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J Biol Phys. 2009 May; 35(2): 197–207.
Published online 2009 March 27. doi:  10.1007/s10867-009-9142-3
PMCID: PMC2669123

An evolving model of undirected networks based on microscopic biological interaction systems

Abstract

With protein or gene interaction systems as the background, this paper proposes an evolving model of biological undirected networks, which are consistent with some plausible mechanisms in biology. Through introducing a rule of preferential duplication of a node inversely proportional to the degree of existing nodes and an attribute of the age of the node (the older, the more influence), by which the probability of a node receiving re-wiring links is chosen, the model networks generated in certain parameter conditions could reproduce series of statistic topological characteristics of real biological graphs, including the scale-free feature, small world effect, hierarchical modularity, limited structural robustness, and disassortativity of degree–degree correlation.

Keywords: Biological undirected networks, Preferential duplication, Nodes, Disassortativity

Introduction

Complex networks abstracted from real complex systems, from social networks, technological networks to biological networks, are not absolutely stochastic. Instead, there are some organized ways and rules existing in their structure, which can be described by all kinds of network attributes, including the distribution of degree, the average of clustering coefficient, the average of shortest distance, the degree–degree correlation, etc. [15]. Different networks are endowed with different values to describe these properties. Nevertheless, experimental data demonstrates that a large number of real networks share similar characteristics qualitatively, for example, real-world biological graphs widely follow the power law distribution of degree, relatively high mean clustering coefficients, small mean shortest path length, and negative degree–degree correlation of node pairs [68]. Since real biological networks may exhibit common properties, it is meaningful to build a model to manifest their evolutionary mechanism; thus we would comprehend biological systems more thoroughly when theories or experimental results cannot give good explanations for the time being.

Up to now, we usually take five types of biological networks into account: transcription factor binding, protein–protein interactions, protein phosphorylation, metabolic interactions, and genetic interaction networks. Of those, protein and gene interaction networks are undirected (or called bi-directional) and all kinds of evolving models have been developed to explain the formation of network topology of this sort of biological maps.

The most classic undirected network model is the BA model [9], which starts at a time when growth and preferential attachment become the essential characteristics in network evolution. Based on the BA model, a great number of evolving models were developed to demonstrate the topological characteristics of undirected networks. Some representative models for biological maps include: the duplication–divergence (DD) model presented by Vázquez et al. and modified by Solé et al. [10, 11]; evolving protein interaction networks put forward by Romualdo et al. [12]; the local-world evolving network model proposed by Sun et al. [13], and so on. Although each of these models has its own speciality in emphasizing some particular properties of biological networks, they can not present the overall topology of this kind of biological system, especially the feature of disassortativity (negative correlation between adjacent node degrees), which is specific to real biological graphs.

Recently, the work of Takemoto and Oosawa has raised considerable interest [14]. They proposed a model maintaining statistic details of biological networks with scale-free features, hierarchical modularity, and disassortativity, as a modification of the MM (merging module) model [15]. But the replication of nodes in networks rooted in the MM model takes a module as a unit and appends nodes in a manner corresponding to the biological mechanism of multiple-gene duplication. Since the multiple-gene duplication (and even whole genome duplication) is not universal in gene duplications [16], single-gene duplication should be taken as a more typical scenario.

In addition, each node in the network of Takemoto and Oosawa is assigned to a variable of fitness in a fitness-driven mechanism. The problem is that the kind of characteristic of a unit (e.g. a gene) the fitness stands for in a network is unknown and what concept the fitness corresponds to in biology is not explicit either. Besides, their model is sensitively dependent on the values of parameters in its presentation of disassortativity. Due to the above disadvantages, we propose a more reasonable evolving model of biological undirected networks based upon some experimental studies.

In this paper, the age attribute of a node (gene or protein) is applied to each unit of network which grows with single-gene duplication. Starting from a completely connected network and through the process of node preferential duplication to degree and link re-wiring due to the age of the node, a model which displays the crucial statistic characteristics of biological systems is obtained.

The model

The principle of biological network growth (the adding of nodes) is based on gene duplications. It is the single-gene duplication in our model, or rather, at every time step, a new node, which copies from an existing node with its set of connections, is appended to the network. A careful experimental observation of protein maps suggests that older nodes (components having orthologs in evolutionarily distant organisms) tend to have higher degrees than newer nodes (components having orthologs in evolutionarily close organisms) [17]. And the same idea is reflected in the genetic regulatory network presented by Yoram Louzoun et al. [18]. Data shows that one of the significant differences between the WWW and the genetic regulatory network is whether there is an obvious correlation between the age of node and its degree. In the genetic regulatory network, older nodes will gain more incoming links and thus have higher degrees or not, while the WWW has a very weak correlation between the node’s age and its degree [19]. Consequently, we naturally consider taking the age of node as a typical attribute of biological networks for further research.

