Degree distribution

In the past decade, a large number of studies emphasized the heterogeneous scale-free degree distribution of metabolic networks Most substrates participate in only a few reactions, whereas a small number of metabolites ( hubs ) participate in a very large number of reactions [19,45,52]. Not surprisingly, the list of highly connected metabolites is headed by the ubiquitous cofactors, such as adenosine triphosphate (ATP), adenosine diphosphate (ADP), and nicotinamide adenine dinucleotide (NAD) in its various forms, as well as by intermediates of glycolysis and the tricarboxylic acid (TCA) cycle. 

Kulicke W-M, Horl HH (1985) Preparation and characterization of a series of poly(acrylamide-co-acrylates), with a copolymer composition between 0-96.3 mol% acrylate units with the same degree distribution of polymerization Colloid Polym Sci 263 530... 

One of the simple but important stochastic characterizations of random networks is the node degree distribution function P(k) which is the probability of a node having k nearest neighbors. 

One of the earliest features that characterizes metabolic compound networks is the scale-free property, which was derived from the finding that the probability that a node can interact with k other nodes, which is the degree distribution P(k) of a... 

The degree distribution P(k) is the probability that a node is linked to k other nodes. The P(k) of random networks exhibits a Poisson distribution, whereas that of scale-free networks approximates a power law of the form m An interesting suggestion is that most cellular networks approximate a scale-free topology" " with an exponent y between 2 and The... 

Where P is the number-average degree of polymerisation (the polymerisation degree distribution numerical function). 

Some interesting features of networks that can be used for network analysis are, for instance, degree of a node and degree distributions. The degree of a node (fe) is the number of edges that are connected to that node. On the other hand, the degree... 

Figure 8.10 Degree distributions for the two networks (A) simpie modei (B) intermediate model (C) integrated model [19].

The degree distribution is one of the important characteristics of this kind of networks because it affects their properties and behavior (R ka Barabasi, 2000). The random graph Gn,p has a binomial degree distribution. The probability pk that a randomly chosen vertex is cormected to exactly k others is (Newman et al., 2001) ... 

The degree distribution can be expressed via the cumulative degree function (Newman, 2002), (Erdos Renyi, 1959) ... 

The degree distribution is a power law distribution P(k) k-° over a part of its range. 

Amaral and al (Amaral et al., 2000) have studied networks whose cumulative degree distribution shape lets appear three kinds of networks. First, scale-free networks whose distribution decays as a power law with an exposant a satisfying bounds seen above. Second, see Fig. 4, broad-scale or truncated scale-free networks whose the degree distribution has a power law regime followed by a sharp cutoff. Third, single-scale networks whose degree distribution decays fast like an exponential. 

Fig. 4. Degree distribution described in (Amaral et al., 2000). The red line follows a power law, as for scale-free networks. The green line corresponds to truncated scale-free networks. The black curve corresponds to single-scale networks.

We compute the cumulative degree distribution for all proteins SSE-IN of studied families. A sample of our results is presented on Figure 6. We can remark that the curves follow a power law distribution and can be ap>proximated by the following power-law function ... 

Fig. 6. Cumulative degree distribution for IRXC from Rossman fold, left, and 1HV4 from TIM beta/alpha-barrel, right.

We observe the same results for all studied proteins. To explain this phenomenon, we have to rely on two facts. First, the mean degree of all proteins SSE-IN is nearly constant (see Table 5). Second, the degree distribution, see Figure 7, follows a Poisson distribution whose peak is reached for a degree near z. These two facts imply that for degree lower than the peak the cumulative degree distribution decreases slowly and after the peak its decrease is fast compared to an exponential one. Consequently, all proteins SSE-IN studied have a similar cumulative degree distribution which can be approximated by a unique power-law function. 

We observe the same result for all studied proteins that is a cumulative degree distribution approximed by the function Pt Here, we discuss about characterisitcs or conditions which involve a such behavior. 

Fig. 10. Mean degree distribution according to protein SSE-IN size. It evolves with values enough close, between 6 and 8.

Fig. 11. Cumulative degree distribution of 1SE9 and 1 AON SSE-IN whose size equal 50 and and 4988. Despite their important size difference, their mean degree stay close and worth respectively 6.6 and 7.5...

To recapitulate, we show that the mean degree values constitute a threshold for protein SSE-IN cumulative degree distribution. For degrees lower than the mean degree it decreases slowly and after this threshold its decrease is fast compared to an exponential one, as shown Fig 8, 9 and 10. 

Consequently, we find a way to approximate all proteins SSE-IN cumulative degree distribution by the function Pjt which can be adjusted. This function describes a power law regime followed by a sharp cut-off which arises for degree values exceeding the mean degree. Proteins SSE-IN are so tnmcated scale-free networks. 

Since the degree distribution depends on the mean degree value, we compare for each node its degree as function of z, see Fig 12, to illustrate how nodes interact and in particular to highlight the weak fraction of highly connected nodes, also called hubs, less than 5 % of the total node number. 

Since the mean degree plays the role of a threshold beyond which the cumulative degree distribution decreases exponentially, it is interesting to study its evolution with the size of the network, see Fig. 10. It appears that the mean degree increases very slightly with the size of the network. Even for networks with size ratio of 100, the mean degree ratio is only 1.15, see Fig. 11. 

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