Guido Caldarelli is currently Associate Professor in the Centre for Statistical Mechanics in the University of Rome "Sapienza" Italy. The institute is part of the National Institute for Condensed Matter currently included in the National Research Council (INFM-CNR). He got his degree in physics in the Department of Physics of the same University in 1992 working with L. Pietronero and A. Vespignani. He then moved to SISSA/ISAS in Trieste where he got the PhD in Statistical Physics in 1996 working on Self-Organised Criticality with A. Maritan. He has been postdoc in the Department of Physics in the University of Manchester with A. McKane and in TCM Group in the University of Cambridge with R. Ball. During his scientific activity in Rome he has also been visiting professor in the École Normale Supérieure in Paris, and in the Department of Physics of the University of Barcelona. In 2010 he will be academic guest at ETH Zuerich. After the studies on fractal growth and self-organised criticality he moved his research on the analysis of scale-free networks. On this topic he published a textbook and he coordinated the European Project COSIN.
Abstract: A variety of different social, natural and technological systems can be described by the same mathematical framework. This holds from Internet to the Food Webs and to the connections between different company boards given by common directors. In all these situations a graph of the elements and their connections displays a universal feature of some few elements with many connections and many with few. This book reports the experimental evidence of these ``Scale-free networks'' and provides to students and researchers a corpus of theoretical results and algorithms to analyse and understand these features. The contents of this book and their exposition makes it a clear textbook for the beginners and a reference book for the experts.
Abstract: In this paper we study the properties of the Barabasi model of queuing under the hypothesis that the number of tasks is steadily growing in time. We map this model exactly onto an invasion percolation dynamics on a Cayley tree. This allows us to recover the correct waiting time distribution P-W(tau) similar to tau(-3/2) at the stationary state (as observed in different realistic data) and also to characterize it as a sequence of causally and geometrically connected bursts of activity. We also find that the approach to stationarity is very slow.
Abstract: For many complex networks present in nature only a single instance, usually of large size, is available. Any measurement made on this single instance cannot be repeated on different realizations. In order to detect significant patterns in a real-world network it is therefore crucial to compare the measured results with a null model counterpart. Here we focus on dense and weighted networks, proposing a suitable null model and studying the behaviour of the degree correlations as measured by the rich-club coefficient. Our method solves an existing problem with the randomization of dense unweighted graphs, and at the same time represents a generalization of the rich-club coefficient to weighted networks which is complementary to other recently proposed ones.
Abstract: We show that the PageRank in a network can be represented as the solution of a differential equation discretized over a directed graph. By exploiting a formal relationship with the time-independent Schrodinger equation it is possible to interpret hub formation and related phenomena as a wave-like localization process in the presence of disorder and trapping potentials. The result opens new perspectives in the physics of networks with interdisciplinary connections and opens the way to the employment of various mathematical techniques to the analysis of self-organization in structured systems. Applications are envisaged in the World-Wide Web, traffic, social and biological networks.
Abstract: Recent years have witnessed the emergence of a new class of social networks, which require us to move beyond previously employed representations of complex graph structures. A notable example is that of the folksonomy, an online process where users collaboratively employ tags to resources to impart structure to an otherwise undifferentiated database. In a recent paper, we proposed a mathematical model that represents these structures as tripartite hypergraphs and defined basic topological quantities of interest. In this paper, we extend our model by defining additional quantities such as edge distributions, vertex similarity and correlations as well as clustering. We then empirically measure these quantities on two real life folksonomies, the popular online photo sharing site Flickr and the bookmarking site CiteULike. We find that these systems share similar qualitative features with the majority of complex networks that have been previously studied. We propose that the quantities and methodology described here can be used as a standard tool in measuring the structure of tagged networks.
Abstract: In the last few years we have witnessed the emergence, primarily in online communities, of new types of social networks that require for their representation more complex graph structures than have been employed in the past. One example is the folksonomy, a tripartite structure of users, resources, and tags-labels collaboratively applied by the users to the resources in order to impart meaningful structure on an otherwise undifferentiated database. Here we propose a mathematical model of such tripartite structures that represents them as random hypergraphs. We show that it is possible to calculate many properties of this model exactly in the limit of large network size and we compare the results against observations of a real folksonomy, that of the online photography website Flickr. We show that in some cases the model matches the properties of the observed network well, while in others there are significant differences, which we find to be attributable to the practice of multiple tagging, i.e., the application by a single user of many tags to one resource or one tag to many resources.
Abstract: We study the properties of the Barabási model of queuing [A.-L. Barabási, Nature (London) 435, 207 (2005); J. G. Oliveira and A.-L. Barabási, Nature (London) 437, 1251 (2005)] in the hypothesis that the number of tasks grows with time steadily. Our analytical approach is based on two ingredients. First we map exactly this model into an invasion percolation dynamics on a Cayley tree. Second we use the theory of biased random walks. In this way we obtain the following results: the stationary-state dynamics is a sequence of causally and geometrically connected bursts of execution activities with scale-invariant size distribution. We recover the correct waiting-time distribution PW(tau) approximately tau(-3/2) at the stationary state (as observed in different realistic data). Finally we describe quantitatively the dynamics out of the stationary state quantifying the power-law slow approach to stability both in single dynamical realization and in average. These results can be generalized to the case of a stochastic increase in the queue length in time with limited fluctuations. As a limit case we recover the situation in which the queue length fluctuates around a constant average value.
Abstract: In this paper we investigate the nature and structure of the relation between imposed classifications and real clustering in a particular case of a scale-free network given by the on-line encyclopedia Wikipedia. We find a statistical similarity in the distributions of community sizes both by using the top-down approach of the categories division present in the archive and in the bottom-up procedure of community detection given by an algorithm based on the spectral properties of the graph. Regardless of the statistically similar behaviour, the two methods provide a rather different division of the articles, thereby signaling that the nature and presence of power laws is a general feature for these systems and cannot be used as a benchmark to evaluate the suitability of a clustering method. Copyright (C) EPLA, 2008.
