The process of coevolutionary competitive exclusion: speciation, multifractality and power-laws in correlation

Competitive exclusion, a key principle of ecology, can be generalized to understand many other complex systems. Individuals under surviving pressure tend to be different from others, and correlations among them change correspondingly to the updating of their states. We show with numerical simulation that these aptitudes can contribute to group formation or speciation in social fields. Moreover, they can lead to power-law topological correlations of complex networks. By coupling updating states of nodes with variation of connections in a network, structural properties with power-laws and functions like multifractality, spontaneous ranking and evolutionary branching of node states can emerge out simultaneously from the present self-organized model of coevolutionary process.

Competitive exclusion, a key principle of ecology, can be generalized to understand many other complex systems. Individuals under surviving pressure tend to be different from others, and correlations among them change correspondingly to the updating of their states. We show with numerical simulation that these aptitudes can contribute to group formation or speciation in social fields. Moreover, they can lead to power-law topological correlations of complex networks. By coupling updating states of nodes with variation of connections in a network, structural properties with power-laws and functions like multifractality, spontaneous ranking and evolutionary branching of node states can emerge out simultaneously from the present self-organized model of coevolutionary process. Process of competitive exclusion [1] occurs in some real systems-evolutionary branching of species in ecosystems, citations in scientific research and designation of consumer goods are examples among many others. It is actually a fundamental ingredient governing main property of dynamical behaviors of systems which are often described with complex networks [2] nowadays. However, the contribution of competitive exclusion to the interactional structure of networks and to their functional features is not widely realized up till now. In modeling a system, individuals are represented as nodes and correlations among them are represented as edges of a graph. Scale-free property [3], characterized by power-law degree distribution, has attracted extensive attention since it reflects a general feature of diverse systems such as the Internet, citation networks, protein-protein interaction, and so on [2]. In most previous models, dynamics of networks and dynamics on the networks are separated. The interplay between the formation of topological structure and functions emerge from the network is usually neglected, which is reasonable when the structure is independent of the dynamical states of nodes, or when these two sides vary in rather different speeds. However, in many practical phenomena like academic and art creation, financial transactions, global climate fluctuation and synaptic plasticity of neuron network in the brain [4], both the structure and functions emerge from the identical process, and time-dependent variations of both individual states and local connections * Electronic address: sjxiong@nju.edu.cn of nodes feed back with each other. Therefore, novel models with coevolution mechanisms [5] underlying them appeared to fit for the necessary. Unfortunately, scarcely could one produce both scale-free structure and collective dynamics of nodes simultaneously. On the other side, new nodes are often assumed to know the global information of whole the growing network, which is usually impossible for huge-size systems. In this sense it is needed to set up models based on local interactions to see if structure and functions at system level will emerge from self-organized dynamics [6].
As it is well known, competitive exclusion plays key role in the formation of species. There is strong competition among species occupying the same or nearest loci, surviving pressure force them drift their traits away from the local average level, and gradually induces evolutionary branching of species. Sympatric speciation [7] in an ecosystem is a recent focus of naturalists. It refers to the origin of two or more species from a single local population. Seceder model [8] based on a simple rule of local third-order collision succeeded in mimicking such a process and capturing its similarity to group formation in society. A network version [9] of it has been reported, giving rise to a possible mechanism of community structure and clustering in social networks.
In this paper the principle of competitive exclusion is generalized outside the realm of ecology, the seceder model is modified to describe temporally updated states of nodes and corresponding variation of connections among them together. We show that generic natures of members in diverse systems, i.e., to be different from others under the pressure of competition, and coevolution between updating node states and varying connection among nodes, can lead to simultaneous emergences of evolutionary branching of individual traits, spontaneous ranking and multifractality of node states and, power-law topological structure of correlations in a system. In this way we are able to understand scale-free phenomena and other characteristics in various fields with a novel common mechanism. Such self-organized coevolution models of scale-free network with both structural and functional properties integrated are still few to the best of our knowledge.
We set up the present model through three iteration rules. (1) Network growth starts from a primitive complete graph with m 0 nodes. Each node i on joining the network was assigned an initial state with a random real number w(i) uniformly distributed in (0, 1). At each time step, a new node i ′ is added to the preexisting network. It gives out m edges (m < m 0 ) to old nodes arbitrarily.
