtepwrlwt the to power-law the from changing ons, r eemndb h oitcpopulation Logistic the by determined are P a l h a-alddge distributions degree fat-tailed the All hs eut a eepce oshed to expected be can results These rdltn oewt probability with node a deleting or elsses oevr ti found is it Moreover, systems. real -ol networks. l-world a h ehns fadto and addition of mechanism the hat y pie h diinaddeletion and addition the mprises P a 2 rdltn oewt the with node a deleting or xsigoe.I hsmdl h EDD the model, this In ones. existing 115 China 411105, n 00 China 6000, P a sa dutbeprmtrin parameter adjustable an is P a and P d 1 = P d − were P a 2 tailed description to the evolution of real systems. How- and ever, this evolution can be even more complex. For ex- ample, as real systems grow, limited resources will cause Pa + Pd =1, (3) interelement competition which, in turn, has a tendency to retard the growth of these systems. Such competition which yield is intensified when the number of element is increased. a K As a result, growth rate of real systems usually decreases Pa(t)= = (4) a + cN(t) K + N(t) with the increase of the system size, exhibiting the so- called density-dependent growth [26]. In fact, competi- and tive dynamics has been found to dominate the evolution cN(t) N(t) of variety of social [27], biological [28] and economic [29] P (t)= = . (5) network systems. Thus, with respect to the AD growth d a + cN(t) K + N(t) rule, to be more realistic, Pa should be a decreasing function of the number of nodes in the network, rather than Where N(t) stands for the number of nodes at time t and a constant [17]. In this paper, we introduce the Logis- K is the parameter which denotes the maximum nodes tic population equation [26] into the AD growth rule. in the network. As a result, the probabilities of addition (deletion) be- Then the Logistic AD model can be defined as follows: comes a decreasing (increasing) function of the network We start from m0 isolated nodes, which act as the nu- size. Based on this dynamical AD growth rule, grow- clear of a growing network. At each time step, either a ing networks models with and without PA are investi- new node is added to the network with probability Pa or gated, respectively. In the present models, networks ex- a randomly selected old node is removed from the net- hibit the density-dependent growth and all the observed work with probability Pd =1 − Pa, where Pa and Pd are FTDDs are created. Moreover, it is found that the net- determined by Eq. (4) and Eq. (5), respectively. works exhibit nonstationary degree distributions, chang- When a new node is added to the network, there are ing from power-law to exponential one or from exponen- still two ways for it to attach to the existing nodes in the tial to Gaussian one. These results can be expected to network. One way is to randomly choose m nodes to set shed some light on the formation and evolution of real up connections (growing network without PA) [11, 12], complex real-world networks. and the other way is to preferentially select m nodes to connect by means of preferential probability in the BA model [5], which reads II. NETWORK MODELS WITH LOGISTIC AD ki +1 GROWTH RULE Πi = , (6) j (kj + 1) P The Logistic equation [26] was proposed by Pierre Ver- where ki is the degree of the ith node (growing network hulst for the analysis of population competitive dynam- with PA). Here, we should note that in order to give ics. Verhulst assumed that during the population growth, chance for isolated nodes to receive a new edge we choose as a result of the competition for the limited resources, Πi proportional to ki + 1 [13]. the death rate per individual is a lineally increasing func- One can find from Eq. (4) and Eq. (5) that when tion of the number of population. Then the Logistic N(t) ≪ K, Pa(t) ≫ Pd(t). This means that the networks equation can be written as grow rapidly at the initial stages of their evolution. In fact, in a Logistic model [26]: dN(t) N(t) − − = [a cN(t)]N(t)= a[1 ]N(t) , (1) dN(t) K − N(t) dt K = a N(t), (7) dt K where N(t) denotes the number of population at time t; a and c are constants; a is the birth rate per individual when N(t) ≪ K, and cN(t) stands for the death rate per individual at dN(t) time t; K = a/c is called carrying capacity [30, 31] which ≈ aN(t), (8) represents the largest population the environment can dt support due to limited resources. the solution is: A more realistic description to the real-network’s evolution can be achieved by the introduction of such Lo- N (t)= C0 exp(at) , (9) gistic dynamics into the AD growth rule. To do this, we obtain the probability of addition Pa and the probability where C0 is a integral constant. Indeed, this kind of of deletion Pd by the following two equations exponential growth has been observed in many newly emerged real-world networks, such as the World-Wide- Pa aN(t) Web (WWW) and the Internet [19, 20]. This indicates = 2 (2) Pd cN(t) that the rapid growth of some real systems in their young 3 age is well described by the Logistic AD model. As N(t) increase, Pa(t) decrease while Pd(t) increase, the growth power-law fit t=12000 of network is slowed down. In the limit of large t, t=31000

t=72000

-1 t=150000

1 10 lim Pa(t) = lim Pd(t)= , (10) t=260000 t→∞ t→∞ 2 t=500000

t=1000000

t=2000000

and P(k)

exponential fit

-3 lim N (t)= K, (11) 10 t→∞ i.e., the network reach a steady state and the number of nodes has the upper limit K. The above analysis (a)

-5

10 imply that networks in the Logistic AD model exhibit 0 1 2 3

10 10 10 10 density-dependent growth characterizing the evolution of

many real systems, not captured in most previous net- exponential fit work models. t=13780

t=50045

We investigated the degree distribution of the networks t=86973 -1

10 by extensive computer simulations. In the simulation, t=165728 t=218474 we set m0 = m = 5 and K = 100000. Network is t=416298

t=1045700

P(k) left to evolve until the steady state is reached. Cumu- t=2184780

Gauss fit

lative degree distributions of growing networks with and -3 without PA at diﬀerent time step are given in Fig. 1(a) 10 and Fig. 1(b), respectively. We found that in the Logis-

tic AD model, the networks exhibit nonstationary degree (b)

distribution before the steady state is reached. This is -5

10 illustrated in Fig. 1. For the growing network with PA, 0 10 20 30 40 50 60 70 80 as time goes on, P (k) of the network undergos a pro- k cess of transition, which can be roughly separated into several stages: (1) At the earlier stage of network evo- FIG. 1: (a) Cumulative degree distribution of the growing ≤ lution, i.e., when t 72000, the network exhibits vari- network with PA, for different time steps t, in logarithmic ous PLDDs with different power-law exponents. In ad- scales. The dash line is power-law fit and the solid line is dition, the exponent increases with time. Particularly, exponential fit. (b) Cumulative degree distribution of the in the asymptotic cases of Pa being close to 1, the net- growing network without PA, for different time steps t, in work is almost equivalent to the well-known BA model, semi-logarithmic scales. The dash line is exponential fit and thus it have PLDD with power-law exponents γ = 3. (2) the solid line is Gaussian fit. Results in both (a) and (b) are 72000

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