Scale-free network

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that divides it randomly into four blocks. The generator thereafter is sequentially applied over and over again to only one of the available blocks picked preferentially with respect to their areas. It results in the partitioning of the square into ever smaller mutually exclusive rectangular blocks. The dual of the WPSL (DWPSL), which is obtained by replacing each block with a node at its center, and each common border between blocks with an edge joining the two corresponding vertices, emerges as a network whose degree distribution follows a power-law. The reason for it is that it grows following

1604: 1695:: the growth and the preferential attachment. By "growth" is meant a growth process where, over an extended period of time, new nodes join an already existing system, a network (like the World Wide Web which has grown by billions of web pages over 10 years). Finally, by "preferential attachment" is meant that new nodes prefer to connect to nodes that already have a high number of links with others. Thus, there is a higher probability that more and more nodes will link themselves to that one which has already many links, leading this node to a hub 1455: 61: 1660:). According to this process, a page with many in-links will attract more in-links than a regular page. This generates a power-law but the resulting graph differs from the actual Web graph in other properties such as the presence of small tightly connected communities. More general models and network characteristics have been proposed and studied. For example, Pachon et al. (2018) proposed a variant of the 2637:. It implies that the higher the links (degree) a node has, the higher its chance of gaining more links since they can be reached in a larger number of ways through mediators which essentially embodies the intuitive idea of rich get richer mechanism (or the preferential attachment rule of the Barabasi–Albert model). Therefore, the MDA network can be seen to follow the PA rule but in disguise. 917:, and thus that the citation network is scale-free. He did not however use the term "scale-free network", which was not coined until some decades later. In a later paper in 1976, Price also proposed a mechanism to explain the occurrence of power laws in citation networks, which he called "cumulative advantage" but which is today more commonly known under the name 1699:. Depending on the network, the hubs might either be assortative or disassortative. Assortativity would be found in social networks in which well-connected/famous people would tend to know better each other. Disassortativity would be found in technological (Internet, World Wide Web) and biological (protein interaction, metabolism) networks. 3542:. These networks are able to reproduce city-size distributions and electoral results by unraveling the size distribution of social groups with information theory on complex networks when a competitive cluster growth process is applied to the network. In models of scale-free ideal networks it is possible to demonstrate that 1500:, while targeted attacks destroys the connectedness very quickly. Other scale-free networks, which place the high-degree vertices at the periphery, do not exhibit these properties. Similarly, the clustering coefficient of scale-free networks can vary significantly depending on other topological details. 3720:

with the degrees of a few uniformly sampled nodes. However, since uniform sampling does not obtain enough samples from the important heavy-tail of the power law degree distribution, this method can yield a large bias and a variance. It has been recently proposed to sample random friends (i.e., random

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has recently been proposed whose coordination number distribution follow a power-law. It implies that the lattice has a few blocks which have astonishingly large number neighbors with whom they share common borders. Its construction starts with an initiator, say a square of unit area, and a generator

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At present, the more specific characteristics of scale-free networks vary with the generative mechanism used to create them. For instance, networks generated by preferential attachment typically place the high-degree vertices in the middle of the network, connecting them together to form a core, with

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Note that some models (see Dangalchev and Fitness model below) can work also statically, without changing the number of nodes. It should also be kept in mind that the fact that "preferential attachment" models give rise to scale-free networks does not prove that this is the mechanism underlying the

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A somewhat different generative model for Web links has been suggested by Pennock et al. (2002). They examined communities with interests in a specific topic such as the home pages of universities, public companies, newspapers or scientists, and discarded the major hubs of the Web. In this case, the

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coined the term "scale-free network" to describe the class of networks that exhibit a power-law degree distribution. However, studying seven examples of networks in social, economic, technological, biological, and physical systems, Amaral et al. were not able to find a scale-free network among these

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of the networks (adding new nodes) is not a necessary condition for creating a scale-free network (see Dangalchev). One possibility (Caldarelli et al. 2002) is to consider the structure as static and draw a link between vertices according to a particular property of the two vertices involved. Once

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et al., for example, demonstrated that a transformation which converts random graphs to their edge-dual graphs (or line graphs) produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient. Scale free graphs,

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distribution, which decreases as the node degree increases. This distribution also follows a power law. This implies that the low-degree nodes belong to very dense sub-graphs and those sub-graphs are connected to each other through hubs. Consider a social network in which nodes are people and links

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The most notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and are thought to serve specific purposes in their networks, although this depends greatly on the domain.

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Scale free topology has been also found in high temperature superconductors. The qualities of a high-temperature superconductor — a compound in which electrons obey the laws of quantum physics, and flow in perfect synchrony, without friction — appear linked to the fractal arrangements of seemingly

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is a variant of the preferential attachment model (proposed by Pachon et al.) which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1−p) that stresses the rich get richer system, and a uniform choice (with probability p) for the most recent

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A variant of the 2-L model, the k2 model, where first and second neighbour nodes contribute equally to a target node's attractiveness, demonstrates the emergence of transient scale-free networks. In the k2 model, the degree distribution appears approximately scale-free as long as the network is

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Although many real-world networks are thought to be scale-free, the evidence often remains inconclusive, primarily due to the developing awareness of more rigorous data analysis techniques. As such, the scale-free nature of many networks is still being debated by the scientific community. A few

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who mapped the topology of a portion of the World Wide Web, finding that some nodes, which they called "hubs", had many more connections than others and that the network as a whole had a power-law distribution of the number of links connecting to a node. After finding that a few other networks,

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Dangalchev (see ) builds a 2-L model by considering the importance of each of the neighbours of a target node in preferential attachment. The attractiveness of a node in the 2-L model depends not only on the number of nodes linked to it but also on the number of links in each of these nodes.

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The power-law degree distribution enables us to make "scale-free" assertions about the prevalence of high-degree nodes. For instance, we can say that "nodes with triple the average connectivity occur half as frequently as nodes with average connectivity." The specific numerical value of what

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The question of how to immunize efficiently scale free networks which represent realistic networks such as the Internet and social networks has been studied extensively. One such strategy is to immunize the largest degree nodes, i.e., targeted (intentional) attacks since for this case

1488:). In addition, the members of a community also have a few acquaintance relationships to people outside that community. Some people, however, are connected to a large number of communities (e.g., celebrities, politicians). Those people may be considered the hubs responsible for the 872:

is infinite but the first moment is finite), although occasionally it may lie outside these bounds. The name "scale-free" could be explained by the fact that some moments of the degree distribution are not defined, so that the network does not have a characteristic scale or "size".

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construction leads to a hierarchical network. Starting from a fully connected cluster of five nodes, we create four identical replicas connecting the peripheral nodes of each cluster to the central node of the original cluster. From this, we get a network of 25 nodes

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progressively lower-degree nodes making up the regions between the core and the periphery. The random removal of even a large fraction of vertices impacts the overall connectedness of the network very little, suggesting that such topologies could be useful for

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Krapivsky, Redner, and Leyvraz demonstrate that the scale-free nature of the network is destroyed for nonlinear preferential attachment. The only case in which the topology of the network is scale free is that in which the preferential attachment is

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relatively small, but significant deviations from the scale-free regime emerge as the network grows larger. This results in the relative attractiveness of nodes with different degrees changing over time, a feature also observed in real networks.

2191: 2419: 3217: 5067:

Poccia, Nicola; Ricci, Alessandro; Campi, Gaetano; Fratini, Michela; Puri, Alessandro; Di Gioacchino, Daniele; Marcelli, Augusto; Reynolds, Michael; Burghammer, Manfred; Saini, Naurang L.; Aeppli, Gabriel; Bianconi, Antonio (2012).

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generative model in which each new Web page creates links to existing Web pages with a probability distribution which is not uniform, but proportional to the current in-degree of Web pages. This model was originally invented by

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Many networks have been reported to be scale-free, although statistical analysis has refuted many of these claims and seriously questioned others. Additionally, some have argued that simply knowing that a degree-distribution is

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The history of scale-free networks also includes some disagreement. On an empirical level, the scale-free nature of several networks has been called into question. For instance, the three brothers Faloutsos believed that the

3470: 3265: = 25). Repeating the same process, we can create four more replicas of the original cluster – the four peripheral nodes of each one connect to the central node of the nodes created in the first step. This gives 3498:

Starting with scale free graphs with low degree correlation and clustering coefficient, one can generate new graphs with much higher degree correlations and clustering coefficients by applying edge-dual transformation.

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nodes. This modification is interesting to study the robustness of the scale-free behavior of the degree distribution. It is proved analytically that the asymptotically power-law degree distribution is preserved.

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is a measure of connectivity, generally quantified by a node's degree—that is, the number of links attached to it. Networks featuring a higher number of high-degree nodes are deemed to have greater connectivity.

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Fratini, Michela; Poccia, Nicola; Ricci, Alessandro; Campi, Gaetano; Burghammer, Manfred; Aeppli, Gabriel; Bianconi, Antonio (2010). "Scale-free structural organization of oxygen interstitials in La2CuO4+y".

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In today's terms, if a complex network has a power-law distribution of any of its metrics, it's generally considered a scale-free network. Similarly, any model with this feature is called a scale-free model.

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generative model which takes into account two different attachment rules: a preferential attachment mechanism and a uniform choice only for the most recent nodes. For a review see the book by Dorogovtsev and

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On a theoretical level, refinements to the abstract definition of scale-free have been proposed. For example, Li et al. (2005) offered a potentially more precise "scale-free metric". Briefly, let

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are acquaintance relationships between people. It is easy to see that people tend to form communities, i.e., small groups in which everyone knows everyone (one can think of such community as a

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to generate scale-free degree distributions. This heterogeneous degree distribution then simply reflects the negative curvature and metric properties of the underlying hyperbolic geometry.

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Many real networks are (approximately) scale-free and hence require scale-free models to describe them. In Price's scheme, there are two ingredients needed to build up a scale-free model:

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and second-neighbour preferential attachment may appear to generate transient scale-free networks, but the degree distribution deviates from a power law as networks become very large.

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and Rényi (1960) studied a model of growth for graphs in which, at each step, two nodes are chosen uniformly at random and a link is inserted between them. The properties of these

969:" and which is essentially the same as that proposed by Price. Analytic solutions for this mechanism (also similar to the solution of Price) were presented in 2000 by Dorogovtsev, 2635: 1688:

model studied by Kumar et al. (2000), in which new nodes choose an existent node at random and copy a fraction of the links of the existent node. This also generates a power law.

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is relatively high and less nodes are needed to be immunized. However, in many realistic cases the global structure is not available and the largest degree nodes are not known.

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When the concept of "scale-free" was initially introduced in the context of networks, it primarily referred to a specific trait: a power-law distribution for a given variable

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Hassan, M. K.; Islam, Liana; Arefinul Haque, Syed (2017). "Degree distribution, rank-size distribution, and leadership persistence in mediation-driven attachment networks".

3352: 836: 1933: 1707:

specified the statistical distribution for these vertex properties (fitnesses), it turns out that in some circumstances also static networks develop scale-free properties.

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Li, L.; Alderson, D.; Tanaka, R.; Doyle, J.C.; Willinger, W. (2005). "Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)".

3725:. Theoretically, maximum likelihood estimation with random friends lead to a smaller bias and a smaller variance compared to classical approach based on uniform sampling. 3602: 5763:

A. Hernando; D. Villuendas; C. Vesperinas; M. Abad; A. Plastino (2009). "Unravelling the size distribution of social groups with information theory on complex networks".

