Abstract
The past two decades have seen significant successes in our understanding of networked systems, from the mapping of realworld networks to the establishment of generative models recovering their observed macroscopic patterns. These advances, however, are restricted to pairwise interactions and provide limited insight into higherorder structures. Such multicomponent interactions can only be grasped through simplicial complexes, which have recently found applications in social, technological, and biological contexts. Here we introduce a model to grow simplicial complexes of order two, i.e., nodes, links, and triangles, that can be straightforwardly extended to structures containing hyperedges of larger order. Specifically, through a combination of preferential and/or nonpreferential attachment mechanisms, the model constructs networks with a scalefree degree distribution and an either bounded or scalefree generalized degree distribution. We arrive at a highly general scheme with analytical control of the scaling exponents to construct ensembles of synthetic complexes displaying desired statistical properties.
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Introduction
All beauty, richness, and harmony in the emergent dynamics of a complex system largely depend on the specific way in which its elementary components interact. The last 20 years have seen the birth and development of the multidisciplinary field of network science, wherein a variety of systems in physics, biology, social sciences, and engineering has been modeled by networks of coupled units, in the attempt to unveil the mechanisms underneath the observed systems’ functionality^{1,2,3,4,5,6}.
But the fundamental limit of such a representation is that networks capture only pairwise interactions, whereas the function of many realworld systems not only involves dyadic connections, but rather is the outcome of collective actions at the level of groups of nodes^{7}. For instance, in ecological systems, three or more species compete for food or territory^{8}. Similar multicomponent interactions appear in functional^{9,10,11,12,13} and structural^{14} brain networks, protein interaction networks^{15}, semantic networks^{16}, multiauthors scientific collaborations^{17}, offline and online social networks^{18,19}, trigenic interactions in gene regulatory networks^{20,21}, and spreading of social behavior due to multiple, simultaneous, social interactions^{22}.
Simplicial complexes (SCs), being structures formed by simplices of different dimensions (nodes, links, triangles, tetrahedra, etc.), can effectively map the relationships between any number of components. Originally introduced over two decades ago^{23}, SCs are becoming increasingly relevant, thanks to the enhanced resolution of current datasets and the recent advances in data analysis techniques^{24,25}. As real data are being accumulated, we encounter a theoretical challenge: how to synthesize SCs that faithfully reproduce the observed structural features. Significant progresses were made in extending to SCs static graph models, such as random graphs^{26,27,28}, configuration models^{29,30,31}, or activitydriven models^{32}. The fact is however that in many circumstances (such as in the case of scientists collaborating with and citing each other), the network is the result of a growing process, and therefore another approach to model SCs is that of outofequilibrium models. Most of the models falling within this category have a geometrical interpretation^{33} and are variations of the socalled “network geometry with flavor” (NGF) model^{34,35,36,37,38}, which aims at providing a theoretical basis to characterize the underlying geometry of complex networks^{39} or to explore hidden geometries in complex materials by aggregating SCs as fundamental building blocks^{40}. While the original NGF model adds one^{34} or more^{37} maximal simplices of fixed dimension at each time step of the growth process, Fountoulakis et al.^{41} introduced even the option to remove an existing simplex when a new one is added.
In this paper, we discuss the NGF model proposed by Courtney and Bianconi^{37} that is able to grow SCs of order two, i.e., structures made of nodes, links and triangles, making use of preferential and nonpreferential rules, and we generalize this model by proposing a model that makes a combination of the two mechanisms. The resulting SCs are characterized by two distributions, the classic degree distribution P(k), capturing the fraction of nodes with degree k, and the generalized degree distribution P(k_{ℓ}), where k_{ℓ} characterizes the number of triangles supported by each link l = (i, j). We show that our generative model always yields a powerlaw scaling in P(k), recovering the ubiquitously observed scalefree property^{42,43}, and, at the same time, it allows full control over P(k_{ℓ}), i.e., bounded or scalefree with any desired scaling exponent. Indeed, P(k_{ℓ}) has been shown to play a crucial role in the emergence of collective behavior, such as synchronization^{44}. We will furthermore show that our methods can be straightforwardly extended to grow structures containing hyperedges of size larger than two.
Results and discussion
SCs growth models
Let us, for the time being and unless otherwise specified, concentrate on NGFs models of SCs of order two proposed by Courtney and Bianconi^{37}. The purpose is to grow a network of N nodes featuring a transitivity coefficient T = 1^{45,46}, thus implying that each link is part of a connected triplet of nodes (a triad) which forms a triangle. This is done by starting at t = 0 with an elementary network seed, consisting in an initial clique (an alltoall connected network) of size N_{0} ≪ N. Then, at each successive times t = 1, 2, 3, ..., N − N_{0}, a new node is added to the graph. The added node selects m_{tri} already existing links, and forms connections with the 2m_{tri} nodes located at the ends of such links, thus generating m_{tri} new triangles (when m_{tri} > 1, a further condition is enforced that the m_{tri} selected links are not pairwise adjacent, to avoid multiple links from the added node to single existing nodes in the graph).
