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All figures (50)
Fig. 5.4. Reprinted figure with permission from Ref. [450]. 2005 by the American Physical Society. (a) N/ 2 (in logarithmic scale) for random graphs vs. . (b) c (see text for definition) vs. the parameter space ( ,B). Random networks have 〈k〉 = 2m = 4, identical to that of scale free networks. In the domain where c < 0 scale free networks synchronize better than random networks. Same conditions as in the caption of Fig. 5.3.
Fig. 6.8. (a) In- and out-degree distributions averaged over a set of 43 different organisms. (b) The diameter of the metabolic network as a function of the number N of metabolites in the E. Coli. (c) The average number of outgoing edges vs. the number of nodes.
Fig. 2.1. Graphical representation of a undirected (a), a directed (b), and a weighted undirected (c) graph with N = 7 nodes and K = 14 links. In the directed graph, adjacent nodes are connected by arrows, indicating the direction of each link. In the weighted graph, the values wi,j reported on each link indicate the weights of the links, and are graphically represented by the link thicknesses.
Fig. 2.3. Communities can be defined as groups of nodes such that there is a higher density of edges within groups than between them. In the case shown in figure there are three communities, denoted by the dashed circles. Reprinted figure with permission from Ref. [51]. 2004 by the American Physical Society.
Fig. 2.13. Color coded maps representing the spatial distributions of centrality in Cairo, an example of a largely self-organized city. The four indices of node centrality, (a) closeness; (b) betweenness; (c) straightness and (d) information, are visually compared over the spatial graph. Different colors represent classes of nodes with different values of the centrality index. The classes are defined in terms of multiples of standard deviations from the average, as reported in the color legend. Figure taken from Ref. [47].
Fig. 3.5. Total number of active packets as a function of time steps in the model by Echenique et al. [336] with steady input of packets. Panels (a) and (b) correspond to the standard protocol, while (c) and (d) have been obtained for the traffic-aware routing with h = 0.85. In each panel, the continuous line stands for subcritical values of p ((a) and (b) p = 3.0, (d) p = 8.0) and the dotted line corresponds to p>pc ((a) and (b) p = 4.0, (c) and (d) p = 13.0). Reprinted figure with permission from Ref. [336]. 2005 by the European Physical Society.
Fig. 3.6. Jamming transitions in the model by Echenique et al. [336]. The order parameter is reported as a function of p. Note that h=1 corresponds to the standard strategy in which traffic awareness is absent. As soon as traffic conditions are taken into account, the jamming transition is reminiscent of a first-order phase transition and the critical point shifts rightward. Figure taken from Ref. [336].
Fig. 2.4. Cumulative degree distributions of the Internet AS graph representation for three different years. The power-law behavior is clear, as well as the fact that, regardless of the very dynamic nature of the Internet, the exponent is constant with time. Reprinted figure with permission from Ref. [25]. 2001 by the American Physical Society.
Fig. 7.6. (a) Components of the first non-trivial eigenvector v2 for a computer-generated network with four communities (see main text). Two communities are clearly identified while the other two overlap. (b) All communities can be clearly identified when the components of v3 are plotted versus those of v2. Figure taken from Ref. [859].
Fig. 3.2. Size of the largest connected component as a function of the tolerance parameter in the model for cascading failures by Motter et al. [298]. Cascades are triggered by the removal of a node chosen at random (squares), or by the removal of the node with largest load (circles) or with the largest degree (asterisks). Homogeneous networks (random graphs with all nodes having k = 3 links) are considered in the main frame, while heterogeneous networks (scale-free networks with an average degree equal to 3) are considered in the inset. Both the networks considered have N = 5000. Reprinted figure with permission from Ref. [298]. 2002 by the American Physical Society.
Fig. 6.1. Political and friendship structure between the elite florentine families during the Renaissance state in Florence.
Fig. 2.7. Reprinted figure with permission from Ref. [214] BBV model, 2004 by the American Physical Society. Local rearrangement of weights due to the presence of the new node j and the new link lj i . The weight of the new edge is w0 and the total weight on the existing edges connected to i is modified by an amount equal to .