To simplify the description, we assign a copy generation index g to all nodes of network during evolution. Any node in the initial network is assumed as the first generation, and g of each is set to 1. For every new node appended to the network, the value of its copy generation is equal to g of the copied node plus 1. In this way, all nodes of network have an indicator describing the relationship among them with their copy originations.

The values of g increase as a natural array in the order of replication. For a single agent, the value of age is inversely proportional to its generation. To emphasize the fact that the change region of generation is greater than that of age in the duplicate process of biological agents, we choose a relatively restricted range of age. Thereby an exponential and reversed association is designed between the value of age and the generation index. Next we evaluate the age of a node i by taking the reciprocal value of the exponential function of its copy generation index gi:

equation M1
1

Then, we can calculate the relative magnitude of the age of all nodes in the network. Obviously, older nodes situated upstream of copy generations retain lower g, but larger age; on the contrary, newer nodes have higher g but smaller age.

As a rule, core components of a network, such as the key nodes of high degree, tend to be conservative for their important function in the network, whereas components at the periphery of the interactions are not [20]. In our opinion, a new node does not randomly select an object to copy, but inclines to choose one of those nodes which play non-central parts in network. Nodes of lower degree, in this point of view, are easy to be copied for their weak effect on the network function. Thus, our model adopts the policy of preferential duplication: the probability of a node to be copied is inversely proportional to its degree.

After the duplication of a node, next is the linkage of a new node in the evolving process of biological networks. We preserved the set of connections copied from the original node initially, but genome-wide studies on available data reveal that most gene copies will lose their function or experience functional divergence after duplication [21]. This phenomenon is reflected in our model by deleting and re-wiring the links of a newly created node. We remove some original links of the new node with a certain probability and then choose nodes without links to it to reconnect according to the values of their age: the probability of a node gaining a new link is directly proportional to its age.

Our evolving model experiences a series of stages at every time step of network development, including single-gene duplication of with inverse preference to degree, stochastic deletion of original links of a new node, and re-wiring preferentially with the age of existing nodes. Then we randomly eliminate a small quantity of connections in the whole network with a certain probability, to control the total number of links and adjust the structure of the network macroscopically.

All in all, the evolution of our biological undirected network proceeds as follows:

  1. Starting from a complete graph of initial size S, at every time step, a node i of network is copied with a probability equation M2 (di is the degree of node i), and the set of connections of i is preserved by the new node.
  2. After deleting a number of original links of the new node with probability a, the new node is reconnected to nodes existing in the network without prior links, with probability equation M3 (yi is the age of node i), until it has the same link number with node i, i.e. the number of re-wirings is the product of probability a and the original link number of node i.
  3. At the end of every time step, all connections in current network are chosen stochastically to be deleted with probability b.

Referring back to Eq. 1, new nodes are added to network successively, until the size of network reaches M.

Results and Discussion

For the complicated form of the analytic expression of our evolving model, which can be described by some differential and difference equations (including variables di and yi and parameters S, M, a, and b), the model is discussed by numerical simulations directly.

Generally, we take the initial size of network as S = 20 and the final size as M = 5,000. According to experimental data [21], normal limits on the two parameters are taken as a  0.5 (the probability of removing new copied links) and b  0.3 (the probability of removing any links of network).

The distribution of degree and the age of the node

The degree distribution of model networks exhibits a power law pattern in various parameter conditions as indicated in Fig. 1, showing that the network has a scale-free feature. And we can easily compute that the power-law exponents of different evolved results are close in number, and converge in the range of (1, 2), in accordance with the characteristics of real biological networks, which demonstrates that nodes of high degree have connections with a majority of other nodes, that is to say, key nodes play important roles in the network. In addition, we can see that the scale-free feature in this model is more sensitive to the value of parameter b than a, and when b increases over 0.2, the scale-free feature is not as significant as before.

Fig. 1
The power law pattern of degree distributions: equation M4, α [set membership] (1, 2). (A) shows networks with different a from 0.5 to 0.8, with b = 0.1; (B) shows networks with different b from 0.1 to 0.3, with a = 0.5. The scale-free ...

Apart from the degree, the age of the node is a particular attribute of our network model. We take the network with a = 0.5, b = 0.1 as an illustration and investigate the distribution of generation index g and the relationship between the age and the degree of the nodes. The results are as follows. The values of g are depicted by a multi-peak distribution and the peak value is about 23 (Fig. 2a). More than half the nodes fall into the range of g [set membership] [20,26], indicating that nodes whose g is around 23 are duplicated most frequently. As a whole, the values of the age interval are proportional to degrees of a node: a minority of nodes with larger age gain relative high degree, while most newly-created nodes come together in node groups with low degree (Fig. 2b).

