Multifractal Network Generator

As our methods of studying the features of our environment are becoming more and more sophisticated, we also learn to appreciate the complexity of the world surrounding us. The corresponding systems (including natural, social and technological phenomena) are made of many units each having an important role from the suitable functioning of the whole. An increasingly popular way of grabbing the intricate structure behind such complex systems is a network or graph representation in which the nodes correspond to the units and the edges to the connections between the units of the original system [1,2,3]. It has turned out that networks corresponding to realistic systems can be highly non-trivial, characterized by a low average distance combined with a high average clustering coefficient [4], anomalous degree distributions [5,6] and an intricate modular structure [7,8,9]. A better understanding of these graphs is expected and, in many cases have been shown, to be efficient in designing and controlling complex systems ranging from power lines to disease networks [10]. As increasingly complex graphs are considered a need for a better representation of the graphs themselves has arisen as well. Sophisticated visualization techniques emerged [11] and a series of new parameters have been introduced over the years [1,2,3]. Very recently one of us (L.L.) proved that in the infinite network size limit, a dense graph's adjacency matrix can be well represented by a continuous function W (x, y) on the unit square [12,13]. A similar approach was introduced by Bollobás et al. [14,15] and used to obtain convergence and phase transition results for inhomogeneous random (including sparse) graphs. This two variables symmetric function (which can have a very simple form for a variety of interesting graphs, and was supposed to be either continuous or almost everywhere continuous) predicts the probability whether two nodes are connected or not. (The non-trivial relations between the limiting objects of graph sequences and 2d functions are discussed in more details in the Supporting Information). In this paper we develop the above ideas further in order to obtain simple and analytically treatable models of random graphs with a level of complexity growing together with their size. This is an important conceptual step acknowledging a rather natural expectation: the internal organization of larger networks is more complex than those of the smallest ones. (E.g., the social contacts in large universities are much more structured than in an elementary school, which is in part due to the underlying hierarchical organization of almost every large networks we know of.) In a sense, using a function to represent a network is very much like using a model to describe a network. Models in the context of networks have been playing a crucial role since they are ideal from the point of grabbing the simplest aspects of complex structures and thus, are extremely useful in un-derstanding the underlying principles. Models are also very useful from the point of testing hypotheses about measured data. Indeed, many important and successful models have been proposed over the past 10 years to interpret the various aspects of real world networks. However, a considerable limitation of these models is that they typically explain a particular aspect of the network (clustering, a given degree distribution, etc), and for each new -to be explained-feature a new model had to be constructed. In the recent years, generating graphs with desired properties has attracted great interest. A few remarkable methods have been proposed, including the systematic approach for analyzing network topologies by Mahadevan et al., using the dK-series of probability distributions [16]. These distributions specify all degree correlations within d sized subgraphs of a given graph, with 0K reproducing the average degree, 1K the degree distribution, 2K the joint degree distribution, etc. Several methods for generating random graphs having a predefined finite dK-series were also given in [16], (with typically d ≤ 3). Most important of these techniques is based on rewiring of the links, as this turned out to be the only efficient tool in practice. The concept of characterizing a network via the frequencies of given subgraphs (forming a series with increasing size) is at the heart of the exponential random graph model as well [17,18,19]. In this approach a possible subgraph g (e.g., a pair of connected nodes, a wedge of a pair of links sharing a node, a triangle, etc.) is assigned a parameter η g related to the frequency of the sub-graph, and the probability of a given network configuration is assumed to be proportional to exp( η g n g ), where n g denotes the number of sub-graphs occurring in the network. The η parameters for a studied network are usually estimated using maximal likelihood techniques. The dK-series method and the exponential random graph model can be viewed as bottom-up approaches: in the first order approximation of the studied network we concentrate on the frequency of the most simple object (an edge), when this is reproduced correctly we move on to a slightly more complex sub-graph and so on. The series of sub-graphs from small/simple to large/complex are ordered into a sort of hierarchy. However, in a realistic scenario we stop in the above process at a relatively early stage, since on one hand most important properties of the networks are usually reproduced already, on the other hand including "higher