(2) At every step, each node i countsw(i) -the average of state values w(j)(j = i) over its nearest linked neighbors, from them it picks up the one whose w(j) makes the maximum distance from averagew(i), i.e. J max (i) corresponds to max|w(j) −w(i)|, then, a randomly selected node j among the nearest neighbors of i is chosen as the offspring of J max (i), called J sed (i). Different from original seceder model [8], it is kept at its own site and, with its state variable updated as w(J sed (i)) = w(J max (i)) + δ, where random number δ ∈ (0, 1) is also uniformly distributed and with positive numerical range for wider applications. Obviously w(i) here can be accounted as a time-dependent non-decreasing fitness [10]. (3) For the new comer node i ′ at every step, together with its 'young' enough fellows (i.e. i ′ − i ≤ ∆I, with ∆I a given integer constant implicating aging effect [11], hereafter we call them I altogether for convenience). Search seceders for all I's neighbors j. When w(J sed (j))/w(I) ≥ h, where h is a given value of threshold, a new edge is added between node J sed (j) and I(Double links and self-loops are forbidden). Meanwhile, an edge linking such node I and its neighbor j is removed if the condition w(j)/w(I) < h or w(I)/w(j) < h is satisfied. Finally, if any node i becomes isolated due to edge-cutting, directly link it to its seceder J sed (i). The threshold description of correlation adopted here is widely used in modeling complex systems [12].
Actually the iteration rules of the model are abstracted from observation to real systems. In art creation and scientific research, people have generic tendency to create new works so that they behave differently from others. Sparkles from collision of opinions with large difference often result in creation. As well known, scholars are often under the pressure of publication. Papers with the same or very similar viewpoint, method and results to existing ones have less chance to get published. Here we see the competition exclusion promotes prosperity of scientific research. Suppose a graduate student just start his academic career by joining the research on a certain topic, usually he has to focus on some papers after extensive searching due to limited time, and often he extends his reading to references of them. Generally speaking, he needs to pay more attention to ones with sharp contrasts against his knowledge background(w(i)), and understand lately published literature (w(jsed(j))) to inspire new ideas for his own paper. But in the reading he may be restrained within the ability of his understanding. Therefore, it is natural to predict a suitable range of threshold ratios within which papers with state values w(Jsed(j)) would be cited (connected). And papers in selective reading based on one's local sight are likely to be cited, forming increased in-degree of those ones. On the opposite, papers(on the node state w(j) ) have small difference (too low ratio of w(j)/w(i)) with w(i) are less cited(the link between node i and j is trimmed). Anyway, a recently updated node state(w(jsed(i))) would be more attractive to a failure(an isolated node). Artists update themselves by continuous creation,therefor the cooccurrence network of musicians serves another example of competitive exclusion. We know that musicians with similar genre are competitors for performance. Managers usually do not intend to arrange opportunity for them to appear on the same stage since audience prefer performance with diversity. It is assumed that whoever created a playlist was using a certain criterion to group artists in them. One does not normally find concerts with a mixture of heavy rock, jazz and piano sonata, therefore a range of thresholds is used to balance the homogeneity and heterogeneity. As the results of coevolution, both citation network [13,14,15] and musician network [16] display the topology of scale-free structure although most foodwebs do not [17]. Suppose a man faces to job crisis, he has to refresh himself to become non-trivial for going out of dilemma. And he may attempt to learn from, even coalesce to a succeeded person by recommendation of a common friend. But whether they can sustain a close relation, it depends on whether they are mutually needed and compensate in a proper measure (e.g. w(i)). In all these cases states of nodes keep varying with time and correlations among them change corresponding to such variations along an optimal gradient. Coevolution of node states and topological connection yields most structural properties of complex networks by self-organization. Numerical simulation reveals out power-law distribution of node degree: p(k) ∼ k −γ , which is illustrated in Fig. 1 a. Without ensemble average on network configurations, it is shown that in the case of h = 3.0 the distribution keep invariant for all values of m, with the slope γ = 2.39. In-degree is counted by a node to its accepted edges from younger ones. The distribution also shows essential power-law as shown in the inset of Fig.1a. The slope of the double-logarithmic line p i (k) ∼ k −β is around β = 2.0, which is in accordance with numerical results of another model [13] and empirical studies [14,15]. In Fig.1b we show the variation of power exponents γ depending on correlation thresholds h. They lay in the range of (2.0, 3.0), which fits well to real complex systems. And the inset of it displays Other parameters are the same as those in Fig.1a.
