The Problem of Action at a Distance in Networks and the Emergence of Preferential Attachment from Triadic Closure
TThe Problem of Action at a Distance in Networks and the Emergence ofPreferential Attachment from Triadic Closure
J´erˆome KUNEGIS *1,2 , Fariba KARIMI , SUN Jun University of Namur, Belgium University of Koblenz–Landau, Germany GESIS – Leibniz Institute for the Social Sciences, Germany*Corresponding author: [email protected]
DOI: 10.18713/JIMIS-140417-2-4Submitted: September 6 2016 - Published: April 14 2017
Volume: - Year: Issue:
Graphs & Social Systems
Editors:
Rosa Figueiredo & Vincent Labatut
Abstract
In this paper, we characterise the notion of preferential attachment in networks as action at a dis-tance, and argue that it can only be an emergent phenomenon – the actual mechanism by whichnetworks grow always being the closing of triangles. After a review of the concepts of triangleclosing and preferential attachment, we present our argument, as well as a simplified model inwhich preferential attachment can be derived mathematically from triangle closing. Additionally,we perform experiments on synthetic graphs to demonstrate the emergence of preferential attach-ment in graph growth models based only on triangle closing.
Keywords networks; preferential attachment; triangle closing; action at a distance
I INTRODUCTION
Many natural and man-made phenomena are networks – i.e., ensembles of interconnected en-tities. To understand such structures is to understand their creation, their evolution and theirdecay. In fact, many models have been proposed for the evolution of networks, for the simplereason that a very large number of real-world systems can be modelled as networks. Rules forthe evolution of networks can be broadly classified into two classes: those postulating localgrowth, and those postulating global growth. An example for a mechanism of local growth istriangle closing: When two people become friends because they have a common friend, then anew triangle is formed, consisting of three persons. This tendency of networks to form trian- In this paper, we use the terms triangle closing and triadic closure exchangably. The notion of triadic closure
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Open-access journal: jimis.episciences.org c (cid:13) JIMIS, Creative CommonsVolume: - Year: ,, DOI: 10.18713/JIMIS-140417-2-4 a r X i v : . [ c s . S I] A p r a) Triangle closing (b) Preferential attachment Figure 1: The two network growth mechanisms considered in this article: triangle closing and pref-erential attachment. In both models, new edges appear (shown as dashed lines), based on the networkenvironment of the current graph. (a) Triangle closing: an edge is more likely to appear between nodesthat have common neighbours, (b) Preferential attachment: A node with higher degree is more likely toreceive an edge. gles is a natural model not only for social networks, but for almost all types of networked data.For instance, if Alice likes a movie and Bob is a friend of Alice, Bob might also come to likethat movie. In this case, the triangle consists of two persons and one movie. In general, net-works can contain any type of object being connected by many different types of connections,and thus many different types of such triangle closings are possible. We call this type of growth local because it only depends on the immediate neighbourhood of the two connected nodes; therest of the network does not play a role.In contrast to local graph growth rules, there is the phenomenon of preferential attachment.When, for instance, two people become friends with each other, not because they have a com-mon friend, or go to the same class, but because one of them or both of them are popular. Givena popular person, i.e. with many friends, it is more likely that he will be chosen as a friend, thanan unpopular person, all else being equal. This phenomenon is referred to as preferential attach-ment. Preferential attachment is an often-used strategy to predict new connections, not only insocial networks: a frequent movie-goer is much more likely to watch a popular film, than some-one who almost never goes out to the movies watching an obscure film almost nobody knowsor has seen. These types of statements seem obviously true and indeed they are used widelyin application systems: recommender systems give a big preference to popular movies, searchengines give higher weight to well-connected web pages, and Facebook or Twitter will make apoint to show you pictures that already have many likes. In that sense, preferential attachmentis true empirically, and has been verified many times in experiments. However, preferentialattachment has one problematic property: It relies on connecting any two completely unrelatednodes, merely because of their degree, without considering their interconnections. Preferen-tial attachment can thus be labelled as “action at a distance”. For this reason, we argue thatpreferential attachment is never a primitive phenomenon, but always a derived phenomenon,emerging as a result of more basic network evolution rules, which themselves do not involveaction at a distance.So, if preferential attachment is not a primitive network evolution mechanism, which networkevolution rules should then be considered as primitive in our network growth model? We willpresent in this paper arguments for the thesis that only the principle of triangle closing is funda- has been alluded to multiple times in the history of the social sciences; and became mainstream with the work ofMark Granovetter (1985). Preferential attachment, too, is a concept with a long history, having been alluded to under multiple names.See the references in (Kunegis et al. , 2013) for an account of the early work on it.