order" sub-graphs becomes computationally very expensive. Hierarchy, self-similarity and fractality are very important concepts when describing complex systems in nature and society, and turned out to be relevant in network theory as well [20,21,22]. Very recently two important network models have been introduced which are intrinsicly hierarchical, yet show general features. Avetisov et al. proposed in [23] the construction of random graphs having an adjacency matrix equivalent to a p-adic randomized locally constant Parisi matrix, one of the key objects in the theory of spin glasses [24]. This symmetric matrix has a hierarchic structure, and its elements are Bernoulli distributed random variables (taking the value of 1 with probability q γ and the value 0 with probability 1 -q γ , where γ counts the hierarchy levels). An interesting feature of this construction is that any sub-graph belonging to a specific hierarchy level γ is equivalent to an Erdös-Rényi random graph [25], nevertheless the overall degree distribution can be scale-free. The Kronecker-graph approach introduced by Leskovec et al. is centered around hierarchic adjacency matrices as well, however in this case the self similar structure is achieved by Kronecker multiplication as follows [26]. Starting from a small adjacency matrix A 1 , (where A 1 ij = 1 if nodes i and j are linked, otherwise A 1 ij = 0), at every iteration we replace each current matrix element by A 1 multiplied by the matrix element itself, hence enlarging the matrix by a factor given by the size of A 1 . In the stochastic version of this model the elements of A 1 are replaced by real numbers between 0 and 1, and at the final stage of the multiplication process we draw a link for each pair of nodes with a probability given by corresponding element in the obtained stochastic adjacency matrix. According to the results, the Kronecker-graphs obtained in this approach can mimic several properties of real networks (heavy tails in the degree distribution, and in the eigenvalue spectra, small diameter, densification power law) simultaneously. Furthermore, in [27] Leskovec and Faloutsos presented a scalable method for fitting real networks with Kronecker graphs. We note that link probability matrices similar to the previous examples can be also used for community detection as pointed out by Nepusz et al. in [28,29]. In their approach (inspired by Szemerédi's regularity lemma [30]) the diagonal elements of the matrix give the the link density inside the corresponding communities, whereas the off-diagonal elements correspond to the link probabilities between the groups. In summary, a plausible classification of the emerging graph generating procedures/approaches involves the following types. Generating graphs as i) stochastic growth processes (e.g., [1]), ii) as a process of connecting or rewiring nodes according to prescribed probabilities ([4, 31, 32, 33, 34]), iii) accepting varying configurations with a prescribed probability [16,17,18,19], iv) by deterministically or stochastically obtaining its adjacency matrix from simpler initial matrix, [23], [26], [27], v) from a function W (x, y) on the unit square providing a value for the probabilities of node pair connections [12,13,14]. Rewiring and the related construction techniques do not provide a clue how a complex network emerges from a simple rule. On the other hand, generating a graph from a fixed function/measure does not result in networks with increasing complexity. Our approach can be considered as a combination of iv) and v) (thus, combining their advantages), assuming that in the infinitely large network limit the right representation is a singular measure (nowhere continuous function). Thus, here we introduce a new method to constructing random graphs inheriting features from real networks. The main idea of our approach is to replace W (x, y) by a fractal (singular) measure (also called multifractal), and go to the limit of infinitely fine resolution of the measure and the infinitely large size of the generated graph simultaneously. Consequently, the complexity of the obtained network is increasing with the size. Another advantage of this approach is that the statistical features characterizing the network topology, e.g., the degree distribution, clustering coefficient, degree correlations, etc, can be simply calculated analytically. For generating networks with a given prescribed statistical feature (e.g., a given degree distribution), the optimal parameters of the generating measure defining the multifractal can be determined from a simple simulated annealing process. The network generation has three main stages in our approach: we start by defining a generating measure on the unit square, next we transform the generating measure through a couple of iterations into a link probability measure, and finally, we draw links between the nodes using the link probability measure. The generating measure is defined as follows. We identically divide both x and y axis of the unit square to m (not necessarily equal) intervals, splitting it to m 2 rectangles, and assign a probability p ij to each rectangle (i, j ∈ [1, m] denote the row and column indices). The probabilities must be normalized, p ij = 1 and symmetric p ij = p ji . Next, the link probabil-ity measure is obtained by recursively multiplying each rectangle with the generating measure k times (which is equivalent to taking the k-th tensorial product of the generating measure). This results in m 2k rectangles, each associated with a