The topological relationship between the large-scale attributes and local interaction patterns of complex networks - PubMed (original) (raw)

The topological relationship between the large-scale attributes and local interaction patterns of complex networks

A Vázquez et al. Proc Natl Acad Sci U S A. 2004.

Abstract

Recent evidence indicates that the abundance of recurring elementary interaction patterns in complex networks, often called subgraphs or motifs, carry significant information about their function and overall organization. Yet, the underlying reasons for the variable quantity of different subgraph types, their propensity to form clusters, and their relationship with the networks' global organization remain poorly understood. Here we show that a network's large-scale topological organization and its local subgraph structure mutually define and predict each other, as confirmed by direct measurements in five well studied cellular networks. We also demonstrate the inherent existence of two distinct classes of subgraphs, and show that, in contrast to the low-density type II subgraphs, the highly abundant type I subgraphs cannot exist in isolation but must naturally aggregate into subgraph clusters. The identified topological framework may have important implications for our understanding of the origin and function of subgraphs in all complex networks.

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Figures

Fig. 1.

Fig. 1.

Subgraph phase diagrams. The phase diagrams organize the subgraphs based on the number of nodes (n, horizontal axis) and the number of links (m, vertical axis), each discrete point explicitly depicting the corresponding subgraph. The stepped yellow line corresponds to the predicted phase boundary separating the abundant type I subgraphs (below the line) from the constant density type II subgraphs (above the line). The background color is proportional to the relative subgraph count Cnm = Nnm/Σ_sNns_ of each _n_-node subgraph, the color code being shown in the upper right corner. Note that some (n, m) points in the phase diagram may correspond to several topologically distinguishable subgraphs. For simplicity, we depict only one representative topology in such cases. Because the yellow phase boundary depends on the γ and α exponents of the corresponding network, each phase diagram is slightly different. Yet, there is a visible similarity between the networks of the same kind: The phase diagrams of the two transcription or the two metabolic networks are almost indistinguishable.

Fig. 2.

Fig. 2.

Subgraph distributions in cellular networks. a and b show all nodes in the S. cerevisiae transcription regulatory network that participate in triangle (3,3) and (5,5) subgraphs (depicted in Insets of c and d). The size (area) of each node is drawn proportional to its degree, k, in the full network, indicating that subgraphs tend to aggregate around the hubs. Indeed, although there are hubs that have only a few subgraphs around them, in most cases subgraph aggregation is seen only around highly connected nodes. Note that the (3,3) subgraphs of S. cerevisiae is above the percolation boundary (Fig. 1_e_), and therefore they are broken into small islands. In contrast, the (5,5) subgraph is well below the boundary, forming a fully connected giant component, with no isolated subgraphs, as predicted. c and d show the P(T) distribution of the number of (3,3) and (5,5) subgraphs, respectively, passing by a node, where T denotes the number of subgraphs of a selected kind passing by a given node. The plot indicates that for both subgraphs, P(T) approximates a power law P(T) ∼ _T_–δ. Note the quite extended scaling regimes for some networks; e.g., for the (5,5) subgraph the scaling extends over four to five orders of magnitude. The δ exponents measured and predicted for each network are summarized in Table 2 and the

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Fig. 3.

Fig. 3.

Subgraph aggregation and percolation. The horizontal axis shows the sequence of n = 6 subgraphs, the number of links (m) increasing from left to right. The vertical axis corresponds to the relative size of the largest cluster for the subgraphs shown on the horizontal axis, being determined by the S(6,m)/S(6,5) ratio, where S(6,m) represents the number of nodes participating in (n = 6,m) subgraphs and S(6,5) represents the total number of nodes participating in the first and most abundant subgraph of the n = 6 subgraph family. The square symbols represent the measured value of the S(6,m)/S(6,5) ratio for the S. cerevisiae networks listed in the upper right corner, indicating that the relative size of the subgraph cluster shrinks from close to one to zero as we move from the highly abundant type I subgraphs to the low-density type II subgraphs. The topological consequences of the predicted transition can be seen on the insert network maps, each corresponding, in order, to the four filled symbols. The sequence of maps demonstrates that although the type I subgraphs all aggregate into a giant subgraph cluster, as we move toward the type II subgraphs, the cluster shrinks rapidly in the vicinity of the predicted percolation transition and disappears by either shrinking to close to zero size (see, e.g., the metabolic network) or by breaking into many small islands, which also disappear by further shrinking (see, e.g., the transitional regulatory and protein interaction networks). The continuous line, corresponding to our analytical prediction for the relative cluster size, is in quite good agreement with the measured curve for the relatively large protein interaction and metabolic networks. The particular shape of the curve depends, however, on the functional form we use for C(k). For example, the continuous curves were obtained by using the analytic approximation C(k) = _C_0/[1 + (k/_k_0)α]. In contrast, the agreement for the transcriptional regulatory network can be significantly improved by replacing this fit with the directly measured C(k) (dashed line), reproducing even the sharp drop for the relative density of the least connected cluster (first symbol in Top).

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