Such a procedure can be conducted with or without adopting a preferential attachment rule. In the latter case, the probability that the node added at time t selects the specific link (i,j) to form its connections is taken to be \({P}_{ij}(t)=\frac{1}{{N}_{0}\left({N}_{0}1\right)/2+2{m}_{{\rm{tri}}}(t1)}\), i.e., the m_{tri} links are randomly selected among all those which already exist in the graph at time t − 1 with equal probability (and therefore without applying a preferential rule). In the former case, instead, one has that \({P}_{ij}(t)=\frac{{k}_{ij}(t1)}{{\sum }_{i,j}{k}_{ij}(t1)}\), which implies that the larger is the number of triangles a given edge (i,j) is part of at time t − 1 the larger the probability for that edge of being selected to form a new triangle with the added node. These models are the NGF models proposed by Courtney and Bianconi^{37} for d = 2, s = 0 (uniform attachment) and d = 2, s = 1 (preferential attachment).
Figure 1a, c reports the degree distribution P(k) of the networks (of size N = 10,000 nodes) generated by the two methods. One immediately sees that in both cases the graphs feature a clear powerlaw scaling, i.e., the scalefreeness which is indeed characterizing the vast majority of realworld networks^{1,2,3,4,5,6,42,43}. In Fig. 1b, d, we report, instead, a visualization of a typical synthesized network with N = 200 and m_{tri} = 1.
We also measure the generalized degree k_{ℓ}. Given a graph of N nodes and its adjacency matrix A (the N × N matrix with entries a_{ij} = 1 if nodes i and j are connected by a link, and a_{ij} = 0 otherwise), the generalized degree k_{ij} of the link between node i and j is the (i, j)entry of the matrix A ⊘ A^{2} (with the symbol ⊘ standing for the Hadamard product), that is, \({k}_{ij}={(A)}_{ij}\oslash {({A}^{2})}_{ij}\), which accounts for the number of triangles in which the link ij participates. Looking at Fig. 1, one can immediately see that while the two generated networks are essentially heterogeneous in the node degree, the properties of k_{ℓ} appear instead to be very different. In particular, the nonpreferential case leads to a much more restricted range of k_{ℓ} values as compared to that generated by the preferential rule (see the two color bars at the right of the panels). Moreover, the structure obtained with nonpreferential attachment is homogeneous in terms of k_{ℓ} (almost all the six colors, each one representing a given value of k_{ℓ}, are visible). On the opposite, the preferential rule generates an SC with a high heterogeneity in k_{ℓ}: almost all links are red (they have the lowest as possible value of k_{ℓ}), and only one link (the black one) displays a value of k_{ℓ} equal to the maximum in the distribution.
These latter features are more quantitatively visible in Fig. 2, which reports the distribution P(k_{ℓ}) of the generalized degree k_{ℓ} for the two cases. Comparing Fig. 2a, b, one immediately realizes that while P(k_{ℓ}) is exponentially decaying when the growth is realized with nonpreferential attachment (Fig. 2a), its scaling is a clear power law in the presence of preferential attachment. A relevant conclusion at this stage is that the two cases are imprinting a completely different topology in the triangular structures (reflected by completely different scaling properties in the distribution of the generalized degree k_{ℓ}).
Nonpreferential model (NGF^{37} with d = 2, s = 0)
We now furnish a full analytic treatment, and provide the rigorous expressions for the distributions P(k) and P(k_{ℓ}). We start with the case of nonpreferential attachment, and we call N(k,t) the number of nodes with degree k at time t. Its rate equation reads
where \(\frac{dN(k,t)}{dt}\equiv N(k,t+1)N(k,t)\), and δ is the Kronecker delta function.
For N(t) ≡ ∑_{k}N(k,t) ≃ t, one seeks a solution of the form N(k,t) = tP(k), where P(k) is assumed to be time independent. Since the total number of edges is ~2m_{tri}t, one has ∑_{k}kN(k,t) ≃ 4m_{tri}t, and the rate equation for P(k) becomes
Such a latter equation is the same as Eq. (6) introduced by Boccaletti et al.^{47}, and its solution (for k ≥ 2m_{tri}) is
which perfectly fits the data (see the dotted, dasheddotted, and dashed black lines in Fig. 1a). A similar, exponential, degree distribution has also been reported in refs. ^{36,39,48}.
Then, one can consider N_{e}(k_{ℓ},t) as the number of edges participating in k_{ℓ} triangles at time t. Its rate equation is
where N_{e}(t) ≃ 2m_{tri}t and the unwritten terms account for the formation of triangles from two or more linked edges. For the distribution of the generalized degree k_{ℓ}, one has that P(k_{ℓ}) = N_{e}(k_{ℓ},t)/N_{e}(t), and the recursive equation is
admitting the following solution:
The exponential function (6) is reported as a dashed line in Fig. 2a, and one can see that the fit with numerical simulations is rather good, especially for the case m_{tri} = 1.