2047: 2923: 19: 6143: 2311: 4609: 3714: 3240: 3033: 3004: 2952: 2868: 2776: 2733: 2573: 2283: 1962: 1898: 1086: 804: 2527: 981:. Notably, however, this mechanism only produces a specific subset of networks in the scale-free class, and many alternative mechanisms have been discovered since. 5188:

Hassan, M. K.; Hassan, M. Z.; Pavel, N. I. (2010). "Scale-free coordination number disorder and multifractal size disorder in weighted planar stochastic lattice".

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evolution of real-world scale-free networks, as there might exist different mechanisms at work in real-world systems that nevertheless give rise to scaling.

5212:

Pachon, Angelica; Sacerdote, Laura; Yang, Shuyi (2018). "Scale-free behavior of networks with the copresence of preferential and uniform attachment rules".

1681:. Based on these observations, the authors proposed a generative model that mixes preferential attachment with a baseline probability of gaining a link. 4590: 4222:

Falkenberg, Max; Lee, Jong-Hyeok; Amano, Shun-ichi; Ogawa, Ken-ichiro; Yano, Kazuo; Miyake, Yoshihiro; Evans, Tim S.; Christensen, Kim (18 June 2020).

577: 3360: 6238: 4952:

Steyvers, Mark; Joshua B. Tenenbaum (2005). "The Large-Scale Structure of Semantic Networks: Statistical Analyses and a Model of Semantic Growth".

4355:, the movie-actor network had a power law regime followed by a sharp cutoff. None of Amaral et al's examples obeyed the power law regime for large 1719:. The recipe of Barabási and Albert has been followed by several variations and generalizations and the revamping of previous mathematical works. 1421:

However, there's a key difference. In statistical field theory, the term "scale" often pertains to system size. In the realm of networks, "scale"

3751: 6702: 5788:

André A. Moreira; Demétrius R. Paula; Raimundo N. Costa Filho; José S. Andrade, Jr. (2006). "Competitive cluster growth in complex networks".

3354:. In the case of World Trade Web it is possible to reconstruct all the properties by using as fitnesses of the country their GDP, and taking 5702:

Krioukov, Dmitri; Papadopoulos, Fragkiskos; Kitsak, Maksim; Vahdat, Amin; Boguñá, Marián (2010). "Hyperbolic geometry of complex networks".

2427: 4156:

Krapivsky, Paul; Krioukov, Dmitri (21 August 2008). "Scale-free networks as preasymptotic regimes of superlinear preferential attachment".

6325:

Dorogovtsev, S.N.; Mendes, J.F.F.; Samukhin, A.N. (2000). "Structure of Growing Networks: Exact Solution of the Barabási—Albert's Model".

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and second neighbour attachment generate networks which are transiently scale-free, but deviate from a power law as networks grow large.

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have been proposed as mechanisms to explain conjectured power law degree distributions in real networks. Alternative models such as

1094: 5283: 1796: 3006:

is not linear, and recent studies have demonstrated that the degree distribution depends strongly on the shape of the function

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Caldarelli G.; Capocci A.; De Los Rios P.; Muñoz M.A. (2002). "Scale-free networks from varying vertex intrinsic fitness".

4594: 4359:, i.e. none of these seven examples were shown to be scale-free. See especially the beginning of the discussion section of 6006: 1640:

are different from the properties found in scale-free networks, and therefore a model for this growth process is needed.

6567: 3746: 2744: 1670: 890: 567: 296: 3045: 6631: 6389: 6132: 2213: 1621: 1196: 6694: 4660:

Statistical Field Theory: Volume 2, Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems

641: 224: 6187: 1666: 970: 3808:

Onnela, J.-P.; Saramaki, J.; Hyvonen, J.; Szabo, G.; Lazer, D.; Kaski, K.; Kertesz, J.; Barabasi, A. -L. (2007).

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is more important than knowing whether a network is scale-free according to statistically rigorous definitions.

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Ramezanpour, A.; Karimipour, V.; Mashaghi, A. (2003). "Generating correlated networks from uncorrelated ones".

1462: 886: 527: 5310: 4278: 3784: 1341: 925: 522: 6509:. Vol. 5. Publication of the Mathematical Institute of the Hungarian Academy of Science. pp. 17–61. 5787: 4430:

Dorogovtsev, S.; Mendes, J.; Samukhin, A. (2000). "Structure of Growing Networks with Preferential Linking".

954:, though eventually this power law regime was followed by a sharp cutoff showing exponential decay for large 677: 636: 153: 4494:; Riordan, O.; Spencer, J.; Tusnády, G. (2001). "The degree sequence of a scale-free random graph process". 6514:

Faloutsos, M.; Faloutsos, P.; Faloutsos, C. (1999). "On power-law relationships of the internet topology".

5872:"Tail-scope: Using friends to estimate heavy tails of degree distributions in large-scale complex networks" 5849:

Heydari, H.; Taheri, S.M.; Kaveh, K. (2018). "Distributed Maximal Independent Set on Scale-Free Networks".

3923:

Clauset, Aaron; Cosma Rohilla Shalizi; M. E. J Newman (2009). "Power-law distributions in empirical data".

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ends of random links) who are more likely come from the tail of the degree distribution as a result of the

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The most widely known generative model for a subset of scale-free networks is Barabási and Albert's (1999)

1002: 482: 326: 273: 88: 3778: 2598: 1786: 1692: 517: 24: 6730: 5641:

Garlaschelli, D.; et al. (2004). "Fitness-Dependent Topological Properties of the World Trade Web".

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proposed a generative mechanism to explain the appearance of power-law distributions, which they called "

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including some social and biological networks, also had heavy-tailed degree distributions, Barabási and

5952:"Maximum Likelihood Estimation of Power-law Degree Distributions via Friendship Paradox-based Sampling" 3740: 3481: 3250: 2236:

edges picks an existing connected node at random and then connects itself, not with that one, but with

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Statistical Field Theory: Volume 1, From Brownian Motion to Renormalization and Lattice Gauge Theory

3308: 2186:{\displaystyle \Pi (k_{i})={\frac {k_{i}+C\sum _{(i,j)}k_{j}}{\sum _{j}k_{j}+C\sum _{j}k_{j}^{2}}},} 1935:, i.e. there is a nonzero probability that a new node attaches to an isolated node. Thus in general 1777:

There have been several attempts to generate scale-free network properties. Here are some examples:

1903: 933: 905:

showed in 1965 that the number of links to papers—i.e., the number of citations they receive—had a

809: 670: 572: 532: 39: 6399:

Dorogovtsev, S.N.; Goltsev A.V.; Mendes, J.F.F. (2008). "Critical phenomena in complex networks".

5592: 5313:; Zoltán N., Oltvai. (2004). "Network biology: understanding the cell's functional organization". 4351:

Among the seven examples studied by Amaral et al, six of them where single-scale and only example

3581: 430: 6811: 3493: 2414:{\displaystyle \Pi (i)={\frac {k_{i}}{N}}{\frac {\sum _{j=1}^{k_{i}}{\frac {1}{k_{j}}}}{k_{i}}}.} 1742: 1649: 966: 918: 902: 882: 878: 653: 472: 239: 173: 128: 3763: – Features that do not change if length or energy scales are multiplied by a common factor 3656: 6793:

Kasthurirathna, D.; Piraveenan, M. (2015). "Complex Network Study of Brazilian Soccer Player".

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There are two major components that explain the emergence of the power-law distribution in the

1656:, but did not reach popularity until Barabási rediscovered the results under its current name ( 1489: 1480: 557: 542: 457: 1190:

This is maximized when high-degree nodes are connected to other high-degree nodes. Now define

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De Masi, Giulia; et al. (2006). "Fitness model for the Italian interbank money market".

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Meng, Xiangyi; Zhou, Bin (2023). "Scale-Free Networks beyond Power-Law Degree Distribution".

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illusion created by routers, which appear as high-degree nodes while concealing the internal

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seven examples. Only one of these examples, the movie-actor network, had degree distribution

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Degree distribution for a network with 150000 vertices and mean degree = 6 created using the

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Assuming that a network has an underlying hyperbolic geometry, one can use the framework of

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constitutes "average connectivity" becomes irrelevant, whether it's a hundred or a million.

6757: 6654: 6594: 6465: 6418: 6344: 6305: 6264: 6202: 6158: 6079: 6028: 5893: 5807: 5721: 5660: 5607: 5549: 5496: 5374: 5231: 5160: 5091: 5025: 4916: 4865: 4804: 4748: 4705: 4556: 4515: 4449: 4384: 4322: 4302: 4245: 4175: 4111: 4056: 3999: 3942: 3890: 3831: 3536: 3277:

The idea is that the link between two vertices is assigned not randomly with a probability

2505: 1418:, evoking parallels with the renormalization group techniques in statistical field theory. 425: 306: 6551: 3212:{\displaystyle P(k)\sim k^{-\gamma }{\text{ with }}\gamma =1+{\frac {\mu }{a_{\infty }}}.} 2669: 1748: 1603: 8: 6738:

Onody, R.N.; de Castro, P.A. (2004). "Complex Network Study of Brazilian Soccer Player".

5139:"Scale-free network topology and multifractality in a weighted planar stochastic lattice" 3532: 3257: 3222:

This way the exponent of the degree distribution can be tuned to any value between 2 and

2643: 1716: 1678: 910: 706: 462: 331: 321: 316: 168: 113: 103: 6761: 6658: 6598: 6550:

Kumar, R.; Raghavan, P.; Rajagopalan, S.; Sivakumar, D.; Tomkins, A.; Upfal, E. (2000).

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Stumpf, M. P. H.; Porter, M. A. (10 February 2012). "Critical Truths About Power Laws".

4060: 4003: 3946: 3894: 3835: 1554:, including collaboration networks. Two examples that have been studied extensively are 6781: 6747: 6610: 6584: 6536: 6481: 6455: 6434: 6408: 6368: 6334: 6254: 6226: 6069: 6044: 6018: 5963: 5924: 5883: 5871: 5850: 5831: 5797: 5764: 5745: 5711: 5684: 5650: 5573: 5539: 5512: 5486: 5338: 5247: 5221: 5150: 5114: 5081: 5069: 5049: 5015: 4987: 4961: 4889: 4855: 4828: 4794: 4721: 4695: 4572: 4546: 4519: 4491: 4473: 4439: 4374: 4334: 4292: 4235: 4199: 4165: 4135: 4079: 4044: 4020: 3989: 3977: 3958: 3932: 3854: 3821: 3809: 3722: 3561: 3543: 2957: 2873: 2824: 2804: 2781: 2692: 2578: 2532: 2288: 2239: 2219: 2015: 1532:

Properties of random graph may change or remain invariant under graph transformations.

