Divisibility patterns of natural numbers on a complex network
aa r X i v : . [ c s . S I] D ec Divisibility patterns of natural numbers on acomplex network
Snehal M. Shekatkar , Chandrasheel Bhagwat , and G. Ambika Indian Institute of Science Education and Research, Pune, 411008, India * [email protected] ABSTRACT
Investigation of divisibility properties of natural numbers is one of the most important themes in the theory of numbers. Varioustools have been developed over the centuries to discover and study the various patterns in the sequence of natural numbersin the context of divisibility. In the present paper, we study the divisibility of natural numbers using the framework of a growingcomplex network. In particular, using tools from the field of statistical inference, we show that the network is scale-free buthas a non-stationary degree distribution. Along with this, we report a new kind of similarity pattern for the local clustering,which we call “stretching similarity”, in this network. We also show that the various characteristics like average degree, globalclustering coefficient and assortativity coefficient of the network vary smoothly with the size of the network. Using analyticalarguments we estimate the asymptotic behavior of global clustering and average degree which is validated using numericalanalysis.
Introduction
The study of complex networks has become a very important part of many disciplines like information, technology, socialsciences, ecology and biology. The characterization of structure of real networks is an indispensable part of this study.Despite being random, real networks show certain statistical properties which set them apart from their completely randommathematical counterparts. This hints towards underlying organizing principles which shape the structures of real networks. In particular, many real networks are scale-free which means that the distribution of degrees of their nodes follows a powerlaw.
The density of triangles in the network is another important characteristic of networks measured using a quantitycalled clustering coefficient. Empirical studies show that the real networks are highly clustered as compared to completelyrandom mathematical models like Erdos-Renyi graph.
In the present paper, we report an analysis for a particular deterministic network that resembles real networks in manyaspects. This network consists of natural numbers 1 , , , · · · as nodes and if a given number divides another, then theircorresponding nodes are connected by an undirected link. The network thus constructed, though deterministic, can be studiedon an equal footing with the other random networks because of the irregular distribution of primes which makes divisibilityrelations themselves irregular. It is helpful to view this network as a growing network where nodes are added one at a time.A similar network with nodes as composite numbers has already been studied. Also, a directed network of natural numbersbased on the divisibility which includes only the multiples in the pattern has been reported by Ding-hua et al. A bipartitestructure separating composite and prime numbers with weighted links between them based on divisibility has been analyzedby Garc´ıa-P´erez et al. In the present work we consider a more general set up where we put all the natural numbers on a complex network withtheir divisibility relations as the underlying deterministic rule of connections. Here the network is undirected with links toboth divisors and multiples. Using tools from statistical inference, we confirm that this network is scale-free and show thataverage degree, global clustering coefficient and assortativity coefficient vary smoothly with the size of the network. Thisis surprising in view of the fact that distribution of primes is quite irregular in the sequence of natural numbers. We provideanalytical results for the asymptotic behavior of average degree and global clustering coefficient for this network. In particular,we show that the global clustering coefficient of this network decays to zero whereas average degree increases logarithmically.We also report an interesting and novel similarity exhibited by local clustering coefficients of nodes in this network which wecall “stretching similarity”.The remaining paper is organized as follows: In the next section we describe the construction of the network and showthat the network is scale-free. We then describe the existence of stretching similarity in this network. Finally we show thebehavior of average degree, global clustering coefficient and assortativity coefficient as a function of size of the network andanalytically obtain the asymptotic trends for average degree and clustering. esults
Construction of the network and its scaling properties.
The nodes of the present network are natural numbers 1 , , , · · · and there is a link between two nodes if either divides the other. We avoid self-links and all the links are undirected. Since thesequence of natural numbers has natural ordering, it is helpful to view this network as a growing network with the addition ofa new node at each discrete time as follows:1. At time t = n = t , a node with the number n = t is added to thenetwork.2. This node connects to all the existing nodes whose numbers divide it.The network thus constructed is shown in Fig. 1 at two different times t =
16 and t =
32 which would correspond to networksof size N =
16 and N =
32 respectively. To find the distribution of degrees of this network, we grow the network till the
Figure 1.