that essential power-law behavior of in-degree distributions also exist for different thresholds. The calculated Pearson coefficients r [18] which describe degree-degree correlation of the network are shown in Fig.2a. They are positive reflecting statistical feature of social networks. Moreover, they also behave asymptotic power-law decay in the size of the system, i.e. r(N ) ∼ N −α , which is, to our knowledge, a specific feature and first predicted by the present model. It is expected to be verified by empirical data from real complex systems. Fig.2b displays size-dependent decay of clustering coefficients [2]:  Fig.3). However, when we allow a small portion (ten to twenty percent) of cut-off operations not to carry rule 3, scalefree properties can be retrieved promptly (see Fig.3). Moreover, degree-degree correlations restore assortativity corresponding to it. This implies that randomness may play an essential role in the origin of scale-free behaviors since there should be more or less relaxation on deterministic rules in complex systems [19].
Ranking behavior of node states also emerge spontaneously from coevolution. Whole the range of node states is divided into 100 intervals in Fig.4 to show that the values are distributed quite discontinuously. This is drastically different from uniform initial distribution and, is comparable to group formation in original seceder model(see Fig.1 of ref. [8]). Inherited from seceder model, two prominent traits(see Fig.4) at both ends can be regarded as the result of evolutionary branching [7] with the tendency of elimination for mediate genotypes. Here, species in sympatry seem to likely drift their traits away from local average level since the strongest competition exists between similar genotypes [20]. Anyway, to make a scrutiny into applicability of co-evolutionary mechanism to sympatric speciation would be valuable. Applied to citation networks, it means that the long term coevolution gradually eliminate the publishing chance of a paper at middle level, instead, the population of quality tends to be divided and shift approaching both ends. Beyond seceder model [8,9], our numerical results also give support to the assumption of the ranking model [21] of SFN with self-organization mechanism. It is noticeable, scalefree property as a result of coevolution can be obtained without the prerequisite of preferential attachment on the power-law function of prestige ranks of nodes.
The updating process of node states induced by competitive exclusion coupled with topological variation re-sults in collective behavior of nodes, which reflects characteristics of functional aspects apart from structural ones of the network. Based on simulated data of node states w(i) which are put in order of the time sequence as node's participation in the network, we calculate function V (q, d) = l µ l (q, d)lnµ l (q, d) with standard boxcounting technique [22] for different moment q versus x =ln d, where d represents scales of boxes, and µ l is normalized measure of the summation over states in box l. Essential linearity can be seen for at least 4-5 center lines in Fig.5 so that the verified singularity spectrum f (α) of multifractal is shown in its inset. Interestingly, the present work gives another example of long-range correlated gradient-driven growth of a multifractal entity [23] with scale-free network as its inherent skeleton. The multifractality of node states is found to emerge accompanied with scale-free property of the structure and vanishes correspondingly. We have verified the correspondence between two properties in the range of m 0 ∈ [15,50], and ∆I ∈ [5,15]. Therefore, the present model suggests a common mechanism of scale-free structure of social systems together with their multifractality and assortativity as well.
Simultaneous emergences of macroscopic properties on both structural and functional sides also enable us to understand functions in coordination with the Internet, world wide spatial distribution of population [23] with all kinds of transport and communication networks connecting resident sites being complex networks among which some ones are coevolution SFNs, middle latitude climate network [12], citation network [13,14,15], number distribution of species in ecological networks [24], musician networks [16], and diversity maintenance method for evolutionary optimization algorithms [15,25], on a novel platform of coevolution with alterable details. Actually, it relies on the mechanism with another type of preferential attachment of node state correlation but does not explicitly depend on node degree [3,13,26], which distinguishes itself from previous ones. Starting from but outreaching seceder model, we can account generic natures of individuals-to update states to selfadapt the competitive exclusion, and correlations among them change correspondingly-as driving force in selforganization of some evolutionary complex systems characterized by power-law distributions of various topological quantities and specific functions.