1512: 1424: 1383:. This property maintains its form when subjected to a continuous scale transformation 1321: 1039: 1019: 978: 301: 254: 229: 118: 6677: 6170: 5266: 3879:"Scale-Free Graph with Preferential Attachment and Evolving Internal Vertex Structure" 6816: 6773: 6722: 6682: 6627: 6614: 6497: 6385: 6372: 6360: 6280: 6218: 6174: 6128: 6121: 6107: 6102: 6057: 5983: 5929: 5911: 5823: 5737: 5676: 5623: 5565: 5330: 5251: 5119: 5041: 4979: 4881: 4820: 4764: 4725: 4663: 4638: 4477: 4465: 4412: 4407: 4362: 4326: 4283: 4258: 4223: 4191: 4139: 4127: 4084: 4025: 3859: 1735: 598: 264: 214: 123: 98: 6785: 5835: 5688: 5577: 5516: 5342: 4893: 4832: 4203: 6765: 6714: 6672: 6662: 6602: 6523: 6473: 6438: 6426: 6352: 6313: 6272: 6230: 6210: 6166: 6097: 6087: 6036: 5973: 5919: 5901: 5815: 5749: 5729: 5668: 5615: 5557: 5504: 5382: 5322: 5275: 5239: 5168: 5109: 5099: 5053: 5033: 4991: 4971: 4932: 4924: 4873: 4812: 4756: 4713: 4564: 4523: 4503: 4457: 4402: 4392: 4310: 4253: 4183: 4119: 4074: 4064: 4015: 4007: 3962: 3950: 3898: 3849: 3839: 3760: 1790: 1593: 1566: 1533: 1497: 357: 346: 244: 204: 188: 27:(blue dots). The distribution follows an analytical form given by the ratio of two 6559:

Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS)

6485: 6276: 5672: 5619: 4760: 4595:"Mathematics and the Internet: A Source of Enormous Confusion and Great Potential" 4576: 4338: 6549: 6048: 4511: 4318: 4314: 3766: 3487: 1661: 1644: 1307: 839: 702: 593: 374: 249: 158: 93: 47: 6640: 6356: 6317: 5508: 5386: 5243: 4928: 4907:

Soramäki, Kimmo; et al. (2007). "The topology of interbank payment flows".

4717: 4461: 3465:{\displaystyle p(x_{i},x_{j})={\frac {\delta x_{i}x_{j}}{1+\delta x_{i}x_{j}}}.} 1559: 6769: 6501: 6430: 6244: 6214: 5819: 5733: 5561: 4975: 4877: 4816: 4281:; Albert, Réka. (October 15, 1999). "Emergence of scaling in random networks". 4187: 4069: 4011: 3528: 3520: 3039: 2709:

value increases the disparity between the super rich and poor decreases and as

1578: 1551: 1485: 603: 409: 384: 379: 353: 342: 219: 183: 178: 138: 76: 28: 6477: 5530:

Ravasz, E.; Barabási (2003). "Hierarchical organization in complex networks".

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Edge dual transformation to generate scale free graphs with desired properties

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we find a transition from rich get super richer to rich get richer mechanism.

962: 938: 929: 6805: 6643:"Winners don't take all: Characterizing the competition for links on the web" 6606: 6324: 6040: 5987: 5915: 5279: 5274:. Foundations of Computer Science, 41st Annual Symposium on. pp. 57–65. 3685: 994: 974: 562: 467: 452: 394: 143: 133: 6641:

Pennock, D.M.; Flake, G.W.; Lawrence, S.; Glover, E.J.; Giles, C.L. (2002).

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Newman, Mark E.J. (2003). "The structure and function of complex networks".

6493: 5762: 5104: 4123: 3844: 1633: 1454: 1278:

in the set of all graphs with degree distribution identical to that of

6777: 6726: 6686: 6667: 6398: 6364: 6284: 6222: 6178: 6111: 6092: 5933: 5827: 5741: 5680: 5627: 5569: 5334: 5123: 5045: 4983: 4885: 4824: 4768: 4469: 4416: 4397: 4330: 4195: 4131: 4088: 4029: 3863: 3734: 1637: 507: 404: 259: 6527: 3922: 6752: 6589: 6541: 6460: 6339: 6259: 6074: 6023: 5802: 5655: 5544: 5425:

S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhim, cond-mat/0011115.

4966: 4799: 4551: 4444: 4379: 4297: 5037: 4860: 3826: 3604:, the induced subgraph constructed by vertices with degrees larger than 6718: 6624:

Evolution and Structure of the Internet: A Statistical Physics Approach

6534: 6513: 6382:

Evolution of Networks: from biological networks to the Internet and WWW

4937: 1624:

rule which also embodies preferential attachment rule but in disguise.

990: 442: 399: 389: 5906: 4537:

Dorogovtsev, S. N.; Mendes, J. F. F. (2002). "Evolution of networks".

3954: 2689:

of the total nodes has degree one and one is super-rich in degree. As

2666:

it describes the winner takes it all mechanism as we find that almost

2495:{\displaystyle {\frac {\sum _{j=1}^{k_{i}}{\frac {1}{k_{j}}}}{k_{i}}}} 5305: 5303: 4507: 4273: 4271: 4269: 4151: 4149: 3507: 1306:) close to 1 is "scale-free". This definition captures the notion of 914: 710: 607: 163: 6293: 5978: 5951: 5362: 5326: 4589: 3502: 1738:. Usually we concentrate on growing the network, i.e. adding nodes. 924:

Recent interest in scale-free networks started in 1999 with work by

60: 6446:

Dorogovtsev, S.N.; Mendes, J.F.F. (2002). "Evolution of networks".

5968: 5855: 5226: 4700: 4240: 3994: 3772: 2549:. Extensive numerical investigation suggest that for approximately 1657: 1574: 1570: 986: 901:

In studies of the networks of citations between scientific papers,

6413: 5888: 5769: 5716: 5491: 5300: 5155: 5086: 5020: 4266: 4170: 4146: 3937: 6642: 6621: 5463:

S. Bomholdt and H. Ebel, cond-mat/0008465; H.A. Simon, Bimetrika

5070:"Optimum inhomogeneity of local lattice distortions in La2CuO4+y" 998: 6792: 6055: 5848: 5701: 4360: 3281:

equal for all the couple of vertices. Rather, for every vertex

1677:

distribution of links was no longer a power law but resembled a

6445: 6379: 6141: 3743: – Two closely related models for generating random graphs 1479:

Another important characteristic of scale-free networks is the

6188:"Topology of Large-Scale Engineering Problem-Solving Networks" 3810:"Structure and tie strengths in mobile communication networks" 3253:

are, by design, scale free and have high clustering of nodes.

4784: 4490: 3269: = 125, and the process can continue indefinitely. 5437:

P.L. Krapivsky, S. Redner, and F. Leyvraz, Phys. Rev. Lett.

2502:

is the inverse of the harmonic mean (IHM) of degrees of the

1466:

Complex network degree distribution of random and scale-free

1180:{\displaystyle s(G)=\sum _{(u,v)\in E}\deg(u)\cdot \deg(v).} 977:, and Leyvraz, and later rigorously proved by mathematician 5950:

Nettasinghe, Buddhika; Krishnamurthy, Vikram (2021-05-19).

5593:"Scale-free networks from varying vertex intrinsic fitness" 4951: 1861:{\displaystyle \Pi (k_{i})={\frac {k_{i}}{\sum _{j}k_{j}}}} 1607:

A snapshot of the weighted planar stochastic lattice (WPSL)

1584:

Some financial networks such as interbank payment networks

6737: 4045:"Rare and everywhere: Perspectives on scale-free networks" 3876: 3807: 2207: 5476: 4429: 2256:

of its neighbors, also chosen at random. The probability

5949: 6185: 6118: 6004: 5066: 5004: 4739:

Tanaka, Reiko (2005). "Scale-Rich Metabolic Networks".

4221: 3716:

of a scale-free network is typically done by using the

2841:. This assumption involves two hypotheses: first, that 2749:

The Barabási–Albert model assumes that the probability

1546:

examples of networks claimed to be scale-free include:

1537:

as such, remain scale free under such transformations.

776:{\displaystyle P(k)\ \sim \ k^{\boldsymbol {-\gamma }}} 6123:

Linked: How Everything is Connected to Everything Else

6056:

Amaral LAN, Scala A, Barthelemy M, Stanley HE (2000).

4662:(1st ed.). New York: Cambridge University Press. 4637:(1st ed.). New York: Cambridge University Press. 4361:

Amaral LAN, Scala A, Barthelemy M, Stanley HE (2000).

1715:

There has been a burst of activity in the modeling of

812: 5367:

Physica A: Statistical Mechanics and Its Applications

4909:

Physica A: Statistical Mechanics and Its Applications

3769: – Network with non-trivial topological features 3702: 3659: 3610: 3584: 3564: 3363: 3311: 3228: 3143: 3107: 3048: 3012: 2983: 2960: 2931: 2896: 2876: 2847: 2827: 2807: 2784: 2755: 2738: 2715: 2695: 2672: 2646: 2601: 2581: 2555: 2535: 2508: 2430: 2314: 2291: 2262: 2242: 2222: 2050: 2018: 1970: 1941: 1906: 1877: 1799: 1751: 1515: 1427: 1389: 1344: 1324: 1282:. This gives a metric between 0 and 1, where a graph 1199: 1097: 1062: 1042: 1022: 848: 806:

is a parameter whose value is typically in the range

792: 738: 16:

Network whose degree distribution follows a power law

6142:

Barabási, Albert-László; Bonabeau, Eric (May 2003).

3789:

Pages displaying wikidata descriptions as a fallback

3756:

Pages displaying wikidata descriptions as a fallback

3691: 1765:

that new nodes will be connected to the "old" node.

993:

data; however, it has been suggested that this is a

989:

had a power law degree distribution on the basis of

725:

