Dynamical Processes on Complex Networks . PDF; Export citation. Contents. pp vii-x 4 - Introduction to dynamical processes: theory and simulation. pp S. Boccaletti et al. Physics Reports , (). • Dynamical processes on complex networks. A. Barrat, M. Barthélemy, A. Vespignani. Cambridge Univ. To cite this article: THOMAS U. GRUND (): Dynamical Processes on Complex Networks (4th ed.) by A. Barrat, M. Barthélemy, & A. Vespignani, The Journal of Mathematical Sociology, , At the core of this development stands the analysis of complex networks and.
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Dynamical Processes on Complex Networks. Lecture 1: • Introduction to Networks: o Applications, examples of dynamical processes on networks. download Dynamical Processes on Complex Networks on medical-site.info ✓ FREE SHIPPING on qualified orders. ➢Dynamical Processes on Complex Networks. Alain Barrat, Marc Barthélemy, Alessandro Vespignani. Cambridge University Press (). ➢ Complex networks.
We show that for a system of units connected by a network of interaction potentials with an arbitrary degree distribution, highly connected units have less impact on the system dynamics when compared with intermediately connected units. In an equilibrium setting, the hubs are often found to dictate the long-term behaviour.
However, we find both analytically and experimentally that the instantaneous states of these units have a short-lasting effect on the state trajectory of the entire system. We present qualitative evidence of this phenomenon from empirical findings about a social network of product recommendations, a protein—protein interaction network and a neural network, suggesting that it might indeed be a widespread property in nature.
Keywords: complex networks, out-of-equilibrium dynamics, information theory, word-of-mouth marketing, gene regulatory networks, neural networks 1. Introduction Many non-equilibrium systems consist of dynamical units that interact through a network to produce complex behaviour as a whole. This assumption is also known as the local thermodynamic equilibrium LTE , originally formulated to describe radiative transfer inside stars [ 1 , 2 ].
Examples of systems of coupled units that have been described in this manner include brain networks [ 3 — 6 ], cellular regulatory networks [ 7 — 11 ], immune networks [ 12 , 13 ], social interaction networks [ 14 — 20 ] and financial trading markets [ 15 , 21 , 22 ].
A state change of one unit may subsequently cause a neighbour unit to change its state, which may, in turn, cause other units to change, and so on. The core problem of understanding the system's behaviour is that the topology of interactions mixes cause and effect of units in a complex manner, making it hard to tell which units drive the system dynamics. The main goal of complex systems research is to understand how the dynamics of individual units combine to produce the behaviour of the system as a whole.
A common method to dissect the collective behaviour into its individual components is to remove a unit and observe the effect [ 23 — 32 ]. In this manner, it has been shown, for instance, that highly connected units or hubs are crucial for the structural integrity of many real-world systems [ 28 ], i. On the other hand, Tanaka et al. Less attention has been paid to study the interplay of the unit dynamics and network topology, from which the system's behaviour emerges, in a non-perturbative and unified manner.
We introduce an information-theoretical approach to quantify to what extent the system's state is actually a representation of an instantaneous state of an individual unit. If a system state St can be in state i with probability pi, then its Shannon entropy is 1.
The more bits of a system's state St are determined by a prior state of a unit si at time t0, the more the system state depends on that unit's state.
This quantity can be measured using the mutual information between and St, defined as 1. This mutual information integrated over time t is a generic measure of the extent that the system state trajectory is dictated by a unit. We consider large static networks of identical units whose dynamics can be described by the Gibbs measure.
The Gibbs measure describes how a unit changes its state subject to the combined potential of its interacting neighbours, in case the LTE is appropriate and using the maximum-entropy principle [ 34 , 35 ] to avoid assuming any additional structure. In fact, in our LTE description, each unit may even be a subsystem in its own right in a multi-scale setting, such as a cell in a tissue or a person in a social network.
In this viewpoint, each unit can actually be in a large number of unobservable microstates which translate many-to-one to the observable macrostates of the unit. We consider that at a small timescale, each unit probabilistically chooses its next state depending on the current states of its neighbours, termed discrete-time Markov networks [ 36 ]. Furthermore, we consider random interaction networks with a given degree distribution p k , which denotes the probability that a randomly selected unit has k interactions with other units, and which have a maximum degree kmax that grows less than linear in the network size N.
Self-loops are not allowed. No additional topological features are imposed, such as degree—degree correlations or community structures. We show analytically that for this class of systems, the impact of a unit's state on the short-term behaviour of the whole system is a decreasing function of the degree k of the unit for sufficiently high k. That is, it takes a relatively short time-period for the information about the instantaneous state of such a high-degree unit to be no longer present in the information stored by the system.
