Complex networks: Structure and dynamics

TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.

About: This article is published in Physics Reports.The article was published on 2006-02-01 and is currently open access. It has received 9441 citations till now. The article focuses on the topics: Network dynamics & Complex network.

Summary

5.1. Introduction to synchronization

• The origin of such word comes from a greek root ( ̀ ó o which means “to share the common time”).
• Historically, synchronization phenomena have been actively investigated since the earlier days of physics.
• Initially, the attention was mainly devoted to synchronization of periodic systems, while recently the search for synchronization has moved to chaotic systems [392].
• This latter phenomenon can also occur intermittently, giving rise to the intermittent lag synchronization, where the coupled systems are most of the time verifying the condition for lag synchronization, but persistent bursts of local non-synchronous behavior may intermittently affect their dynamics [398,401].
• Finally, almost synchronization is meant as the asymptotic boundedness of the difference between a subset of the variables of one system and the corresponding subset of variables of another system [403].

5.2. The Master stability function approach

• The authors start with discussing the so called Master Stability Function approach.
• As the authors will see in the following sections, the coupling matrix C is suitably related with the classical matrices defining the topology of the network (introduced in Sections.
• The synchronization manifold S is an invariant set, due to the zero row-sum condition of the coupling matrix C and due to the fact that the coupling function H(x) is the same for all network nodes.
• In the following, the authors will describe the master stability function approach in the context of Lyapunov exponents.
• It is easy to understand that both cases (I) and (II) of Fig. 5.1 correspond to rather trivial situations.

5.3. Network propensity for synchronization

• A basic assumption characterizing most of the early works on synchronization in complex networks is that the local units are symmetrically coupled with uniform undirected coupling strengths (unweighted links).
• As the authors have already discussed in Section 2.4, this simplification does not retain in general the full information on the structure of real networks.
• There are, indeed, paradigmatic cases where a weighting in the connections has relevant consequences in determining the network’s dynamics.
• Again, the natural differences of neurons and their dendritic connections may result in distinct capabilities of transmission and information processing in neural networks [444,445].
• This was also motivated by the fact that asymmetry in the coupling was shown to play a fundamental role in connection with synchronization of coupled spatially extended fields [446,447].

5.3.1. Synchronization in weighted networks: coupling matrices with real spectra

• The relevant results of Ref. [448] is the tremendous improvement on the propensity of synchronization, obtained with a weighting procedure that retains information on the local features of the network (the node degree).
• In the following the authors will summarize this latter approach.
• The load ij of the link connecting nodes i and j quantifies the traffic of shortest paths that are making use of that link [42], this way reflecting a measure of the network structure at a global scale.
• Scalefree networks were obtained by the procedure introduced in Refs. [3,146], and discussed in Section 2.3.5.

5.3.2. Synchronization in weighted networks: coupling matrices with complex spectra

• Both the approaches developed in Refs. [448,450] delt with situations wherein the coupling matrix C had a real spectrum of eigenvalues.
• In growing networks, such age ordering will be naturally related to the appearance order of the node during the growing process.
• Ref. [454] also investigated the propensity for synchronization in scale free networks at different m and B values.
• Indeed, for positive values, the dominant coupling direction is from to nodes.
• This is in general accounted for by a proper normalization in the off diagonal elements of C, assuring that hubs receive an input from a connected node scaling with the inverse of their degree, and therefore the structure of hubs is connected always with the rest of the network in a way that is independent on the network size.

5.4. Synchronization of coupled oscillators

• After having discussed the general stability properties of synchronization states in complex networks, the authors review the most significant obtained results in the study of synchronization of different networked systems.
• The first works on conditions and effects of synchronization in complex networks were reported by Watts [5] and Barahona and Pecora [433].
• Specifically, they studied the Kuramoto model on top of Barabási–Albert networks and on top of small structures that were relevant in different biological and social networks, with the aim of inspecting the critical point associated to the onset of synchronization, i.e., when small groups of synchronized oscillators first appear in the system.
• Moreover, as it was discovered later on, the choice of the order parameter seems to be a crucial point when analyzing the conditions for the existence of the transition threshold.
• Hence, the more connected a node is, the more stable it is.