Fig. 7.4. Average fraction of correctly identified vertices as a function of zout . Each point represents an average over 100–500 computer generated modular random graphs with 128 vertices and 1024 edges. The results of the GN algorithm are compared with those obtained by using the random betweenness, the optimization of Q, and the Fortunato et al. algorithm (See Section 7.1.4).
Table 6.2 Basic properties of the WWW
Fig. 6.9. (a) Degree distribution of networks from different data sets (specified in the legend). (b) Clustering coefficient as a function of the degree for the same data sets. In both plots, the straight lines have slope −2. Reprinted figures with permission from Ref. [647]. 2004 by Wiley.
Table 6.3 In-degree distribution exponent in Web pages of homogeneous category
Fig. 7.1. Finding community structures in the karate club network of Zachary. The numbered vertices of the network represent the members of the club, while the edges represent friendships, as determined by the observation of the interactions [837]. The two groups into which the club split during the course of the study are indicated by the squares and circles, while the dark grey and white show the divisions of the network found by (a) the spectral bisection algorithm of Section 7.1.1, (b) the hierarchical clustering method of Section 7.1.2 and (c) the Monte Carlo sampled version of the algorithm of Girvan and Newman proposed by Tyler et al. and discussed in Section 7.1.4. In (b) the lightly shaded vertices are those not assigned by the algorithm to either of the two principal communities. In (c) shades intermediate between the dark grey and white indicate ambiguously assigned vertices that fall in one community or the other, or neither, on different runs of the algorithm. Reprinted figure with permission from [829]. 2004 by the European Physical Society.
Fig. 5.6. Reprinted figure with permission from Ref. [454]. 2005 by the American Physical Society. rN / r 2 vs. for (a) scale free networks with B = 0 and m= 2 (circles), 5 (squares), 10 (diamonds); (b) scale free networks with m= 5 and B = 0 (circles), 5 (squares), and 10 (diamonds); (c) Erdös–Rényi random network with arbitrary age order (circles), Erdös–Rényi random network with age depending on degree (squares), scale free networks with m= 5 and B = 0 without connection between nodes 1 and 5 (diamonds), and scale free networks with m= 5 and B = 0 (triangles).
Fig. 5.5. Reprinted with permission from Ref. [454]. 2005 by the American Physical Society. rN and r 2 (a), r N / r 2(b), and M (c) vs. for scale free (m= 5 and B = 0, solid lines) and random networks (dashed lines).
Fig. 2.11. Phase diagram of D-dimensional lattices supplemented with long range links whose lengths are distributed according to q( ) ∼ − . SW denotes the small-world phase in the −D plane. Reprinted figure with permission from Ref. [240]. 2002 by the American Physical Society.
Fig. 2.2. All the 13 types of motifs consisting in three-nodes directed connected subgraphs.
Table 2.1 Basic characteristics of a number of information/communication, biological and social networks from the real-world
Fig. 7.3. Modularity Q (top panel) and dendrogram (bottom panel) obtained by the application of the GN algorithm to a computer generated modular random graph with 64 vertices and 256 edges. The random graph has been generated, as described in the text, by dividing the nodes into 4 groups of 16 nodes each (respectively empty circles, full circles, triangles and squares) and by considering zin = 6, zout = 2. The peak in the modularity (dotted line) corresponds to a perfect identification of the 4 communities.
Fig. 2.10. (a) The box counting method [95] applied to the distributions shown in Fig. 6.8. The log–log plot shows the number of boxes of size l × l with non-zero routers/AS/inhabitants as a function of l for North America. The slope of the straight line indicates that Df = 1.5± 0.1 for each data set. (b) The plot of the number of router/AS nodes in a 1◦ × 1◦ box as a function of the number of people living in the same area, shows that router density is strongly correlated with economic factors. Figure taken from Ref. [230]. Courtesy of A.L. Barabási.
Fig. 5.1. Possible classes of master stability function for networked chaotic systems. In all cases ( = 0)> 0 is the maximum Lyapunov exponent of the single uncoupled system. The case I (II) corresponds to a monotonically increasing (decreasing) master stability function. Case III admits a finite range of negative values for ( ).