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Open-access journal: jimis.episciences.org c (cid:13) JIMIS, Creative CommonsVolume: - Year: ,, DOI: 10.18713/JIMIS-140417-2-4 ental, all forms of preferential attachment being derived from it. To give an argument in favourof our thesis, we will first review basic notions of networks and network evolution models, andthen review preferential attachment, proposing various mechanisms by which it can arise fromtriangle closing, a fundamental notion in the evolution of networks. Finally, we perform experi-ments on synthetic graphs to test to what extent preferential attachment may emerge from graphgrowth models that include only triangle closing and/or random addition of edges.
II RELATED WORK
The debate over the nature of preferential attachment mechanisms dates back to the 1960s, whenthe economist Herbert Simon defended the role of randomness and the mathematician BenoˆıtMandelbrot defended the role of optimisation (Barab´asi, 2012). The concept of preferentialattachment is also used to explain the nature of scale-free degree distributions in biological net-works such as metabolic networks (Jeong et al. , 2000) and protein networks (Jeong et al. , 2001).There are various suggestions to explain the nature of preferential attachment for instance byintroducing hidden variable models in which nodes possess an intrinsic fitness to other nodesin unipartite (Bogu˜n´a and Pastor-Satorras, 2003) or bipartite networks (Kitsak and Krioukov,2011). In a recent
Nature paper, Papadopoulos and colleagues proposed a model based on ge-ometric optimisation of homophily space (2012). However, in these models, triadic closure isnot defined as the main principle for the formation of edges.Triadic closure, a tendency to connect to the friend of a friend (Rapoport, 1953), has been ob-served undeniably in many social networks such as friendship at a university (Kossinets andWatts, 2006), scientific collaborations (Newman, 2001) and in the World Wide Web (Adamic,1999). The concept of triadic closure was first suggested by German sociologist Georg Simmeland colleagues (1950) and later on popularised by Fritz Heider and Mark Granovetter as thetheory of cognitive balance in which if two individuals feel the same way about an object ora person, they seek closure by closing the triad between themselves (Heider, 2013). Since theclassic preferential attachment model fails to explain the number of clusters in many social net-works, many attempts have been made to include triadic closure to the model (Holme and Kim,2002; V´azquez, 2003), in which nodes connect with certain probabilities based on the principleof triadic closure. These works have shown that the scaling law for the degree distribution andclustering coefficient can be reproduced based on these models (Klimek and Thurner, 2013).Similarly, models based on random walks as local processes have been proposed, too, of whichtriangle closing is a special case (Evans and Saram¨aki, 2005).Hence, the scale-free nature of networks and the abundance of triangles in socio-technical net-works beg for a more fundamental explanation. Moreover, the observable part of these sys-tems is not necessarily completely representative for the entire system. Networks are generallymulti-layered or multiplex, in which some layers can be hidden or simply not possible to ob-serve (Kivel¨a et al. , 2014). For instance, the creation of a new Facebook tie can be caused byattending the same class, sharing the same hobby or living in a same neighbourhood, which ishidden from the observable data. Consequently, these focal points contribute to the tie creationknown as focal closure and need to be considered in modelling realistic networks, as argued byKossinets and Watts (2006).