connections to other nodes goes for large values of

6561:. Redondo Beach, CA: IEEE CS Press. pp. 57–65. 5211: 5137:Hassan, M. K.; Hassan, M. Z.; Pavel, N. I. (2010). 4780: 4778: 3975: 3877:Choromański, K.; Matuszak, M.; MiȩKisz, J. (2013). 2977:In non-linear preferential attachment, the form of 6120: 4681: 4679: 4651: 4626: 3708: 3676: 3645: 3596: 3570: 3464: 3346: 3234: 3211: 3126: 3093: 3027: 2998: 2966: 2946: 2917: 2882: 2862: 2833: 2813: 2790: 2770: 2727: 2701: 2681: 2658: 2629: 2587: 2567: 2541: 2521: 2494: 2413: 2297: 2277: 2248: 2228: 2185: 2024: 2004: 1956: 1927: 1892: 1860: 1757: 1632:Scale-free networks do not arise by chance alone. 1521: 1433: 1410: 1375: 1330: 1248: 1179: 1080: 1048: 1028: 864: 830: 798: 775: 6574: 5956:ACM Transactions on Knowledge Discovery from Data 5309: 5187: 5136: 4536: 4155: 3503:Uniform-preferential-attachment model (UPA model) 713:, at least asymptotically. That is, the fraction 6803: 4775: 3094:{\displaystyle \Pi (k_{i})\sim a_{\infty }k_{i}} 1236: 6291: 5412:R. Albert, and A.L. Barabási, Phys. Rev. Lett. 4676: 4658:Itzykson, Claude; Drouffe, Jean-Michel (1989). 4657: 4633:Itzykson, Claude; Drouffe, Jean-Michel (1989). 4632: 4224:"Identifying time dependence in network growth" 3814:Proceedings of the National Academy of Sciences 3781: – Scale-free network generation algorithm 1710: 1249:{\displaystyle S(G)={\frac {s(G)}{s_{\max }}},} 31:(black line) which approximates as a power-law. 5529: 4732: 4277: 3475: 5264: 4608:(5). American Mathematical Society: 586–599. 3546:is the cause of the phenomenon known as the ' 3245: 1560:the co-authorship by mathematicians of papers 1458:Random network (a) and scale-free network (b) 973:and Samukhin and independently by Krapivsky, 678: 6700: 6622:Pastor-Satorras, R.; Vespignani, A. (2004). 5640: 4593:; David Alderson; John C. Doyle (May 2009). 4101: 3514: 2032:is the initial attractiveness of the node.) 1780: 1612:random oxygen atoms and lattice distortion. 950:) following a power law regime for moderate 6565: 6294:"Generation models for scale-free networks" 6007:"Statistical mechanics of complex networks" 5870:Eom, Young-Ho; Jo, Hang-Hyun (2015-05-11). 5363:"Generation models for scale-free networks" 4217: 4215: 4213: 3752:Bose–Einstein condensation (network theory) 3737: – Graph generated by a random process 3646:{\displaystyle \log {n}\times \log ^{*}{n}} 6492: 6380:Dorogovtsev, S.N.; Mendes, J.F.F. (2003). 5695: 5590: 5360: 3976:Broido, Anna; Aaron Clauset (2019-03-04). 3134:. In this case the rate equation leads to 2925:, and second, that the functional form of 2035: 1868:and adds one new node at every time step. 1556:the collaboration of movie actors in films 1036:(that is, the number of edges incident to 865:{\displaystyle k^{\boldsymbol {-\gamma }}} 685: 671: 6751: 6676: 6666: 6588: 6540: 6459: 6412: 6338: 6258: 6101: 6091: 6073: 6022: 5977: 5967: 5923: 5905: 5887: 5854: 5801: 5768: 5715: 5654: 5543: 5490: 5265:Kumar, Ravi; Raghavan, Prabhakar (2000). 5225: 5172: 5154: 5113: 5103: 5085: 5019: 4965: 4936: 4859: 4798: 4699: 4583: 4550: 4443: 4406: 4396: 4378: 4296: 4257: 4239: 4169: 4078: 4068: 4019: 3993: 3936: 3918: 3916: 3914: 3902: 3853: 3843: 3825: 1617:weighted planar stochastic lattice (WPSL) 4906: 4685: 4210: 3553: 2890:, in contrast to random graphs in which 1602: 1461: 1453: 1376:{\displaystyle f(k)\propto k^{-\gamma }} 18: 6241:Oxford University Press, Oxford (2007). 5433: 5431: 5399:Barabási, A.-L. and R. Albert, Science 4845: 2214:mediation-driven attachment (MDA) model 2208:Mediation-driven attachment (MDA) model 2200:is a coefficient between 0 and 1. 857: 768: 6804: 5356: 5354: 5352: 5207: 5205: 5203: 4738: 3911: 6552:"Stochastic models for the web graph" 5945: 5943: 5869: 5444: 4042: 3969: 2595:limit becomes a constant which means 2005:{\displaystyle \Pi (k)=A+k^{\alpha }} 1793:has a linear preferential attachment 5591:Caldarelli, G.; et al. (2002). 5428: 5419: 3775: – Graph of connected web pages 2630:{\displaystyle \Pi (i)\propto k_{i}} 1671:super-linear preferential attachment 1627: 1615:A space-filling cellular structure, 1016:, and denote the degree of a vertex 891:super-linear preferential attachment 5634: 5584: 5523: 5457: 5349: 5268:Stochastic Models for the Web Graph 5200: 13: 6703:"Revisiting "scale-free" networks" 6186:Dan Braha; Yaneer Bar-Yam (2004). 6005:Albert R.; Barabási A.-L. (2002). 5998: 5940: 5406: 5393: 3747:Non-linear preferential attachment 3696:Estimating the power-law exponent 3229: 3199: 3121: 3076: 3049: 3013: 2984: 2932: 2897: 2848: 2756: 2745:Non-linear preferential attachment 2739:Non-linear preferential attachment 2676: 2602: 2315: 2263: 2051: 1971: 1942: 1907: 1878: 1871:(Note, another general feature of 1800: 1752: 1449: 1310:implied in the name "scale-free". 14: 6828: 6695:Scale-Free Networks and Terrorism 6503:On the Evolution of Random Graphs 6171:10.1038/scientificamerican0503-60 6058:"Classes of small-world networks" 5361:Dangalchev, Chavdar (July 2004). 4363:"Classes of small-world networks" 3692:Estimating the power law exponent 1622:mediation-driven attachment model 1411:{\displaystyle k\to k+\epsilon k} 721:) of nodes in the network having 6566:Matlis, Jan (November 4, 2002). 6119:Barabási, Albert-László (2004). 4496:Random Structures and Algorithms 4259:10.1103/PhysRevResearch.2.023352 3787: – model in network science 3754: – model in network science 3272: 3127:{\displaystyle k_{i}\to \infty } 2575:the mean IHM value in the large 1684:Another generative model is the 854: 765: 59: 5863: 5842: 5781: 5756: 5470: 5289:from the original on 2016-03-03 5258: 5181: 5130: 5060: 4998: 4945: 4900: 4839: 4615:from the original on 2011-05-15 4530: 4484: 4423: 4345: 4043:Holme, Petter (December 2019). 3305:is created with a probability 2305:of the existing node picked is 1503: 1294:) is "scale-rich", and a graph 6626:. Cambridge University Press. 5214:Physica D: Nonlinear Phenomena 4688:Chaos, Solitons & Fractals 4095: 4036: 3978:"Scale-free networks are rare" 3883:Journal of Statistical Physics 3870: 3801: 3558:For a scale-free network with 3393: 3367: 3347:{\displaystyle p(x_{i},x_{j})} 3341: 3315: 3153: 3147: 3118: 3065: 3052: 3022: 3016: 2993: 2987: 2941: 2935: 2906: 2900: 2857: 2851: 2765: 2759: 2611: 2605: 2324: 2318: 2272: 2266: 2109: 2097: 2067: 2054: 1980: 1974: 1951: 1945: 1916: 1910: 1887: 1881: 1816: 1803: 1393: 1354: 1348: 1227: 1221: 1209: 1203: 1171: 1165: 1153: 1147: 1130: 1118: 1107: 1101: 1075: 1069: 831:{\textstyle 2<\gamma <3} 748: 742: 1: 6277:10.1103/PhysRevLett.89.258702 5673:10.1103/physrevlett.93.188701 5620:10.1103/physrevlett.89.258702 5174:10.1088/1367-2630/12/9/093045 4761:10.1103/PhysRevLett.94.168101 3794: 3718:maximum likelihood estimation 3653:is a scale-free network with 3578:nodes and power-law exponent 2778:that a node attaches to node 1928:{\displaystyle \Pi (0)\neq 0} 1772: 1474: 4315:10.1126/science.286.5439.509 3597:{\displaystyle \gamma >3} 1711:Generalized scale-free model 838:(wherein the second moment ( 7: 6384:. Oxford University Press. 6357:10.1103/PhysRevLett.85.4633 6318:10.1016/j.physa.2004.01.056 5777:European Physical Journal B 5509:10.1016/j.physa.2016.11.001 5387:10.1016/j.physa.2004.01.056 5244:10.1016/j.physd.2018.01.005 4929:10.1016/j.physa.2006.11.093 4718:10.1016/j.chaos.2023.114173 4462:10.1103/PhysRevLett.85.4633 3728: 3476:Hyperbolic geometric graphs 3251:Hierarchical network models 1789:, an undirected version of 1726: 1717:scale-free complex networks 1588:Protein–protein interaction 1540: 1313: 10: 6833: 6770:10.1103/PhysRevE.70.037103 6431:10.1103/RevModPhys.80.1275 6215:10.1103/PhysRevE.69.016113 5820:10.1103/PhysRevE.73.065101 5734:10.1103/PhysRevE.82.036106 5562:10.1103/physreve.67.026112 4976:10.1207/s15516709cog2901_3 4878:10.1103/PhysRevE.74.066112 4817:10.1103/PhysRevE.67.046107 4188:10.1103/PhysRevE.78.026114 4070:10.1038/s41467-019-09038-8 4012:10.1038/s41467-019-08746-5 3677:{\displaystyle \gamma '=2} 3482:Hyperbolic geometric graph 3479: 3297:and a link between vertex 3246:Hierarchical network model 2742: 1669:. Some mechanisms such as 896: 6478:10.1080/00018730110112519 4569:10.1080/00018730110112519 3904:10.1007/s10955-013-0749-1 3548:six degrees of separation 3515:Scale-free ideal networks 2918:{\displaystyle \Pi (k)=p} 2216:, a new node coming with 1900:in real networks is that 1781:The Barabási–Albert model 1012:be a graph with edge set 907:heavy-tailed distribution 538:Exponential random (ERGM) 205:Informational (computing) 6607:10.1137/S003614450342480 6292:Dangalchev, Ch. (2004). 