Network of natural numbers with two different sizes. (a) t =
16 nodes and (b) t =
32 nodes. In each panel, thesize of each node is proportional to its degree and color of each node is graded according to its clustering coefficient withmore white nodes as nodes with higher value of local clustering.size reaches N = = , , , p ( k ) ∼ k − a )asymptotically. Using the method of maximum likelihood we find that the scaling-index a ∼
2. We establish the existenceof power-law in the distribution (and hence the fact that this network is scale-free) using the approach described in Clausetet al (see Methods). We also study the scaling behavior of the local clustering coefficient with degree. The local clusteringof a node in the network is defined as the fraction of number of edges that are present among its neighbors. For node i withdegree k i this can be written as: c i = E ik C (1)where E i is the actual number of edges among the neighbors of node i .In Fig. 3 we show the dependence of local clustering coefficient of nodes in the network on the degree. It can be seenthat the asymptotic behavior is compatible with a power law with exponent 1. This behavior is similar to one that is usuallyobserved in real networks. Stretching similarity of local clustering.
We now discuss an interesting behavior that setsnetwork of natural numbers apart from other complex networks. In the network presented here, each node has an identitywhich is the number attached to it and this defines a natural order on the nodes. This means that we can study variousproperties of nodes as a function of their labels. This is not possible for other networks because no such unique labeling existsfor the nodes. Here we specifically consider local clustering coefficient of nodes and study its behavior as a function of node -16 -14 -12 -10 -8 -6 -4 -2 p ( k ) k Figure 2.
Degree distribution of network of natural numbers with logarithmic binning.
Sizes of successive bins areequal to successive positive powers of 2 and count in each bin is normalized by dividing by a bin width. The dotted line inthe graph has slope a = − method of maximum likelihood . The existence of the underlyingpower law is established by calculating p -value using Kolmogorov-Smirnov statistic for smaller sizes of the same network(see Methods).index. We find that the clustering coefficient c i of node i varies seemingly irregularly. However, when c i is plotted against i , a global pattern is seen. In Fig. 4 we show this pattern for three different network sizes. For better visualization, the plotsare shown only for relatively small network sizes. From the figure, it is clear that the global pattern of the local clusteringcoefficient gets stretched as the size of the network increases such that the nature of the pattern remains the same. We callthis new kind of similarity as “stretching similarity” and this seems to be a unique feature of this network, not so far reportedfor any other complex network. We note from plots in Fig. 4 that for a network with size N some discontinuous verticalsteps occur approximately at values N / , N / , N / , · · · . Also, we observe a band of numbers with clustering coefficient 1between N / N / p in the interval ( N / , N / ) . On the lower side, it is connected only to 1while on the upper side, it would be connected only to its multiples. However, all the numbers in this range would have onlyone multiple 2 p up to N . Thus, three numbers 1 , p , p form a triangle and hence clustering coefficient of number p must be1. A similar argument for prime powers in this range tells that they also have clustering coefficient 1. There is another bandof numbers with clustering coefficient exactly 0 between N / N which are also prime numbers. This is because all theprimes in this range are connected only to 1 making their clustering 0.Now we discuss the local clustering coefficient for the composite numbers between N / N . For a vertex n , the onlyneighbors are the proper divisors of n i.e. m such that 1 ≤ m < n and n is divisible by m . -6 -5 -4 -3 -2 -1 c ( k ) k Figure 3.
Dependence of local clustering coefficient on degree.