A corollary of this finding is that if one would observe the system's state trajectory for a short amount of time, then the out-of-equilibrium behaviour of the system cannot be explained by the behaviour of the hubs. In this paper we resolve the baseline SSCN deficiencies with some simple adjustments.
Connections to spatially closest neighbors are added to mimic the clustering effect in real social networks, and relocations turn out to be crucial in reducing the average path length between nodes. This demonstrates the suitability of the model as a substrate for simulations of other dynamic social processes that depend on both the contacts and the geographical locations of the agents. Materials and Methods 2. The model produces a spatially embedded network where is a set of nodes and is a set of undirected edges.
The spatial embedding of the network is encoded in the set of coordinates , where for 2D spatial networks such as in our case. Consider first the creation of nodes and edges in the baseline model, defining and. Nodes are added to the network one at a time, each contributing to a new edge, according to a variant of the Krapivsky-Redner KR model [ 31 — 33 ] with a single parameter —the redirection probability. Approximations and computational techniques are discussed as well.
Chapter 4 provides an introduction to the basic theoretical concepts and tools needed for the analysis of dynamical processes taking place on networks. All these introductory chapters will allow researchers not familiar with mathematical and computational modeling to get acquainted with the approaches and techniques used in the book.
In Chapter 5, we report the analysis of equilibrium processes and we review the behaviors of basic equilibrium physical models such as the Ising model in complex networks. Chapter 6 addresses the analysis of damage and attack processes in complex networks by mapping those processes into percolation phase transitions. Chapter 7 is devoted to synchronization in coupled systems with complex connectivity patterns. This is a transition chapter in that some of the phenomena analyzed and their respective models can be considered in the equilibrium processes domains while others fall into the non-equilibrium class.
The following four chapters of the book are devoted to non-equilibrium processes. One chapter considers search, navigation, and exploration processes of complex networks; the three other chapters concern epidemic spreading, the emergence of social behavior, and traffic avalanche and congestion, respectively. These chapters, therefore, are considering far from equilibrium systems with absorbing states and driven-dissipative xiv Preface dynamics.
Finally, Chapter 12 is devoted to a discussion of the recent application of network science to biological systems. A final set of convenient appendices are used to detail very technical calculations or discussions in order to make the reading of the main chapters more accessible to non-expert readers.
The postface occupies a special place at the end of the book as it expresses our view on the value of complex network science.
We hope that our work will result in a first coherent and comprehensive exposition of the vast research activity concerning dynamical processes in complex networks. The large number of references and research focus areas that find room in this book make us believe that the present volume will be a convenient entry reference to all researchers and students who consider working in this exciting area of interdisciplinary research.
Although this book reviews or mentions more than scientific works, it is impossible to discuss in detail all relevant contributions to the field. We have therefore used our author privilege and made choices based on our perception of what is more relevant to the focus and the structure of the present book.
This does not imply that work not reported or cited here is less valuable, and we apologize in advance to all the colleagues who feel that their contributions have been overlooked. Acknowledgements There are many people we must thank for their help in the preparation and completion of this book. First of all, we wish to thank all the colleagues who helped us shape our view and knowledge of complex networks through invaluable scientific collaborations: Alvarez-Hamelin, F.
Alvarez, L. Amaral, D. Balcan, A. Baronchelli, M. Caldarelli, C. Caretta Cartozo, C. Castellano, F. Cecconi, A. Chessa, E. Chow, V. Colizza, P. De Montis, L. Eliassi-Rad, A. Flammini, S. Fortunato, F. Gargiulo, D. Garlaschelli, A. Gautreau, B. Gondran, E.
Guichard, S. Havlin, H.
Hu, C. Herrmann, W. Ke, E. Kolaczyk, B. Kozma, M. Leone, V. Loreto, F. Menczer, Y. Moreno, M. Nekovee, A. Maritan, M. Marsili, A. Maguitman, M. Meiss, S. Mossa, S. Myers, C. Nardini, M. Nekovee, D. Parisi, R. Pastor-Satorras, R. Percacci, P. Provero, F. Radicchi, J. Ramasco, C. Ricotta, A. Scala, M. Angeles Serrano, H. Stanley, A. Valleron, A. Vergassola, F. Viger, D. Vilone, M. Weigt, R. Zecchina, and C. We also acknowledge with pleasure the many discussions with, encouragements, and useful criticisms from: Albert, E.