III NETWORKS
The assertion that networks are to be found everywhere has become a clich´e because it is true.Social networks, knowledge networks, information networks, communication networks – manypapers in the field of network science motivate their use by enumerating fields in which they
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Open-access journal: jimis.episciences.org c (cid:13) JIMIS, Creative CommonsVolume: - Year: ,, DOI: 10.18713/JIMIS-140417-2-4 lay a central role. Biological networks, molecules, lexical networks, Feynman diagrams –hardly a scientific field exists in which networks do not play a fundamental role. Instead ofgiving a hopelessly incomplete enumeration of examples, we will simply refer the reader to theintroductory section of our Handbook of Network Analysis (Kunegis, 2017), in case she wishesto convince herself of this fact. In case this is not enough, we may point to the existence ofentire fields of research incorporating the word network and synonyms that have emerged in thelast decade: network science (B ¨orner et al. , 2007; Newman, 2010), web science (Hendler et al. ,2008), and others (Tiropanis et al. , 2015). There are many ways to justify the ubiquitous use ofnetworks as a model. As an example, we may consider their use in the field of machine learning.Most classical machine learning algorithms deal with datasets consisting of data points, eachconsisting of the same features. Mathematically, we may model such a dataset as a set of pointsin a space whose dimensions are the individual features (Salton et al. , 1975). This formalismis very powerful, and still constitutes the backbone of many machine learning and data miningmethods to this day. The standard formulation of classification, clustering and other learningproblems all rely on the set-of-points-in-a-space model. However, not all machine learningproblems are well described by the set of points model. While the set of words contained intext documents are well represented by the bag of words model (Baeza-Yates and Ribeiro-Neto,1999), a social network is not. We may try to represent a social network as a bag of friends, butthis representation is very unsatisfactory: each person has a set of friends, but the model doesnot reflect the fact that a person contained in one of these bags is the same person as one having a bag of friends. Thus, the vector space model cannot find connections such as “the friend ofmy friend” – it can only find “a person that has the same friend as me”. In other words, thevector space model disconnects the role of having friends and that of being a friend . Instead,the natural way to represent friendships is as a network. Using a network model, the symmetryof the friend relationship is included automatically in the model, and relationships such as thefriend of my friend arise as the natural way to create new edges in the network, i.e., triangleclosing. In fact, we will argue that this is the only way new edges can be created in a network,and that other models are merely consequences of it, such as preferential attachment.As an additional remark, the terms network and graph are often used interchangeably. Strictlyspeaking, a network is the real-world object to be analysed, such as a social network, while agraph is a mathematical structure used to model it.
IV PREFERENTIAL ATTACHMENT
Preferential attachment, also referred to by the phrase “the rich get richer”, or as the Mattheweffect, is observed empirically in many social networks (Kunegis et al. , 2013). In fact, thephenomenon of preferential attachment is known by many other names in different contexts;see the references within (Kunegis et al. , 2013) for an account. In other words, who has manyfriends, will get more new friends than who has few. Movies that have been seen by manypeople will be seen by more people than movies that have not. Websites that have been linkedto many times will receive more new links because of this. These statements seem true, andindeed, they are true empirically for many different network types.In fact, preferential attachment is the basis for a whole class of network models. The most basicof these, the model of Barab´asi and Albert (1999), describes the growth of a network, whichproceeds as follow: Start with a small graph, and at each step, add a node, and connect thatnode to k existing nodes with a probability proportional to the number of neighbours for eachexisting node. In the limit where many nodes have been added in that way, the network tendsto become scale-free , i.e. tends to have a distribution of neighbour counts that follow a power J. of Interd. Method. and Issues in Science
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Open-access journal: jimis.episciences.org c (cid:13) JIMIS, Creative CommonsVolume: - Year: ,, DOI: 10.18713/JIMIS-140417-2-4 aw. Since power law degree distributions are observed in many natural networks, the usualconclusion is that preferential attachment is correct.Preferential attachment is thus undeniably real. Why then, are we arguing against it? The reasonis that preferential attachment cannot be a fundamental driving force for tie creation. How aretwo nodes, completely unconnected from each other, be supposed to choose to connect witheach other? How can two completely disconnected nodes even know of each other’s existence?This is a fundamental problem with all nonlocal interactions. For instance, the classical theoryof gravitation as defined and used by Isaac Newton (1687) includes nonlocal interactions. Inthat theory, two masses exert a force on each other, regardless of their position. While the forcedecreases with distance, it is always nonzero, and instantaneous. The conceptual problem withthis type of interaction had been identified already by Newton himself (Hesse, 1955). In modernphysics, Newton’s formalism is replaced by more precise theories that do not include any actionat a distance. The theory of general relativity as defined by Albert Einstein (1916) for instance,only includes local interactions in the form of the Einstein field equations. Einstein’s generalrelativity is thus free from any problematic action at a distance , and has been verified at manyexperimental scales. This is also true for other types of physical interactions – instead of aforce that acts at a distance between matter particles, quantum field theory models bosons thatconnect particles. In fact, such interactions can be represented by Feynman diagrams: graph-like representations of particles in which edges are particles and nodes are interactions – anyinteracting particles must be connected in one diagram, directly or indirectly. In this light, wemay interpret preferential attachment as a theory that is true superficially, but must be explainedby an underlying phenomenon. Specifically, an underlying phenomenon that does not rely onaction at a distance. As this phenomenon, we propose the known mechanism of triangle closing.