The plot is created using a logarithmic binning.Asymptotically, the local clustering is seen to follow a power law with exponent ∼ n = k (cid:213) i = p j i i be the factorization of n as the product of (distinct) prime powers. The fundamental theorem of arithmeticstates that such a factorization is unique up to a reordering of the primes p i ’s. It can be observed that every divisor m of n isof the form m = k (cid:213) i = p ℓ i i where 0 ≤ ℓ i ≤ j i for every 1 ≤ i ≤ k .Any two neighbors m = k (cid:213) i = p ℓ i i and m ′ = k (cid:213) i = p ℓ ′ i i such that m < m ′ are adjacent to each other if and only if ℓ i ≤ ℓ ′ i for all i .Thus the clustering coefficient of n in the network of size N is given by, c n = (cid:18) s − (cid:19) − " j (cid:229) ℓ = j (cid:229) ℓ = · · · j k (cid:229) ℓ k = [( ℓ + )( ℓ + ) · · · ( ℓ k + ) − ] ! − [ s − ] . (2)where s = ( j + )( j + ) · · · ( j k + ) . ∴ c n = (cid:18) s − (cid:19) − k (cid:213) i = (cid:18) j i + (cid:19) − s + ! (3)From the above expression it follows that value of c n depends only on the number of distinct prime factors of n and thepowers j i ’s which appear in the prime factorization of n ; but not on the actual primes which appear there. Thus for any given c i (a) 0 0.2 0.4 0.6 0.8 1 0 2000 4000 6000 8000 10000 12000 14000 16000 c i (b) 0 0.2 0.4 0.6 0.8 1 0 5000 10000 15000 20000 25000 30000 c i i (c) Figure 4.
Local clustering coefficient as a function of node index for three different sizes of network. (a) N = , (b) N = and (c) N = . In any local region of the plot, the values c i seem to be scattered irregularly. However, with theincrease in the network size, the whole pattern is stretched on a global scale. We call this similarity as “stretching similarity”. j , j , . . . j k , the value c n is constant for every n in the range N / < n ≤ N such that n = k (cid:213) i = p j i i for some set of k distinct primes p , p , · · · p k . This explains the occurrence of horizontal dotted lines in the plot for local clustering coefficients.Similarly, the clustering coefficients for other n can be computed and it can be observed that they depend on the powersand the number of distinct prime factors of n as well as the range in which n belongs that is r such that N / ( r + ) < n ≤ N / r .Here one has to also consider the number of multiples of n in the range 1 , , · · · , N . This leads to possibly different values ofclustering coefficients. This explains the occurrence of demarked regions like N / N , N / N / N / N / N there will be sufficient number of primes in the range [ , N / ] and choicesfor j i such that the pattern of horizontal lines between N / N remains the same. Also, the demarked regions have similarstructures. This provides a possible explanation for the observed stretching similarity in the clustering coefficients as N ischanged (Fig. 4).We also observe an interesting pattern when we plot the difference △ c = c i − c i + as a function of i in Fig. 5. We findthat this pattern is symmetric about △ c = Topological characteristics ofthe network.
In the present section, we discuss how three of the most important quantities average degree, global clusteringand assortativity coefficient vary with the size of the network. D c = c i - c i + i Figure 5.
Difference between clustering coefficients of successive nodes i and i + as a function of index i . Thispattern is symmetric about the line △ c = Average degree
Here we derive an approximate expression for the average degree of the network as a function of its size. By definition, theaverage degree of the network is given by: < k > n = mn (4)where m is the total number of edges in the network and n is the size of the network. The value of m is also equal to the sumof the elements in lower (or upper) triangular part of the adjacency matrix. To find this sum, we interpret the second index ofelement A i j of adjacency matrix to be the divisor of first index if A i j =
1. In other words, let A i j = i > j and j | i . Then the sum of the elements in the lower triangular part of the matrix is equal to the number of integers of the form k j with k ≥ k j ≤ n . However, whenever j > n all the entries in the in the j th column of the lower triangular part of A arezero. Let ⌊ x ⌋ denote the greatest integer ≤ x . Then m is given by: m = n / (cid:229) j = (cid:18)(cid:22) nj (cid:23) − (cid:19) = n (cid:229) j = (cid:22) nj (cid:23) − (cid:229) n / < j ≤ n (cid:22) nj (cid:23) − n
18 20 22 24 26 28 30 32 340 x 10
10 x 10
15 x 10
20 x 10
25 x 10
30 x 10 < k > n n Figure 6.
Average degree of the network as a function of size.