6041:10.1103/RevModPhys.74.47 5280:10.1109/SFCS.2000.892065 4228:Physical Review Research 3525:scale-free ideal network 1266:is the maximum value of 934:University of Notre Dame 225:Scientific collaboration 6247:Physical Review Letters 5315:Nature Reviews Genetics 5311:Barabási, Albert-László 5105:10.1073/pnas.1208492109 4432:Physical Review Letters 4279:Barabási, Albert-László 4124:10.1126/science.1216142 3845:10.1073/pnas.0610245104 3785:Bianconi–Barabási model 3709:{\displaystyle \gamma } 3235:{\displaystyle \infty } 3028:{\displaystyle \Pi (k)} 2999:{\displaystyle \Pi (k)} 2947:{\displaystyle \Pi (k)} 2863:{\displaystyle \Pi (k)} 2798:is proportional to the 2771:{\displaystyle \Pi (k)} 2728:{\displaystyle m>14} 2568:{\displaystyle m>14} 2278:{\displaystyle \Pi (i)} 2036:Two-level network model 1957:{\displaystyle \Pi (k)} 1893:{\displaystyle \Pi (k)} 1743:Preferential attachment 1652:in 1965 under the term 1650:Derek J. de Solla Price 1081:{\displaystyle \deg(v)} 967:preferential attachment 919:preferential attachment 883:Preferential attachment 799:{\displaystyle \gamma } 654:Category:Network theory 174:Preferential attachment 6668:10.1073/pnas.032085699 6093:10.1073/pnas.200327197 5143:New Journal of Physics 4398:10.1073/pnas.200327197 3710: 3678: 3647: 3598: 3572: 3466: 3348: 3285:there is an intrinsic 3236: 3213: 3128: 3095: 3029: 3000: 2968: 2948: 2919: 2884: 2864: 2835: 2815: 2792: 2772: 2729: 2703: 2683: 2660: 2631: 2589: 2569: 2543: 2523: 2496: 2461: 2415: 2377: 2299: 2279: 2250: 2230: 2187: 2026: 2006: 1958: 1929: 1894: 1862: 1759: 1734:1. Adding or removing 1608: 1523: 1490:small-world phenomenon 1481:clustering coefficient 1467: 1459: 1435: 1412: 1377: 1332: 1250: 1181: 1082: 1050: 1030: 926:Albert-László Barabási 866: 832: 800: 777: 543:Random geometric (RGG) 32: 6701:Keller, E.F. (2005). 6568:"Scale-Free Networks" 6528:10.1145/316194.316229 6144:"Scale-Free Networks" 4049:Nature Communications 3982:Nature Communications 3779:Barabási–Albert model 3711: 3679: 3648: 3599: 3573: 3554:Novel characteristics 3467: 3349: 3237: 3214: 3129: 3096: 3030: 3001: 2969: 2949: 2920: 2885: 2865: 2836: 2816: 2793: 2773: 2730: 2704: 2684: 2661: 2632: 2590: 2570: 2544: 2524: 2522:{\displaystyle k_{i}} 2497: 2434: 2416: 2350: 2300: 2280: 2251: 2231: 2188: 2027: 2007: 1959: 1930: 1895: 1863: 1787:Barabási–Albert model 1760: 1693:Barabási–Albert model 1606: 1524: 1465: 1457: 1436: 1413: 1378: 1333: 1251: 1182: 1083: 1051: 1031: 867: 833: 801: 778: 659:Category:Graph theory 25:Barabási–Albert model 22: 6239:Scale-Free Networks" 5450:B. Tadic, Physica A 3700: 3657: 3608: 3582: 3562: 3540:density distribution 3537:scale-free ideal gas 3361: 3309: 3226: 3141: 3105: 3046: 3010: 2981: 2958: 2929: 2894: 2874: 2845: 2825: 2805: 2782: 2753: 2713: 2693: 2682:{\displaystyle 99\%} 2670: 2644: 2599: 2579: 2553: 2533: 2529:neighbors of a node 2506: 2428: 2312: 2289: 2260: 2240: 2220: 2048: 2016: 1968: 1939: 1904: 1875: 1797: 1758:{\displaystyle \Pi } 1749: 1654:cumulative advantage 1513: 1425: 1387: 1342: 1322: 1197: 1095: 1060: 1040: 1020: 903:Derek de Solla Price 846: 810: 790: 736: 6762:2004PhRvE..70c7103O 6659:2002PNAS...99.5207P 6599:2003SIAMR..45..167N 6470:2002AdPhy..51.1079D 6448:Advances in Physics 6423:2008RvMP...80.1275D 6349:2000PhRvL..85.4633D 6310:2004PhyA..338..659D 6269:2002PhRvL..89y8702C 6207:2004PhRvE..69a6113B 6163:2003SciAm.288e..60B 6151:Scientific American 6084:2000PNAS...9711149A 6033:2002RvMP...74...47A 5898:2015NatSR...5E9752E 5812:2006PhRvE..73f5101M 5726:2010PhRvE..82c6106K 5665:2004PhRvL..93r8701G 5612:2002PhRvL..89y8702C 5554:2003PhRvE..67b6112R 5501:2017PhyA..469...23H 5379:2004PhyA..338..659D 5236:2018PhyD..371....1P 5190:J. Phys.: Conf. Ser 5165:2010NJPh...12i3045H 5096:2012PNAS..10915685P 5080:(39): 15685–15690. 5038:10.1038/nature09260 5030:2010Natur.466..841F 4921:2007PhyA..379..317S 4870:2006PhRvE..74f6112D 4809:2003PhRvE..67d6107R 4753:2005PhRvL..94p8101T 4710:2023CSF...17614173M 4561:2002AdPhy..51.1079D 4539:Advances in Physics 4454:2000PhRvL..85.4633D 4389:2000PNAS...9711149A 4307:1999Sci...286..509B 4250:2020PhRvR...2b3352F 4180:2008PhRvE..78b6114K 4116:2012Sci...335..665S 4061:2019NatCo..10.1016H 4004:2019NatCo..10.1017B 3947:2009SIAMR..51..661C 3895:2013JSP...151.1175C 3836:2007PNAS..104.7332O 3533:degree distribution 2659:{\displaystyle m=1} 2176: 1679:normal distribution 1005:they interconnect. 911:Pareto distribution 707:degree distribution 463:Degree distribution 114:Community structure 6719:10.1002/bies.20294 5876:Scientific Reports 4602:Notices of the AMS 3723:friendship paradox 3706: 3674: 3643: 3594: 3568: 3519:In the context of 3462: 3344: 3232: 3209: 3124: 3091: 3025: 2996: 2964: 2944: 2915: 2880: 2860: 2831: 2811: 2788: 2768: 2725: 2699: 2679: 2656: 2627: 2585: 2565: 2539: 2519: 2492: 2411: 2295: 2275: 2246: 2226: 2183: 2162: 2161: 2135: 2113: 2022: 2002: 1954: 1925: 1890: 1858: 1844: 1755: 1745:: The probability 1609: 1519: 1468: 1460: 1431: 1408: 1373: 1328: 1246: 1177: 1140: 1078: 1046: 1026: 862: 828: 796: 773: 699:scale-free network 647:Network scientists 573:Soft configuration 33: 5907:10.1038/srep09752 5790:Physical Review E 5704:Physical Review E 4954:Cognitive Science 4848:Physical Review E 4669:978-0-521-37012-7 4644:978-0-521-34058-8 4591:Willinger, Walter 4438:(21): 4633–4636. 4291:(5439): 509–512. 4158:Physical Review E 4110:(6069): 665–666. 3955:10.1137/070710111 3820:(18): 7332–7336. 3741:Erdős–Rényi model 3571:{\displaystyle n} 3457: 3204: 3175: 2967:{\displaystyle k} 2883:{\displaystyle k} 2834:{\displaystyle i} 2814:{\displaystyle k} 2791:{\displaystyle i} 2702:{\displaystyle m} 2588:{\displaystyle N} 2542:{\displaystyle i} 2490: 2477: 2406: 2393: 2345: 2298:{\displaystyle i} 2249:{\displaystyle m} 2229:{\displaystyle m} 2178: 2152: 2126: 2092: 2025:{\displaystyle A} 1856: 1835: 1628:Generative models 1599:Airline networks. 1594:Semantic networks 1567:computer networks 1522:{\displaystyle c} 1434:{\displaystyle k} 1331:{\displaystyle k} 1241: 1113: 1049:{\displaystyle v} 1029:{\displaystyle v} 1001:structure of the 759: 753: 695: 694: 615: 614: 523:Bianconi–Barabási 417: 416: 235:Artificial neural 210:Telecommunication 6824: 6798: 6789: 6755: 6753:cond-mat/0409609 6734: 6729:. Archived from 6690: 6680: 6670: 6637: 6618: 6592: 6590:cond-mat/0303516 6571: 6562: 6556: 6546: 6544: 6542:cond-mat/0501169 6531: 6510: 6508: 6489: 6463: 6461:cond-mat/0106144 6454:(4): 1079–1187. 6442: 6416: 6407:(4): 1275–1335. 6395: 6376: 6342: 6340:cond-mat/0004434 6321: 6304:(3–4): 659–671. 6288: 6262: 6260:cond-mat/0207366 6234: 6192: 6182: 6148: 6138: 6126: 6115: 6105: 6095: 6077: 6075:cond-mat/0001458 6068:(21): 11149–52. 6052: 6026: 6024:cond-mat/0106096 5992: 5991: 5981: 5971: 5947: 5938: 5937: 5927: 5909: 5891: 5867: 5861: 5860: 5858: 5846: 5840: 5839: 5805: 5803:cond-mat/0603272 5785: 5779: 5774: 5772: 5760: 5754: 5753: 5719: 5699: 5693: 5692: 5658: 5656:cond-mat/0403051 5638: 5632: 5631: 5597: 5588: 5582: 5581: 5547: 5545:cond-mat/0206130 5527: 5521: 5520: 5494: 5474: 5468: 5461: 5455: 5448: 5442: 5435: 5426: 5423: 5417: 5410: 5404: 5397: 5391: 5390: 5373:(3–4): 659–671. 