Almaas, L. Adamic, A. Arenas, A. Banavar, C. Cattuto, kc claffy, P. De Los Rios, S. Dorogovtsev, A. Erzan, R. Goldstone, S. Havlin, C. Hidalgo, D. Krioukov, H. Orland, N. Martinez, R. May, J. Mendes, A. Motter, M. Mungan, M. Newman, A. Pagnani, T. Petermann, D.
Rittel, L. Rocha, G. Schehr, S. Schnell, O. Sporns, E. Trizac, Z. Toroczkai, S. Wasserman, F. Willinger, L. Yaeger, and S. Special thanks go to E. Almaas, C. Castellano, V. Colizza, J.
Dunne, S. Fortunato, H. Jeong, Y. Moreno, N. Martinez, M.
Newman, M. Angeles Serrano, and O. Sporns for their kind help in retrieving data and adapting figures from their work. Baronchelli, C. Castellano, L. Fortunato, and R. Pastor-Satorras reviewed earlier versions of this book and made many suggestions for improvement.
Hook deserves a special acknowledgement for xv xvi Acknowledgements proofreading and editing the first version of the manuscript. Simon Capelin and his outstanding editorial staff at Cambridge University Press were a source of continuous encouragement and help and greatly contributed to make the writing of this book a truly exciting and enjoyable experience.
We also thank all those institutions that helped in making this book a reality. We also acknowledge the generous funding support at various stages of our research by the following awards: Finally, A. We define metrics such as the shortest path length, the clustering coefficient, and the degree distribution, which provide a basic characterization of network systems.
The large size of many networks makes statistical analysis the proper tool for a useful mathematical characterization of these systems. We therefore describe the many statistical quantities characterizing the structural and hierarchical ordering of networks including multipoint degree correlation functions, clustering spectrum, and several other local and non-local quantities, hierarchical measures and weighted properties.
This chapter will give the reader a crash course on the basic notions of network analysis which are prerequisites for understanding later chapters of the book. Needless to say the expert reader can freely skip this chapter and use it later as a reference if needed. In very general terms a network is any system that admits an abstract mathematical representation as a graph whose nodes vertices identify the elements of the system and in which the set of connecting links edges represent the presence of a relation or interaction among those elements.
Clearly such a high level of abstraction generally applies to a wide array of systems. In this sense, networks provide a theoretical framework that allows a convenient conceptual representation of interrelations in complex systems where the system level characterization implies the mapping of interactions among a large number of individuals.
The study of networks has a long tradition in graph theory, discrete mathematics, sociology, and communication research and has recently infiltrated physics and biology.
While each field concerned with networks has introduced, in many cases, its own nomenclature, the rigorous language for the description of networks 1 2 Preliminaries: On the other hand, the study of very large networks has spurred the definitions of new metrics and statistical observables specifically aimed at the study of large-scale systems.
In the following we provide an introduction to the basic notions and notations used in network theory and set the cross-disciplinary language that will be used throughout this book.
Our intention is to select those notions and notations which will be used throughout the rest of this book. Throughout the book we will refer to a vertex by its order i in the set V. The edge i, j joins the vertices i and j, which are said to be adjacent or connected.
It is also common to call connected vertices neighbors or nearest neighbors. The total number of vertices in the graph the cardinality of the set V is denoted as N and defines the order of the graph. It is worth remarking that in many biological and physical contexts, N defines the physical size of the network since it identifies the number of distinct elements composing the system. However, in graph theory, the size of the graph is identified by the total number of edges E. Unless specified in the following, we will refer to N as the size of the network.
Undirected graphs are depicted graphically as a set of dots, representing the vertices, joined by lines between pairs of vertices, representing the corresponding edges. An interesting class of undirected graph is formed by hierarchical graphs where each edge known as a child has exactly one parent node from which it originates.
Such a structure defines a tree and if there is a parent node, or root, from which the whole structure arises, then it is known as a rooted tree. It is easy to prove that the number of nodes in a tree equals the number of edges plus one, i. A directed graph D, or digraph, is defined by a non-empty countable set of vertices V and a set of ordered pairs of different vertices E that are called directed edges.
In a graphical representation, the directed nature of the edges is depicted by means of an arrow, indicating the direction of each edge.
The main difference between directed and undirected graphs is represented in Figure 1. In an undirected graph the presence of an edge between vertices i and j connects the vertices in both directions. On the other hand, the presence of an edge from i and j in a directed graph does not necessarily imply the presence of the reverse edge between j and i. This fact has important consequences for the connectedness of a directed graph, as will be discussed in more detail in Section 1.
Adjacency matrix and graphical representation of different networks. In the graphical representation of an undirected graph, the dots represent the vertices and pairs of adjacent vertices are connected by a line edge. In directed graphs, adjacent vertices are connected by arrows, indicating the direction of the corresponding edge.