The solid dots represent the actual values calculated bydirect numerical simulations while the solid line is plotted using the analytic expression (9).It is well known that the first term on the right satisfies an estimate as follows: n (cid:229) j = (cid:22) nj (cid:23) = n ln n + n ( g − ) + O ( √ n ) (6)where g is Euler-Mascheroni constant. Also we observe that: (cid:22) nj (cid:23) = ∀ n < j ≤ n (7)From Eqs.(4),(5),(6),(7), it follows that: < k > n = n + ( g − ) − + O ( √ n ) as n → ¥ (8)Since g ≈ . n , we get, < k > n ∼ n − . N = and the results obtained,shown by solid dots in Fig. 6, are found to agree exactly with analytic expression (9). Since the average degree of the networkincreases with size, the degree distribution of the network is not stationary though as shown in the previous section, thenetwork is scale-free at each stage (see Methods). lobal clustering coefficient The global clustering coefficient of the network quantifies the density of closed triplets in the network. A connected triplet inthe network is the set of 3 nodes connected to each other with exactly 2 links. A closed triplet is the set of 3 nodes connectedto each other with exactly 3 links. A triangle in the network counts as three closed triplets (one centered at each node of thetriangle). The global clustering coefficient of the network is then defined as: C = × Number of trianglesNumber of connected triplets (10)We estimate the number of triangles T n in the network using the following strategy. Let us fix a vertex i and calculate thenumber of triangles in which i is the smallest vertex. The number i has (cid:4) ni (cid:5) − [ , n ] . Each ofthem is of the form ki where k = , , ..., (cid:4) ni (cid:5) . Thus, T n is given by: T n = n (cid:229) i = ⌊ ni ⌋ (cid:229) k = j nki k (11)Using the integral approximation for the above:
10 x 10
15 x 10
20 x 10
25 x 10
30 x 10 C n a -0.16-0.15-0.14-0.13-0.12-0.11-0.1-0.09-0.08-0.070 x 10
10 x 10
15 x 10
20 x 10
25 x 10
30 x 10 r n b Figure 7.
Global clustering coefficient and assortativity coefficient as a function of size of the network. ( a ) The globalclustering coefficient (see Eq.(10)) decays to 0 as the size of network increases. ( b ) The assortativity coefficient r (seeEq.(18)) also seems to reach 0 asymptotically though it always remains negative. T n ∼ n n (cid:229) i = n / i Z x = ix dx ∼ n n (cid:229) i = i (cid:16) ln ni − A (cid:17) (12) he above is bounded by, n n (cid:229) i = i (cid:18)r ni − A (cid:19) ∼ n √ n n Z x = dxx / − An n Z x = dxx ∼ Bn − An ln n (13)Here A and B are constants. Hence we see that: T n ≤ O ( n ) + O ( n ln n ) + o ( n ) (14)In particular, T n = o ( n ) (15)Let U ( n ) be the the number of connected triplets in the network after n th stage. Then U ( n ) is given by: U n = n (cid:229) i = ( k i − k i ) = n (cid:10) k (cid:11) − n h k i ∼ O ( n (cid:10) k (cid:11) ) (16)Since we have (Fig. 2) observed that the degree distribution of the network follows a power law k − a with a ∼
2, we seethat the proportion p ( k ) of nodes with degree k is ∼ k − .Thus, the expectation of the variable k satisfies: (cid:10) k (cid:11) = n (cid:229) k = k p ( k ) ∼ n Hence we see that U n ∼ n (cid:10) k (cid:11) = O ( n ) (17)From Eqs.(10), (14) and (17), the global clustering coefficient decays to zero as the network size goes to infinity. Weverify this by numerically computing the global clustering coefficient and this is shown in Fig. 7. a . However, we note that theWatts-Strogatz clustering coefficient C WS of the network (which is defined as the average of all local clustering coefficientsover all the nodes of the network ) does not decay to zero and instead reaches to a constant value ∼ .