5358: 5347: 5346: 5307: 5298: 5297: 5295: 5294: 5288: 5273: 5262: 5256: 5255: 5229: 5209: 5198: 5197: 5185: 5179: 5178: 5176: 5158: 5134: 5128: 5127: 5117: 5107: 5089: 5064: 5058: 5057: 5023: 5002: 4996: 4995: 4969: 4967:cond-mat/0110012 4949: 4943: 4942: 4940: 4904: 4898: 4897: 4863: 4843: 4837: 4836: 4802: 4800:cond-mat/0212469 4782: 4773: 4772: 4736: 4730: 4729: 4703: 4683: 4674: 4673: 4655: 4649: 4648: 4630: 4624: 4623: 4621: 4620: 4614: 4599: 4587: 4581: 4580: 4554: 4552:cond-mat/0106144 4545:(4): 1079–1187. 4534: 4528: 4527: 4508:10.1002/rsa.1009 4488: 4482: 4481: 4447: 4445:cond-mat/0004434 4427: 4421: 4420: 4410: 4400: 4382: 4380:cond-mat/0001458 4373:(21): 11149–52. 4349: 4343: 4342: 4300: 4298:cond-mat/9910332 4275: 4264: 4263: 4261: 4243: 4219: 4208: 4207: 4173: 4153: 4144: 4143: 4099: 4093: 4092: 4082: 4072: 4040: 4034: 4033: 4023: 3997: 3973: 3967: 3966: 3940: 3920: 3909: 3908: 3906: 3889:(6): 1175–1183. 3874: 3868: 3867: 3857: 3847: 3829: 3805: 3790: 3761:Scale invariance 3757: 3715: 3713: 3712: 3707: 3683: 3681: 3680: 3675: 3667: 3652: 3650: 3649: 3644: 3642: 3634: 3633: 3621: 3603: 3601: 3600: 3595: 3577: 3575: 3574: 3569: 3488:spatial networks 3471: 3469: 3468: 3463: 3458: 3456: 3455: 3454: 3445: 3444: 3425: 3424: 3423: 3414: 3413: 3400: 3392: 3391: 3379: 3378: 3353: 3351: 3350: 3345: 3340: 3339: 3327: 3326: 3241: 3239: 3238: 3233: 3218: 3216: 3215: 3210: 3205: 3203: 3202: 3190: 3176: 3174: with 3173: 3171: 3170: 3133: 3131: 3130: 3125: 3117: 3116: 3100: 3098: 3097: 3092: 3090: 3089: 3080: 3079: 3064: 3063: 3034: 3032: 3031: 3026: 3005: 3003: 3002: 2997: 2973: 2971: 2970: 2965: 2953: 2951: 2950: 2945: 2924: 2922: 2921: 2916: 2889: 2887: 2886: 2881: 2869: 2867: 2866: 2861: 2840: 2838: 2837: 2832: 2820: 2818: 2817: 2812: 2797: 2795: 2794: 2789: 2777: 2775: 2774: 2769: 2734: 2732: 2731: 2726: 2708: 2706: 2705: 2700: 2688: 2686: 2685: 2680: 2665: 2663: 2662: 2657: 2636: 2634: 2633: 2628: 2626: 2625: 2594: 2592: 2591: 2586: 2574: 2572: 2571: 2566: 2548: 2546: 2545: 2540: 2528: 2526: 2525: 2520: 2518: 2517: 2501: 2499: 2498: 2493: 2491: 2489: 2488: 2479: 2478: 2476: 2475: 2463: 2460: 2459: 2458: 2448: 2432: 2420: 2418: 2417: 2412: 2407: 2405: 2404: 2395: 2394: 2392: 2391: 2379: 2376: 2375: 2374: 2364: 2348: 2346: 2341: 2340: 2331: 2304: 2302: 2301: 2296: 2284: 2282: 2281: 2276: 2255: 2253: 2252: 2247: 2235: 2233: 2232: 2227: 2192: 2190: 2189: 2184: 2179: 2177: 2175: 2170: 2160: 2145: 2144: 2134: 2124: 2123: 2122: 2112: 2085: 2084: 2074: 2066: 2065: 2031: 2029: 2028: 2023: 2011: 2009: 2008: 2003: 2001: 2000: 1963: 1961: 1960: 1955: 1934: 1932: 1931: 1926: 1899: 1897: 1896: 1891: 1867: 1865: 1864: 1859: 1857: 1855: 1854: 1853: 1843: 1833: 1832: 1823: 1815: 1814: 1764: 1762: 1761: 1756: 1569:, including the 1528: 1526: 1525: 1520: 1440: 1438: 1437: 1432: 1417: 1415: 1414: 1409: 1382: 1380: 1379: 1374: 1372: 1371: 1337: 1335: 1334: 1329: 1255: 1253: 1252: 1247: 1242: 1240: 1239: 1230: 1216: 1186: 1184: 1183: 1178: 1139: 1087: 1085: 1084: 1079: 1055: 1053: 1052: 1047: 1035: 1033: 1032: 1027: 871: 869: 868: 863: 861: 860: 837: 835: 834: 829: 805: 803: 802: 797: 782: 780: 779: 774: 772: 771: 757: 751: 687: 680: 673: 558:Stochastic block 548:Hyperbolic (HGN) 497: 496: 360: 349: 281: 280: 189:Social influence 63: 35: 34: 6832: 6831: 6827: 6826: 6825: 6823: 6822: 6821: 6802: 6801: 6634: 6554: 6516:Comp. Comm. Rev 6506: 6392: 6327:Phys. Rev. Lett 6237:Caldarelli G. " 6190: 6146: 6135: 6127:. Perseus Pub. 6001: 5999:Further reading 5996: 5995: 5979:10.1145/3451166 5948: 5941: 5868: 5864: 5847: 5843: 5786: 5782: 5775:, submitted to 5761: 5757: 5700: 5696: 5643:Phys. Rev. Lett 5639: 5635: 5600:Phys. Rev. Lett 5595: 5589: 5585: 5528: 5524: 5475: 5471: 5462: 5458: 5449: 5445: 5436: 5429: 5424: 5420: 5411: 5407: 5398: 5394: 5359: 5350: 5327:10.1038/nrg1272 5308: 5301: 5292: 5290: 5286: 5271: 5263: 5259: 5210: 5201: 5186: 5182: 5135: 5131: 5065: 5061: 5014:(7308): 841–4. 5003: 4999: 4950: 4946: 4905: 4901: 4861:physics/0610108 4844: 4840: 4783: 4776: 4741:Phys. Rev. Lett 4737: 4733: 4684: 4677: 4670: 4656: 4652: 4645: 4631: 4627: 4618: 4616: 4612: 4597: 4588: 4584: 4535: 4531: 4489: 4485: 4428: 4424: 4350: 4346: 4276: 4267: 4220: 4211: 4154: 4147: 4100: 4096: 4041: 4037: 3974: 3970: 3921: 3912: 3875: 3871: 3827:physics/0610104 3806: 3802: 3797: 3788: 3767:Complex network 3755: 3731: 3701: 3698: 3697: 3694: 3660: 3658: 3655: 3654: 3638: 3629: 3625: 3617: 3609: 3606: 3605: 3583: 3580: 3579: 3563: 3560: 3559: 3556: 3544:Dunbar's number 3517: 3505: 3496: 3484: 3478: 3450: 3446: 3440: 3436: 3426: 3419: 3415: 3409: 3405: 3401: 3399: 3387: 3383: 3374: 3370: 3362: 3359: 3358: 3335: 3331: 3322: 3318: 3310: 3307: 3306: 3296: 3275: 3248: 3227: 3224: 3223: 3198: 3194: 3189: 3172: 3163: 3159: 3142: 3139: 3138: 3112: 3108: 3106: 3103: 3102: 3085: 3081: 3075: 3071: 3059: 3055: 3047: 3044: 3043: 3011: 3008: 3007: 2982: 2979: 2978: 2959: 2956: 2955: 2930: 2927: 2926: 2895: 2892: 2891: 2875: 2872: 2871: 2846: 2843: 2842: 2826: 2823: 2822: 2806: 2803: 2802: 2783: 2780: 2779: 2754: 2751: 2750: 2747: 2741: 2714: 2711: 2710: 2694: 2691: 2690: 2671: 2668: 2667: 2645: 2642: 2641: 2621: 2617: 2600: 2597: 2596: 2580: 2577: 2576: 2554: 2551: 2550: 2534: 2531: 2530: 2513: 2509: 2507: 2504: 2503: 2484: 2480: 2471: 2467: 2462: 2454: 2450: 2449: 2438: 2433: 2431: 2429: 2426: 2425: 2400: 2396: 2387: 2383: 2378: 2370: 2366: 2365: 2354: 2349: 2347: 2336: 2332: 2330: 2313: 2310: 2309: 2290: 2287: 2286: 2261: 2258: 2257: 2241: 2238: 2237: 2221: 2218: 2217: 2210: 2171: 2166: 2156: 2140: 2136: 2130: 2125: 2118: 2114: 2096: 2080: 2076: 2075: 2073: 2061: 2057: 2049: 2046: 2045: 2038: 2017: 2014: 2013: 1996: 1992: 1969: 1966: 1965: 1940: 1937: 1936: 1905: 1902: 1901: 1876: 1873: 1872: 1849: 1845: 1839: 1834: 1828: 1824: 1822: 1810: 1806: 1798: 1795: 1794: 1783: 1775: 1750: 1747: 1746: 1729: 1713: 1662:rich get richer 1645:rich get richer 1630: 1552:Social networks 1543: 1514: 1511: 1510: 1506: 1477: 1452: 1450:Characteristics 1426: 1423: 1422: 1388: 1385: 1384: 1364: 1360: 1343: 1340: 1339: 1338:, expressed as 1323: 1320: 1319: 1316: 1308:self-similarity 1265: 1235: 1231: 1217: 1215: 1198: 1195: 1194: 1117: 1096: 1093: 1092: 1061: 1058: 1057: 1041: 1038: 1037: 1021: 1018: 1017: 899: 853: 849: 847: 844: 843: 840:scale parameter 811: 808: 807: 791: 788: 787: 764: 760: 737: 734: 733: 691: 629: 594:Boolean network 568:Maximum entropy 518:Barabási–Albert 435: 352: 341: 129:Controllability 94:Complex network 81: 68: 67: 66: 65: 64: 48:Network science 29:gamma functions 17: 12: 11: 5: 6830: 6820: 6819: 6814: 6812:Graph families 6800: 6799: 6790: 6735: 6733:on 2011-08-13. 6713:(10): 1060–8. 6698: 6691: 6653:(8): 5207–11. 6638: 6632: 6619: 6583:(2): 167–256. 6572: 6563: 6547: 6532: 6522:(4): 251–262. 6511: 6490: 6443: 6401:Rev. Mod. Phys 6396: 6390: 6377: 6333:(21): 4633–6. 6322: 6289: 6253:(25): 258702. 6242: 6235: 6183: 6139: 6133: 6116: 6053: 6011:Rev. Mod. Phys 6000: 5997: 5994: 5993: 5939: 5862: 5841: 5780: 5755: 5694: 5649:(18): 188701. 5633: 5606:(25): 258702. 5583: 5522: 5469: 5456: 5443: 5441:, 4629 (2000). 5427: 5418: 5405: 5392: 5348: 5321:(2): 101–113. 5299: 5257: 5199: 5180: 5129: 5059: 4997: 4944: 4915:(1): 317–333. 4899: 4838: 4774: 4747:(16): 168101. 4731: 4675: 4668: 4650: 4643: 4625: 4582: 4529: 4502:(3): 279–290. 4483: 4422: 4344: 4265: 4209: 4145: 4094: 4035: 3968: 3931:(4): 661–703. 