In Figure 1. An important feature of many graphs, which helps in dealing with their structure, is their sparseness.
The presence of subgraphs and communities raises the issue of modularity in networks. Modularity in a network is determined by the existence of specific subgraphs, called modules or communities. Clustering techniques can be employed to determine major clusters.
They comprise non-hierarchical methods e. Non-hierarchical and hierarchical clustering methods typically work on attribute value information. For example, the similarity of social actors might be judged based on their hobbies, ages, etc. Nonhierarchical clustering typically starts with information on the number of clusters that a data set is expected to have and sorts the data items into clusters such that an optimality criterion is satisfied.
Hierarchical clustering algorithms create a hierarchy of clusters grouping similar data items. Connectivity-based approaches exploit the topological information of a network to identify dense subgraphs. They comprise measures such as betweenness centrality of nodes and edges Girvan and Newman, ; Newman, , superparamagnetic clustering Blatt, Wiseman and Domany, ; Domany, , hubs and bridging edges Jungnickel, , and others.
Recently, a series of sophisticated overlapping and non-overlapping clustering methods has been developed, aiming to uncover the modular structure of real networks Reichardt and Bornholdt, ; Palla et al. In a connected network every vertex is reachable from any other vertex.
The connected components of a graph thus define many properties of its physical structure. The path Pi0 ,in is said to connect the vertices i 0 and i n. The length of the path Pi0 ,in is n. The number Ni j of paths of length n between two nodes i and j is given by the i j element of the nth power of the adjacency matrix: A graph is called connected if there exists a path connecting any two vertices in the graph. A component C of a graph is defined 6 Preliminaries: It is clear that for a given number of nodes the number of loops increases with the number of edges.
A most interesting property of random graphs Section 3. The presence of a giant component implies that a macroscopic fraction of the graph is connected. The structure of the components of directed graphs is somewhat more complex as the presence of a path from the node i to the node j does not necessarily guarantee the presence of a corresponding path from j to i. Therefore, the definition of a giant component needs to be adapted to this case.
In general, the component structure of a directed network can be decomposed into a giant weakly connected component GWCC , corresponding to the giant component of the same graph in which the edges are considered as undirected, plus a set of smaller disconnected components, as sketched in Figure 1. The GWCC is itself composed of several parts because of the directed nature of its edges: Component structure of a directed graph.
Figure adapted from Dorogovtsev et al. Indeed, while graphs usually lack a metric, the natural distance measure between two vertices i and j is defined as the number of edges traversed by the shortest connecting path see Figure 1.
Basic metrics characterizing a vertex i in the network. A, The degree k quantifies the vertex connectivity. B, The shortest path length identifies the minimum connecting path dashed line between two different vertices. There are also other measures of interest which are related to the characterization of the linear size of a graph. In most random graphs Sections 2. The fact that any pair of nodes is connected by a small shortest path constitutes the so-called small-world effect.
The importance of a node or edge is commonly defined as its centrality and this depends on the characteristics or specific properties we are interested in. Various measurements exist to characterize the centrality of a node in a network. Among those characterizations, the most 2 It is worth stressing that the average shortest path length has also been referred to in the physics literature as another definition for the diameter of the graph.
Edges are frequently characterized by their betweenness centrality. Degree centrality The degree ki of a vertex i is defined as the number of edges in the graph incident on the vertex i. While this definition is clear for undirected graphs, it needs some refinement for the case of directed graphs. Thus, we define the in-degree kin,i of the vertex i as the number of edges arriving at i, while its out-degree kout,i is defined as the number of edges departing from i.
The degree of a vertex has an immediate interpretation in terms of centrality quantifying how well an element is connected to other elements in the graph. The Bonacich power index takes into account not only the degree of a node but also the degrees of its neighbors.
Betweenness centrality While the previous measures consider nodes which are topologically better connected to the rest of the network, they overlook vertices which may be crucial for connecting different regions of the network by acting as bridges.
In order to account quantitatively for the role of such nodes, the concept of betweenness centrality has been introduced Freeman, ; Newman, a: The calculation of this measure is computationally very expensive.
The basic algorithm for its computation would lead to a complexity of order O N 2 E , which is prohibitive for large networks. An efficient algorithm to compute betweenness centrality is reported by Brandes and reduces the complexity to O N E for unweighted networks. According to these definitions, central nodes are therefore part of more shortest paths within the network than less important nodes.
Moreover, the betweenness centrality of a node is often used in transport networks to provide an estimate of the traffic handled by the vertices, assuming that the number of shortest paths is a zero-th order approximation to the frequency of use of a given node.