5.5. Synchronization of chaotic dynamics

• At the beginning of this chapter the authors have largely discussed the master stability approach for the assessment of the synchronizability of a given network.
• This conjecture, on its turn, implies that a network of diffusively coupled identical oscillators can always be synchronized for sufficiently high coupling strengths [469–471].
• All of them agree with the main results of Section 5.3 that the ability of a given network to synchronize is strongly ruled by the structure of connections.
• In scale free networks, for instance, the clustered organization of nodes as well as the connectivity patterns between such clusters, render the network structure tolerant to random removal of nodes but very vulnerable to targeted attacks [275].
• A direct consequence of this phenomenon is the existence of an upper limit in the number of chaotic oscillators that can be synchronized, upon which the synchronized state of a given network becomes unstable [472,473].

5.6. Other collective behaviors in networks of ordinary differential equations

• The authors finally review the other collective behaviors that have been studied in networks of ordinary differential equations .
• In the study of interacting oscillators, three types of coupling schemes are currently considered: global coupling where each unit interacts with all the others, local coupling where an element interacts with its neighbors (defined by a given metric), and non-local or intermediate couplings.
• In networks of both limit-cycle and chaotic oscillators with slightly different oscillation modes, a phase transition associated to a collective and coherent behavior was produced by an increasing of the coupling strength [459,489–491].
• This approach presents severe theoretical difficulties for its analysis [460,492,493].
• Interestingly, the properties of these states and the corresponding phase transitions are very different from the phases observed in regular magnets [510].

6. Applications

• These include both issues concerning the structure of the networks and their dynamics.
• The authors shall review some structural aspects of social networks and consider two kind of dynamics involving social networks: opinion formation and game models.
• The authors shall then discuss the statistical properties of the Internet and of the World Wide Web.
• Finally the authors shall focus of complex networks of interest to biology and medicine, such as networks describing the interactions between cell components, and neural networks.

6.1. Social networks

• A social network is a set of actors (individuals or social groups) and relationships of different kinds (friendship, kinship, status, sexual, business or political) among them [18,19].
• The quantitative analysis of social interactions goes back to the early 1920s [513].
• The cross-interaction of researchers form diverse disciplines as sociology, applied anthropology, social psychology and statistics has raised over the years a solid scientific discipline with its own textbooks [18,19], and specialized journals as Social Networks published by Elsevier.
• Centralities based on degree or on betweenness are some examples of such indices [38,39,517–520].
• Some current problems in network analysis, as the characterization of a node by its relations, have also been raised in sociometric studies: several methods have been developed to quantify similarities between actors, based exclusively on the topology [521,522] (see Section 7.1.2).

6.1.1. Structure

• Graph theory has been largely used to describe different social systems as friendship, affiliation and collaboration networks.
• The quantification of social interactions has even suggested an underlying network organization of conspiracy [526] or of terrorist cells [527,528].
• Since then, the statistical properties of the distribution of citations have been discussed in different manuscripts [548–550], and recently the power law behavior of the in-degree distribution suggested the introduction of a new indicator (the index h) quantifying the impact of a scientist on the modern scientific community [551,552].
• Different studies have reported a non-random structure, characterized by small-world behavior, scaling in the degree distribution and disassortative mixing, in different semantic webs in languages as Portuguese, English, Czech German and Rumanian [554–560].
• Results from real data have confirmed the key role of the topology of social contacts in the spread dynamics of different pathogens as hepatitis B virus (HBV), human immunodeficiency viruses (HIV), bacterial pneumonia, syphilis, chlamydia and gonorrhea.