Abstract: The objective of this paper is to analyse the network topology of the Italian segment of the European overnight money market through methods of statistical mechanics applied to complex networks. We investigate differences in the activities of banks of different sizes and the evolution of their connectivity structure over the maintenance period. The main purpose of the analysis is to establish the potential implications of the current institutional arrangements on the stability of the banking system and to assess the efficiency of the interbank market in terms of absence of speculative and preferential trading relationships. (C) 2007 Elsevier B.V. All rights reserved.
Abstract: Here we provide a detailed analysis, along with some extensions and additonal investigations, of a recently proposed [1] self-organized model for the evolution of complex networks. Vertices of the network are characterized by a fitness variable evolving through an extremal dynamics process, as in the Bak-Sneppen [2] model representing a prototype of Self-Organized Criticality. The network topology is in turn shaped by the fitness variable itself, as in the fitness network model [3]. The system self-organizes to a nontrivial state, characterized by a power-law decay of dynamical and topological quantities above a critical threshold. The interplay between topology and dynamics in the system is the key ingredient leading to an unexpected behaviour of these quantities.
Abstract: We analyze CiteULike, an online collaborative tagging system where users bookmark and annotate scientific papers. Such a system can be naturally represented as a tri-partite graph whose nodes represent papers, users and tags connected by individual tag assignments. The semantics of tags is studied here, in order to uncover the hidden relationships between tags. We find that the clustering coefficient can be used to analyze the semantical patterns among tags.
Abstract: We analyze several florae (collections of plant species populating specific areas) in different geographic and climatic regions. For every list of species we produce a taxonomic classification tree and we consider its statistical properties. We find that regardless of the geographical location, the climate and the environment all species collections have universal statistical properties that we show to be also robust in time. We then compare observed data sets with simulated communities obtained by randomly sampling a large pool of species from all over the world. We find differences in the behavior of the statistical properties of the corresponding taxonomic trees. Our results suggest that it is possible to distinguish quantitatively real species assemblages from random collections and thus demonstrate the existence of correlations between species.
Abstract: In recent work we presented a new approach to the analysis of weighted networks, by providing a straightforward generalization of any network measure defined on unweighted networks. This approach is based on the translation of a weighted network into an ensemble of edges, and is particularly suited to the analysis of fully connected weighted networks. Here we apply our method to several such networks including distance matrices, and show that the clustering coefficient, constructed by using the ensemble approach, provides meaningful insights into the systems studied. In the particular case of two datasets from microarray experiments the clustering coefficient identifies a number of biologically significant genes, outperforming existing identification approaches.
Abstract: We analyze here a particular kind of linguistic network where vertices represent words and edges stand for syntactic relationships between words. The statistical properties of these networks have been recently studied and various features such as the small-world phenomenon and a scale-free distribution of degrees have been found. Our work focuses on four classes of words: verbs, nouns, adverbs and adjectives. Here, we use spectral methods sorting vertices. We show that the ordering clusters words of the same class. For nouns and verbs, the cluster size distribution clearly follows a power-law distribution that cannot be explained by a null hypothesis. Long-range correlations are found between vertices in the ordering provided by the spectral method. The findings support the use of spectral methods for detecting community structure.
Abstract: We introduce an exact probabilistic description for L=2 of the Barabási model for the dynamics of a list of L tasks. This permits us to study the problem out of the stationary state and to solve explicitly the extremal limit case where a critical behavior for the waiting time distribution is observed. This behavior deviates at any finite time from that of the stationary state. We study also the characteristic relaxation time for finite time deviations from stationarity in all cases showing that it diverges in the extremal limit, confirming that these deviations are important at all time.
Abstract: The interplay between topology and dynamics in complex networks is a fundamental but widely unexplored problem. Here, we study this phenomenon on a prototype model in which the network is shaped by a dynamical variable. We couple the dynamics of the Bak-Sneppen evolution model with the rules of the so-called fitness network model for establishing the topology of a network; each vertex is assigned a 'fitness', and the vertex with minimum fitness and its neighbours are updated in each iteration. At the same time, the links between the updated vertices and all other vertices are drawn anew with a fitness-dependent connection probability. We show analytically and numerically that the system self-organizes to a non-trivial state that differs from what is obtained when the two processes are decoupled. A power-law decay of dynamical and topological quantities above a threshold emerges spontaneously, as well as a feedback between different dynamical regimes and the underlying correlation and percolation properties of the network.
Abstract: Using a data set which includes all transactions among banks in the Italian money market, we study their trading strategies and the dependence among them. We use the Fourier method to compute the variance-covariance matrix of trading strategies. Our results indicate that well defined patterns arise. Two main communities of banks, which can be coarsely identified as small and large banks, emerge. (c) 2006 Elsevier B.V. All rights reserved.
Abstract: We present an approach to the analysis of weighted networks, by providing a straightforward generalization of any network measure defined on unweighted networks, such as the average degree of the nearest neighbors, the clustering coefficient, the "betweenness," the distance between two nodes, and the diameter of a network. All these measures are well established for unweighted networks but have hitherto proven difficult to define for weighted networks. Our approach is based on the translation of a weighted network into an ensemble of edges. Further introducing this approach we demonstrate its advantages by applying the clustering coefficient constructed in this way to two real-world weighted networks.