V TRIANGLE CLOSING
How do we make new friends? By meeting the friends of our friends. This represents a triangleformed by ourselves, our previous friend, and our new friend. What if we meet our new friendin another way – maybe at a party, or a concert, or at work . . . in any case, there is always some element in common. If we meet our new friend at a party, then we are both connected tothe party, and by modelling the party as a node in our network, that new friendship is indeedcreated by the closing of a person–person–party triangle. Of course, we may continue to askhow our connection to the party arose. After all, we did not come to a party randomly. No, wecame to the party because a friend invited us, or for any other reason, as long as there is someconnection. This game of connections can be played to any desired degree of precision. Maybewe really went from door to door until we found a party with many people. But then, how didwe get from door to door? We surely must have started somewhere, likely near to our home,and have then gone on to the next door, and to the next door, and so on. In doing this, we haveonly followed links: We are connected to our home by living there; our home is connected tothe neighbouring house, which itself is connected to the next house, and so on. This example isof course exaggerated, but serves to illustrate the principle: in order for a new edge to appear,a path has to exist from one node to another; this can go over nodes representing any type ofentity, and these nodes may be visible or hidden. All in all, there is no escaping the principleof triangle closing. However we arrived at the party, it must have been by a series of triangleclosings.Thus, triangle closing fulfils the expected role as a fundamental mechanism of network growth,as it is purely local. However, we cannot deny the existence of preferential attachment, forwhich we must now find suitable explanations.
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I EXPLANATIONS
In recommender systems, such as those used on web sites that recommend movies to watch,preferential attachment is often taken as a solution to the cold start problem. The cold startproblem in recommender systems refers to the situation in which a user has not yet entered anyinformation about herself, and thus triangle closing cannot be used to recommend her anything.If the user has watched only a single movie, then we can find similar movies and recommendthem. If a user has added only a single friend, then we can take movies liked by that friendand recommend them. But if the user is completely new, as has no friends and no ratings yet,then this strategy will not work. How then, do recommender systems give recommendationsto new users? The solution is simple: they recommend the most popular items. If you sub-scribe to Twitter, you will be recommended popular accounts to follow. If you subscribe toLast.fm, you will be recommended popular music. For these sites, this strategy is better thannot recommending anything, and in fact is a form of preferential attachment: Create, or ratherrecommend, links to nodes with many neighbours. How can we interpret this in terms of tri-angle closing? If a node has no connections yet, then surely it cannot acquire new nodes bytriangle closing. How then will a node ever acquire new edges, if it starts without neighbours?The answer is that a node does not start without any neighbours. Everything is connected. Achild when it is born does not start without connections; it is already connected to its parentsand to its birthplace. Likewise, a user on the Web never starts from scratch: every page has areferrer, and thus the user can be connected to another website. Even if the referring web pageis not known, there has to be a referrer. If a user types in a URL by hand, she has to have takenit somewhere: maybe a friend gave it to her, maybe she read it in a magazine, on a billboard, oron a truck . . . in all cases, the newly created connection is not created ex nihilo – it is createdby triangle closing.The explanation for preferential attachment thus lies in hidden nodes: Nodes that make indi-rect connections between things, but do not appear in the model. On Facebook for instance,many new friendships are created between people who do not have common friends. Thesenew friendships seemingly appear without the help of triangle closing. However, that is alwaysdue to the fact that Facebook does not know everything. Some people are simply not on Face-book, which means that if one meets a new friend through a friend that is not on Facebook andthen connects the new friend via Facebook, then from the point of view of Facebook a new edgewas created without triangle closing. But that is only true because Facebook does not know myinitial friend. If it did, it could correctly infer the new friendship via triangle closing. Thus,any two nodes in a network can potentially be linked, even if they do not share common neigh-bours in the network at hand , because they may share a hidden common neighbour. The sameargumentation applies to hidden nodes that represent non-actors, such as classes, hometowns,parties, etc.In order to justify preferential attachment as an emergent phenomenon, we must thus derive themechanism that leads to edges being created specifically between nodes of high degrees. Con-sider a network, for instance a social network. Call this the known network. Then, consider acertain number of nodes outside of that network, that are connected at random to the nodes in theknown network. Call these the unknown nodes. How many common neighbours do two mem-bers of the known network have outside of the known network? Without knowing the distribu-tion of hidden edges, this question cannot be answered. But consider that triangle closing actsnot only on known–unknown–known paths, but also on known–known–unknown paths. Start-ing with an equal probability for all known–unknown edges, performing triangle closing willlead to the creation of known–known–unknown triangles. The newly created known–unknown
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Open-access journal: jimis.episciences.org c (cid:13) JIMIS, Creative CommonsVolume: - Year: ,, DOI: 10.18713/JIMIS-140417-2-4 dges can then be combined with other unknown–known edges to perform, again, triangle clos-ing, leading to new known–known edges. The result are new edges in the observed socialnetwork, with a probability proportional to the number of the initial known node’s neighbours.Thus, preferential attachment emerges as a necessary consequence of iterated triangle closing,if hidden nodes are admitted. The next section will make this heuristic argument precise.