6.1.2. Dynamics I: opinion formation

• The application of Ising models and other tools of computational or statistical physics to social systems has a long tradition since the work by Majorana [580].
• Here the authors shall discuss some of the models of opinion formation, such as those by Sznajd (S) [581], Deffuant et al. (D) [582], and Krause and Hegselmann (KH) [583], that have been recently studied in the context of complex networks.
• Other models, as the voter model [584–587], Galam’s majority rule and the Axelrod multicultural model [588] will not be discussed.
• Eventually, for a critical value = c, all the individuals reach a complete consensus.
• Yu-Song et al. [595] have analyzed the transition from the state with no consensus to the state with complete consensus in the S model on small-world and scale free networks.

6.1.3. Dynamics II: strategic games

• More precisely a game consists of the following ingredients [603,604]: (i) at least two decision-makers (also known as players or agents) are involved in the game.
• Depending on whether or not the strategies are the same for each player, different game models can be defined.
• If a player cooperates and the other defects, the cooperator scores zero points while the defector scores b points, with b >.
• The authors then argued that the crucial structural ingredients causing the observed behavior are the inhomogeneous degree distribution and the fact that the connections within vertices of highest degree are rather sparse.

6.2. The Internet and the World Wide Web

• The former is used to pass information within the same AS, while the latter serves for exchanging topological, path information and conditions on the network between peer routers in different AS.
• The Internet and the WWW are interesting networks because of their large sizes that allow for a reliable statistical analysis of their topological properties.
• The need for information spreading pervades their lives and its efficient handling and delivery is becoming one of the most important practical problems.
• The scalability of current protocols as well as their performance for larger system sizes and heavier loads are critical issues to be addressed in order to warrant the networks’ functioning.

6.2.1. Structure of the Internet

• Faloutsos et al. have analyzed the Internet AS maps collected from 1997 to 1998 by the National Laboratory for Applied Network Research [617].
• The distribution probability P(L) is characterized by a sharp peak around its average value 〈L〉 ≈ 4 and its shape remains essentially unchanged from the AS97 to the AS99 maps.
• Concerning the degree, Table 6.1 indicates that the average degree is time independent, even if the number of nodes is more than doubled in the three-year period considered.
• As shown in Table 6.1, the Internet has a relatively high clustering coefficient, about 100 times larger than that of a ER random graph with the same size.
• All the previous quantities can also be measured at the router level.

6.2.2. Structure of the World Wide Web

• As mentioned before, the WWW can be studied at the web-page level, where a node corresponds to a web-page and the hyper-links are mapped into directed links between nodes.
• Another possible resolution is the site level, where a node corresponds to a site having a collection of Web pages, and two nodes are connected by undirected edges when there exist hyper-links between Web pages in the corresponding sites.
• Albert et al. were the first to find powerlaws in the degree distributions of Web pages in the domain *.nd.edu [623].
• In general, the out-degree distributions appear to have a bending from a pure power-law behavior.
• Similar results were observed by an independent analysis [625].

6.2.3. Dynamics

• Some important aspects on the dynamics of the Internet, as routing strategies and congestion effects, have already been treated in Section 3.2.2.
• Recently, the topological properties of the WWW have been studied from the viewpoint of data mining [630,631].
• The authors will come back to some general ideas on searching processes in Section 7.2.
• Here the authors focus instead on a different aspect of the dynamics that concerns the fluctuation of fluxes over the networks.
• A direct consequence of such an observation is that, in the case of the Internet, the fluctuations solely come from the local dynamics, whereas the WWW is strongly influenced by the external environment.

6.3. Metabolic, protein, and genetic networks

• In 1999, Hartwell and collaborators published an influential paper discussing the new challenges of modern biology [634].
• It is then natural to ask what these biological networks at the cell organization level look like and how their structure couples to the dynamics.
• Similarly to p53 network case, several other observations prove that some functional activities of the cell emerge from interactions between different cell’s components through complex webs.
• Metabolic reactions are catalyzed and regulated by enzymes.
• As a result, transcriptions and protein synthesis are also regulated by the interaction among genes and proteins.

