We show that to explain the growth of the citation network by preferential attachment (PA), one has to accept that individual nodes exhibit heterogeneous fitness values that decay with time. While previous PAbased models assumed either heterogeneity or decay in isolation, we propose a simple analytically treatable model that combines these two factors. Depending on the input assumptions, the resulting degree distribution shows an exponential, log-normal or power-law decay, which makes the model an apt candidate for modeling a wide range of real systems.Over the years, models with preferential attachment (PA) were independently proposed to explain the distribution of the number of species in a genus [1], the power-law distribution of the number of citations received by scientific papers [2], and the number of links pointing to World Wide Web (WWW) pages [3]. A theoretical description of this class of processes and the observation that they generally lead to power-law distributions are due to Simon [4]. Notably, the application of PA to WWW data by Barabási and Albert helped to initiate the lively field of complex networks [5]. Their network model, which stands at the center of attention of this work, was much studied and generalized to include effects such as presence of purely random connections [6], nonlinear dependence on the degree [7], node fitness [8], and others ([9], Chap. 8).Despite its success in providing a common roof for many theoretical models and empirical data sets, preferential attachment is still little developed to take into account the temporal effects of network growth. For example, it predicts a strong relation between a node's age and its degree. While such first-mover advantage [10] plays a fundamental role for the emergence of scale-free topologies in the model, it is a rather unrealistic feature for several real systems (e.g., it is entirely absent in the WWW [11] and significant deviations are found in citation data [10,12]). This motivates us to study a model of a growing network where a broad degree distribution does not result from strong time bias in the system. To this end we assign fitness to each node and assume that this fitness decays with time-we refer it as relevance henceforth. Instead of simply classifying the vertices as active or inactive, as done in [13,14], we use real data to investigate the relevance distribution and decay therein and build a model where decaying and heterogeneous relevance are combined.Models with decaying fitness values (''aging'') were shown to produce narrow degree distributions (except for very slow decay) [15] and widely distributed fitness values were shown to produce extremely broad distributions or even a condensation phenomenon where a single node attracts a macroscopic fraction of all links [16]. We show that when these two effects act together, they produce various classes of behavior, many of which are compatible with structures observed in real data sets.Before specifying a model and attempting to solve it, we turn to data to provide sup...

The aim of this work is to explore the possible types of phenomena that simple macroeconomic Agent-Based models (ABM) can reproduce. We propose a methodology, inspired by statistical physics, that characterizes a model through its "phase diagram" in the space of parameters. Our first motivation is to understand the large macro-economic fluctuations observed in the "Mark I" ABM devised by D. Delli Gatti and collaborators. In this regard, our major finding is the generic existence of a phase transition between a "good economy" where unemployment is low, and a "bad economy" where unemployment is high. We then introduce a simpler framework that allows us to show that this transition is robust against many modifications of the model, and is generically induced by an asymmetry between the rate of hiring and the rate of firing of the firms. The unemployment level remains small until a tipping point, beyond which the economy suddenly collapses. If the parameters are such that the system is close to this transition, any small fluctuation is amplified as the system jumps between the two equilibria. We have explored several natural extensions of the model. One is to introduce a bankruptcy threshold, limiting the firms maximum level of debt-tosales ratio. This leads to a rich phase diagram with, in particular, a region where acute endogenous crises occur, during which the unemployment rate shoots up before the economy can recover. We also introduce simple wage policies. This leads to inflation (in the "good" phase) or deflation (in the "bad" phase), but leaves the overall phase diagram of the model essentially unchanged. We have also explored the effect of simple monetary policies that attempt to contain rising unemployment and defang crises. We end the paper with general comments on the usefulness of ABMs to model macroeconomic phenomena, in particular in view of the time needed to reach a steady state that raises the issue of ergodicity in these models.It is human nature to think wisely and to act absurdly