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The spectral dimension d-of an infinite graph, defined according to the asymptotic behavior of the Laplacian operator spectral density, seems to be the right generalization of the Euclidean dimension d of lattices to non translationally invariant networks when dealing with dynamical and thermodynamical properties. In fact d-exactly replaces d in most laws where dimensional dependence explicitly appears : the spectrum of harmonic oscillations, the average autocorrelation function of random walks, the critical exponents of the spherical model, the low temperature specific heat, the generalized Mermin-Wagner theorem, the infrared singularities of the Gaussian model and many other. Still, d-would be a rather unsatisfactory generalization of d if it hadn't a second fundamental property : the independence of geometrical details at any finite scale (or geometrical universality). Here we show that d-is invariant under al1 geometrical transformations affecting only finite scale topology. In particular we prove that d-is left unchanged by any quasi-isometry (including coarse-graining and addition of finite range couplings), by local rescaling of couplings and by addition of infinite range of couplings provided they decay faster than a given power law.

In this article, we implement and compare 10 node removal (attack) strategies from the literature over the photosystem I (PSI) complex network of the common pea plant (Pisum sativum), representing the FRET energy transfer among its nodes/chromophores. We measure the network robustness (functioning) with four indicators. The node attack strategies and the network robustness indicators consider both the binary-topological and the weighted structure of the network. First, we find that the well-known node betweenness centrality attack, which has proven highly effective in dismantling most real-world networks’ topological connectivity, is ineffective over the PSI network. Second, the degeneracy of the node properties caused by the PSI’s higher network connectivity level induces a random-like node removal even when nodes are removed according to a specific node centrality measure. This phenomenon triggers a very low decrease of the PSI network functioning even when subjected to node attack. Such an outcome would indicate that the node attack strategies based on classic node properties, such as the degree or the betweenness centrality, may show low efficacy in dismantling real-world networks with very high connectivity levels.Last, the PSI network can be built by tuning a cut-off distance (CD) that defines the viable energy transfers among nodes/chromophores and progressively discards the lower energy transfer links among distant nodes/chromophores. This represents a ‘weight thresholding’ procedure allowing us to investigate the efficacy of the node attack strategies when links of lower weight are progressively pruned from the PSI network. We find that the best node attack strategies change by decreasing the CD, showing that the weight thresholding procedure affects the network response to node removal. This last outcome outlines the importance of investigating the stability of the system response for real-world weighted complex networks subjected to the weight thresholding procedure.

Abstract:: When averages over all starting points are considered, the type problem for the recurrence or transience of a simple random walk on an inhomogeneous network in general differs from the usual “local" type problem. This difference leads to a new classification of inhomogeneous discrete structures in terms of recurrence and transience on the average, describing their large scale topology from a “statistical" point of view. In this paper we analyze this classification and the properties connected to it, showing how the average behavior affects the thermodynamic properties of statistical models on graphs.

In this paper, we model the excitation energy transfer (EET) of photosystem I (PSI) of the common pea plant Pisum sativumas a complex interacting network. The magnitude of the link energy transfer between nodes/chromophores is computed by Forster resonant energy transfer (FRET) using the pairwise physical distances between chromophores from the PDB 5L8R (Protein Data Bank). We measure the global PSI network EET efficiency adopting well-known network theory indicators: the network efficiency (Eff) and the largest connected component (LCC). We also account the number of connected nodes/chromophores to P700 (CN), a new ad hoc measure we introduce here to indicate how many nodes in the network can actually transfer energy to the P700 reaction centre. We find that when progressively removing the weak links of lower EET, the Eff decreases, while the EET paths integrity (LCC and CN) is still preserved. This finding would show that the PSI is a resilient system owning a large window of functioning feasibility and it is completely impaired only when removing most of the network links. From the study of different types of chromophore, we propose different primary functions within the PSI system: chlorophyll a(CLA) molecules are the central nodes in the EET process, while other chromophore types have different primary functions. Furthermore, we perform nodes removal simulations to understand how the nodes/chromophores malfunctioning may affect PSI functioning. We discover that the removal of the CLA triggers the fastest decrease in the Eff, confirming that CAL is the main contributors to the high EET efficiency. Our outcomes open new perspectives of research, such comparing the PSI energy transfer efficiency of different natural and agricultural plant species and investigating the light-harvesting mechanisms of artificial photosynthesis both in plant agriculture and in the field of solar energy applications.