6.3.1. Structure

• Publicly available database reporting integrated metabolic pathways, such as KEGG (http://www.genome.ad.jp/kegg), WIT (wit.mcs.anl.gov/wit) [639], EcoCyc (biocyc.org/ecocyc) [640], have allowed for a study of the structure of metabolic networks.
• It has been shown that the hierarchical structure resulting from the graph topology matched well to the biologically identified modules [645,646].
• There, it is shown that the metabolites in a metabolic network are far from being fully connected.
• Another class of well-studied cellular networks is that of protein–protein and protein–gene interaction networks.
• The eigenvalue spectrum of the adjacency matrix shows similarity to that of scale-free networks.

6.3.2. Dynamics

• During the last several years a wealth of experimental data, obtained with technological advances such as cDNA microarrays, have allowed the dynamical characterization of several biological processes both on a genome-wide and on a multi-gene scales and with fine time resolution.
• The authors think that it is important to provide at least some ideas about the research lines that relate the structure and the function of biological systems.
• The results obtained for elementary gene circuits certainly provide answers to intriguing questions about how gene circuits could be organized, but at the same time pose new ones.
• Recently, Gómez-Gardeñes et al. [686] have analyzed a model with activatory and inhibitory interactions that generically describes biological processes (as metabolic reactions and gene expression) and that can be used to explore the conditions for the existence of stable fixed points and periodic states in heterogeneous networks.
• This is achieved by regulatory mechanisms that necessarily require some kind of feedback control as that emerging in studies like [686].

6.4. Brain networks

• Neural assemblies (i.e. local networks of neurons transiently linked by selective interactions) are considered to be largely distributed and linked to form a Web-like structure of the brain [696].
• The resulting network, extremely sparse, is capable to coordinate and integrate distributed brain activities in a unified neural process.
• The interplay between neuron organizations and neural functions was pointed out in the early work of Ramon y’ Cajal (for a review see [697,698]).
• Evidence exists that human cerebral cortex is an ensemble of clusters of densely and reciprocally coupled cortical areas that are globally interconnected to form a large-scale cortical circuit [696,706].
• This plasticity renders neurons able to continuously change connections, or establish new ones according to the computational and communication needs.

6.4.1. Structure

• Most of the recent works aim to quantify the role of connectivity in the computational and communication abilities of neural networks.
• The aim was to classify connections as ascending, descending, or lateral, according to their patterns of origin and termination in the cortical layers.
• Other topological indices as the global efficiency [30], the fraction of links with a reciprocal connection, or the relative abundance of cycles of a given length within the network, were also computed to support the idea of a small-world wiring in cortical networks [191,742].
• Such local connectivity patterns are then related to functional contributions of each brain area to the whole cortical network [750,754].
• The authors must notice that correlation-based connectivity patterns lead to undirected graphs.

6.4.2. Dynamics

• The dynamical interactions between different neural assemblies allows to integrate different sources of information into a coherent brain task [696,706,708,709].
• Results suggest that cortical networks tend to maximize the interplay between local interactions and a global integration, which is reflected in high complexity values [191,742].
• If the connectivity per node is increased, random networks may display a coherent and synchronous dynamics.
• Different studies have shown that sparse connectivities may modify the capacity of storage and retrieval of patterns in different models of associative memory [799–801].
• Results showed that a diversity of neural dynamics as spontaneous, bursts, or seizure-like activities can be obtained by changing the coupling strength, the connectivity degree and/or the degree of randomness in the connections [810].

6.4.3. Open questions

• However it is reasonable that weighted and/or asymmetric couplings might play a role in neural dynamics [817]. .
• Most of connectivity schemes used in neural modelling do not take into account the spatial distance in the wiring procedure.
• Experimental evidence suggests that learning could induce changes in the wiring scheme [819].