linking probability p ij (k) equivalent to a product of k factors from the original generating p ij . In our convention k = 1 stands for the generating measure, thus, a link probability measure at k = 1 is equivalent to the generating measure itself. Finally, we distribute N points independently, uniformly at random on the [0, 1] interval, and link each pair with a probability given by the p ij (k) at the given coordinates. The above process of network generation is illustrated in Fig. 1, whereas in Fig. 2. we show a small network obtained with this method. We note that our construction could be made even more general by replacing the "standard" multifractal with the k-th tensorial product of a symmetric 2d function 0 ≤ W (x, y) ≤ 1 defined on the unit square. Although the resulting with the help of a measure preserving bijection between [0, 1] and [0, 1] k it could be used to generate random graphs in the same manner as with our multifractal. The diversity of the linking probabilities p ij (k) (and correspondingly, the complexity of the generated graph) is increasing with the number of iterations, just like in case of a standard multifractal. In order to keep the generated networks sparse, we must ensure that the average degree, d of the nodes does not change between subsequent iterations. This can be achieved by an appropriate choice of the number of nodes as a function of k, using the following relation: where a ij (k) denotes the area of the box i, j at iteration k. In the special case of equal sized boxes a ij (k) = m -2k , and due to the normalization of the linking probabilities the above expression simplifies to d = Nm -2k . Thus, to keep the average degree constant when increasing the number of iterations for a given generating measure, the number of nodes have to be increased exponentially with k. The construction of the link probability measure. We start from a symmetric generating measure on the unit square defined by a set of probabilities p ij = p ji associated to m × m rectangles (shown on the left). In the example shown here m = 2, the length of the intervals defining the rectangles is given by l 1 and l 2 respectively, and the magnitude of the of the probabilities is indicated by both the height and the color of the corresponding boxes. The generating measure is iterated by recursively multiplying each box with the generating measure itself as shown in the middle and on the right, yielding m k × m k boxes at iteration k. The variance of the height of the boxes (corresponding to the probabilities associated to the rectangles) becomes larger at each step, producing a surface which is getting rougher and rougher, meanwhile the symmetry and the self similar nature of the multifractal is preserved. b) Drawing linking probabilities from the obtained measure. We assign random coordinates in the unit interval to the nodes in the graph, and link each node pair I, J with a probability given by the probability measure at the corresponding coordinates. One of the main advantages of our model is that the statistical properties characterizing the network topology can be calculated analytically. An important observation concerning our model is that nodes having coordinates falling into the same row (column) of the link probability measure are statistically identical. This means that e.g., the expected degree or clustering coefficient of the nodes in a given row is the same. Consequently, the distributions related to the topology are composed of sub-distributions associated with the individual rows. Let us concentrate on the degree distribution first, which can be expressed as i (d) denotes the sub-distribution of the nodes in row i, and l i (k) corresponds to the width of the row (giving the ratio of nodes in row i compared to the number of total nodes). These ρ (k) i can be calculated using the generating function formalism as shown in the Appendix, resulting in where d i (k) = N j p ij (k)l j (k) denotes the average degree of nodes in row i. Even though the degree distribution of nodes in a given row follows a Poisson-distribution according to (3), the overall degree distribution of the generated graph can show non-trivial features, as will be demonstrated later. Similarly to the degree distribution, the clustering coefficient and the average nearest neighbors degree can be calculated analytically as well in a rather simple way (as given in the Supporting Information). According to Fig. 3b-d, the analytical results for the quantities above are in very good agreement with the empirical distributions, (obtained by generating a number of sample graphs for the chosen parameters). The use of analytic formulas instead of empirical distributions can significantly speed up the optimization of the generating measure with respect to some prescribed target property. Depending on the choice of the generating measure and the box boundaries, our method is capable of producing graphs with diverse properties. However, to generate a random graph with prescribed features in our approach we need to optimize the generating measure with respect to the given requirements. Let us suppose that the number of nodes in the graph to be generated is given. In this case we have two parameters: the number of boxes in the generating measure (given by m 2 ), and the number of iterations, k. The actual p ij and box boundaries are "self adjusting", as we shall describe in the following. Let us denote the property to which we are optimizing the generating measure by F . A conceptually simple example is