3910: 3869: 3799: 3798: 3796: 3793: 3792: 3791: 3782: 3776: 3770: 3764: 3758: 3749: 3744: 3738: 3730: 3727: 3705: 3693: 3690: 3673: 3670: 3666: 3663: 3641: 3637: 3632: 3628: 3624: 3620: 3616: 3613: 3593: 3590: 3587: 3567: 3555: 3552: 3535:following the 3529:random network 3521:network theory 3516: 3513: 3504: 3501: 3495: 3492: 3480:Main article: 3477: 3474: 3473: 3472: 3461: 3453: 3449: 3443: 3439: 3435: 3432: 3429: 3422: 3418: 3412: 3408: 3404: 3398: 3395: 3390: 3386: 3382: 3377: 3373: 3369: 3366: 3343: 3338: 3334: 3330: 3325: 3321: 3317: 3314: 3292: 3274: 3271: 3247: 3244: 3231: 3220: 3219: 3208: 3201: 3197: 3193: 3188: 3185: 3182: 3179: 3169: 3166: 3162: 3158: 3155: 3152: 3149: 3146: 3123: 3120: 3115: 3111: 3088: 3084: 3078: 3074: 3070: 3067: 3062: 3058: 3054: 3051: 3040:asymptotically 3024: 3021: 3018: 3015: 2995: 2992: 2989: 2986: 2963: 2943: 2940: 2937: 2934: 2914: 2911: 2908: 2905: 2902: 2899: 2879: 2859: 2856: 2853: 2850: 2830: 2810: 2787: 2767: 2764: 2761: 2758: 2740: 2737: 2724: 2721: 2718: 2698: 2678: 2675: 2655: 2652: 2649: 2624: 2620: 2616: 2613: 2610: 2607: 2604: 2584: 2564: 2561: 2558: 2538: 2516: 2512: 2487: 2483: 2474: 2470: 2466: 2457: 2453: 2447: 2444: 2441: 2437: 2422: 2421: 2410: 2403: 2399: 2390: 2386: 2382: 2373: 2369: 2363: 2360: 2357: 2353: 2344: 2339: 2335: 2329: 2326: 2323: 2320: 2317: 2294: 2285:that the node 2274: 2271: 2268: 2265: 2245: 2225: 2209: 2206: 2194: 2193: 2182: 2174: 2169: 2165: 2159: 2155: 2151: 2148: 2143: 2139: 2133: 2129: 2121: 2117: 2111: 2108: 2105: 2102: 2099: 2095: 2091: 2088: 2083: 2079: 2072: 2069: 2064: 2060: 2056: 2053: 2037: 2034: 2021: 1999: 1995: 1991: 1988: 1985: 1982: 1979: 1976: 1973: 1953: 1950: 1947: 1944: 1924: 1921: 1918: 1915: 1912: 1909: 1889: 1886: 1883: 1880: 1852: 1848: 1842: 1838: 1831: 1827: 1821: 1818: 1813: 1809: 1805: 1802: 1782: 1779: 1774: 1771: 1754: 1728: 1725: 1712: 1709: 1629: 1626: 1601: 1600: 1597: 1591: 1585: 1582: 1579:World Wide Web 1565:Many kinds of 1563: 1542: 1539: 1518: 1505: 1502: 1486:complete graph 1476: 1473: 1451: 1448: 1430: 1407: 1404: 1401: 1398: 1395: 1392: 1370: 1367: 1363: 1359: 1356: 1353: 1350: 1347: 1327: 1315: 1312: 1263: 1257: 1256: 1245: 1238: 1234: 1229: 1226: 1223: 1220: 1214: 1211: 1208: 1205: 1202: 1188: 1187: 1176: 1173: 1170: 1167: 1164: 1161: 1158: 1155: 1152: 1149: 1146: 1143: 1138: 1135: 1132: 1129: 1126: 1123: 1120: 1116: 1112: 1109: 1106: 1103: 1100: 1077: 1074: 1071: 1068: 1065: 1045: 1025: 898: 895: 859: 856: 852: 827: 824: 821: 818: 815: 795: 784: 783: 770: 767: 763: 756: 750: 747: 744: 741: 693: 692: 690: 689: 682: 675: 667: 664: 663: 662: 661: 656: 650: 649: 644: 639: 631: 630: 628: 627: 624: 620: 617: 616: 613: 612: 611: 610: 601: 596: 588: 587: 583: 582: 581: 580: 575: 570: 565: 560: 555: 550: 545: 540: 535: 533:Watts–Strogatz 530: 525: 520: 515: 510: 502: 501: 493: 492: 488: 487: 486: 485: 480: 475: 470: 465: 460: 455: 450: 445: 437: 436: 434: 433: 428: 422: 419: 418: 415: 414: 413: 412: 407: 402: 397: 392: 387: 382: 377: 369: 368: 364: 363: 362: 361: 354:Incidence list 350: 343:Adjacency list 339: 334: 329: 324: 319: 314: 312:Data structure 309: 304: 299: 294: 286: 285: 277: 276: 270: 269: 268: 267: 262: 257: 252: 247: 242: 240:Interdependent 237: 232: 227: 222: 217: 212: 207: 199: 198: 194: 193: 192: 191: 186: 184:Network effect 181: 179:Balance theory 176: 171: 166: 161: 156: 151: 146: 141: 139:Social capital 136: 131: 126: 121: 116: 111: 106: 101: 96: 91: 83: 82: 80: 79: 73: 70: 69: 58: 57: 56: 55: 54: 51: 50: 44: 43: 15: 9: 6: 4: 3: 2: 6829: 6818: 6815: 6813: 6810: 6809: 6807: 6796: 6791: 6787: 6783: 6779: 6775: 6771: 6767: 6763: 6759: 6754: 6749: 6746:(3): 037103. 6745: 6741: 6736: 6732: 6728: 6724: 6720: 6716: 6712: 6708: 6704: 6699: 6696: 6692: 6688: 6684: 6679: 6674: 6669: 6664: 6660: 6656: 6652: 6648: 6644: 6639: 6635: 6633:0-521-82698-5 6629: 6625: 6620: 6616: 6612: 6608: 6604: 6600: 6596: 6591: 6586: 6582: 6578: 6573: 6569: 6564: 6560: 6553: 6548: 6543: 6538: 6533: 6529: 6525: 6521: 6517: 6512: 6505: 6504: 6499: 6495: 6491: 6487: 6483: 6479: 6475: 6471: 6467: 6462: 6457: 6453: 6449: 6444: 6440: 6436: 6432: 6428: 6424: 6420: 6415: 6410: 6406: 6402: 6397: 6393: 6391:0-19-851590-1 6387: 6383: 6378: 6374: 6370: 6366: 6362: 6358: 6354: 6350: 6346: 6341: 6336: 6332: 6328: 6323: 6319: 6315: 6311: 6307: 6303: 6299: 6295: 6290: 6286: 6282: 6278: 6274: 6270: 6266: 6261: 6256: 6252: 6248: 6243: 6240: 6236: 6232: 6228: 6224: 6220: 6216: 6212: 6208: 6204: 6201:(1): 016113. 6200: 6196: 6189: 6184: 6180: 6176: 6172: 6168: 6164: 6160: 6156: 6152: 6145: 6140: 6136: 6134:0-452-28439-2 6130: 6125: 6124: 6117: 6113: 6109: 6104: 6099: 6094: 6089: 6085: 6081: 6076: 6071: 6067: 6063: 6059: 6054: 6050: 6046: 6042: 6038: 6034: 6030: 6025: 6020: 6016: 6012: 6008: 6003: 6002: 5989: 5985: 5980: 5975: 5970: 5965: 5961: 5957: 5953: 5946: 5944: 5935: 5931: 5926: 5921: 5917: 5913: 5908: 5903: 5899: 5895: 5890: 5885: 5881: 5877: 5873: 5866: 5857: 5852: 5845: 5837: 5833: 5829: 5825: 5821: 5817: 5813: 5809: 5804: 5799: 5796:(6): 065101. 5795: 5791: 5784: 5778: 5771: 5766: 5759: 5751: 5747: 5743: 5739: 5735: 5731: 5727: 5723: 5718: 5713: 5710:(3): 036106. 5709: 5705: 5698: 5690: 5686: 5682: 5678: 5674: 5670: 5666: 5662: 5657: 5652: 5648: 5644: 5637: 5629: 5625: 5621: 5617: 5613: 5609: 5605: 5601: 5594: 5587: 5579: 5575: 5571: 5567: 5563: 5559: 5555: 5551: 5546: 5541: 5538:(2): 026112. 5537: 5533: 5526: 5518: 5514: 5510: 5506: 5502: 5498: 5493: 5488: 5484: 5480: 5473: 5466: 5460: 5453: 5447: 5440: 5434: 5432: 5422: 5416:, 5234(2000). 5415: 5409: 5403:, 509 (1999). 5402: 5396: 5388: 5384: 5380: 5376: 5372: 5368: 5364: 5357: 5355: 5353: 5344: 5340: 5336: 5332: 5328: 5324: 5320: 5316: 5312: 5306: 5304: 5285: 5281: 5277: 5270: 5269: 5261: 5253: 5249: 5245: 5241: 5237: 5233: 5228: 5223: 5219: 5215: 5208: 5206: 5204: 5195: 5191: 5184: 5175: 5170: 5166: 5162: 5157: 5152: 5149:(9): 093045. 5148: 5144: 5140: 5133: 5125: 5121: 5116: 5111: 5106: 5101: 5097: 5093: 5088: 5083: 5079: 5075: 5071: 5063: 5055: 5051: 5047: 5043: 5039: 5035: 5031: 5027: 5022: 5017: 5013: 5009: 5001: 4993: 4989: 4985: 4981: 4977: 4973: 4968: 4963: 4959: 4955: 4948: 4939: 4934: 4930: 4926: 4922: 4918: 4914: 4910: 4903: 4895: 4891: 4887: 4883: 4879: 4875: 4871: 4867: 4862: 4857: 4854:(6): 066112. 4853: 4849: 4842: 4834: 4830: 4826: 4822: 4818: 4814: 4810: 4806: 4801: 4796: 4793:(4): 046107. 4792: 4788: 4781: 4779: 4770: 4766: 4762: 4758: 4754: 4750: 4746: 4742: 4735: 4727: 4723: 4719: 4715: 4711: 4707: 4702: 4697: 4693: 4689: 4682: 4680: 4671: 4665: 4661: 4654: 4646: 4640: 4636: 4629: 4611: 4607: 4603: 4596: 4592: 4586: 4578: 4574: 4570: 4566: 4562: 4558: 4553: 4548: 4544: 4540: 4533: 4525: 4521: 4517: 4513: 4509: 4505: 4501: 4497: 4493: 4487: 4479: 4475: 4471: 4467: 4463: 4459: 4455: 4451: 4446: 4441: 4437: 4433: 4426: 4418: 4414: 4409: 4404: 4399: 4394: 4390: 4386: 4381: 4376: 4372: 4368: 4364: 4358: 4354: 4348: 4340: 4336: 4332: 4328: 4324: 4320: 4316: 4312: 4308: 4304: 4299: 4294: 4290: 4286: 4285: 4280: 4274: 4272: 4270: 4260: 4255: 4251: 4247: 4242: 4237: 4234:(2): 023352. 4233: 4229: 4225: 4218: 4216: 4214: 4205: 4201: 4197: 4193: 4189: 4185: 4181: 4177: 4172: 4167: 4164:(2): 026114. 4163: 4159: 4152: 4150: 4141: 4137: 4133: 4129: 4125: 4121: 4117: 4113: 4109: 4105: 4098: 4090: 4086: 4081: 4076: 4071: 4066: 4062: 4058: 4054: 4050: 4046: 4039: 4031: 4027: 4022: 4017: 4013: 4009: 4005: 4001: 3996: 3991: 3987: 3983: 3979: 3972: 3964: 3960: 3956: 3952: 3948: 3944: 3939: 3934: 3930: 3926: 3919: 3917: 3915: 3905: 3900: 3896: 3892: 3888: 3884: 3880: 3873: 3865: 3861: 3856: 3851: 3846: 3841: 3837: 3833: 3828: 3823: 3819: 3815: 3811: 3804: 3800: 3786: 3783: 3780: 3777: 3774: 3771: 3768: 3765: 3762: 3759: 3753: 3750: 3748: 3745: 3742: 3739: 3736: 3733: 3732: 3726: 3724: 3719: 3703: 3689: 3687: 3686:almost surely 3671: 3668: 3664: 3661: 3639: 3635: 3630: 3626: 3622: 3618: 3614: 3611: 3591: 3588: 3585: 3565: 3551: 3549: 3545: 3541: 3538: 3534: 3530: 3526: 3522: 3512: 3509: 3500: 3491: 3489: 3483: 3459: 3451: 3447: 3441: 3437: 3433: 3430: 3427: 3420: 3416: 3410: 3406: 3402: 3396: 3388: 3384: 3380: 3375: 3371: 3364: 3357: 3356: 3355: 3336: 3332: 3328: 3323: 3319: 3312: 3304: 3300: 3295: 3291: 3288: 3284: 3280: 3273:Fitness model 3270: 3268: 3264: 3259: 3254: 3252: 3243: 3206: 3195: 3191: 3186: 3183: 3180: 3177: 3167: 3164: 3160: 3156: 3150: 3144: 3137: 3136: 3135: 3113: 3109: 3086: 3082: 3072: 3068: 3060: 3056: 3042:linear, i.e. 3041: 3035: 3019: 2990: 2975: 2961: 2954:is linear in 2938: 2912: 2909: 2903: 2877: 2854: 2828: 2808: 2801: 2785: 2762: 2746: 2736: 2722: 