VII DERIVATION
This section gives an exemplary derivation of a simplified model that we introduce to illustratethat preferential attachment arises as a consequence of triangle closing in the presence of hid-den nodes. The given scenario is very general and may be generalised easily for instance byconsidering multiple node types or multiple edge types. In this model, we distinguish two typesof nodes: visible nodes in the set V , and hidden nodes in the set W . We will assume that thereis a given, fixed number of visible nodes | V | , and a possibly very large number of hidden nodes | W | . In particular, we will consider the limit | W | → ∞ .Let G = ( V ∪ W, E ) be the graph representing the complete system, in which V is the set ofvisible nodes, and W the set of hidden nodes. Additionally, E is the set of edges connectingnodes in V with nodes in W . While we assume that the individual edges in E are hidden, thedegree of the nodes in V is not hidden. In other words, the number of edges of E incident toeach node in V is known. Edges between nodes in V will not be considered. Likewise, edgesbetween nodes in W need not be considered, since they do not contribute to the degree of nodesin V . Thus, the considered network G is bipartite. We will use the convention that n = | W | ,and the degree of a node u is denoted by d ( u ) . We now assume that the graph G will receivenew edges according to the principle of triangle closing. Thus, two nodes in V will connectwith a probability proportional to the number of common neighbours they have. Seeing onlynodes in V and their degree, preferential attachment can then be observed as described in thefollowing.In order to make our derivation, we need to make two assumptions: • The triangle closing process is random in the sense that new edges are added between anypossible node pairs with equal probability. • The typical degree of nodes is significantly smaller than the number of nodes, i.e., d ( x ) (cid:28) n . This is precise when n goes to infinity.Let u, v ∈ V be two nodes of the network. Under the assumption that the edges are distributedrandomly in the graph, the probability p that u and v are connected can be derived combi-natorically by considering the number of configurations in which the two nodes do not share acommon neighbour. Given that u and v have degree d ( u ) and d ( v ) respectively, the total numberof configurations for the edges connected to the nodes is (cid:18) nd ( u ) (cid:19)(cid:18) nd ( v ) (cid:19) . (1)Out of those, the number of configurations in which the neighbours of the two nodes are disjointis given by (cid:18) nd ( u ) (cid:19)(cid:18) n − d ( u ) d ( v ) (cid:19) . (2) J. of Interd. Method. and Issues in Science
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Open-access journal: jimis.episciences.org c (cid:13) JIMIS, Creative CommonsVolume: - Year: ,, DOI: 10.18713/JIMIS-140417-2-4 hus, the probability that the two nodes share a common neighbour is given by p = 1 − (cid:0) nd ( u ) (cid:1)(cid:0) n − d ( u ) d ( v ) (cid:1)(cid:0) nd ( u ) (cid:1)(cid:0) nd ( v ) (cid:1) = 1 − (cid:0) n − d ( u ) d ( v ) (cid:1)(cid:0) nd ( v ) (cid:1) . (3)We now use the falling factorial to express binomial coefficients, i.e., n a = n ( n − n − · · · ( n − a + 1) . (4)The falling factorial has the property that in the limit where a is constant and n goes to infinity,we have lim n →∞ n a n a = 1 (5)and also, (cid:18) na (cid:19) = n a a ! , (6)and thus p = 1 − ( n − d ( u )) d ( v ) d ( v )! d ( v )! n d ( v ) = 1 − ( n − d ( u )) d ( v ) n d ( v ) . (7)In the limit when n goes to infinity we may thus assume that p = 1 − ( n − d ( u )) d ( v ) n d ( v ) = 1 − (cid:18) − d ( u ) n (cid:19) d ( v ) (8)and using again the limit n → ∞ , and the property that in the limit where ε goes to zero, (1 − ε ) k goes to (1 − kε ) , p = d ( u ) d ( v ) n . (9)It thus follows that p ∼ d ( u ) d ( v ) , i.e., the probability of the nodes u and v being connected isproportional to both d ( u ) and d ( v ) . Thus, we find that preferential attachment is a consequenceof the triangle closing model. Preferential attachment itself then leads to a scale-free degreedistribution, as per Barab´asi and Albert (1999). VIII EXPERIMENTS
