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. 2008 Dec 9;105(49):19235-40.
doi: 10.1073/pnas.0805344105. Epub 2008 Nov 25.

Ordered Cyclic Motifs Contribute to Dynamic Stability in Biological and Engineered Networks

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Free PMC article

Ordered Cyclic Motifs Contribute to Dynamic Stability in Biological and Engineered Networks

Avi Ma'ayan et al. Proc Natl Acad Sci U S A. .
Free PMC article

Abstract

Representation and analysis of complex biological and engineered systems as directed networks is useful for understanding their global structure/function organization. Enrichment of network motifs, which are over-represented subgraphs in real networks, can be used for topological analysis. Because counting network motifs is computationally expensive, only characterization of 3- to 5-node motifs has been previously reported. In this study we used a supercomputer to analyze cyclic motifs made of 3-20 nodes for 6 biological and 3 technological networks. Using tools from statistical physics, we developed a theoretical framework for characterizing the ensemble of cyclic motifs in real networks. We have identified a generic property of real complex networks, antiferromagnetic organization, which is characterized by minimal directional coherence of edges along cyclic subgraphs, such that consecutive links tend to have opposing direction. As a consequence, we find that the lack of directional coherence in cyclic motifs leads to depletion in feedback loops, where the number of nodes affected by feedback loops appears to be at a local minimum compared with surrogate shuffled networks. This topology provides more dynamic stability in large networks.

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Cycle characterization. The spins assigned to clockwise, counterclockwise, and neutral links are 1, −1, and 0, respectively. The directional coherence of the cycle on the right is characterized by having absolute value of magnetization |M| = 3, 2 nodes with link–link correlation ρ = 1 (PT nodes), 2 nodes with ρ = −1 (source and sink nodes), and 2 nodes with ρ = 0 (those with impinging neutral links).
Fig. 2.
Fig. 2.
Directional coherence of cyclical motifs as measured by the magnetization. (A–D) Solid symbols correspond to the real networks and open symbols to the spin model fit. (E and F) The magnetization for the foodweb and signaling networks, the IODP randomization, the TDP randomization, and the fit from the independent-link model from Eq. 1 (E) and its generalization (F).
Fig. 3.
Fig. 3.
Nodes in feedback loops. (A and B) Linear stability conditions for generic cycles without (A) and with (B) feedback loop connectivity. (C) The number of nodes that participate in feedback loops for the 9 networks (purple bars) and their IODP-randomized versions (gray bars). The stems topped with solid triangles indicate the number of nodes in the corresponding network. (Insets) Enlargements of the yeast and E. coli results. For the latter, the only feedback loop found is shown.
Fig. 4.
Fig. 4.
Dynamics of the signaling network. (A and B) Simulations of the dynamics (Eq. 3) endowed to the actual signaling network (A) and the IODP-randomized version of this network (B). au, Arbitrary units. The enhanced instabilities in B are typical, as quantified in C, with the parameter S2 defined as the sum of the variances over all of the dynamic variables for the actual network (black line) and the IODP-randomized (blue line) as a function the interaction strength between nodes, λ. The error bars in C correspond to the standard error in 100 simulations. For λ = 2.5, the distribution of S2 shows a bimodal distribution both for the actual (D) and the IODP-randomized (E) networks.
Fig. 5.
Fig. 5.
Average of the disorder coefficient S2 as a function of Nfb>1, the number of nodes in at least 1 feedback loop (open symbols), and a logistic fit (dotted line, r = 0.93), for the signaling network and its IODP-randomized surrogates. Error bars indicate the standard error around the mean indicated by the open symbols.

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