Abstract: The configuration space network (CSN) of a dynamical system is an effective approach to represent the ensemble of configurations sampled during a simulation and their dynamic connectivity. To elucidate the connection between the CSN topology and the underlying free-energy landscape governing the system dynamics and thermodynamics, an analytical solution is provided to explain the heavy tail of the degree distribution, neighbor connectivity, and clustering coefficient. This derivation allows us to understand the universal CSN topology observed in systems ranging from a simple quadratic well to the native state of the beta3s peptide and a two-dimensional lattice heteropolymer. Moreover, CSNs are shown to fall in the general class of complex networks described by the fitness model.
Abstract: We present an empirical analysis of the network formed by the trade relationships between all world countries, or World Trade Web (WTW). Each (directed) link is weighted by the amount of wealth flowing between two countries, and each country is characterized by the value of its Gross Domestic Product (GDP). By analysing a set of year-by-year data covering the time interval 1950-2000, we show that the dynamics of all GDP values and the evolution of the WTW (trade flow and topology) are tightly coupled. The probability that two countries are connected depends on their GDP values, supporting recent theoretical models relating network topology to the presence of a `hidden' variable (or fitness). On the other hand, the topology is shown to determine the GDP values due to the exchange between countries. This leads us to a new framework where the fitness value is a dynamical variable determining, and at the same time depending on, network topology in a continuous feedback.
Abstract: We present an analysis of the statistical properties and growth of the free on-line encyclopedia Wikipedia. By describing topics by vertices and hyperlinks between them as edges, we can represent this encyclopedia as a directed graph. The topological properties of this graph are in close analogy with those of the World Wide Web, despite the very different growth mechanism. In particular, we measure a scale-invariant distribution of the in and out degree and we are able to reproduce these features by means of a simple statistical model. As a major consequence, Wikipedia growth can be described by local rules such as the preferential attachment mechanism, though users, who are responsible of its evolution, can act globally on the network.
Abstract: We use the theory of complex networks in order to quantitatively characterize the formation of communities in a particular financial market. The system is composed by different banks exchanging on a daily basis loans and debts of liquidity. Through topological analysis and by means of a model of network growth we can determine the formation of different group of banks characterized by different business strategy. The model based on Pareto's law makes no use of growth or preferential attachment and it reproduces correctly all the various statistical properties of the system. We believe that this network modeling of the market could be an efficient way to evaluate the impact of different policies in the market of liquidity.
Abstract: We present here a study of the clustering and loops in a graph of the Internet at the autonomous systems level. We show that, even if the whole structure is changing with time, the statistical distributions of loops of order 3, 4, and 5 remain stable during the evolution. Moreover, we will bring evidence that the Internet graphs show characteristic Markovian signatures, since the structure is very well described by two-point correlations between the degrees of the vertices. This indeed proves that the Internet belongs to a class of network in which the two-point correlation is sufficient to describe their whole local (and thus global) structure. Data are also compared to present Internet models.
Abstract: We propose a network description of large market investments, where both stocks and shareholders are represented as vertices connected by weighted links corresponding to shareholdings. In this framework, the in-degree (k(in)) and the sum of incoming link weights (nu) of an investor correspond to the number of assets held (portfolio diversification) and to the invested wealth (portfolio volume), respectively. An empirical analysis of three different real markets reveals that the distributions of both k(in), and nu display power-law tails with exponents gamma and alpha. Moreover, we find that k(in), scales as a power-law function of nu with an exponent beta. Remarkably, despite the values of alpha, beta and gamma differ across the three markets, they are always governed by the scaling relation beta = (1 - alpha)/(1 - gamma). We show that these empirical findings can be reproduced by a recent model relating the emergence of scale-free networks to an underlying Paretian distribution of 'hidden' vertex properties. (c) 2004 Elsevier B.V. All rights reserved.
Abstract: We develop an algorithm to detect community structure in complex networks. The algorithm is based on spectral methods and takes into account weights and link orientation. Since the method detects efficiently clustered nodes in large networks even when these are not sharply partitioned, it turns to be specially suitable for the analysis of social and information networks. We test the algorithm on a large-scale data-set from a psychological experiment of word association. In this case, it proves to be successful both in clustering words, and in uncovering mental association patterns. © 2005 Elsevier B.V. All rights reserved.
Abstract: Networks representing social systems display specific features that put them apart from biological and technological ones. In particular, the number of links attached to a node is positively correlated to that of its nearest neighbours. We develop a model that reproduces this feature, starting from microscopical mechanisms of growth. The statistical properties arising from the simulations are in good agreement with those of the real-world social networks of scientists co-authoring papers in condensed matter physics. Moreover, the model highlights the determinant role of correlations in shaping the network's topology. (C) 2004 Elsevier B.V. All rights reserved.
Abstract: We develop an algorithm to detect community structure in complex networks. The algorithm is based on spectral methods and takes into account weights and links orientations. Since the method detects efficiently clustered nodes in large networks even when these are not sharply partitioned, it turns to be specially suitable to the analysis of social and information networks. We test the algorithm on a large-scale data-set from a psychological experiment of word association. In this case, it proves to be successful both in clustering words, and in uncovering mental association patterns.
Abstract: We present here a brief summary of the various possible applications of network theory in the field of finance. Since we want to characterize different systems by means of simple and universal features, graph theory could represent a rather powerful methodology. In the following we report our activity in three different subfields, namely the board and director networks, the networks formed by prices correlations and the stock ownership networks. In most of the cases these three kind of networks display scale-free properties making them interesting in their own. Nevertheless, we want to stress here that the main utility of this methodology is to provide new measures of the real data sets in order to validate the different models.
Abstract: Erosion by flowing water is one of the major forces shaping the surface of Earth. Studies in the last decade have shown, in particular, that the drainage region of rivers, where water is collected, exhibits scale invariant features characterized by exponents that are the same for rivers around the world. Here we show that from the data obtained by the MOLA altimeter of the Mars Global Surveyor one can perform the same analysis for mountain sides on Mars. We then show that in some regions fluid erosion might have played a role in the present martian landscape.