In this section, we give empirical evidence for the emergence of preferential attachment in graphgrowth models that do not include it. In the experiments, we generate synthetic networks via arandom growth process that does not include preferential attachment, as well as using randomgrowth processes that do include preferential attachment. In all generated networks, effectsof preferential attachment are then measured empirically. All generated networks have 1,000nodes and 10,000 edges, and are undirected, loopless, and do not allow multiple edges. In allcases, the graphs are generated by starting with a graph of 1,000 nodes and without edges, andadding edges one by one. For each edge that is added, one of the following three methods ischosen at random:
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Random : With probability p r , an edge is added randomly between two unconnectednodes. All pairs of distinct unconnected nodes are chosen with equal probability. • Triangle closing : With probability p tc , among all unclosed triads, one is chosen randomlywith equal probability, and the third edge is added. An unclosed triad is a triple of nodes ( u, v, w ) such that ( u, v ) and ( u, w ) are connected, but v and w are not connected. Ifchosen, the triangle is completed by adding the edge ( v, w ) . If no unclosed triads arepresent, an edge is added at random as described in the previous case. • Preferential attachment : With probability p pa , a node is chosen with a probability pro-portional to the node’s current degree. Then, out of all nodes not connected to that node,one is chosen randomly and with equal probability, and an edge is added between the twoselected nodes. If there is not at least one unconnected pair of nodes with nonzero degree,an edge is added at random as described in the first case.In each experimental trial, the three probabilities are chosen such that p r + p tc + p pa = 1 . Eachof these probabilities is varied from 0 to 1 in increments of 1/11, excluding the case p r = 0 inorder to avoid the runaway case of an individual node accumulating all edges. First, in order to verify whether a graph created by the process of triangle closing display scale-free behaviour, we compare the generated distribution of the triangle closing case with thedegree distributions for the random and preferential attachment cases. All three degree dis-tributions are shown in Figure 2. In the plot, several observations can be made. The degreedistribution for the triangle closing case displays power law-like behaviour over multiple ordersof magnitude, from the smallest degrees of one, to approximately one hundred. While the net-works generated by triangle closing and preferential attachment have similar power-like degreedistribution, both with comparable exponent, we must note that the maximum degree in the pref-erential attachment case is larger than in the triangle closing case. However, the triangle closingmodel displays a power law degree distribution with exponential cut-off that has been observedin many real networks due to finite size effect (Bogu˜n´a et al. , 2004; Clauset et al. , 2009). Forcomparison, the preferential attachment case also displays power law-like behaviour, althoughnot for very small degrees (under about 10), and additionally has a well-defined long tail. Thepurely random case leads to a degree distribution that shows no scale-free behaviour.We measure the equality of the distribution of edges, or its opposite, its skewness, as the primaryconsequence of the preferential attachment process. As a measure, we use the Gini coefficientof the degree distribution, as defined in (Kunegis and Preusse, 2012). The Gini coefficient iszero when all nodes have equal degree, and attains its theoretical maximum of one when allnodes except a single one have degree zero. The experimental results are shown in Figure 3. In the triangle shown in the figure, the top-to-bottom-right edge shows the cases in which preferential attachment is excluded, while thebottom-left corner (
Pref. att. = 100% ) represents the case of exclusive preferential attachment.As expected, the 100% random case results in an Erd ˝os–R´enyi graph in which the degreeshave a Poisson distribution, and thus a very uniform number of edges over all nodes, givinga small Gini coefficient of 17.7%. The pure preferential attachment case gives a higher value In the degenerate case of p r = 0 , almost all edges will be attached to a single node in the n → ∞ limit. As described in the previous paragraph, the cases of pure triangle closing and preferential attachment alsoinclude 1/11 of edges based on random assignment. Since an edge always connects two nodes, the actual maximum is attained in star graphs, in which all edgesattach to a single node, and other nodes have a degree of zero and one. In the large-graph limit, the Gini coefficientin such graphs tends to one.