when our goal is to obtain a network with a given degree distribution, in this case F is equivalent to p(d). In principle, F depends on p ij , l i , k, N (and in an implicit way on m, through the box sizes and linking probabilities). However, as m, k and N are kept constant, we discard them from the notation and write the "value" of the property corresponding to a given choice of p ij and l i as F (p ij , l i ). (Note that in most cases F (p ij , l i ) is actually a high dimensional object, e.g, a degree distribution, and not a real number). The target value of the property to which we would like the system to converge is denoted by F * . In order to be able to make the studied property of the generated network converge to the goal F * , we have to define a way to judge the quality of the actual F (p ij , l i ). In other words, we have to define a sort of distance or similarity between F (p ij , l i ) and F * . This distance/similarity measure can be used as an energy function during a so called simulated annealing procedure, and we shall denote it by E[F (p ij , l i ), F * ]. The actual form of this function depends on the actual choice of the property, e.g., in case of optimizing the degree distribution a plausible choice is the sum of the relative differences between the degree distributions: where d runs over the degrees, ρ (k) (d) is the value of the actual degree distribution at degree d and ρ * (d) is the value of the target degree distribution at the same degree. In the simulating annealing we also define a temperature, T , which is decreased slowly during the process. The process itself consist of many Monte-Carlo steps, and in one step we try to change one of the linking probabilities or one of the box boundaries by a small amount, following the Metropolis algorithm [35]. If the energy E 2 after the change is smaller than the energy E 1 before, the change is accepted. In the opposite case, the change is accepted by a probability given by P The above procedure can be generalized in principle to optimizing with respect to multiple properties simultaneously as well. However, for simplicity here we consider the optimization of the different properties separately. In Fig. 4a we show the results for optimizing the generating measure with respect to various target degree distributions. Although the three chosen targets are rather different, (a scale-free distribution, a log-normal one, and a bimodal distribution), our method succeeded in finding a setting of p ij and l i producing a degree distribution sufficiently close to the target. Similarly, in Fig. 4b the results from optimizing with respect to three clustering coefficient distributions are displayed showing again a reasonable agreement between the targets and the results. Our approach raises a number of fundamental graph theoretical and practical questions. Should we expect that large real graphs converge to some limiting network in a strict sense of the convergence? Or, alternatively, their structure cannot be mapped onto a fixed function, and only an ever changing (with the size of the network) measure (in the infinite network size limit becoming singular) can be used to reflect the underlying structural complexity? This would be in contrast with the consequences of the renowned Szemerédi Lemma [30] valid for arbitrary dense graphs. Although it can be shown analytically (see SI) that in the infinitely large network size limit our construction converges to a relatively simple graph, the convergence to this structure is extremely slow. According to our numerical studies, there is a very extensive region between the small and infinite regimes in which a well defined, increasingly complex structure emerges as our method is applied. Details about aspects of the slowness of convergence involving an extremely slow growth of the relative number of isolated nodes and the appearance of oscillations are given in the SI. In summary, our results demonstrate that it is possible to use simple models to construct large graphs with arbitrary distributions of their essential The black symbols come from an experiment where the target was a powerlaw degree distribution (the inset shows this on log-scale), the gray symbols correspond to a setting with a log-normal target, whereas the white symbols show the results of an experiment with a bi-modal target distribution. b) Optimizing with respect to different clustering coefficient distributions. characteristics, such as degree distribution, clustering coefficient distribution or assortativity. In turn, these graphs can be used to test hypotheses, or, as models of actual data. The combination of the tensorial product of a simple generating measure and simulated annealing technique leads to small (in practice 3x3 to 5x5) matrices representing the most relevant statistical features of observed networks. A very unique feature of this construction is that the complexity of the generated network is increasing with the size. In addition, the multifractal measure we propose is likely to result in networks displaying aspects of self-similarity in the spirit of the related findings by Song et al. [21]. . Therefore, we let inj(F, G) denote the number of injective homomorphisms from graph F to graph G and define (Thus, s(F, G) is the average number of labeled copies of F , such that a specified node of F goes on a specified node of G.) Convergent graph sequences are related to 2d functions in a non-trivial way [12,13]. First of all, we can construct a convergent graph sequence using a