Assortativity in random line graphs
aa r X i v : . [ c ond - m a t . d i s - nn ] A ug Assortativity in random line graphs ∗ Anna Ma´nka-Kraso´n and Krzysztof Ku lakowski
Faculty of Physics and Applied Computer Science, AGH University of Scienceand Technology, al. Mickiewicza 30, PL-30059 Krak´ow, PolandWe investigate the degree-degree correlations in the Erd¨os-R´enyi net-works, the growing exponential networks and the scale-free networks. Wedemonstrate that these correlations are the largest for the exponential net-works. We calculate also these correlations in the line graphs, formed fromthe considered networks. Theoretical and numerical results indicate that allthe line graphs are assortative, i.e. the degree-degree correlation is positive.PACS numbers: 64.60.aq; 02.10.Ox; 05.10.Ln
1. Introduction
A network of tennis players is formed when we link two players who metin the same game. Alternatively we can form a network of tennis games;two games are linked if the same competitor played in both of them. Thesame can be told on boxers and football teams. This construction is knownas a line graph [1, 2, 3]. Each graph can be converted to its line graph.Under this transformation links become nodes, and two nodes of the linegraph are linked if the respective links in the original graph share a node.The mathematical representation of a network by its line graph can be ofinterest in the science of complex networks [4]; for some applications of linegraphs see [5, 6, 7, 8, 9, 10, 11].Our concern in line graphs is due to their specific topology. Recently weshown that line graphs formed from the Erd¨os-R´enyi networks, the growingexponential networks and the Barab´asi-Albert scale-free networks are highlyclustered, with the clustering coefficient C higher than 0.5 [12]. This makesthe line graphs to be potentially attractive for modeling of social networks, ∗ Presented on the Summer Solstice 2009 International Conference on Discrete Mod-els of Complex Systems, Gda´nsk, Poland, 22-24 June 2009. Correspondence to: [email protected] (1) which are also highly clustered [13]. Here we focus on the degree-degree cor-relation in the line graphs, formed from the three kinds of networks listedabove. Once this correlation is positive, nodes of high degree are more fre-quently linked to nodes of high degree; such networks are termed to showassortative mixing [14]. If the degree-degree correlation is negative, themixing is termed disassortative. < k ’ ( k ) > k Fig. 1. Degree-degree correlations in the Erd¨os-R´enyi networks, measured by thecurves h k ′ ( k ) i , where k ′ is the degree of a neighbor of a node of degree k . Thedata shown are obtained for h k i = 5 , ,
20 and 50 (squares, triangles, rhombs andcircles, respectively). The data increase with h k i . In our former calculations [12], theoretical calculations of the clusteringcoefficient C in the line graphs were based on the assumption, that there isno degree-degree correlations in the initial networks. The accordance of thetheoretical results with the simulations was quite reasonable at least for wellconnected networks, with mean degree larger than 10. Still, some differencescould be observed for the exponential networks (Fig. 4 in [12]). Our aimhere is i ) to compare the degree-degree correlations in the the Erd¨os-R´enyinetworks, the exponential networks and the scale-free networks, ii ) to cal-culate these correlations in the line graphs, obtained from the three kinds of networks. The results are presented in the form of h k ′ ( k ) i , where k ′ ( k ) isthe degree of a neighbour of a node of degree k .Next section is devoted to the numerical calculations of the degree-degreecorrelations in the initial networks. In section 3, analytical calculations of h k ′ ( k ) i for the line graphs are presented. In Section 4 we show the corre-lations in the line graphs, obtained numerically. Last section is devoted toconclusions.