6. This is clear fromFig. 4 since the pattern repeats with stretching similarity as the network size increases. To the best of our knowledge, there isno other network in which C WS saturates to a high non-zero value but the global clustering coefficient decays to 0. Assortativity coefficient
The correlation of degrees in the network is an important quantifier of the network structure. If in a network the high degreenodes tend to connect to low degree nodes (i.e. if the network has negative degree correlations), then the network is said to bedissortative in structure whereas if similar degree nodes tend to connect to each other, network is said to be assortative. Allthe real networks except social networks are dissortative and this has been explained using the fact that the dissortative stateis the most likely state of scale-free networks. The assortative/dissortative nature of networks can be quantified using theassortativity coefficient: r = (cid:229) i j ( A i j − k i k j / m ) k i k j (cid:229) i j ( k i d i j − k i k j / m ) k i k j (18)where k i is the degree of the i th node, A i j is the ( i , j ) th element of the adjacency matrix, m is the total number of edges in thenetwork and d i j is the Kronecker delta.In Fig. 7. b we show the dependence of r on the size of the network and in spite of irregularity in the divisibility pattern, r has a smooth behavior with n . It can be seen that r always remains negative though asymptotically it seems to reach the value0 implying that the network is dissortative. The dissortative nature of the network of natural numbers is understandable fromthe following argument. For any link in this network, the one end of the link is divisor (node A ) and other is multiple (node B ). Hence node A is also connected to all the nodes which are multiples of B but the reverse is not true. This means that thedegree of node A always tends to be very high as compared to degree of node B for a given size of the network giving thenegative value for the overall correlation coefficient.We also find that all the important statistical properties of the network like stretching similarity, degree distribution,clustering-degree correlation etc. are very robust to the removal of even the biggest hubs like numbers 1 , , , .. . This showsthat the global divisibility pattern of natural numbers does not depend only on the few nodes but instead is built by contribu-tions from all the nodes. (See Methods) iscussion The network of natural numbers constructed using divisibility relations looks like real networks in many characteristics likedegree distribution, clustering and degree correlations. We show how insights into the divisibility patterns of natural numberscan be obtained using the framework of complex networks, where we consider both composite and prime numbers in a singleundirected network with links generated using both multiples and divisors. Some of the interesting results that we get are thescale-free nature of the network with a non-stationary distribution and the existence of stretching similarity. We validate theexistence of power-law in the distribution and estimate the corresponding power-law index using rigorous techniques fromstatistical inference advocated by Clauset et al. We find that the average degree of the network grows logarithmically withthe size of the network and we find the exact formula for its behaviour analytically. We also find that the global clusteringcoefficient of the network reaches to the value 0 while the average clustering coefficient C WS saturates to a high value. Allthese results are validated by extensive numerical calculations for network up to size 2 .We also find that there exists a pattern in the local clustering coefficients that reflects universality in the organizationof natural numbers in terms of their prime constituents. We observe that this pattern has a stretching similarity which is areflection of the nature of prime factorization of natural numbers. Also, the behavior of characteristics like average degree,global clustering and assortativity coefficients for this network vary quite smoothly and hence may help us to understandbetter the divisibility relations between natural numbers. In conclusion, the work presented here describes an interestingperspective on the divisibility relations of natural numbers and has potential to become an important tool in the investigationof the properties of natural numbers. Methods
Establishing the scale-free nature of the network.
The shape of the degree distribution of the network in Fig. 2 hints at theexistence of asymptotic power law in the distribution ( p ( k ) ∼ k − a for k ≥ k min ). However a visual inspection to find k min andleast square fit and related methods to find the exponent a of the power law are known to produce very bad estimates. Hencewe use the method of maximum likelihood for the degree sequence of the network to find scaling index a of the power-lawdistribution. For this, we initially assume that the sequence is drawn from a distribution that follows a power law k − a for all k after k ≥ k min . To find this k min , we use the approach proposed by Clauset et al. The idea behind this method is to choosethat value of k as k min which makes the probability distribution of the data and best-fit power-law model as similar as possibleabove k min where we use Kolmogorov-Smirnov statistic as the distance between two distributions. After finding k min usingthis method, the best estimation for scaling exponent a is given by: a = + N " N (cid:229) i = ln k i k i − − (19)where k i , i = , · · · , N are values of k such that k i ≥ k min . For the network of size 2 , the value a is obtained here as ∼ In this approach we generatemany synthetic data sets from a true power-law distribution and measure how far they fluctuate from the power-law type ofbehavior. We then compare the results of similar measurements on the observed data. If the observed data set is much furtherfrom the power-law form than the synthetic one, the power-law is rejected. The p − value is defined as the fraction of thesynthetic distances that are larger than the empirical distance. A large p − value is indicative of existence of power law in thedata. In the present work we calculate the p − values for three different sizes of the network: N = , , p − values accurate up to two decimal places as 0 .
