# Coevolutionary dynamics on scale-free networks

###### Abstract

We investigate Bak-Sneppen coevolution models on scale-free networks with various degree exponents including random networks. For , the critical fitness value approaches to a nonzero finite value in the limit , whereas approaches to zero as . These results are explained by showing analytically on the networks with size . The avalanche size distribution shows the normal power-law behavior for . In contrast, for has two power-law regimes. One is a short regime for small with a large exponent and the other is a long regime for large with a small exponent (). The origin of the two power-regimes is explained by the dynamics on an artificially-made star-linked network.

###### pacs:

87.10.+e, 05.40.-a, 87.23.-nBak and Sneppen (BS) BS has introduced an excellent model to explain the evolution of bio-species which exhibits the punctuated equilibrium behavior KJ . BS model has two important features, coevolution of the interacting species and the intermittent bursts of activity separating relatively long periods of the stasis. In BS model the ecosystem evolves into a self-organized criticality with avalanches of mutations occurring all scales. Aside from its importance for the evolution BS model has been also shown to have rich scaling behaviors PMB .

Since BS model was suggested, the model has been extensively studied on regular lattices or networks PMB . However, many important bio-systems have been elucidated to form nontrivial networks by the recently developed network theories ABDM . Important examples are metabolic network, cellular network, and protein network JTAOB ; JMBO ; WM ; CGA . Especially the important bio-networks are scale-free networks (SFNs) ABDM , in which the degree distribution satisfies a power law ABDM . Thus it is important to study the BS dynamics on SFNs or to find out how the base structure of interacting biological elements (cells, proteins, or species) affects the evolutionary change or dynamics of the given bio-system. Until now BS models on the nontrivial networks were not investigated extensively. Christensen et al. CDKS have studied BS model on random networks (RNs). Kulkani et al. KAS studied BS model on small-world networks. Slania and Kotrla SK studied the forward avalanches of a sort of extremal dynamics with evolving networks. Moreno and Vazquez MV studied BS model only on a SFN with .

In this letter, we will study BS models on SFNs in complete and comprehensive ways. One of the main purposes of this study is to find which structure of interacting species is the most stable network or most close to mutation-free network under the coevolationary change with interacting species. As is well-known, SFNs with the degree exponent are physically much different from those with ABDM . We study BS models not only on SFNs with but also on SFNs with including random networks (or SFN with ). As we shall see, two important results are found in this study. First, the critical fitness value of BS models for is shown to have the limiting behavior when the number of nodes of the network goes to infinity. In contrast, approaches finite nonzero value as for . Furthermore, on SFNs with finite is shown to satisfy the relation , which is also directly supported by simulation. Second, for the distribution of avalanches is shown to have two power-law regimes. To find the origin of this anomalous behavior of avalanches we also study BS models on an artificially-made star-linked network and find the similar two power-law regimes.

We now explain the model treated in this letter. All the models are defined on a graph , where is the number of nodes and is the number of degrees with the average degree . Initially, a random fitness value is assigned to each node . At each time step, the system is updated by the following two rules: (I) first assign new fitness value to the node with the smallest fitness value . (II) Second assign new fitness values to the nodes which are directly connected to the node with . We use SFNs with the various degree exponent as . To generate SFNs, we use the static model SNUGroup instead of preferential attachment algorithm ABDM .

To understand the dependence of the critical fitness value on , we generate SFNs with . To exclude the effects of finite percolation clusters CDKS and to see the effect of network structure itself, all the networks are made to have average degree . To understand the dependence on number of nodes , the networks with the sizes are generated for each . To determine the critical fitness value , we consider as a function of the total number of updates PMB . Initially, is the gap , where is the maximum of all for PMB . When jumps to a new higher value, there are no nodes in the system with . Thus .

We measure on the various SFNs. Fig. 1 shows the plot of versus for SFNs with various . The values of critical fitness evaluated from data in Fig. 1 are , , , and for , , , and . The results in Fig. 1 mean that for , .

Fig. 2 shows the plot of versus for . For , nicely satisfies the relation, MV . For , ’s seem to follow a power-law and approach to zero as goes to . In contrast to the results in Fig. 1, for .

In the RN, every pair of nodes are randomly connected and the degree distribution is a Poisson distribution ABDM ; CDKS . So the BS model on RN CDKS is a good realization of the mean-field-type random neighbor model. In the random neighbor model, the fitness values of the randomly selected nodes as well as the node with are updated and FSB . The result on RN is very close to CDKS . In the steady state of BS model, the probability measure is . Suppose the case that the number of updates for each step is fixed as , as in the random neighbor model. To sustain the steady state in the case, at most one new fitness value should be less than and the other new values should be larger than FSB . Therefore we can easily see or . , which is expected one from the random neighbor model by setting

On a network the number of updats depends on the degree of the node with and the probability which a node with degree is connected to the node with should be proportional to . For an updating step the probability that a node with degree is updated is proportional to , because the node itself can be the node with . Therefore, after an arbitrary update, the probability of a node with degree being the node with is proportional to . This means that in the steady state should be proportional to , or of the nodes updated for one updating process is therefore . The average number

(1) |

and thus is

(2) |

When the number of updates is fixed as , Eq. (2) reproduces the mean-field result . In SFNs with , Eq. (2) becomes

(3) |

Eq. (3) explains the results in Figs. 1 and 2 including the result for . For , measured is fitted to the relation , where is constant and is for the network with the size . The fitted lines in Fig. 2 show that the relation holds well and directly supports Eq. (3).