2. Numerical calculations for the initial networks
The original Erd¨os-R´enyi network is generated from N = 10 nodes; alink is placed between two nodes with the probability p . For the exponentialand scale-free networks the algorithm starts from a fully connected clusterof M nodes. In a series of steps new nodes are added, each with M . Eachedge of this node is connected to a randomly chosen node. For the scale-freenetwork we have to use preferential attachment; nodes are selected propor-tionally to their degree. The final size of the exponential and scale-freenetworks is again N = 10 nodes.To evaluate the degree-degree correlation we check how the average de-gree k ′ of the nearest neighbours of nodes with degree k depends on k .Numerical calculation begins with a search for nodes with degree k . Then,the average degree is calculated of all nearest neighbours of these nodes.Those steps are repeated for subsequent values of k .As it is shown in Fig. 1, the slope of obtained curves is close to zero. Thismeans, that the Erd¨os-R´enyi networks show no degree-degree correlations,i.e. no assortativity at all. This result is a natural consequance of theconstruction of this kind of networks. On the contrary, the results for theexponential networks (Fig. 2) indicate that the degree-degree correlationsare positive: more connected nodes are nearest neighbours of also moreconnected ones. The result is in accordance with analytical calculations[15, 16]. The results for the Barab´asi-Albert networks (Fig. 3) are morefuzzy. Still, except perhaps the case of small k ’s, the correlations are notobserved. This observation coincides with the conclusion of [14], obtainedfrom analytical method.
3. Analytical calculations for the line graphs
The assortativity of the line graph is to be investigated by the calculationof the mean degree of a node, converted from a link, which is a neighbour of < k ’ ( k ) > k Fig. 2. Degree-degree correlations in the growing exponential networks, measuredby the curves h k ′ ( k ) i , where k ′ is the degree of a neighbor of a node of degree k .The data shown are obtained for h k i = 4 , ,
20 and 50 (rhombs, triangles, circlesand squares, respectively). The data increase with h k i . another node of degree k , converted also from a link. These two links shareda node in the initial graph. The notation is as follows: the first link joinednodes of degrees k and k , and the second link joined nodes of degrees k and k . Now these links are nodes, with degrees k + k − k + k − h k ′ ( k ) i of a neighbour of a node ofdegree k in the line graph can be found as (cid:10) k ′ ( k ) (cid:11) = P k ,k ,k k P ( k ) k P ( k ) k P ( k )( k + k − δ k,k + k − P k ,k ,k k P ( k ) k P ( k ) k P ( k ) δ k,k + k − (1)where P ( k ) is the degree distribution for the initial graph. We use the Kro-necker delta to eliminate the sums over k . Then, the sums over k are fromone to infinity, and the sums over k from one to k . < k ’ ( k ) > k Fig. 3. Degree-degree correlations in the growing Barab´asi-Albert networks, mea-sured by the curves < k ′ ( k ) > , where k ′ is the degree of a neighbor of a nodeof degree k . The data shown are obtained for h k i = 4 , ,
20 and 50 (rhombs,triangles, circles and squares, respectively). The data increase with h k i . For the Erd¨os-R´enyi networks P ( k ) is Poissonian; let us denote < k > = λ . We get (cid:10) k ′ ( k ) (cid:11) = λ + 1 + k − k − (2 + k ) − − k k − λ + k/ k .For the exponential networks with the minimal degree M the degreedistribution is P ( k ) ∝ c k , what gives < k > = 2 M , c = M/ (1 + M ) and < k ′ ( k ) > = 2 k + 5 < k > −
24 (3)In this case the sums in Eq. 1 start from k i = M , i = 1 , ,
3. After elimi-nating the sum over k , the sum over k ends at k = k − M + 2. < k ’ ( k ) > k Fig. 4. Degree-degree correlations in the line graphs constructed from the Erd¨os-R´enyi networks. The data shown are obtained for h k i = 10 ,
20 and 50 (circles,triangles and rhombs, respectively). Lines are obtained from Eq. 2.
For the scale-free networks P ( k ) ∝ k − and the obtained series does notconverge. For finite networks we can use Eq. 1 with an upper cut-off of k , determined by the system size [17]. The obtained plot is practically thesame for the cut-off between 10 and 10 . The limits of summations are thesame as for the exponential networks.