62, 0 .
95 and 0 .
98 respectively.The existence of power-law degree distribution for this network is thus confirmed by the fact that p − values rapidly convergeto 1 as the network size increases.The distribution in Fig. 2 is plotted with logarithmic binning with the successive bin sizes equal to successive powers of2 and the count in each bin is normalized by dividing the count by the bin-width. The same strategy is used to show thedependence of local clustering coefficient c ( k ) on degree k in Fig. 3. Symmetry in difference of successive local clustering coefficients.
To establish the global symmetry of difference inlocal clustering values △ c around the horizontal axis △ c = N , we calculate the local density ofpoints in the plot. For this, we divide the horizontal axis into 2 =
128 cells and vertical axis into 200 cells. The whole plotthen gets divided into pixels of dimension 0 . × N − . We define density r ( x , y ) of a particular pixel ( x , y ) as the ratio ofthe number of points present in the pixel to the maximum number that can be there which is equal to 2 N − (all the points ony-axis with difference less than 0 .
01 are to be considered same so the vertical dimension of each pixel is just 1). For each x we alculate the absolute difference between the corresponding pixels on each side of the line △ c =
0. If the pattern is symmetricthen these absolute differences are expected to be small. We calculate the average of such differences as: f ( x ) = (cid:229) y = | r ( x , y ) − r ( x , − y ) | (20)In Fig. 8 we show f ( x ) as a function of x and as is clear from the figure, all f values are very close to 0 confirming that thepattern is indeed symmetric. Removal of hubs from the network.
To test the robustness of the various statistical properties f ( x ) x Figure 8.
Symmetry quantifier for the Fig. 5 as given by Eq.(20).
The values of f are very close to zero for allhorizontal pixel indices establishing the approximate symmetry for the pattern.of the network against the removal of hubs from the network, we simulated the network of natural numbers removing numbers1 to 4 step by step. When number 1 is removed from the network, all the prime numbers between N / N become isolatedand these remain as the only isolated nodes. This means that in this case the network consists of a giant component along withmany isolated nodes. We find that such a removal does not affect the degree distribution and clustering-degree correlationtoo much and qualitatively the network remains scale-free with the same power-law index as for the original network. Theother properties like average degree, clustering coefficients and assortativity do change to some extent by this removal butqualitatively remain the same. The plot of degree distributions after removing hubs is shown in Fig. 9. Acknowledgements
S.M.S. is supported by Senior Research Fellowship from University Grants Commission, Delhi, India. C.B. is supported byDST-INSPIRE faculty scheme, award number [IFA-11MA-05]. Authors acknowledge Joel Ornstein for making the pythonimplementations of some of the methods used in this paper available to us. -12 -10 -8 -6 -4 -2 p ( k ) k Node 1 removed 10 -12 -10 -8 -6 -4 -2 p ( k ) k Nodes 1 to 4 removed10 -5 -4 -3 -2 -1 c ( k ) k Node 1 removed 10 -5 -4 -3 -2 -1 c ( k ) k Nodes 1 to 4 removed
Figure 9.
The degree distributions of the network of natural numbers after removing nodes from 1 to 4. The distributionsfollow a power-law similar to the original network. uthor contributions statement
S.M.S. proposed the idea and performed the numerical simulations. C.B. derived the results analytically. G.A. supervised thestudy. All authors discussed the results and prepared the manuscript.
Additional information
The authors declare no competing financial interests.
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