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RandomTr. clos.Pref. att.
Figure 2: The cumulative degree distribution for the three extremal generated networks.
Random = 100%Pref. att. = 100% Tr. cl. = 100%Not computed0.740.550.370.180.00gini =
Figure 3: Experimental results: Each cell shows one experimental run with a different probability ofadding each edge at random (top), via triangle closing (bottom right), and preferential attachment (bottomleft). The bottom row was not executed due to the tendency of models with random edges to attach alledges to a single node, giving values of the Gini coefficient very close to the theoretical maximum ofone.
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Open-access journal: jimis.episciences.org c (cid:13) JIMIS, Creative CommonsVolume: - Year: ,, DOI: 10.18713/JIMIS-140417-2-4 f about 51.5%. The pure triangle closing method results in a value of the Gini coefficient of65.1%, a value similar to (and even superior to) the value in the pure preferential attachmentcase. Thus, it is indeed the case that a skewed degree distribution is generated by a purelylocal process of triangle closing, without the need for explicit preferential attachment. We notealso that preferential attachment is observed even though the number of nodes in the network( n = 1 , ) is relatively small when compared to the theoretical model described in the previoussection in which the limit n → ∞ is taken. IX DISCUSSION
Our experiments have allowed us to observed that triangle closing leads to skewed and scale-free degree distributions. However, the status of a mechanism as fundamental is not clear cut.When a phenomenon is explained by another, more fundamental phenomenon, we can considerit as derived. But how can we be sure that a phenomenon is not explained by a more basicphenomenon? What does it mean for a phenomenon to be fundamental? Just as physics cannotdeclare one theory to be final, we cannot declare one network growth mechanism to be final.Thus, individual instances of triangle closing can for instance be explained by several layersof triangle closing, just as in physics a direct interaction can be explained by a new mediatingparticle. In the end however, this applies only to specific instances of triangle closing, as itreplaces them with other, more detailed instances of triangle closing. Thus, triangle closing does play a fundamental role in growing network models, only that it cannot always be derivedwhich three nodes are taking part in it, as one of the three nodes is often hidden. In the end, theonly judge of the validity of a model remains the experiment, and in practice, used models donot have to be fundamental – recommenders and information retrieval systems have had enoughsuccess by applying preferential attachment directly.As mentioned in the introduction, triangle closing is itself a general phenomenon that not onlyapplies to pure social networks, but also to other types of networks. In the case of propertynetworks, i.e., networks containing edges between persons and the properties they have, triangleclosing can be identified with the concept of homophily, i.e., the concept that friends tend to besimilar. As an example, the fact that two smokers become friends can be modelled as the closureof the (person A)–(colleague + smoker)–(person B) triangle, in which “colleague + smoker” isa non-person node of the network representing the property of being a colleague and a smoker .Thus, the fact that friends of smokers are more likely to be smokers too (a classical exampleof homophily) can be analysed as a form of triangle closing in a graph that is not purely asocial network, as it contains non-person nodes. Homophily is thus consistent with the viewthat triangle closing is fundamental (Shalizi and Thomas, 2011).The problem posed in this paper can be generalised to other graph growth mechanisms. Forinstance, we may ask whether assortativity (the tendency of connected nodes to have correlateddegrees) or community structures emerge from triangle closing alone. In the case of commu-nity structures, triangle closing trivially plays a role, as triangle closing by construction leadsto tightly connected graphs. As for assortativity, the fact that both assortativity (a positive cor-relation between degrees) and dissortativity (a negative correlation between degrees) have beenobserved in social networks points to the fact that a single model such as triangle closing cannot(and is not expected to) explain all properties of a social network, and other phenomena mustbe at work, which may or may not be local. In this and all subsequent cases labelled as pure , the method in question has a probability of p tc , pa = 10 / while a random edge is added with a probability of p r = 1 / . J. of Interd. Method. and Issues in Science
Open-access journal: jimis.episciences.org c (cid:13) JIMIS, Creative CommonsVolume: - Year: ,,
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