Scalefree network
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A scalefree network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as
where is a parameter whose value is typically in the range 2 < < 3, although occasionally it may lie outside these bounds.^{[1]}^{[2]}
Many networks have been reported to be scalefree, although statistical analysis has refuted many of these claims and seriously questioned others.^{[3]} Preferential attachment and the fitness model have been proposed as mechanisms to explain conjectured power law degree distributions in real networks.
History
In studies of the networks of citations between scientific papers, Derek de Solla Price showed in 1965 that the number of links to papers—i.e., the number of citations they receive—had a heavytailed distribution following a Pareto distribution or power law, and thus that the citation network is scalefree. He did not however use the term "scalefree 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 preferential attachment.
Recent interest in scalefree networks started in 1999 with work by AlbertLászló Barabási and colleagues at the University of Notre Dame who mapped the topology of a portion of the World Wide Web,^{[4]} finding that some nodes, which they called "hubs", had many more connections than others and that the network as a whole had a powerlaw distribution of the number of links connecting to a node. After finding that a few other networks, including some social and biological networks, also had heavytailed degree distributions, Barabási and collaborators coined the term "scalefree network" to describe the class of networks that exhibit a powerlaw degree distribution. Amaral et al. showed that most of the realworld networks can be classified into two large categories according to the decay of degree distribution P(k) for large k.
Barabási and Albert proposed a generative mechanism to explain the appearance of powerlaw distributions, which they called "preferential attachment" 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, Mendes and Samukhin ^{[5]} and independently by Krapivsky, Redner, and Leyvraz, and later rigorously proved by mathematician Béla Bollobás.^{[6]} Notably, however, this mechanism only produces a specific subset of networks in the scalefree class, and many alternative mechanisms have been discovered since.^{[7]}
The history of scalefree networks also includes some disagreement. On an empirical level, the scalefree nature of several networks has been called into question. For instance, the three brothers Faloutsos believed that the Internet had a power law degree distribution on the basis of traceroute data; however, it has been suggested that this is a layer 3 illusion created by routers, which appear as highdegree nodes while concealing the internal layer 2 structure of the ASes they interconnect. ^{[8]}
On a theoretical level, refinements to the abstract definition of scalefree have been proposed. For example, Li et al. (2005) recently offered a potentially more precise "scalefree metric". Briefly, let G be a graph with edge set E, and denote the degree of a vertex (that is, the number of edges incident to ) by . Define
This is maximized when highdegree nodes are connected to other highdegree nodes. Now define
where s_{max} is the maximum value of s(H) for H in the set of all graphs with degree distribution identical to that of G. This gives a metric between 0 and 1, where a graph G with small S(G) is "scalerich", and a graph G with S(G) close to 1 is "scalefree". This definition captures the notion of selfsimilarity implied in the name "scalefree".
Characteristics
The most notable characteristic in a scalefree network is the relative commonness of vertices with a degree that greatly exceeds the average. The highestdegree nodes are often called "hubs", and are thought to serve specific purposes in their networks, although this depends greatly on the domain.
The scalefree property strongly correlates with the network's robustness to failure. It turns out that the major hubs are closely followed by smaller ones. These smaller hubs, in turn, are followed by other nodes with an even smaller degree and so on. This hierarchy allows for a fault tolerant behavior. If failures occur at random and the vast majority of nodes are those with small degree, the likelihood that a hub would be affected is almost negligible. Even if a hubfailure occurs, the network will generally not lose its connectedness, due to the remaining hubs. On the other hand, if we choose a few major hubs and take them out of the network, the network is turned into a set of rather isolated graphs. Thus, hubs are both a strength and a weakness of scalefree networks. These properties have been studied analytically using percolation theory by Cohen et al.^{[9]}^{[10]} and by Callaway et al.^{[11]} It was proven by Cohen ^{[12]} that for a broad range of scale free networks the critical percolation threshold, p_c=0. This means that removing randomly any fraction of nodes from the network will not destroy the network. This is in contrast to Erdős–Rényi graph where p_c =1/ , where is the average degree.
Another important characteristic of scalefree networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. This implies that the lowdegree nodes belong to very dense subgraphs and those subgraphs are connected to each other through hubs. Consider a social network in which nodes are people and links 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 complete graph). 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 smallworld phenomenon.
At present, the more specific characteristics of scalefree networks vary with the generative mechanism used to create them. For instance, networks generated by preferential attachment typically place the highdegree vertices in the middle of the network, connecting them together to form a core, with progressively lowerdegree 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 security, while targeted attacks destroys the connectedness very quickly. Other scalefree networks, which place the highdegree vertices at the periphery, do not exhibit these properties. Similarly, the clustering coefficient of scalefree networks can vary significantly depending on other topological details.
A final characteristic concerns the average distance between two vertices in a network. As with most disordered networks, such as the small world network model, this distance is very small relative to a highly ordered network such as a lattice graph. Notably, an uncorrelated powerlaw graph having 2 < γ < 3 will have ultrasmall diameter d ~ ln ln N where N is the number of nodes in the network, as proved by Cohen and Havlin.^{[13]} The diameter of a growing scalefree network might be considered almost constant in practice.