Fig. 2.6. From Ref. [194]. Comparison of topological and weighted quantities for the scientific collaborations [panel (a) and (c)], and the world-wide airport network [panel (b) and (d)], considered in Refs. [194,196]. In panel (a) the weighted clustering coefficient CW(k) separates from the topological one around k 10, while in panel (b) CW(k) is larger than C(k) in the whole degree spectrum. In panel (c) the assortative behavior is shown both by the unweighted and the weighted definition of the average nearest neighbors degree. In panel (d), knn(k) is reaching a plateau for k > 10 denoting the absence of marked topological correlations while, on the contrary, kwnn(k) exhibits an assortative behavior. Courtesy of A. Vespignani.
Fig. 7.8. Centrality coefficient C vs. node degree K for BA scale-free network, in a double logarithmic plot. The straight line is a guide for the eyes. Reprinted figure with permission from Ref. [863]. 2002 by the American Physical Society.
Fig. 6.3. Schematic diagram of the Internet. The large circles denote the domains, the small filled circles are the routers, and each unfilled square corresponds to a router host.
Fig. 3.1. Network robustness under random failures and attacks on nodes. Average shortest path length L and global efficiency E are reported as a function of the fraction of removed nodes f . An ER random graph and a BA scale-free network, both with N=5000 and K=10, 000 are considered. Reprinted from Ref. [282], 2003, with permission from Elsevier.
Fig. 7.2. The hierarchical tree (dendrogram) depicting the results (from bottom to top) of a single linkage agglomerative hierarchical clustering of the karate club network based on Euclidean distance defined in Eq. (7.1). A cross-section of the tree at any level will give the communities at that level. The cross-section indicated by the dotted line corresponds to the community division shown in Fig. 7.1b. The vertical height of the branching points in the tree is indicative only of the order in which the joins between vertices take place. Note that the heights of some joins coincide, indicating that the vertices joined at that level have identical similarities. Reprinted figure with permission from [829]. 2004 by the European Physical Society.
Fig. 2.12. (a) The cumulative length distribution, defined as ∫ R d P (x) dx, with d being the Euclidean distance between two Internet routers and R = 6378 km the radius of the Earth, is plotted as a function of the dimensionless variable d/R. (b) The cumulative change k in the connectivity of nodes with k links. The dotted line has slope 2 and indicates that k is linear in k. Figure taken from Ref. [230]. Courtesy of A.L. Barabási.
Fig. 7.7. Schematic diagram of (a) the breadth-first search algorithm and (b) the maximum degree strategy for searching from node p to node q. In all cases d denotes the step number of the algorithm. Reprinted figure with permission from Ref. [861]. 2002 by the American Physical Society.
Fig. 3.4. Modelling cascading failure in the North American electrical power grid [301]. Global efficiency of the power grid after the removal of random (triangles) or high-load (circles) generators or transmission substations. As the overload tolerance of the substations increases, the final efficiency approaches the unperturbed value. The random disruption curves were obtained by averaging over 10–100 individual removals. The load-based disruption curve is obtained by removing the highest load generator and transmission node, respectively. Reprinted figure with permission from Ref. [301]. 2005 by the European Physical Society.
Fig. 3.3. Model for cascading failures by Crucitti et al. [300]. Cascading failure in (a) ER random graphs and (b) BA scale-free networks as triggered by the removal of a node chosen at random (squares), or by the removal of the node with largest load (circles). Both the networks considered have N = 2000 and K = 10, 000. The final efficiency E of the network is reported as a function of the tolerance parameter . In the case triggered by the removal of a node chosen at random the curve corresponds to an average over 10 triggers. Reprinted figure with permission from Ref. [300]. 2004 by the American Physical Society.
Table 6.4 Number of nodes N , characteristic path length L, clustering coefficient C, and global efficiency Eglob, for some neural networks
Fig. 5.2. Reprinted figure with permission from Ref. [448]. 2005 by the European Physical Society. Eigenratio R ≡ N/ 2 vs. the weighting parameter in Eq. (5.5). The figure reports data coming from random networks having a scale free distribution with = 3 (circles), = 5 (squares), = 7 (upper triangles), and =∞ (solid line).
Fig. 2.9. (a) Worldwide router density map obtained by locating the geographical position of the 228265 routers of the router-level Internet database from Ref. [231]; (b) population density map. In either maps the resolution consists of boxes of 1o × 1o. The color code indicates the density values. The highest population density within this resolution is of the order 107 people/box, while the highest router density is of the order of 104 routers/box. Figure taken from Ref. [230]. Courtesy of A.L. Barabási.