4. Numerical calculations for the line graphs
The line graphs are constructed from the initial networks as follows. Inthe connectivity matrix of the initial network, the number of units above themain diagonal are substituded by their consecutive numbers. The maximalnumber is equal to the number of nodes in the line graph. In the connec-tivity matrix of the line graph, two nodes i and j are linked if the numbers i and j are in the same row or the same column in the renumbered connec-tivity matrix of the initial network. The same algorithm of construction ofthe line graphs was applied in [12]. < k ’ ( k ) > k Fig. 5. Degree-degree correlations in the line graphs constructed from the growingexponential networks. The data shown are obtained for h k i = 10 and 50 (circlesand triangles, respectively). Lines are obtained from Eq. 3. The size of the initial network is equal 10 . The calculations are per-formed for the line graphs of size dependent on the size, type and connec-tivity of the initial network. Then, the line graphs constructed from theErd¨os-R´enyi networks of the mean degree h k i = 5 , ,
20 and 50 are of sizeof 25 , ,
100 and 250 thousands, respectively. For the initial exponentialand the Barab´asi-Albert networks of degree h k i = 4 , ,
20 and 50 the sizesof the line graphs are respectively 20 , ,
100 and 250 thousands. The de-gree distribution of the obtained line graphs was described in details in [12];briefly, the line graphs retain the degree distributions of the initial networks.The degree-degree correlations in the line graphs, obtained numerically,are shown in Figs. 4, 5 and 6 for the initial networks of three kinds: theErd¨os-R´enyi networks, the exponential networks and the Barab´asi-Albertnetworks, respectively. In the same graphs the theoretical curves are shown,derived from Eq. 1 with an assumption, that there is no degree-degree cor-relations in the initial networks. However, as we see in Figs. 1, 2 and 3, thisassumption is perfectly true only in the case of the Erd¨os-R´enyi networks. < k ’ ( k ) > k Fig. 6. Degree-degree correlations in the line graphs constructed from the growingBarab´asi-Albert networks. The data shown are obtained for h k i = 10 and 50(circles and triangles, respectively). Lines are obtained from Eq. 1. Then it is not surprising, that the numerical results on the degree-degreecorrelations agree perfectly with theory only for this kind of networks (Fig.4). As the exponential networks show assortativity (Fig. 2), the degree-degree correlations in the line graphs formed from the exponential networksdiffer from the theoretical data (Fig. 5). Finally, the noisy character of thecorrelations in the initial scale-free networks, observed in Fig. 3, has somecounterpart in Fig. 6. Moreover, in the latter case the numerical curvesshow some systematic deviation from theory till some value of the degree k .One of possible exploanations of the observed deviations could be the influ-ence of hubs. We checked that this part of data differ from one generatedgraph to another.
5. Conclusions
Our numerical results on the h k ′ ( k ) i for the original networks indi-cate that the degree-degree correlations are remarkable for the exponen-tial networks, but they are negligible for the Erd¨os-R´enyi networks andthe Barab´asi-Albert scale-free networks as long as the mean degree is large enough. These results coincide with the former calculations of the cluster-ing coefficient C [12], where the largest difference between theoretical andnumerical results were found for the exponential networks. These resultsagree also with analytical calculations of other authors [14, 15, 16].The degree-degree correlations in the exponential networks allow to in-terpret also the results on the h k ′ ( k ) i dependence in the line graphs. Asbefore, the theoretical calculations are performed with the assumption thatthe correlations are absent in the initial networks. We know that this as-sumption is not true for the exponential networks. As a result, the theoret-ical curves h k ′ ( k ) i for the line graphs formed from the exponential networksdiffer from the same curves obtained from the numerical simulations. On thecontrary, the accordance is quite good for the Erd¨os-R´enyi networks, wherethe degree correlations are absent. For the scale-free networks of finite size,theory gives a linear plot h k ′ ( k ) i . The simulation for these networks givesa broad distribution of points, and therefore the accordance is only quali-tative. Summarizing, all the investigated line graphs are assortative. Thesedegree-degree correlations can be understood as a consequence of the factthat the neighboring nodes in the line graphs are formed from links sharinga common node in the initial graph. The degree of this common node con-tributes to the degree of both neighboring nodes in the line graph.REFERENCES [1] F. Harary, Graph Theory , Addison-Wesley, Reading, MA, 1969.[2] V. K. Balakrishnan,
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