Properties of random graph may change or remain invariant under graph transformations. Mashaghi A. et al., for example, demonstrated that a transformation which converts random graphs to their edgedual 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, as such, remain scale free under such transformations.^{[14]}
Examples
Although many realworld networks are thought to be scalefree, the evidence often remains inconclusive, primarily due to the developing awareness of more rigorous data analysis techniques.^{[3]} As such, the scalefree nature of many networks is still being debated by the scientific community. A few examples of networks claimed to be scalefree include:
 Social networks, including collaboration networks. Two examples that have been studied extensively are the collaboration of movie actors in films and the coauthorship by mathematicians of papers.
 Many kinds of computer networks, including the internet and the webgraph of the World Wide Web.
 Some financial networks such as interbank payment networks ^{[15]}^{[16]}
 Proteinprotein interaction networks.
 Semantic networks.^{[17]}
 Airline networks.
Scale free topology has been also found in high temperature superconductors.^{[18]} The qualities of a hightemperature 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 random oxygen atoms and lattice distortion.^{[19]}
A spacefilling cellular structure, weighted planar stochastic lattice (WPSL) has recently been proposed whose coordination number distribution follow a powerlaw. 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 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) obtained by replacing each block with a node at its center and common border between blocks with an edge joining the two corresponding vertices emerges as a network whose degree distribution follows a powerlaw.^{[20]}^{[21]} The reason for it is that it grows following mediationdriven attachment model rule which also embodies preferential attachment rule but in disguise.
Generative models
Scalefree networks do not arise by chance alone. Erdős 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 random graphs are different from the properties found in scalefree networks, and therefore a model for this growth process is needed.
The most widely known generative model for a subset of scalefree networks is Barabási and Albert's (1999) rich get richer 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 indegree of Web pages. This model was originally invented by Derek J. de Solla Price in 1965 under the term cumulative advantage, but did not reach popularity until Barabási rediscovered the results under its current name (BA Model). According to this process, a page with many inlinks will attract more inlinks than a regular page. This generates a powerlaw 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 rich get richer generative model which takes into account two different attachment rules: a preferential attachment mechanism and a uniform choice only for the most recent nodes.^{[22]} For a review see the book by Dorogovtsev and Mendes.
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 distribution of links was no longer a power law but resembled a normal distribution. Based on these observations, the authors proposed a generative model that mixes preferential attachment with a baseline probability of gaining a link.
Another generative model is the copy model studied by Kumar et al.^{[23]} (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.
The growth of the networks (adding new nodes) is not a necessary condition for creating a scalefree network. Dangalchev^{[24]} (2004) gives examples of generating static scalefree networks. Another 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 specified the statistical distribution for these vertex properties (fitnesses), it turns out that in some circumstances also static networks develop scalefree properties.
Generalized scalefree model
There has been a burst of activity in the modeling of scalefree complex networks. The recipe of Barabási and Albert^{[25]} has been followed by several variations and generalizations^{[26]}^{[27]}^{[28]}^{[29]}^{[22]} and the revamping of previous mathematical works.^{[30]} As long as there is a power law distribution in a model, it is a scalefree network, and a model of that network is a scalefree model.
Features
Many real networks are (approximately) scalefree and hence require scalefree models to describe them. In Price's scheme, there are two ingredients needed to build up a scalefree model:
1. Adding or removing nodes. Usually we concentrate on growing the network, i.e. adding nodes.
2. Preferential attachment: The probability that new nodes will be connected to the "old" node.
Note that Fitness models (see below) could 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 scalefree networks does not prove that this is the mechanism underlying the evolution of realworld scalefree networks, as there might exist different mechanisms at work in realworld systems that nevertheless give rise to scaling.
Examples
There have been several attempts to generate scalefree network properties. Here are some examples:
The Barabási–Albert model
For example, the first scalefree model, the Barabási–Albert model, has a linear preferential attachment and adds one new node at every time step.
(Note, another general feature of in real networks is that , i.e. there is a nonzero probability that a new node attaches to an isolated node. Thus in general has the form , where is the initial attractiveness of the node.)
Twolevel network model
Dangalchev^{[24]} builds a 2L model by adding a secondorder preferential attachment. The attractiveness of a node in the 2L model depends not only on the number of nodes linked to it but also on the number of links in each of these nodes.
where C is a coefficient between 0 and 1.
Mediationdriven attachment (MDA) model
In the mediationdriven attachment (MDA) model in which a new node coming with edges picks an existing connected node at random and then connects itself not with that one but with of its neighbors chosen also at random. The probability that the node of the existing node picked is
The factor is the inverse of the harmonic mean (IHM) of degrees of the neighbors of a node . Extensive numerical investigation suggest that for a approximately the mean IHM value in the large limit becomes a constant which means . 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.^{[31]}
However, for it describes the winner takes it all mechanism as we find that almost of the total nodes has degree one and one is superrich in degree. As value increases the disparity between the super rich and poor decreases and as we find a transition from rich get super richer to rich get richer mechanism.