Abstract: We study the properties of quantities aimed at the characterization of grid-like ordering in complex networks. These quantities are based on the global and local behavior of cycles of order four, which are the minimal structures able to identify rectangular clustering. The analysis of data from real networks reveals the ubiquitous presence of a statistically high level of grid-like ordering that is non-trivially correlated with the local degree properties. These observations provide new insights on the hierarchical structure of complex networks.
Abstract: The following discussion is an edited summary of the public debate started during the conference "Growing Networks and Graphs in Statistical Physics, Finance, Biology and Social Systems" held in Rome in September 2003. Drafts documents were circulated electronically among experts in the field and additions and follow-up to the original discussion have been included. Among the scientists participating to the discussion L. A. N. Amaral, A. Barrat, A. L. Barabasi, G. Caldarelli, P. De Los Rios, A. Erzan, B. Kahng, R. Mantegna, J. F. F. Mendes, R. Pastor-Satorras, A. Vespignani are acknowledged for their contributions and editing.
Abstract: In this work we apply network theory to detect in a quantitative fashion some of the characters of the system composed by companies and their boards of directors. Modelling this as a bipartite graph, we can derive two networks (one for the companies and one for the directors) and apply to them the standard graph analysis instruments. The emerging picture shows an environment where the exchange of information and mutual influences, conveyed by interlocks between boards, is predominant. Such a result should be taken into account when modelling this system. (C) 2004 Elsevier B.V. All rights reserved.
Abstract: We review the recent approach of correlation based networks of financial equities. We investigate portfolio of stocks at different time horizons, financial indices and volatility time series and we show that meaningful economic information can be extracted from noise dressed correlation matrices. We show that the method can be used to falsify widespread market models by directly comparing the topological properties of networks of real and artificial markets.
Abstract: We study a recent model of random networks based on the presence of an intrinsic character of the vertices called fitness. The vertex fitnesses are drawn from a given probability distribution density. The edges between pairs of vertices are drawn according to a linking probability function depending on the fitnesses of the two vertices involved. We study here different choices for the probability distribution densities and the linking functions. We find that, irrespective of the particular choices, the generation of scale-free networks is straightforward. We then derive the general conditions under which scale-free behavior appears. This model could then represent a possible explanation for the ubiquity and robustness of such structures.
Abstract: Many social, technological, and biological interactions involve network relationships whose outcome intimately depends on the structure of the network and on the strengths of the connections. Yet, although much information is now available concerning the structure of many networks, the strengths are more difficult to measure. Here we show that, for one particular social network, notably the e-mail network, a suitable measure of the strength of the connections can be available. We also propose a simple mechanism, based on positive feedback and reciprocity, that can explain the observed behavior and that hints toward specific dynamics of formation and reinforcement of network connections. Network data from contexts different from social sciences indicate that power-law, and generally broad, distributions of the connection strength are ubiquitous, and the proposed mechanism has a wide range of applicability.
Abstract: In this Brief Report we present a version of a network growth model, generalized in order to describe the behavior of social networks. The case of study considered is the preprint archive at cul.arxiv.org. Each node corresponds to a scientist, and a link is present whenever two authors wrote a paper together. This graph is a nice example of degree-assortative network, that is, to say a network where sites with similar degree are connected to each other. The model presented is one of the few able to reproduce such behavior, giving some insight on the microscopic dynamics at the basis of the graph structure.
Abstract: The widespread occurrence of an inverse square relation in the hierarchical distribution of subcommunities within communities (or subspecies within species) has been recently invoked as a signature of hierarchical self-organization within social and ecological systems. Here we show that, whether such systems are self-organized or not, this behavior is the consequence of the treelike classification method. Different treelike classifications (both of real and truly random systems) display a similar statistical behavior when considering the sizes of their sub-branches.
Abstract: We study the acoustic emission produced by micro-cracks using a two-dimensional disordered lattice model of dynamic fracture, which allows to relate the acoustic response to the internal damage of the sample. We find that the distributions of acoustic energy bursts decays as a power law in agreement with experimental observations. The scaling exponents measured in the present dynamic model can related to those obtained in the quasi-static random fuse model.
Abstract: In addition to traditional properties such as the degree distribution P( k), in this work we propose two other useful quantities that can help in characterizing the topology of food webs quantitatively, namely the allometric scaling relations C( A) and the branch size distribution P(A) A which are defined on the spanning tree of the webs. These quantities, whose use has proved relevant in characterizing other different networks appearing in nature (such as river basins, Internet, and vascular systems), are related (in the context of food webs) to the efficiency in the resource transfer and to the stability against species removal. We present the analysis of the data for both real food webs and numerical simulations of a growing network model. Our results allow us to conclude that real food webs display a high degree of both efficiency and stability due to the evolving character of their topology.
Abstract: We present data analysis and modeling of two particular cases of study in the field of growing networks. We analyze World Wide Web data set and authorship collaboration networks in order to check the presence of correlation in the data. The results are reproduced with good agreement through a suitable modification of the standard Albert-Barabási model of network growth. In particular, intrinsic relevance of sites plays a role in determining the future degree of the vertex.
Abstract: We compare the topological properties of the minimal spanning tree obtained from a large group of stocks traded at the New York Stock Exchange during a 12-year trading period with the one obtained from surrogated data simulated by using simple market models. We find that the empirical tree has features of a complex network that cannot be reproduced, even as a first approximation, by a random market model and by the widespread one-factor model.