symmetric measurable function, 0 ≤ W (x, y) ≤ 1, defined on the unit square as follows. For a given network size N, we distribute N points independently, uniformly at random on the [0, 1] interval. These points correspond to the nodes in the network, and each pair of nodes is linked with the probability given by W (x, y) at the coordinate of the according points. In the N → ∞ limit the obtained graph sequence is converging. What is even more surprising, it can be proven that we can represent any convergent graph sequence by a 2d function, since for any convergent graph sequence one can find a W (x, y) providing the same limiting sub-graph densities. The average degree of nodes in a random graph generated from a given W (x, y) using the construction above can be given simply as Thus, in the N → ∞ limit the obtained network becomes dense. In contrast, real networks are usually sparse in the sense that their average degree is not expected to grow with increasing size. A solution to this problem was proposed by Bollobás et al., by redefining the linking probabilities as W (x, y)/N, resulting in a network with an average degree independent of N. They showed that depending on the choice of W (x, y), a wide range of sparse networks can be generated. Our approach is different from this method in that instead of using a construction into which we build in the level of complexity from the beginning, we generate complexity by using tensorial products of increasing power as N → ∞. This is a qualitatively new picture, corresponding to reality to a higher degree (larger graphs are more complex/inhomogeneous/structured in nature than smaller graphs). In addition, we achieve this using a relatively simple construction. A shortcoming of our model is that it can lead to a network in which the majority of nodes are isolated in the N → ∞ limit. However, as we shall see, this effect is negligable for graphs in the size range of real networks. In general, if W 1 (x, y), W 2 (x, y), . . . , W k (x, y) is a sequence of symmetric measurable functions on the unit square (with 0 ≤ W k (x, y) ≤ 1 for any k), let us define w k (x) as the average linking probability for a node at position x given by Similarly, let ω k denote the average link probability for the whole network, which can be expressed as Let us choose the number of nodes, N k associated to W k (x, y) in such a way that the average degree of nodes converges to a constant (non zero) (This means that the number of links is around d /2.) The degree distribution of a node at position x can be given by a binoimal distribution as In the thermodynamic limit this can be approximated by a Poisson-distribution written as The degree distribution of the whole network is obtained by integrating ρ(d, x), resulting in In particular, the probability that a randomly chosen node will be isolated (having degree zero) is From ( 9) and ( 10) it follows that the average value of w k (x) is around d /N k . In case w k (x) is actually independent of x, then w k (x) = d /N k , and However, if w k (x) is such that its typical value is much smaller than its average, then typically e -N k w k (x) ≃ 1 resulting in which means that the majority of nodes becomes isolated. The condition for avoiding this degeneracy can be formulated as where c < 1 is a constant. In case of the multifractal graph generator, (or a more general "tensoring" construction), the above condition is not fulfilled, unless w k (x) is independent of x. As mentionned in the main text, by using a measure preserving bijection between [0, 1] and [0, 1] k , our model can be formulated in a more general form using the tensorial product The marginals (8) in this representation are given by where w(x) = 1 0 W (x, y)dy. Similarly, ( 9) is transformed into where ω = 1 0 w(x)dx. Thus, according to (10), we should choose Unfortunately, these functions do not satisfy condition (17 almost surely by the Law of Large Numbers. Let α = exp( ln w(x) dx), then α < ω by the Jensen inequality (expect if w is constant), and the value of w k is almost always close to α k , while its average is ω k . Since (α/ω) k → 0 if k → ∞, this shows that if (17) holds, then w(x) is constant. On the other hand if w(x) = ω for any x, then the expected degree of the nodes becomes independent from their position and the degree distribution converges to a Poisson distribution, just like in case of an Erdős-Rényi graph. In this case it is also easy to calculate the number of copies of any connected graph F with l nodes in a graph G k obtained from W k . There are injectively, and for each map, the probability that it is a homomorphism is t(F, W k ) = t(F, W ) k . Hence Since we have a sparse graph, we want to normalize this by N k ; so the normalized number of copies of F is For example, the normalized number of triangles is It is easy to see that if w(x) = ω is constant, then where equality holds if and only if F is a tree or W is an equivalence relation such that there is a partition From ( 23) and (25) we gain Using high concentration inequalities one can prove that this convergence happens almost surely (not just in expectation). We see from (27) that the sequence G k is convergent with probability 1 in the Benjamini-Schramm sense. There are a number of possibilities which one could try to cure the degeneracy of the thermodynamic limit of our model shown here, however these are out of the scope of the present study. We could modify the tensoring construction by adding to W k a constant c k tending to 0 reasonably slowly. Another possibility