Fig. 6.11. Enhancement of synchronization in a network of inhibitory interneurons by the addition of a small number of long-range interneurons. Upper panel: example of connectivity schemes. Middle panel: spike raster of 1000 neurons. Bottom panel: average population firing rate. The red arrow indicates the instant when the shortcuts are added. Figure taken from Ref. [814]. Courtesy of M. Le van Quyen.
Fig. 6.7. An example of cell’s responding activity to an external signal. Pictures taken from [637].
Fig. 6.6. Schematic diagram of interactions between the different cell’s components.
Fig. 7.9. Search timeLs (solid line), and average shortest path lengthL (dashed line), as a function of the rewiring probability p for a WS small-world network with N = 2000 nodes and K = 10 000 links. The inset shows LLUTs for the search method adopted in Ref. [865] (thick line) and for the breadth-first search method (thin line). Reprinted figure with permission from Ref. [865]. 2003 by the American Physical Society.
Table 6.1 Basic properties of the Internet at the AS level for the year 1997, 1998, and 1999
Fig. 2.8. From Ref. [210] DM model. Schematic view of the network growth. At each time step, the weight of a preferentially chosen edge is increased by an amount equal to and a new vertex is attached to the ends of this edge with two edges having weight equal to 1.
Fig. 7.5. Modularity Q (top panel) and dendrogram (bottom panel) obtained by the application of the GN algorithm to the karate club network The modularity has two maxima corresponding respectively to (i) a split into two communities, which matches closely the real split of the club (only node 3 is incorrectly classified), and (ii) a split into five communities. Reprinted figure with permission from Ref. [51]. 2004 by the American Physical Society.
Fig. 6.2. From Ref. [570]. Structure and shape of the largest connected component of a drug users network. The nodes are clustered into the three primary ethnic groups considered in the analysis of Ref. [570].
Fig. 6.10. Reprinted from Ref. [735]: 2000 The Royal Society of London. (A) lateral representation of macaque cortex where the labels denote the approximative position of areas. (B) Example of a metric representation of the connections in the cortical network as obtained by NMDS.
Fig. 2.5. Average degree of the d-nearest neighbors of a root node with degree k in: Internet AS networks (a), and Internet Routers (b). The insets show in both plots the dependency of the slopes of the curves in the main panel as a function of d, indicating that the correlations change as the distance from the root is increased. Reprinted with permission from Ref. [102]. 2005 by the American Physical Society.
Fig. 5.3. Reprinted figure with permission from Ref. [450]. 2005 by the American Physical Society. (a) N/ 2 (in logarithmic scale) for SF networks vs. the parameter space ( ,B). (b) (see text for definition) vs. ( ,B). In all cases m= 2, and the reported values refer to an average over 10 realizations of networks with N = 1000 nodes. The domain with < 0 is outlined by the black contours drawn on the figure.
Fig. 6.4. Path length distribution P(L) (a) and normalized hop plots (b) vs. L for the Internet network at the AS level, for 1997 (AS97, circles), 1998 (AS98, squares), and 1999 (AS99, diamonds). Data collected by the National Laboratory for Applied Network Research (NLANR). Reprinted figure with permission from Ref. [618]. 2002 by the American Physical Society.
Fig. 6.5. Clustering coefficient Ck as a function of degree k (a), average degree of the nearest neighbors of a node with degree k vs. k (b), integrated betweenness probability distribution (c), and normalized betweenness as a function of the degree (d), for the 1997, 1998, and 1999 AS Internet networks. Reprinted figure with permission from Ref. [618]. 2002 by the American Physical Society.
Journal Article
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DOI
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Complex networks: Structure and dynamics
[...]
Stefano Boccaletti
,
Vito Latora
1
,
Vito Latora
2
,
Yamir Moreno
3
,
Mario Chavez
4
,
Dong-Uk Hwang
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+2 more
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Institutions (4)
University of Catania
1
,
Istituto Nazionale di Fisica Nucleare
2
,
University of Zaragoza
3
,
Centre national de la recherche scientifique
4
01 Feb 2006
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Physics Reports