Abstract: The structure of ecological communities is usually represented by food webs. In these webs, we describe species by means of vertices connected by links representing the predations. We can therefore study different webs by considering the shape (topology) of these networks. Comparing food webs by searching for regularities is of fundamental importance, because universal patterns would reveal common principles underlying the organization of different ecosystems. However, features observed in small food webs are different from those found in large ones. Furthermore, food webs (except in isolated cases) do not share general features with other types of network (including the Internet, the World Wide Web and biological webs). These features are a small-world character and a scale-free (power-law) distribution of the degree (the number of links per vertex). Here we propose to describe food webs as transportation networks by extending to them the concept of allometric scaling (how branching properties change with network size). We then decompose food webs in spanning trees and loop-forming links. We show that, whereas the number of loops varies significantly across real webs, spanning trees are characterized by universal scaling relations.
Abstract: We study here the Bak-Sneppen model, a prototype model for the study of self-organized criticality. In this model several species interact and undergo extinction with a power-law distribution of activity bursts. Species are defined through their "fitness" whose distribution in the system is uniform above a certain threshold. Run time statistics is introduced for the analysis of the dynamics in order to explain the peculiar properties of the model. This approach based on conditional probability theory, takes into account the correlations due to memory effects. In this way, we may compute analytically the value of the fitness threshold with the desired precision. This represents a substantial improvement with respect to the traditional mean field approach.
Abstract: We address the problem of the role of the concept of local rigidity in the family of sandpile systems. We define rigidity as the ratio between the critical energy and the amplitude of the external perturbation and we show, in the framework of the dynamically driven renormalization group, that any finite value of the rigidity in a generalized sandpile model renormalizes to an infinite value at the fixed point, i.e., on a large scale. The fixed-point value of the rigidity allows then for a nonambiguous distinction between sandpilelike systems and diffusive systems. Numerical simulations support our analytical results.
Abstract: A new mechanism leading to scale-free networks is proposed in this Letter. It is shown that, in many cases of interest, the connectivity power-law behavior is neither related to dynamical properties nor to preferential attachment. Assigning a quenched fitness value x(i) to every vertex, and drawing links among vertices with a probability depending on the fitnesses of the two involved sites, gives rise to what we call a good-get-richer mechanism, in which sites with larger fitness are more likely to become hubs (i.e., to be highly connected).
Abstract: We study, both with numerical simulations and theoretical methods, a cellular automata model for surface growth in the presence of a local instability, driven by an external flux of particles. The growing tip is selected with probability proportional to the local curvature. A probability p of developing overhangs through lateral growth is also introduced. For small external fluxes, we nd a fractal regime of growth. The value of p determines the fractal dimension of the aggregate. Furthermore, for each value of p a crossover between two different fractal dimensions is observed. The roughness exponent of the aggregates, instead, does not depend on p (chi similar or equal to 5). A Fixed Scale Transformation (FST) approach is applied to compute theoretically the fractal dimension for one of the branches of the structure.
Abstract: We study the Bak-Sneppen model in the probabilistic framework of the run time statistics (RTS). This model has attracted a large interest for its simplicity being a prototype for the whole class of models showing self-organized criticality. The dynamics is characterized by a self-organization of almost all the species fitnesses above a nontrivial threshold value, and by a lack of spatial and temporal characteristic scales. This results in avalanches of activity power law distributed. In this Letter we use the RTS approach to compute the value of x(c), the value of the avalanche exponent tau, and the asymptotic distribution of minimal fitnesses.
Abstract: In the stable marriage problem two sets of agents must be paired according to mutual preferences, which may happen to conflict. We present two generalizations of its sex-oriented version, aiming to take into account correlations between the preferences of agents and costly information. Their effects are investigated both numerically and analytically. (C) 2001 Elsevier Science B.V. All rights reserved.
Abstract: This paper focuses on the statistical properties of wild-land fires and, in particular, investigates if spread dynamics relates to simple invasion model. The fractal dimension and lacunarity of three fire scars classified from satellite imagery are analysed. Results indicate that the burned clusters behave similarly to percolation clusters on boundaries and look denser in their core. We show that Dynamical Percolation reproduces this behaviour and can help to describe the fire evolution. By mapping fire dynamics onto the percolation models, the strategies for re control might be improved.
Abstract: The stable marriage problem has been introduced in order to describe a complex system where individuals attempt to optimise their own satisfaction, subject to mutually conflicting constraints. Due to the potential large applicability of such model to describe all the situation where different objects has to be matched pairwise, the statistical properties of this model have been extensively studied. In this paper, we present a generalisation of this model, introduced in order to take into account the presence of correlations in the lists and the effects of distance when the players are supposed to be represented by a position in space. (C) 2001 Elsevier Science B.V. All rights reserved.
Abstract: We introduce a simplified protein model where the water degrees of freedom appear explicitly (although in an extremely simplified fashion). Using this model we are able to recover both the warm and the cold protein denaturation within a single framework, while addressing important issues about the structure of model proteins.
Abstract: In this paper we present a model for the growth and evolution of Internet providers. The model reproduces the data observed for the Internet connection as probed by tracing routes from different computers. This problem represents a paramount case of study for growth processes in general, but can also help in the understanding the properties of the Internet. Our main result is that this network can be reproduced by a self-organized interaction between users and providers that can rearrange in time. This model can then be considered as a prototype model for the class of phenomena of aggregation processes in social networks.
Abstract: A cellular model introduced for the evolution of the fluvial landscape is revisited using extensive numerical and scaling analyses. The basic network shapes and their recurrence especially in the aggregation structure are then addressed. The roles of boundary and initial conditions are carefully analyzed as well as the key effect of quenched disorder embedded in random pinning of the landscape surface. It is found that the above features strongly affect the scaling behavior of key morphological quantities. In particular, we conclude that randomly pinned regions (whose structural disorder bears much physical meaning mimicking uneven landscape-forming rainfall events, geological diversity or heterogeneity in surficial properties like vegetation, soil cover or type) play a key role for the robust emergence of aggregation patterns bearing much resemblance to real river networks.