is to modify W k to (W k ) a k , where a k → 0. Next, let us investigate the magnitude of the above effect for graphs in the size range of real networks. For this purpose we generated networks from randomly chosen generating measures (with m = 4, equal sized boxes) iterated from k = 1 to k = 11. The number of nodes at k = 1 was set to 1000 and to 5000 respectively, and for k > 1 it was adjusted using eq. () in the main text. (Thus the average degree of the graphs remained the same during the iterations.) In Fig. 5. we show the results obtained by averaging over 1000 samples for both settings by plotting ρ(d = 0) as a function of N. Inspite of the increasing tendency of the curves, at the last iteration with network sizes above 10 9 , the fraction of isolated nodes is still very low. Thus, the effect of isolated nodes becoming dominant is negligible on the scale of real world applications. In spite of the analytical results for the convergence in the thermodynamic limit, the degree distribution often shows an oscillatory behavior in the size range of real networks. This is shown in Fig. 6 for an m = 3 generating measure iterated from k = 1, N = 100 to k = 11, N = 4.24 • 10 8 . As k is getting larger, the more oscillations can be observed in ρ(d) towards the large degrees. A serious advantage of our model is that the statistical properties characterizing the network topology can be calculated analytically. In the Appendix of the main text we give a derivation for the degree distribution, based on the generating function formalism. The definition and the most important properties of the generating functions can be summarized as follows. If a random variable ξ can take non-negative integer values according to some probability distribution P(ξ = n) ≡ ρ(n), then the corresponding generating function is given by The generating-function of a properly normalized distribution is absolute convergent for all |x| ≤ 1 and hence has no singularities in this region. For x = 1 it is simply The original probability distribution and its moments can be obtained from the generating-function as And finally, if η = ξ 1 + ξ 2 + ... + ξ l , where ξ 1 , ξ 2 , ..., ξ l are independent random variables (with non-negative integer values), then the generating function corresponding to P(η = n) ≡ σ(n) is given by The degree distribution of the nodes falling in row i of the link probability measure, ρ i (d), can be calculated as follows. In our construction we draw links for a node in row i pointing to nodes in row j altogether n j (k) times with a probability p ij (k), where n j (k) is the number of nodes in row j, given by n j (k) = Nl j (k). Since the link draws are independent, the distribution of the number of links from a node in row i to nodes in row j is binomial. This can be approximated by a Poisson-distribution when n j is sufficiently large as where d ij denotes the average number of links from a node in row i to nodes in row j given by The degree of a node in row i is given by the sum over the links towards the other rows as Therefore, the generating function of ρ i (d) is the product of the generating functions of the ρ where G (k) ij (x) is defined as (A summary of the most important properties of the generating functions is given in the Supporting Information). By substituting (38) into (37) we arrive to where we used that due to the independence of the links, the expected degree of a node in row i can be expressed as An alternative form for d i can be obtained by substituting (35) into the equation above, yielding The degree distribution of the nodes falling into row i can be obtained by transforming back the generating function in (39), resulting in Similarly to the degree distribution, the clustering coefficient of nodes falling into the same row of the link probability measure is expected to be the same. The clustering coefficient of a node in row i can be obtained by calculating the number of triangles containing the node, divided by the number of link pairs originating from the node. Since the triangles are equivalent to link pairs originating from the node having their other end connected by a third link, the expected clustering coefficient of a node in row i can be given as . (43) The first term in both the numerator and the denominator corresponds to the link pairs for which the other end of the links point to the same row j, whereas the second terms give the contribution from link pairs connecting our node in row i to distinct rows j and q. A2. 4 The average nearest neighbors degree Finally, we mention that the degree correlations can be calculated from p ij (k) (and l i (k)) as well. Here we derive the expression for the average nearest neighbors degree, d N N , as a function of the node-degree. This is one of the most simplest quantity characterizing the degree correlations: an increasing curve corresponds to an assortative network, whereas a decreasing one signals disassortative behavior. The average degree of the neighbors of a node from row i can be given as The average degree of the neighbors of a node with degree d can be given as a sum over the possible d (k) N N,i , multiplied by the conditional probability p (k) (i|d) that the node is from row i, given that its degree is d: These conditional probabilities can be obtained as follows. The number of nodes from row i with degree d is n i (k)ρ (k) i (d), whereas the total number of nodes with degree d is nρ (k) (d). The probability that a node is from row i given that its degree is d is the ratio of these two: (47)