Analogously to the vertex betweenness, the betweenness centrality of edges can be calculated as the number of shortest paths among all possible vertex couples that pass through the given edge. Edges with the maximum score are assumed to be important for the graph to stay interconnected. Removing them frequently leads to unconnected clusters of nodes.
However, networks with many such bridges are more fragile and less clustered. The concept of clustering3 of a graph refers to the tendency observed in many natural networks to form cliques in the neighborhood of any given vertex.
In this sense, clustering implies the property that, if the vertex i is connected to the vertex j, and at the same time j is connected to l, then with a high probability i is also connected to l.
The clustering of an undirected graph can be quantitatively measured by means of the clustering coefficient which measures the local group cohesiveness Watts and Strogatz, Given a vertex i, the clustering C i of a node i is defined as the ratio of the number of links between the neighbors of i and the maximum number of such links.
If the degree of node i is ki and if these nodes have ei edges between them, we have ei , 1. This definition corresponds to the concept of the fraction of transitive triples introduced in sociology Wasserman and Faust, Different definitions give rise to different values of the clustering coefficient for a given graph. Hence, the comparison of clustering coefficients among different graphs must use the very same measure.
In any case, both measures are normalized and bounded to be between 0 and 1.
Indeed, in large systems, asymptotic regularities cannot be detected by looking at local elements or properties. In other words, one has to shift the attention to statistical measures that take into account the global behavior of these quantities. It is obtained by constructing the normalized histogram of the degree of the nodes in a network. In the case of directed graphs, one has to consider instead two distributions, the in-degree P kin and out-degree P kout distributions, defined as the probability that a randomly chosen vertex has in-degree kin and out-degree kout , respectively.
In the next chapters we will see that the properties of the degree distribution will be crucial to identify different classes of networks. Finally, it is clear that in most cases the larger the degree ki of a node, the larger its betweenness centrality bi will be. Such associations are extremely relevant as they correspond to the fact that a large number of shortest paths go through the nodes with large degree the hubs.
These nodes will therefore be visited and discovered easily in processes of network exploration see Chapter 8. They will also typically see high traffic which may result in congestion see Chapter Of course, fluctuations are also observed in real and model networks, and small-degree nodes may also have large values of the betweenness centrality if they connect different regions of the network, acting as bridges.
In particular, it is likely that nodes do not connect to each other irrespective of their property or type. On the contrary, in many cases it is possible to collect empirical evidence of specific mixing patterns in networks. This is common to observe in the social context where people prefer to associate with others who share their interests. Interesting observations about 14 Preliminaries: Mixing patterns have a profound effect on the topological properties of a network as they affect the community formation and the detailed structural arrangements of the connections among nodes.
This type of mixing refers to the likelihood that nodes with a given degree connect with nodes with similar degree, and is investigated through the detailed study of multipoint degree correlation functions. Most real networks do exhibit the presence of non-trivial correlations in their degree connectivity patterns. Empirical measurements provide evidence that high or low degree vertices of the network tend, in many cases, to preferentially connect to other vertices with similar degree.
In this situation, correlations are named assortative. In contrast, connections in many technological and biological networks are more likely to attach vertices of very different degree. Correlations are then referred to as disassortative.
Such quantities might be the simplest theoretical functions that encode degree correlation information from a local perspective. A network is said to be uncorrelated when the conditional probability is structureless, in which case the only relevant function is just the degree distribution P k.
However, such a measure can be misleading when a complicated behavior of the correlation functions non-monotonous behavior is observed. In this case the Pearson coefficient gives a larger weight to the more abundant degree classes, which in many cases might not express the variations of the correlation function behavior. Even in this simple case, however, the direct evaluation 1. On the other hand, such a histogram is highly affected by statistical fluctuations and, thus, it is not a good candidate when the data set is not extremely large and accurate.
In the presence of correlations, the behavior of knn k identifies two general classes of networks see Figure 1. If knn k is an increasing function of k, vertices with high degree have a larger probability of being connected knn k Assortative Disassortative k Fig.
Pictorial representation of the assortative disassortative mixing property of networks as indicated by the behavior of the average degree of the nearest neighbors, knn k , for vertices of degree k. This corresponds to an assortative mixing Newman, a. On the contrary, a decreasing behavior of knn k defines a disassortative mixing, in the sense that high degree vertices have a majority of neighbors with low degree, while the opposite holds for low degree vertices Newman, a.
It is important to stress, however, that given a certain degree distribution, a completely degree—degree uncorrelated network with finite size is not always realizable owing to structural constraints.