Abstract: We introduce a polymer model where the transition from swollen to compact configurations is due to interactions between the monomers and the solvent. These interactions are the origin of the effective attractive interactions between hydrophobic amino acids in proteins. We find that in the low and high temperature phases polymers are swollen, and there is an intermediate phase where the most favorable configurations are compact. We argue that such a model captures in a single framework both the cold and the warm denaturation experimentally detected for thermosensitive polymers and for proteins.
Abstract: We present the result of numerical simulations for a fracturing process in a three-dimensional solid subjected to a mode-I load in a quasi-static regime. The solid is described using the Born model on an FCC lattice with a starting notch. We obtain a value of the roughness exponent zeta similar or equal to 0.5 in agreement with the value measured in microfracturing experiments. Our result supports the idea that at small length scales the fracturing process can be considered as quasi-static, which is the basic of the possible application of the model of line depinning to the case of fractures. (C) 2000 Published by Elsevier Science B.V. All rights reserved.
Abstract: We study the roughness of fracture surfaces of three-dimensional samples through numerical simulations of a model for quasi-static cracks known as Born Model. We find for the roughness exponent a value zeta similar or equal to 0.5 measured for "small length scales" in microfracturing experiments. Our simulations confirm that at small length scales the fracture can be considered as quasi-static. The isotropy of the roughness exponent on the crack surface is also showed. Finally, considering the crack front, we compute the roughness exponents of longitudinal and transverse fluctuations of the crack line (zeta (parallel to) similar to zeta (perpendicular to) similar to 0.5). They result in agreement with experimental data, and support the possible application of the model of line depinning in the case of long-range interactions.
Abstract: We formulate the angular structure of lacunarity in fractals, in terms of a symmetry reduction of the three point correlation function. This provides a rich probe of universality, and first measurements yield new evidence in support of the equivalence between self-avoiding walks (SAW's) and percolation perimeters in two dimensions. We argue that the lacunarity reveals much of the renormalization group in real space. This is supported by exact calculations for random walks and measured data for percolation clusters and SAW's. Relationships follow between exponents governing inward and outward propagating perturbations. and we also find a very general test for the contribution of long-range interactions.
Abstract: We introduce a simplified protein model where the solvent (water) degrees of freedom appear explicitly (although in an extremely simplified fashion). Using this model we are able to recover the thermodynamic phenomenology of proteins over a wide range of temperatures. In particular we describe both the warm and the cold protein denaturation within a single framework, while addressing important issues about the structure of model proteins.
Abstract: We present a detailed study of a two-dimensional lattice model introduced to describe mud cracking in the limit of extremely thin layers. In this model to each bond in the lattice is assigned a (quenched) random breaking threshold. Fractures proceed by selecting the 'weakest' part of the material (i.e. the smallest value of the threshold). A local damage rule is also implemented, by using two different types of weakening of the neighbouring sites, corresponding to different physical situations. We present the results of numerical simulations on this model. We also derive some analytical results through a probabilistic approach known as run time statistics. In particular, we find that the total time to divide the sample scales with the square power L-2 of the linear size L of the lattice. This result is not straightforward since the percolating cluster has a non-trivial fractal dimension. Furthermore, we present here a formula for the mean weakening of the whole sample during the evolution.
Abstract: We study the properties of the "rigid Laplacian" operator; that is we consider solutions of the Laplacian equation in the presence of fixed truncation errors. The dynamics of convergence to the correct analytical solution displays the presence of a metastable set of numerical solutions, whose presence can be related to granularity. We provide some scaling analysis in order to determine the value of the exponents characterizing the process. We believe that this prototype model is also suitable to provide an explanation of the widespread presence of power law in a social and economic system where information and decision diffuse, with errors and delay from agent to agent.
Abstract: In this paper we show that the Internet web, from a user's perspective, manifests robust scaling properties of the type P(n) proportional to n(-tau), where n is the size of the basin connected to a given point, P represents the density of probability of finding n points downhill and tau = 1.9 +/- 0.1 s a characteristic universal exponent. This scale-free structure is a result of the spontaneous growth of the web, but is not necessarily the optimal one for efficient transport. We introduce an appropriate figure of merit and suggest that a planning of few big links, acting as information highways, may noticeably increase the efficiency of the net without affecting its robustness.
Abstract: In this paper we present a theoretical approach that allows us to describe the transition between critical and noncritical behavior when stocastic noise is introduced in extremal models with disorder. Namely, we show that the introduction of thermal noise in invasion percolation (IP) brings the system outside;the critical point. This result suggests a possible definition of self-organized criticality systems as ordinary critical systems where the critical. point corresponds to set to 0 one of the parameters. We recover both the TP and Eden models for T-->0 and T-->infinity, respectively. For small T we fmd a dynamical second-order transition with correlation length diverging when T-->0.
Abstract: It has been recently noticed that heterogeneous media undergoing a fracturing process display a set of properties characteristic of systems at the critical state. in the present work we focus on the way in which the critical regime is reached. It is possible to define a branching ratio, for the breaking processses in the material, that represents the probability to trigger future breakdowns given an initial failure, This probability takes the value 1 when the system is critical thereby representing a measure of the distance of the system from the critical state, We show that, although the models considered in literature become really critical only in correspondence of the global failure, different dynamical rules may drive the system close to the critical state at different rates, such that the duration of the "quasi-critical" stage largely varies from model to model. (C) 1999 Elsevier Science B.V. All rights reserved.
Abstract: We introduce a model for the dynamics of mud cracking in the limit of of extremely thin layers. In this model the growth of fracture proceeds by selecting the part of the material with the smallest (quenched) breaking threshold. In addition, weakening affects the area of the sample neighbour to the crack. Due to the simplicity of the model, it is possible to derive some analytical results. In particular, we find that the total time to break down the sample grows with the dimension L of the lattice as L-2 even though the percolating cluster has a non-trivial fractal dimension. Furthermore, we obtain a formula for the mean weakening with time of the whole sample.