Fig. 2
The age of node in the network with a = 0.5, b = 0.1. (A) is the multi-peak distribution of generation index and the peak value is about 23; (B) is the degree distribution by different ages, showing that the degree is proportional ...

Small world effect and hierarchical modularity

We calculated the average of clustering coefficient equation M5 and the average of shortest distance equation M6 of model networks with certain parameters, and the results are given in Table 1.

Table 1
Values of the average of clustering coefficient equation M7 and shortest distance equation M8 and a robustness index S(m) of networks with different parameters a and b

From Table 1 we can see that the value of equation M11 is sensitive to parameter b. Compared with stochastic networks of the same size, with b = 0.1, the small values of mean path length are close to stochastic results and the mean clustering coefficient is remarkably larger than the stochastic values, showing that the networks have small world properties hold. By contrast, when b increases over 0.1, the small world effect is of no significance.

The probability parameter b controlling the size of connections gives a limitation to the growth of network links by removing links in an indefinite way. We could comprehend the sensitivity of network features to b from two sides. On the one hand, as the integral parameter of adjusting the structure of network, the effect of its fluctuation is obviously more remarkable than local parameters. On the other hand, it may reveal the impacts of random behavior in the process of evolution in our model. The larger the b, the stronger the randomness of the evolving network, thus the connections of the network are more easily disturbed, and the clustering of the network declines.

Another indicator about the clustering coefficient is C(k), i.e. the average of clustering coefficient of all nodes with k links in the network. As shown in Fig. 3, for the largest groups excluding isolated nodes of networks in our model, we find an approximately inverse correlation between C(k) and k, indicating the model network is provided with a generally hierarchical structure of module growth [22]. We can see in the figures that, however, the inverse correlation is somewhat weakened when the values of parameter a or b increases. This again reflects the influence of the change of model parameters on the topology of the network.

Fig. 3
The hierarchical modularity of networks with different parameters. Figures show the inverse proportion between C(k) and k. a = 0.5, a = 0.6, a = 0.7, a = 0.8 from left to right respectively, ...

Structural robustness

Robustness can be divided into structural robustness, functional and dynamical robustness, etc. Now we only talk about structural robustness, referring to the stability of structure in the presence of changes in the surroundings. For a network, structural robustness can be explained from the viewpoint of topology as the maintenance of connections when attacking network nodes or links, which may be captured by the change of degree responding to various conditions. As a matter of fact, there is delicate difference between stability of state and of structure, which could be separately described by the relativity of degree and the distribution of degree in a network, but this slight diversity is often ignored in computational biology, and we do not distinguish them either. An index S(m) derived from graph theory was induced by Doyle et al. [23] to uncover the structural robustness of networks.

With a graph of n nodes, let di denote the degree of node i. Give the degree sequence of the graph D = {d1, d2, ..., dn} (d1  d2  ...  dn). Let M(D) denote the set of all connected simple graphs having the same D. For a graph m of degree sequence D, the graph-theoretic quantity equation M12 is defined, where E(m) is the set of edges in the graph. Normalizing, let equation M13 when equation M14.

Hence, the structural robustness of networks is obtained in terms of the magnitude of S(m) for different degree distributions. Generally, the lower the S(m), the stronger the robustness of the system. Some networks possess small values of S(m) and are of additional robustness, which exhibits not only high tolerance when deleting general nodes, but robustness to key nodes. However, for some particular modes of scale-free evolution (e.g. the BA model), networks tend to generate only high S(m), whose structure are thought to be RYF: robust to random losses of nodes yet fragile to targeted attacks on the highly connected nodes (called hubs) [24, 25].

Here, the model networks in some parameter conditions possess medium values of S(m) (Table 1) and are thought of as structurally robust to a certain extent. Actually, we examine the robustness corresponding to specific attacks, through observing the change in the size of network and the size of the largest connected group when removing parts of the node of highest degree in networks of different parameters, and find that the structural stability and hub dependence coexist in our model networks (Fig. 4). Thus, we conclude that these networks have the RYF feature as general biological networks do.

Fig. 4
The limited structural robustness of network. We separately attack 50, 100, 150, and 200 nodes of highest degree in model networks with different a and b, and measure changes in the size of network (A) and the size of the largest connected group (B)

The correlation of degree

The correlation of degree in complex networks, namely degree–degree correlation, stands for the tendency of connection between adjacent nodes. When nodes of high degree incline to link other nodes of high degree in the network, positive correlation of degree is given, and the corresponding network property is called assortativity; for the opposite situation, when nodes of high degree incline to link nodes of low degree, negative correlation of degree is given, and it is called disassortativity.

