We study scale free simple graphs with an exponent of the degree distribution γ less than two. Generically one expects such extremely skewed networks -which occur very frequently in systems of virtually or logically connected units -to have different properties than those of scale free networks with γ > 2: The number of links grows faster than the number of nodes and they naturally posses the small world property, because the diameter increases by the logarithm of the size of the network and the clustering coefficient is finite. We discuss a simple prototype model of such networks, inspired by real world phenomena, which exhibits these properties and allows for a detailed analytical investigation. [7]. Indeed in each of these systems nodes -web pages or actors -are linked -by hyperlinks or collaboration in the same movie -to a number k of other nodes, which is called the degree of the node [27], and which obeys a power law distribution P (k) ∼ k −γ . In many cases (table I) the exponent γ of such a distribution is larger than two which its occurrence has been related to some interaction mechanismsuch as preferential attachment [3] -in simplified models.Scale-free networks with an exponent γ < 2 have received less attention, despite of their widespread appearance (table I), in the peer-to-peer Gnutella network [28] [8, 9], outgoing E-mails network [10], traffic in networks [11], co-authorship network in high energy physics [12] and in the network of dependency among software packages [13,14].The aim of this letter is to show that simple graphs with γ < 2 have markedly different properties than simple graphs with γ > 2. We shall do this first on the basis of general arguments and then using a prototype model motivated by the above mentioned real networks. This model reproduces all the discussed generic properties. Furthermore we show that its generalization to a weighted network exhibits non-trivial statistical properties.

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Which properties of a molecule define its odor? This is a basic yet unanswered question regarding the olfactory system. The olfactory system of Drosophila has a repertoire of approximately 60 odorant receptors. Molecules bind to odorant receptors with different affinities and activate them with different efficacies, thus providing a combinatorial code that identifies odorants. We hypothesized that the binding affinity of an odorant-receptor pair is affected by their relative sizes. The maximum affinity can be attained when the molecular volume of an odorant matches the volume of the binding pocket. The affinity drops to zero when the sizes are too different, thus obscuring the effects of other molecular properties. We developed a mathematical formulation of this hypothesis and verified it using Drosophila data. We also predicted the volume and structural flexibility of the binding site of each odorant receptor; these features significantly differ between odorant receptors. The differences in the volumes and structural flexibilities of different odorant receptor binding sites may explain the difference in the scents of similar molecules with different sizes.

We have studied the topology of the energy landscape of a spin-glass model and the effect of frustration on it by looking at the connectivity and disconnectivity graphs of the inherent structure. The connectivity network shows the adjacency of energy minima whereas the disconnectivity network tells us about the heights of the energy barriers. Both graphs are constructed by the exact enumeration of a two-dimensional square lattice of a frustrated spin glass with nearest-neighbor interactions up to the size of 27 spins. The enumeration of the energy-landscape minima as well as the analytical mean-field approximation show that these minima have a Gaussian distribution, and the connectivity graph has a log-Weibull degree distribution of shape κ = 8.22 and scale λ = 4.84.To study the effect of frustration on these results, we introduce an unfrustrated spin-glass model and demonstrate that the degree distribution of its connectivity graph shows a power-law behavior with the −3.46 exponent, which is similar to the behavior of proteins and Lennard-Jones clusters in its power-law form.