Abstract: In this paper we investigate the influence of different boundary conditions on the final breakdown of a lattice model for the fracture of heterogeneous media. Experimental evidence shows that disordered media subject to stress display some features that are characteristic of critical systems, therefore suggesting an interpretation of the global breakdown of the system as a kind of critical transition. Many of the observed features are well reproduced at least at a qualitative level by lattice models; however, mechanisms at the base of the onset of criticality are not well understood. Besides disorder, there are many parameters that seem to influence the critical properties of the system. The system size and the boundary conditions are among these. We find that the statistical properties of the final breakdown are strongly influenced by the boundary condition. In particular constant-stress relaxation leads to a final breakdown always involving the breaking of a finite number of bonds, which is also large if compared with the number of bonds broken during the formation of each localized crack preceding the final breakdown. When the lattice undergoes constant-strain relaxation instead, the breakdown may involve a vanishingly small number of bond-breaking events.
Abstract: We study the stationary state of a Poisson problem for a system of N perfectly conducting metal balls driven by electric forces to move within a medium of very low electrical conductivity onto which charges are sprayed from outside. When grounded at a confining boundary, the system of metal balls is experimentally known to self-organize into stable fractal aggregates. We simulate the dynamical conditions leading to the formation of such aggregated patterns and analyse the fractal properties. From our results and those obtained for steady-state systems that obey minimum total energy dissipation land potential energy of the system as a whole), we suggest a possible dynamical rule for the emergence of scale-free structures in nature.
Abstract: A mean field theory is discussed for a sandpile model, a cellular automaton prototype of systems showing self-organized criticality. The previous formulation of the mean field does not take into account the dissipation effects that take place on boundaries. This gives rise to some inconsistencies that are eliminated by carefully considering the boundaries effects, as it is shown in this paper. We present here a revised version of the MF equations. The main result is that criticality arises in the thermodynamic limit for sandpile systems, confirming numerical observations on the behavior of the order parameter. The mean field approach is also generalized by applying it to the more general case of sandpiles in thermal equilibrium where a temperature-like parameter T is introduced. In this case we show that criticality is not destroyed at T > 0. (C) 1998 Published by Elsevier Science B.V. All rights reserved.
Abstract: We introduce a model for fractures in quenched disordered media. This model has a deterministic extremal dynamics, driven by the energy function of a network of springs (Born Hamiltonian). The breakdown is the result of the cooperation between the external field and the quenched disorder. This model can be considered as describing the low-temperature Limit for crack propagation in solids. To describe the memory effects in this dynamics and then to study the resistance properties of the system we realized some numerical simulations of the model. The model exhibits interesting geometric and dynamical properties, with a strong reduction of the fractal dimension of the clusters and of their backbone, with respect to the case in which thermal fluctuations dominate. This result can be explained by a recently introduced theoretical tool as a screening enhancement due to memory effects induced by the quenched disorder.
Abstract: We introduce the Webworld model, which links together the ecological modelling of food web structure with the evolutionary modelling of speciation and extinction events. The model describes dynamics of ecological communities on an evolutionary time-scale. Species are defined as sets of characteristic features, and these features are used to determine interaction scores between species. A simple rule is used to transfer resources from the external environment through the food web to each of the species, and to determine mean population sizes. A time step in the model represents a speciation event. A new species is added with features similar to those of one of the existing species and a new food web structure is then calculated. The new species may (i) add stably to the web, (ii) become extinct immediately because it is poorly adapted, or (iii) cause one or more other species to become extinct due to competition for resources. We measure various properties of the model webs and compare these with data on real food webs. These properties include the proportions of basal, intermediate and top species, the number of links per species and the number of trophic levels. We also study the evolutionary dynamics of the model ecosystem by following the fluctuations in the total number of species in the web. Extinction avalanches occur when novel organisms arise which are significantly better adapted than existing ones. We discuss these results in relation to the observed extinction events in the fossil record, and to the theory of self-organized criticality. (C) 1998 Academic Press
Abstract: We study the boundary effects in invasion percolation (IP) with and without trapping. We And that the presence of boundaries introduces a new set of surface critical exponents, as in the case of standard percolation. Numerical simulations show a fractal dimension, for the region of the percolating cluster near the boundary, remarkably different from the bulk one. In fact, on the surface we find a value of D-sur = 1.65 +/- 0.02 (for IP with trapping D-tr(sur) = 1.59 +/- 0.03), compared with the bulk value of D-bul = 1.88 +/- 0.02 (D-tr(bul) = 1.85 +/- 0.02). We find a logarithmic crossover from surface to bulk fractal properties, as one would expect from the finite-size theory of critical systems. The distribution of the quenched variables on the growing interface near the boundary self-organizes into an asymptotic shape characterized by a discontinuity ata value x(c) = 0.5, which coincides with the bulk critical threshold. The exponent tau(sur) of the boundary avalanche distribution for IP without trapping is tau(sur) = 1.56 +/- 0.05; this value is very near to the bulk one. Then we conclude that only the geometrical properties (fractal dimension) of the model are affected by the presence of a boundary, while other statistical and dynamical properties are unchanged. Furthermore, we are able to present a theoretical computation of the relevant critical exponents near the boundary. This analysis combines two recently introduced theoretical tools, the fixed scale transformation and the run time statistics, which are particularly suited for the study of irreversible self-organized growth models with quenched disorder. Our theoretical results are in rather good agreement with numerical data.
Abstract: A simple scheme for the evolution of a fluvial landscape in heterogeneous environments is critically examined to capture the essential mechanism responsible for the recurrent scale-free landforms in the river basin. It is shown that, regardless of boundary and initial conditions, geomorphological constraints in the form of quenched randomly pinned regions play a key role in the robust emergence of aggregation patterns with a scaling behavior in agreement with that of real river basins.