So far, several methods have been proposed to measure degree–degree correlation. The definition of degree–degree correlation is [26]: equation M15 (nn is the nearest neighbor, and equation M16 is the conditional probability that a node with degree k connects to a node with degree k), and the positive or negative correlation is represented by increasing or decreasing knn(k) of k, respectively. We also calculate the ‘neighborhood connectivity’ of nodes [27] or the Pearson correlation of nearest neighbor degrees [28] to quantify the correlation between node pairs.

The assortative coefficient (AC) is adopted to measure the degree–degree correlation in this paper [7], and the assortative coefficient r is expressed as:

equation M17
2

where ki and kj are the degrees of two nodes at the ends of a link and <...> denotes the average over all edges. The range of r is [  1, 1]. When r = 0, degrees are uncorrelated. In the case of equation M18, the degree–degree correlation is positive, i.e., the network shows assortativity, if r > 0; the correlation is negative, i.e., the network shows disassortativity, if r < 0, and the magnitude of r reflects the strength of the correlation.

The assortative coefficient r of networks with different a and b are evaluated by (2) and the results are depicted in Fig. 5. We find that r <  0.4 in various parameter conditions. Moreover, the values of r of our model are always negative numbers, even if the initial and final size (S and M) of network change, which indicates the model network maintains strong and stable disassortativity regardless of parameters.

Fig. 5
The disassortativity of the network. Horizontal axis stands for the simulated times, and vertical axis stands for the values of r (r <  0.4 in above conditions). Different symbols in three figures correspond to ...

Some discussion about biological directed networks

This paper focuses on the model of undirected networks, but the other three types of directed networks in biological systems are important as well. Our research team once proposed an evolving model of gene regulatory networks, in which the in-degree and out-degree distribution can reproduce the structural characteristics of networks from Saccharomyces cerevisiae and Escherichia coli [29]. In the model, new nodes tend to regulate nodes with small number of downstream nodes [30] and be regulated by nodes with large number of downstream nodes. This scheme emphasizes the influence of some nodes on others from the directional aspect in addition to the role of degree in biological directed networks. Therefore, we conclude that it is not sufficient to take degree only as an index to describe the force of a node in the evolution of either undirected or directed networks. Nevertheless, the attribute of node age in undirected networks and the number of downstream nodes in directed networks demonstrate the influence of time dimension and directional aspects on their own networks, which might be examined as a parallel property of nodes in the two kinds of networks.

The presence of a number of downstream nodes in directed networks highlights the ability of certain nodes in regulating the others, and the ability enhances along with the evolution of network. This mechanism may be seen as ‘the Matthew effect’ in biology, or it may represent a kind of self-optimizing property of the organism. The role of node age in undirected networks demonstrates the influence of some old nodes on new ones along with evolution. The degree of a node affects its ability of reproduction, relating to the growth of network size, while the age of a node indicates the temporal correlation among network agents. From this point of view, the behavior of interaction in undirected networks and the behavior of regulation in directed networks share a common evolutionary character: the older the age of a node, or the higher level of a regulator, the more its impact on new nodes as a function of with time. So the evolutionary mechanism of our model, in which the age of the node is a key factor, combined with the evolutionary mechanism of directed networks based on regulatory capability, contribute to a description of the evolutionary behavior of biological networks.

Summary and expectation

Compared with previous network models which often introduce only one node attribute, such as degree, into the network, our biological undirected network takes the influence of nodes in the time dimension into account, which endows the growth of network with the evolving mechanism of ‘trust in old grandmother’: network units of older age are inclined to have more effect on new ones. Consequently, nodes of older age gain more connections with others and gradually become key components of network. After some developing steps, the model networks manifest a series of characteristics that real biological networks possess, including scale-free topology, small world properties and limited structural robustness, hierarchical modularity in certain conditions, and disassortativity independent of parameters. Furthermore, our evolving model, being in agreement with universal principles of gene duplications, is of more biological sense.

To expand on the present work, we could investigate the effect of different network’s evolutionary processes on topological structure and dynamic characteristics, along with the potential relationships among them. As is well known, the dynamics of scale-free networks is stable when the degree–degree correlation is negative; otherwise, it is unstable [31]. Supposing the degree distribution is fixed, the correlation of degree and the phenomenon of synchronization of the network are relevant as well: the synchronization of overall the network is enhanced (hindered) by variable assortative (disassortative) degree–degree correlations [32]. Since both the dynamic stability and the synchronization of the network are concerned with degree–degree correlation, and experimental study shows that social networks usually possess assortative mixing, while natural networks including biological networks are usually disassortative [33], we could explore the following two questions next: first, whether the diversity in their dynamics originates from some underlying factors contained in networks evolutionary mechanisms, and secondly, what substantial elements arouse the difference between the relationships of their dynamic stability or synchronization and the completely different degree–degree correlations of the two types of networks.

Acknowledgements

We thank all our research group members for their suggestions.

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