Nonlinear preferential attachment
The Barabási–Albert model assumes that the probability that a node attaches to node is proportional to the degree of node . This assumption involves two hypotheses: first, that depends on , in contrast to random graphs in which , and second, that the functional form of is linear in . The precise form of is not necessarily linear, and recent studies have demonstrated that the degree distribution depends strongly on
Krapivsky, Redner, and Leyvraz^{[28]} demonstrate that the scalefree 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 asymptotically linear, i.e. as . In this case the rate equation leads to
This way the exponent of the degree distribution can be tuned to any value between 2 and .
Hierarchical network model
There is another kind of scalefree model, which grows according to some patterns, such as the hierarchical network model.^{[32]}
The iterative construction leading 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 (N = 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 N = 125, and the process can continue indefinitely.
Fitness model
The idea is that the link between two vertices is assigned not randomly with a probability p equal for all the couple of vertices. Rather, for every vertex j there is an intrinsic fitness x_{j} and a link between vertex i and j is created with a probability .^{[33]} 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
 ^{[34]}
Hyperbolic geometric graphs
Assuming that a network has an underlying hyperbolic geometry, one can use the framework of spatial networks to generate scalefree degree distributions. This heterogeneous degree distribution then simply reflects the negative curvature and metric properties of the underlying hyperbolic geometry.^{[35]}
Edge dual transformation to generate scale free graphs with desired properties
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 edgedual transformation.^{[14]}
UniformPreferentialAttachment model (UPA model)
UPA model 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 nodes. This modification is interesting to study the robustness of the scalefree behavior of the degree distribution. It is proved analytically that the asymptotically powerlaw degree distribution is preserved.^{[22]}
Scalefree ideal networks
In the context of network theory a scalefree ideal network is a random network with a degree distribution following the scalefree ideal gas density distribution. These networks are able to reproduce citysize 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.^{[36]}^{[37]} In models of scalefree ideal networks it is possible to demonstrate that Dunbar's number is the cause of the phenomenon known as the 'six degrees of separation' .
Novel characteristics
For a scalefree network with nodes and powerlaw exponent , the induced subgraph is constructed by vertices with degrees larger than is a scalefree network with , almost surely (a.s.).^{[38]}
See also
References
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(help)  1 2 3 Pachon, Angelica; Sacerdote, Laura; Yang, Shuyi (2018). "Scalefree behavior of networks with the copresence of preferential and uniform attachment rules". Physica D: Nonlinear Phenomena. 371: 1. arXiv:1704.08597. Bibcode:2018PhyD..371....1P. doi:10.1016/j.physd.2018.01.005.
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Further reading
 Albert R.; Barabási A.L. (2002). "Statistical mechanics of complex networks". Rev. Mod. Phys. 74: 47–97. arXiv:condmat/0106096. Bibcode:2002RvMP...74...47A. doi:10.1103/RevModPhys.74.47.
 Amaral, LAN, Scala, A., Barthelemy, M., Stanley, HE. (2000). "Classes of behavior of smallworld networks". PNAS. 97 (21): 11149–52. arXiv:condmat/0001458. Bibcode:2000PNAS...9711149A. doi:10.1073/pnas.200327197. PMC 17168. PMID 11005838.
 Barabási, AlbertLászló (2004). Linked: How Everything is Connected to Everything Else. ISBN 0452284392.
 Barabási, AlbertLászló; Bonabeau, Eric (May 2003). "ScaleFree Networks" (PDF). Scientific American. 288 (5): 50–9. Bibcode:2003SciAm.288e..60B. doi:10.1038/scientificamerican050360.
 Dan Braha; Yaneer BarYam (2004). "Topology of LargeScale Engineering ProblemSolving Networks" (PDF). Phys. Rev. E. 69: 016113. Bibcode:2004PhRvE..69a6113B. doi:10.1103/PhysRevE.69.016113.
 Caldarelli G. "ScaleFree Networks" Oxford University Press, Oxford (2007).
 Caldarelli G.; Capocci A.; De Los Rios P.; Muñoz M.A. (2002). "Scalefree networks from varying vertex intrinsic fitness". Physical Review Letters. 89 (25): 258702. arXiv:condmat/0207366. Bibcode:2002PhRvL..89y8702C. doi:10.1103/PhysRevLett.89.258702. PMID 12484927.
 R. Cohen, K. Erez, D. benAvraham and S. Havlin (2000). "Resilience of the Internet to Random Breakdowns". Phys. Rev. Lett. 85 (21): 4626–8. arXiv:condmat/0007048. Bibcode:2000PhRvL..85.4626C. doi:10.1103/PhysRevLett.85.4626. PMID 11082612.
 R. Cohen, K. Erez, D. benAvraham and S. Havlin (2001). "Breakdown of the Internet under Intentional Attack". Phys. Rev. Lett. 86 (16): 3682–5. arXiv:condmat/0010251. Bibcode:2001PhRvL..86.3682C. doi:10.1103/PhysRevLett.86.3682. PMID 11328053.
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