Abstract: A prototype model of stock market is introduced and studied numerically. In this self-organized system, we consider only the interaction among traders without external influences. Agents trade according to their own strategy, to accumulate their assets by speculating on the price's fluctuations which are produced by themselves. The model reproduced rather realistic price histories whose statistical properties are also similar to those observed in real markets.
Abstract: We study the scaling behavior in currency exchange rates. Our results suggest that they satisfy scaling with an exponent close to 0.5, but that it differs qualitatively from that of a simple random walk. Indeed price variations cannot be considered as independent variables and subtle correlations are present. Furthermore, we introduce a novel statistical analysis for economic data which makes the physical properties of a signal more evident and eliminates the systematic effects of time periodicity.
Abstract: Boundary effects for invasion percolation are introduced and discussed here. The presence of boundaries determines a set of critical exponents characteristic of the boundary. In this paper we present numerical simulations showing a remarkably different fractal dimension for the region of the percolating cluster near the boundary. In fact, near the surface we find a value of D-sar= 1.67 +/- 0.03, with respect to the bulk value of D-bul = 1.87 +/- 0.01. Furthermore, we are able to present a theoretical computation of the fractal dimension near the boundary in fairly good agreement with numerical data.
Abstract: An efficient transfer matrix technique is introduced to study directed optimal paths in two and three dimensions. The roughness exponent zeta is 0.6325 +/- 0.0007 for the two-dimensional case and zeta = 0.555 +/- 0.008 for the three-dimensional one, in agreement with the recent conjecture zeta = v(perpendicular to)/v(parallel to), where v(perpendicular to) and v(parallel to) are the correlation length exponents of directed percolation. Exactly solvable examples are also analyzed.
Abstract: A temperature-like parameter is introduced in ordinary sandpiles models. A temperature-dependent probability distribution is assigned for the sand toppling on sites of any height. In mean-field theory criticality is obtained for all the values of temperature and no characteristic avalanche size appears. Numerical simulations support the existence of criticality at any temperature with apparently continuously varying critical exponents.
Abstract: We show that a vectorial model for inhomogeneous elastic media self-organizes under external stress. An onset of crack avalanches of every duration and length scale compatible with the lattice size is observed. The behavior is driven by the introduction of annealed disorder, i.e., by lowering the breaking threshold in the neighborhood of a bond broken by the stress, with a process similar to self-organized criticality. A further comparison with experimental results of acoustic emission (AE), shows that the stability of the elastic potential energy of the system in the AE regime is a sufficient condition for reproducing the algebraic distribution of the energy released during cracks formation.
Abstract: In a critically self-organized model of punctuated equilibrium, boundaries determine peculiar scaling of the size distribution of evolutionary avalanches. This is derived by an inhomogeneous generalization of standard branching processes, extending previous mean field descriptions and yielding nu = 1/2 together with tau' = 7/4, as distribution exponent of avalanches starting from species at the ends of a food chain. For the nearest neighbor chain one obtains numerically tau' = 1.25 +/- 0.01, and tau(first) = 1.35 +/- 0.01 for the first return times of activity, again distinct from bulk exponents.
Abstract: We use a stochastic description of models with a dynamic in quenched disorder to analyze the mechanism of their self-organization to a critical state in terms of memory effects. We introduce a framework to characterize both memory effects and avalanche events which suggests that self-organization can result in general from memory. This issue is settled by the introduction and the analysis of a model that contains explicitly memory and generalizes the corresponding dynamics in quenched disorder. The model displays a rich behavior and self-organized critical properties for a whole range of the exponent that tunes the strength of memory.
Abstract: At dissipative boundaries, models of self-organized criticality show peculiar scalings, different from the bulk ones, in the distributions characterizing avalanches. For Abelian models with Dirichlet boundary conditions, evidence of this is obtained by a mean field approach to semi-infinite sandpiles, and by numerical simulations in two and three dimensions. On the other hand, within the mean field description, closed Neumann conditions restore bulk scaling exponents also at the border. Numerical results are consistent with this property also at finite d.
Abstract: We use the Fixed Scale Transformation theoretical approach to study the problem of fractal growth in fractures generated by using the Born Model. In this case the application of the method is more complex because of the vectorial nature of the model considered. In particular, one needs a careful choice of the lattice path integral for the fracture evolution and the identification of the appropriate way to take effectively into account screening effects. The good agreement of our results with computer simulations shows the validity and flexibility of the FST method in the study of fractal patterns evolution.
Abstract: We use the Fixed Scale Transformation (FST) approach to study the problem of fractal growth in fracture patterns generated by using the Born Model, The application of the method to this model is very complex because of the vectorial nature of the system considered. In particular, the implementation of this scheme requires a careful choice of the fracture path and the identification of the appropriate way to take into account screening effects, The good agreements of our results with computer simulations shows the validity and flexibility of the FST method which represents a general theoretical approach for the study of fractal patterns evolution.
Abstract: We use the Born model for the energy of elastic networks to simulate ''directed'' fracture growth. We define directed fractures as crack patterns showing a preferential evolution direction imposed by the type of stress and boundary conditions applied. This type of fracture allows a more realistic description of some kinds of experimental cracks and presents several advantages in order to distinguish between the various growth regimes. By choosing this growth geometry it is also possible to use without ambiguity the box-counting method to obtain the fractal dimension for different subsets of the patterns and for a wide range of the internal parameters of the model. We find a continuous dependence of the fractal dimension of the whole patterns and of their backbones on the ratio between the central- and noncentral-force contributions. For the chemical distance we find a one-dimensional behavior independent of the relevant parameters, which seems to be a common feature for fractal growth processes.