7. Other topics

• In this final chapter the authors consider three topics that have recently attracted a large interest in the scientific community.
• The authors first discuss the problem of partitioning large graphs into their community structures, i.e. tightly connected subgraphs.
• The authors then review the recent advancements in finding reliable and fast ways for navigation and searching in a complex network.
• The chapter ends with a discussion of adaptive and dynamical wirings.

7.1. Algorithms for finding community structures

• Community structures are a typical feature of social networks, where some of the individuals can be part of a tightly connected group or of a closed social elite, others can be completely isolated, while some others may act as bridges between groups.
• In fact, social subgroups often have their own norms, orientations and subcultures, sometimes running counter to the official culture, and are the most important source of a person’s identity.
• Community structures are also an important property of other complex networks.
• The general aim of such methods is to find meaningful divisions into groups by investigating the structural properties of the whole graph.
• The authors then review some methods proposed in the physics community in the last few years.

7.1.1. Spectral graph partitioning

• Graph partitioning finds many practical applications such as load balancing in parallel computation, circuit partitioning and telephone network design, and is known to be a NP-complete problem [830].
• The network represents the friendship relationships between the members of a karate club and was obtained from data collected by an anthropologist, Wayne Zachary, over a 2-year period of observations.
• The two groups obtained by the spectral bisection method are shown in grey and white colors in Fig. 7.1a.
• If this is not the case, the division into a larger number of communities can be obtained by iterative bisection, although the disadvantage is that repeated bisections are not guaranteed to reach the best partition.

7.1.2. Hierarchical clustering

• A class of algorithms that work much better when there is no prior knowledge on the number of communities is the hierarchical clustering analysis used in social networks analysis [18,19].
• Let the distances between the clusters be the same as the distances between the items they contain.
• An alternative measure of structural equivalence is based on the correlation between rows (or columns) of the adjacency matrix.
• As an example of how the method works in practice, in Fig. 7.2 the authors show the dendrogram resulting from the application to the karate club network of Section 7.1.1.
• They do not tell which is the best division of the network, i.e. at which level the tree should be considered.

7.1.4. Variations and extensions of the GN algorithm

• Following the work of Ref. [673], a series of algorithms based on the idea of iteratively removing edges with high centrality score have been proposed.
• Such methods use different measures of edge centrality, as the random-walk betweenness, the current-flow betweenness [51], and the information centrality [851].
• For 6 zout 8, where the communities are very mixed and hardly detectable, the algorithm performs slightly better than the GN algorithm and the modularity-based algorithm of Section 7.1.5.
• This is not true for the second community.

7.1.5. Fast methods based on the modularity

• In order to deal with large networks, for which some of the previous algorithms turn out to be not viable, Newman has developed in Ref. [856] a fast method directly based on the optimization of the modularity of formula (7.3).
• Starting with N communities, each containing a single node, the communities are repeatedly joined together in pairs, by choosing at each step the join that results in the greatest increase (or smallest decrease) in Q.
• The method appears to work well both in test cases and in real-world situations and can be trivially generalized to weighted networks [856].
• By exploiting some shortcuts in the optimization problem and using more sophisticated data structures, the algorithm runs far more quickly, in time O(KD log N) where D is the depth of the “dendrogram” describing the network’s community structure.
• This is not merely a technical advance but has substantial practical implications, as it allows to study networks with million of nodes in reasonable run times.

7.1.6. Other methods based on spectral analysis

• The study of the eigenvector associated with the second smallest eigenvalue of , discussed in Section 7.1.1, is of practical use only when a clear partition into two parts exists, which is rarely the case.
• In most common occurrences, however, the number of nodes is large and the separation between the different communities is rather smooth so that the communities cannot be simply detected by looking iteratively at the first non-trivial eigenvector.
• The method proposed by Donetti et al. generalizes this idea.
• Then, a quantitative measure of similarity between two nodes of the graph is extracted either from the Euclidean or from the angular distance between the respective points in the D-dimensional space.
• The calculation of eigenvectors is the slowest part of the two previous algorithms.