An avalanche in Bak-Sneppen model is defined as the sequential step for which the minimal site has a fitness value smaller than given PMB . For each network, we choose to satisfy . The probability distribution of avalanche size on the networks with the size are shown in Fig. 3 and Fig. 4. All the data in Figs. 3 and 4 are taken in the steady-states.

As is shown in Fig. 3, in SFNs with including RN satisfy the normal power-law behavior with an exponential cutoff as . The curves in Fig. 3 represent the fitted curves to data for . From those fittings the obtained values for are 1.5 for RN and , and 1.65 for . The result for RN and SFN with is expected from the random neighbor model FSB . As decreases to 4.0 or so increases to . For , however, the best fitting function is with and we cannot find the cut-off-dependent behavior within our data. Instead, it is even observed that tails of measured data for around seem to deviate from the fitting function and are lager than values estimated from the best fitting function. This rather anomalous tail behavior of for should be the signal of the anomalous behavior of for .

In contrast to the simple power-law behavior for , anomalous behavior for shows up for (Fig. 4). We can see two power-law regimes clearly for in Fig. 4. Initially the avalanche size distribution follows about 1 decade or so. After this short initial power-law regime, the long second power-law regime appears as , where . The measured exponents , are summarized in Table I.

3.0 | 2.09 | 1.59 |

2.75 | 2.22 | 1.47 |

2.4 | 2.27 | 1.32 |

2.15 | 2.30 | 1.20 |

Compared to the behavior of the avalanche size distribution for , this anomalous behavior of is very peculiar. In the steady state, it is expected that the node with (the minimal node) is most frequently found among the last updated nodes FSB and then the minimal node locally performs a random walk. However, there can be longer jumps of any length with a very low probability. If this kind of a jumpy random walk is the motion of the minimal node, then a subnetwork consists of a hub node (center node) and many slave nodes directly linked to the hub should be important to decide the behavior of . Due to the jumpy random walk behavior, the more slave nodes the hub node has, the longer stay of the minimal node or the longer avalanche exists at the given subnetwork. This effect explains the second power-law regime with the exponent in Fig. 4, because diverges for , and so the subnetwork of a hub node and many slave nodes should be the main substructure in SFNs with . Evidently, the jumpy steps of the jumpy random walk make the shorter avalanches possible and this effect explains the first power-law regime with the exponent .

To support the qualitative explanation of the two power-law regimes, we consider an artificially-made star-linked network shown in Fig. 5. In the star-linked network, a main subnetwork consists of a center (star) node and many dangling slave nodes linked directly to the star node. Then the center nodes are linked hierarchically to one after another as sketched in Fig. 5(a). We make a star-linked network in which there are base subnetworks with , , , and slave nodes, respectively. In this network, we perform BS dynamics and find . is also measured on the star-linked network and is shown in Fig. 5(b). We find the very two power-law regimes with the exponents and . The plateau between two power-regimes in the data of in Fig. 5(b) is probably from the discrete distribution of the number of slave nodes.

In conclusion, we study BS models on SFNs with various . For , approaches to a nonzero value in the limit and shows normal power-law behavior with . For , approaches to zero as and has two power-law regimes. The origin of the two power-regimes are explained by the dynamics on a star-linked network.

In Ref. MV , BS dynamics only on a SFN with was studied and the only meaningful numerical result was to show . Ref. MV suggested a relation similar to Eq. (2) from a rate equation which was obtained by a naive and immature analogy of BS dynamics to the epidemic dynamics on SFNs PV . However the rate equation should never be the exact one. Even the exact rate equation for the simple random neighbor model FSB is much more complex than that of Ref. MV or the epidemic dynamics. The correct rate equation for BS dynamics on SFNs must be derived by considering all the terms of the rate equation in Ref. FSB and the base network structure simultaneously and correctly. The derivation of the correct rate equation should be a subject for the future study. In Ref. MV they argued for satisfies a simple power-law with . By the brute-forced fit of the relation to our data in Fig. 4, we also obtain for . However, this blind application of the simple power law should be wrong and there should exist the two-power law regimes even for . One can easily identify the two power-law regimes in the data of Ref. MV rather clearly although the tail parts of their data are qualitatively poor and show large fluctuations.

The occurrence of two power-law regimes for was also found in BS dynamics on small-world networks KAS and in an extremal dynamics with evolving networks SK . However the origins of the two power-law regimes were completely different from ours. The origin in the small-world networks was argued to be the long range connectivity of the networks KAS . The extremal dynamics with evolving random networks SK changes the network structure and is not exactly the same as BS dynamics. Furthermore the evolving network develop many disconnected clusters. In the model SK the forward avalanches are mainly measured. The forward avalanchesSK should be affected by the dynamical aggregate and splitting of subnetworks by the extremal dynamics, which should be the origin of the two power-law regimes. In contrast our avalanches of BS dynamics is measured on a fully-connected static scale-free network and should not be directly comparable to the avalanches on dynamically varying networks.

Authors would like to thank Prof. H. Jeong for valuable suggestions. This work is supported by Korea Research Foundation Grant No. KRF-2004-015-C00185.

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