2719: 2716: 2696: 2673: 2653: 2650: 2647: 2640:However, for 2638: 2622: 2618: 2614: 2608: 2582: 2562: 2559: 2556: 2536: 2514: 2510: 2485: 2481: 2472: 2468: 2464: 2455: 2451: 2445: 2442: 2439: 2435: 2408: 2401: 2397: 2388: 2384: 2380: 2371: 2367: 2361: 2358: 2355: 2351: 2342: 2337: 2333: 2327: 2321: 2308: 2307: 2306: 2292: 2269: 2243: 2223: 2215: 2205: 2201: 2199: 2180: 2172: 2167: 2163: 2157: 2153: 2149: 2146: 2141: 2137: 2131: 2127: 2119: 2115: 2106: 2103: 2100: 2093: 2089: 2086: 2081: 2077: 2070: 2062: 2058: 2044: 2043: 2042: 2033: 2019: 1997: 1993: 1989: 1986: 1983: 1977: 1964:has the form 1948: 1922: 1919: 1913: 1884: 1869: 1850: 1846: 1840: 1836: 1829: 1825: 1819: 1811: 1807: 1792: 1791:Price's model 1788: 1778: 1770: 1766: 1744: 1739: 1737: 1732: 1724: 1720: 1718: 1708: 1705: 1702:However, the 1700: 1698: 1694: 1689: 1687: 1682: 1680: 1674: 1672: 1668: 1663: 1659: 1655: 1651: 1646: 1641: 1639: 1638:random graphs 1635: 1625: 1623: 1618: 1613: 1605: 1598: 1595: 1592: 1589: 1586: 1583: 1580: 1576: 1572: 1568: 1564: 1561: 1557: 1553: 1549: 1548: 1547: 1538: 1535: 1530: 1516: 1501: 1499: 1493: 1491: 1487: 1482: 1472: 1464: 1456: 1447: 1443: 1428: 1419: 1405: 1402: 1399: 1396: 1390: 1368: 1365: 1361: 1357: 1351: 1345: 1325: 1311: 1309: 1305: 1301: 1297: 1293: 1289: 1285: 1281: 1277: 1273: 1269: 1262: 1243: 1232: 1224: 1218: 1212: 1206: 1200: 1193: 1192: 1191: 1174: 1168: 1162: 1159: 1156: 1150: 1144: 1141: 1136: 1133: 1127: 1124: 1121: 1114: 1110: 1104: 1098: 1091: 1090: 1089: 1072: 1066: 1063: 1043: 1023: 1015: 1011: 1006: 1004: 1000: 996: 992: 988: 982: 980: 979:Béla Bollobás 976: 972: 968: 964: 961:Barabási and 959: 957: 953: 949: 945: 940: 935: 931: 927: 922: 920: 916: 912: 908: 904: 894: 892: 888: 887:fitness model 884: 880: 874: 850: 841: 825: 822: 819: 816: 813: 793: 761: 754: 745: 739: 732: 731: 730: 728: 724: 720: 716: 712: 708: 704: 700: 688: 683: 681: 676: 674: 669: 668: 666: 665: 660: 657: 655: 652: 651: 648: 645: 643: 640: 638: 635: 634: 633: 632: 625: 622: 621: 619: 618: 609: 605: 602: 600: 597: 595: 592: 591: 590: 589: 585: 584: 579: 578:LFR Benchmark 576: 574: 571: 569: 566: 564: 563:Blockmodeling 561: 559: 556: 554: 551: 549: 546: 544: 541: 539: 536: 534: 531: 529: 528:Fitness model 526: 524: 521: 519: 516: 514: 511: 509: 506: 505: 504: 503: 499: 498: 495: 494: 490: 489: 484: 481: 479: 476: 474: 471: 469: 468:Assortativity 466: 464: 461: 459: 456: 454: 451: 449: 446: 444: 441: 440: 439: 438: 432: 429: 427: 424: 423: 421: 420: 411: 408: 406: 403: 401: 398: 396: 393: 391: 388: 386: 383: 381: 378: 376: 373: 372: 371: 370: 366: 365: 359: 355: 351: 348: 344: 340: 338: 335: 333: 330: 328: 325: 323: 320: 318: 315: 313: 310: 308: 305: 303: 300: 298: 295: 293: 290: 289: 288: 287: 283: 282: 279: 278: 275: 272: 271: 266: 263: 261: 258: 256: 253: 251: 248: 246: 243: 241: 238: 236: 233: 231: 228: 226: 223: 221: 218: 216: 213: 211: 208: 206: 203: 202: 201: 200: 197:Network types 196: 195: 190: 187: 185: 182: 180: 177: 175: 172: 170: 167: 165: 162: 160: 157: 155: 152: 150: 147: 145: 144:Link analysis 142: 140: 137: 135: 134:Graph drawing 132: 130: 127: 125: 122: 120: 117: 115: 112: 110: 107: 105: 102: 100: 97: 95: 92: 90: 87: 86: 85: 84: 78: 75: 74: 72: 71: 62: 53: 52: 49: 46: 45: 41: 37: 36: 30: 26: 21: 6794: 6743: 6740:Phys. Rev. E 6739: 6731:the original 6710: 6706: 6693:Robb, John. 6650: 6646: 6623: 6580: 6576: 6558: 6519: 6515: 6502: 6451: 6447: 6404: 6400: 6381: 6330: 6326: 6301: 6297: 6250: 6246: 6198: 6195:Phys. Rev. E 6194: 6154: 6150: 6122: 6065: 6061: 6017:(1): 47–97. 6014: 6010: 5959: 5955: 5879: 5875: 5865: 5844: 5793: 5789: 5783: 5776: 5758: 5707: 5703: 5697: 5646: 5642: 5636: 5603: 5599: 5586: 5535: 5532:Phys. Rev. E 5531: 5525: 5482: 5478: 5472: 5467:, 425(1955). 5464: 5459: 5454:, 273(2001). 5451: 5446: 5438: 5421: 5413: 5408: 5400: 5395: 5370: 5366: 5318: 5314: 5291:. Retrieved 5267: 5260: 5217: 5213: 5193: 5189: 5183: 5146: 5142: 5132: 5077: 5073: 5062: 5011: 5007: 5000: 4960:(1): 41–78. 4957: 4953: 4947: 4912: 4908: 4902: 4851: 4847: 4841: 4790: 4787:Phys. Rev. E 4786: 4744: 4740: 4734: 4691: 4687: 4659: 4653: 4634: 4628: 4617:. Retrieved 4605: 4601: 4585: 4542: 4538: 4532: 4499: 4495: 4492:Bollobás, B. 4486: 4435: 4431: 4425: 4370: 4366: 4356: 4352: 4347: 4288: 4282: 4231: 4227: 4161: 4157: 4107: 4103: 4097: 4052: 4048: 4038: 3985: 3981: 3971: 3928: 3924: 3886: 3882: 3872: 3817: 3813: 3803: 3735:Random graph 3695: 3557: 3524: 3518: 3506: 3497: 3485: 3302: 3298: 3293: 3289: 3286: 3282: 3278: 3276: 3266: 3262: 3255: 3249: 3221: 3036: 2976: 2748: 2639: 2423: 2211: 2202: 2197: 2195: 2039: 1870: 1784: 1776: 1767: 1740: 1733: 1730: 1721: 1714: 1703: 1701: 1696: 1690: 1685: 1683: 1675: 1653: 1642: 1631: 1614: 1610: 1544: 1531: 1507: 1504:Immunization 1494: 1478: 1469: 1444: 1420: 1317: 1303: 1299: 1295: 1291: 1287: 1283: 1279: 1275: 1271: 1267: 1260: 1258: 1189: 1013: 1009: 1007: 983: 960: 955: 951: 947: 943: 923: 909:following a 900: 875: 785: 726: 722: 718: 714: 698: 696: 553:Hierarchical 508:Random graph 356: / 345: / 327:Neighborhood 169:Transitivity 149:Optimization 108: 6797:. In Press. 6577:SIAM Review 6157:(5): 50–9. 5962:(6): 1–28. 5882:(1): 9752. 4938:10419/60649 4055:(1): 1016. 3988:(1): 1017. 3925:SIAM Review 2870:depends on 2424:The factor 1534:Mashaghi A. 1286:with small 963:Réka Albert 939:Réka Albert 930:Réka Albert 599:agent based 513:Erdős–Rényi 154:Reciprocity 119:Percolation 104:Small-world 6806:Categories 5969:1908.00310 5856:1804.02513 5293:2016-02-10 5227:1704.08597 4701:2310.08110 4694:: 114173. 4619:2011-02-03 4241:2001.09118 3995:1801.03400 3795:References 2743:See also: 1475:Clustering 991:traceroute 879:fat-tailed 709:follows a 626:Categories 483:Efficiency 478:Modularity 458:Clustering 443:Centrality 431:Algorithms 255:Dependency 230:Biological 109:Scale-free 6707:BioEssays 6615:221278130 6498:Rényi, A. 6494:Erdős, P. 6414:0705.0010 6373:118876189 6298:Physica A 5988:1556-4681 5916:2045-2322 5889:1411.6871 5770:0905.3704 5717:1006.5169 5492:1411.3444 5485:: 23–30. 5479:Physica A 5252:119320331 5156:1008.4994 5087:1208.0101 5021:1008.2015 4726:263909425 4478:118876189 4171:0804.1366 4140:206538568 3938:0706.1062 3704:γ 3662:γ 3636:⁡ 3631:∗ 3623:× 3615:⁡ 3586:γ 3508:UPA model 3434:δ 3403:δ 3258:iterative 3230:∞ 3200:∞ 3192:μ 3178:γ 3168:γ 3165:− 3157:∼ 3122:∞ 3119:→ 3077:∞ 3069:∼ 3050:Π 3014:Π 2985:Π 2933:Π 2898:Π 2849:Π 2757:Π 2677:% 2615:∝ 2603:Π 2436:∑ 2352:∑ 2316:Π 2264:Π 2154:∑ 2128:∑ 2094:∑ 2052:Π 1998:α 1972:Π 1943:Π 1920:≠ 1908:Π 1879:Π 1837:∑ 1801:Π 1753:Π 1590:networks. 1403:ϵ 1394:→ 1369:γ 1366:− 1358:∝ 1163:⁡ 1157:⋅ 1145:⁡ 1134:∈ 1115:∑ 1088:. Define 1067:⁡ 915:power law 858:γ 855:− 820:γ 794:γ 769:γ 766:− 755:∼ 711:power law 375:Bipartite 297:Component 215:Transport 164:Homophily 124:Evolution 99:Contagion 6817:Networks 6795:Sci. 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6595:Bibcode 6466:Bibcode 6439:3174463 6419:Bibcode 6345:Bibcode 6306:Bibcode 6265:Bibcode 6231:1001176 6203:Bibcode 6159:Bibcode 6080:Bibcode 6029:Bibcode 5925:4426729 5894:Bibcode 5808:Bibcode 5750:6451908 5722:Bibcode 5661:Bibcode 5608:Bibcode 5550:Bibcode 5497:Bibcode 5375:Bibcode 5232:Bibcode 5161:Bibcode 5115:3465392 5092:Bibcode 5054:4405620 5026:Bibcode 4992:6000627 4917:Bibcode 4866:Bibcode 4805:Bibcode 4749:Bibcode 4706:Bibcode 4557:Bibcode 4524:1486779 4516:1824277 4450:Bibcode 4385:Bibcode 4323:2091634 4303:Bibcode 4284:Science 4246:Bibcode 4176:Bibcode 4112:Bibcode 4104:Science 4080:6399274 4057:Bibcode 4021:6399239 4000:Bibcode 3963:9155618 3943:Bibcode 3891:Bibcode 3855:1863470 3832:Bibcode 3531:with a 3287:fitness 2212:In the 1697:in-fine 1577:of the 999:layer 2 995:layer 3 932:at the 897:History 703:network 426:Metrics 395:Labeled 265:on-Chip 250:Spatial 159:Closure 6784: 6776: 6725: 6685: 6678:122747 6675: 6630: 6613: 6486:429546 6484: 6437: 6388: 6371: 6363: 6283: 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