7.1.7. Other algorithms

• Various other algorithms, based on the most different ideas can be found in the literature.
• The method is based on an approximate iterative algorithm that allows to solve the Kirchoff’s equations for node voltages in time O(N + K), i.e. in linear time (exact methods involve inversion of the Laplacian matrix and take a time O(N3) in the worst case [51,849]).
• Reichardt and Bornholdt have proposed in Ref. [860] a method based on a q-state Potts model associated to the graph.
• The distance matrix thus defined is asymmetric and can be calculated by solving N linear-algebraic equations [852].
• A dissimilarity index (7.7) Thus, A hierarchical clustering algorithm (see Section 7.1.2) is then worked out on the dissimilarity matrix D, and each of the resulting communities is characterized by an upper and a lower dissimilarity threshold [853].

7.2. Navigation and searching

• Milgram’s pioneering experiment and related works discussed in Section 2.2.1 elucidate two interesting aspects of social networks [89,90]: (1) relatively short paths exist in large size networks; (2) individuals are able to find short paths even in the absence of a global knowledge of the network.
• While the former feature is recovered in the various models described in Section 2.3, the latter property is not yet clearly understood.
• Two main approaches have been followed: one is based on finding the best strategy for constructing a path to a target node from an initial node by using local or geometric information; the second aims at assessing the most efficient network structure for path searching, once the way of finding a path is fixed.

7.2.1. Searching with local information

• The authors start by discussing the breadth-first search algorithm [40], also known as burning algorithm [862], that is one of the most used methods to find the shortest path between a source p and a target vertex q in a network [20–22].
• The method is based on the following iterative algorithm.
• If the highest-degree vertex has been already visited by the agent in the past, then the second highest is chosen, and so forth.
• Kim et al. have compared the average path length of BA scale-free networks as actually measured by different searching strategies such as random walk, maximum degree search, and the preferential choice strategy, i.e. a strategy in which the node with the larger degree has the higher probability to be chosen [861].
• The same happens for the looking up time LLUTs , that at each step of the searching is proportional to the number of neighbors of the node currently visited.

7.2.2. Network navigability

• In the previous section, the authors have discussed efficient searching methods that do not require knowledge on global topology.
• Social identities are sets of characteristics attributed to individuals such as their occupation or their geographical location.
• At the subsequent node j along the shortest path to the target, the number of questions to ask to find the right exit link is reduced to log2(kj − 1), since the incoming link is known.
• For some real networks, as, for example the Internet at the autonomous system level, S(l) is even negative at distances shorter than a certain horizon lhorizon, which implies that these real networks are organized to optimize the search at these short distances.
• As increases, the system undergoes a continuous phase transition to a congested phase in which N(t) ∝ t , that is, packets accumulate in the network (for more details see Section 3.2.2).

7.3. Adaptive and dynamical wirings

• Most of the results that have been described so far refer to cases where the wiring topology is static, i.e. it is fixed, or grown, once forever, and the dynamical processes are pertinent to the interactions among elements induced by such static connection schemes.
• This step forward is clearly motivated by the need of suitably modelling specific cases, such as genetic regulatory networks [687], ecosystems [869], financial markets [310,870], mutations and evolutions in social and biological phenomena [871], as well as to properly describe a series of technologically relevant problems emerging, e.g., in mobile and wireless connected units.
• In order to reproduce the essential dynamical features of such a mechanism, Ref. [543] investigates a specific model of evolving e-mail networks.
• Wiring evolution processes have been studied also in models of weighted complex networks.
• The walker maintains unchanged the modulus of its velocity, and assumes a direction for the velocity vector that is the average direction of its neighboring walkers plus an added noisy term chosen with uniform probability within a given interval [− /2, /2].

Figures

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.

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