Number of loops of size h in growing scale-free networks

The hierarchical structure of scale-free networks has been investigated focusing on the scaling of the number $N_h(t)$ of loops of size h as a function of the system size. In particular we have found the analytic expression for the scaling of $N_h(t)$ in the Barab\'asi-Albert (BA) scale-free network. We have performed numerical simulations on the scaling law for $N_h(t)$ in the BA network and in other growing scale free networks, such as the bosonic network (BN) and the aging nodes (AN) network. We show that in the bosonic network and in the aging node network the phase transitions in the topology of the network are accompained by a change in the scaling of the number of loops with the system size.Comment: 4 pages, 3 figure

Temperature-induced crossovers in the static roughness of a one-dimensional interface

At finite temperature and in presence of disorder, a one-dimensional elastic interface displays different scaling regimes at small and large lengthscales. Using a replica approach and a Gaussian Variational Method (GVM), we explore the consequences of a finite interface width $\xi$ on the small-lengthscale fluctuations. We compute analytically the static roughness $B(r)$ of the interface as a function of the distance $r$ between two points on the interface. We focus on the case of short-range elasticity and random-bond disorder. We show that for a finite width $\xi$ two temperature regimes exist. At low temperature, the expected thermal and random-manifold regimes, respectively for small and large scales, connect via an intermediate `modified' Larkin regime, that we determine. This regime ends at a temperature-independent characteristic `Larkin' length. Above a certain `critical' temperature that we identify, this intermediate regime disappears. The thermal and random-manifold regimes connect at a single crossover lengthscale, that we compute. This is also the expected behavior for zero width. Using a directed polymer description, we also study via a second GVM procedure and generic scaling arguments, a modified toy model that provides further insights on this crossover. We discuss the relevance of the two GVM procedures for the roughness at large lengthscale in those regimes. In particular we analyze the scaling of the temperature-dependent prefactor in the roughness B(r)\sim T^{2 \text{\thorn}} r^{2 \zeta} and its corresponding exponent \text{\thorn}. We briefly discuss the consequences of those results for the quasistatic creep law of a driven interface, in connection with previous experimental and numerical studies

Generalized Smoluchowski equation with correlation between clusters

In this paper we compute new reaction rates of the Smoluchowski equation which takes into account correlations. The new rate K = KMF + KC is the sum of two terms. The first term is the known Smoluchowski rate with the mean-field approximation. The second takes into account a correlation between clusters. For this purpose we introduce the average path of a cluster. We relate the length of this path to the reaction rate of the Smoluchowski equation. We solve the implicit dependence between the average path and the density of clusters. We show that this correlation length is the same for all clusters. Our result depends strongly on the spatial dimension d. The mean-field term KMFi,j = (Di + Dj)(rj + ri)d-2, which vanishes for d = 1 and is valid up to logarithmic correction for d = 2, is the usual rate found with the Smoluchowski model without correlation (where ri is the radius and Di is the diffusion constant of the cluster). We compute a new rate: the correlation rate K_{i,j}^{C} (D_i+D_j)(r_j+r_i)^{d-1}M{\big(\frac{d-1}{d_f}}\big) is valid for d \leq 1(where M(\alpha) = \sum+\infty i=1i\alphaNi is the moment of the density of clusters and df is the fractal dimension of the cluster). The result is valid for a large class of diffusion processes and mass radius relations. This approach confirms some analytical solutions in d 1 found with other methods. We also show Monte Carlo simulations which illustrate some exact new solvable models

Temporal networks of face-to-face human interactions

The ever increasing adoption of mobile technologies and ubiquitous services allows to sense human behavior at unprecedented levels of details and scale. Wearable sensors are opening up a new window on human mobility and proximity at the finest resolution of face-to-face proximity. As a consequence, empirical data describing social and behavioral networks are acquiring a longitudinal dimension that brings forth new challenges for analysis and modeling. Here we review recent work on the representation and analysis of temporal networks of face-to-face human proximity, based on large-scale datasets collected in the context of the SocioPatterns collaboration. We show that the raw behavioral data can be studied at various levels of coarse-graining, which turn out to be complementary to one another, with each level exposing different features of the underlying system. We briefly review a generative model of temporal contact networks that reproduces some statistical observables. Then, we shift our focus from surface statistical features to dynamical processes on empirical temporal networks. We discuss how simple dynamical processes can be used as probes to expose important features of the interaction patterns, such as burstiness and causal constraints. We show that simulating dynamical processes on empirical temporal networks can unveil differences between datasets that would otherwise look statistically similar. Moreover, we argue that, due to the temporal heterogeneity of human dynamics, in order to investigate the temporal properties of spreading processes it may be necessary to abandon the notion of wall-clock time in favour of an intrinsic notion of time for each individual node, defined in terms of its activity level. We conclude highlighting several open research questions raised by the nature of the data at hand.Comment: Chapter of the book "Temporal Networks", Springer, 2013. Series: Understanding Complex Systems. Holme, Petter; Saram\"aki, Jari (Eds.

an analysis of recent research on venture capital networks

Purpose – this paper examines recent trends in venture capital network research. Network analysis is a useful approach for analyzing inter-organizational networks, especially for venture capitalists, which are characterized by plenty of connections. Although important steps ahead have been made, several research questions are still unanswered. Research methodology – this brief review analyses deeply three papers which are representative of the novel scientific literature in this field. Scrutinizing these works, I identify their points of strength and weaknesses, in order to understand how to pave the way for further research. Findings – this paper shows that the study of network weak ties and the role of risk management strategies are promising areas for pushing the frontiers of research on venture capital networks. Research limitations – although less recent papers are not considered in this analysis, the in-depth discussion of the latest research provides interesting insights and advice for scholars willing to do research on this field of studies. Practical implications – extending our knowledge on this topic is crucial for understanding the best strategic decisions venture capitalists should take when operating within an inter-organizational network. Originality/Value – this paper critically analyses steam of literature which is important from both scientific and managerial viewpoints. Furthermore, it poses questions to be addressed by future research

Mechanisms and models of human dynamics (Reply)

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