Preferential model (NGF^{37} with d = 2, s = 1)
The analytic treatment of the preferential attachment case is far more complicated, as it implies the demonstration of a couple of theorems and the extensive use of a known lemma. We limit here ourselves to furnish the main results without reporting all the (sometimes cumbersome) formal mathematical steps, whereas the interested reader can find the full details within the “Methods” section. Calling again N(k,t) the number of vertices with degree k at time t, its recurrence relation (see details in “Methods”) can be written as
for k > 2m_{tri}, and
for k = 2m_{tri}.
In order to obtain an expression for P(k), one supposes that \(\frac{N(k,t+1)}{t+1}=\frac{N(k,t)}{t}\) for large t. Therefore, one gets \(P(2{m}_{{\rm{tri}}})=\frac{3}{3+2{m}_{{\rm{tri}}}}\) and \(P(k)=P(k2)\frac{k/2+1}{k/2+1.5}\), which ultimately gives \(P(k)=\frac{3}{3+2{m}_{{\rm{tri}}}}\mathop{\prod }\nolimits_{l = {m}_{{\rm{tri}}}+1}^{k/2}\frac{l1}{l+1.5}\) or alternatively
where Γ is here the gamma function. The powerlaw scaling predicted by Eq. (9) fits remarkably well the numerical data (see the dotted, dasheddotted, and dashed black lines in Fig. 1c) and was also measured in refs. ^{36,39}.
As for P(k_{ℓ}), one calls again N_{e}(k_{ℓ},t) the number of edges participating in k_{ℓ} triangles at time t. The recurrence relation for N_{e}(k_{ℓ},t) (see details in “Methods”) is
for k_{ℓ} > 1, and
for k_{ℓ} = 1. Imposing that \(\frac{{N}_{{\rm{e}}}({k}_{{\rm{\ell }}},t+1)}{t+1}=\frac{{N}_{{\rm{e}}}({k}_{{\rm{\ell }}},t)}{t}\) for large t, one gets an equation for P(k_{ℓ}) which reads as \(P({k}_{{\rm{\ell }}})=P({k}_{{\rm{\ell }}}1)\frac{{k}_{{\rm{\ell }}}1}{{k}_{{\rm{\ell }}}+3}\), with \(P(1)=\frac{3}{4}\). The solution is
The powerlaw function (12) is reported as a dashed line in Fig. 2b, and one can observe that the fit is, once again, extremely good.
Mixed model
Finally, our study can be extended and generalized to a mixed model, through which it is possible to effectively imprint any desired powerlaw scaling in the triangular structure of the network. Contrary to the previous two cases, our mixed model cannot be encompassed within the framework of the NGF^{36}. In particular, we consider the case in which the probability that the node added at time t selects the specific link (i,j) to form its connections is
for some constants A and B. Here, B is nonnegative, N_{ℓ}(t) ~ 2m_{tri}t is the number of links at moment t and ∑_{ij}k_{ij}(t − 1) = 3m_{tri}t is the sum of the generalized degrees of all edges. Notice that A = 0 and \(B=\frac{2}{3}\) (\(A=\frac{1}{2}\) and B = 0) recovers the preferential (nonpreferential) case discussed above.
From the constraint that the sum of all probabilities must be equal to 1, it follows that A and B must obey \(2A+\frac{3}{2}B=1\), so that A = 1/2 − 3/4B. Furthermore, P_{ij}(t) must be nonnegative for all (i,j) (also those for which k_{ij} = 1), and this gives the following bounds for A and B: \(1\le A\le \frac{1}{2}\) and 0 ≤ B ≤ 2. Under these conditions, one can analytically demonstrate (see “Methods” for details) that for strictly positive B values, the resulting networks display scalefree distributions \(P({k}) \sim {k}^{{\gamma _{\ell}}}\) and \(P({k}_{{{\ell }}}) \sim {k}_{{\rm{\ell }}}^{{\gamma _ {\ell}}}\) with exponents given by
and
For B = 0, one has instead (see “Methods” for full details) \(P({k}_{{\rm{\ell }}})=\frac{2}{{3}^{{k}_{{\rm{\ell }}}}}\). Equation (14) implies that γ values are between 2 (B = 2) and 3 (B = 0), whereas γ_{ℓ} is equal to 2 for B = 2, and tends to infinity as B tends to 0 according to Eq. (15). On its turn, this means that choosing B between 1 and 2, γ and γ_{ℓ} can be preselected ad libitum between 2 and 3, i.e., the imprinted structures of links and triangles feature well defined mean values of the degrees, but unbounded fluctuations as the system grows in size.
In Fig. 3, we report P(k) and P(k_{ℓ}) for three distinct values of B. It is seen that the fit between analytic predictions and numerically generated data is always remarkably good. Moreover, the figure demonstrates that our method constitutes a highly general scheme by means of which one can construct, in a fully flexible way, ensembles of synthetic complexes displaying any desired statistical properties [from the condition of Fig. 3a, d featuring a super scalefreeness—where even the mean degrees diverge in the thermodynamic limit—to any milder condition which characterizes in fact many networks from the real world].
Extension to d uniform hypergraphs
The mixed model for SCs can be straightforwardly extended to grow uniform d hypergraphs. For simplicity, let us start by setting N_{0} = d + 1 nodes in a ring, such that, for d = 2 we have a triangle (2hyperlink), for d = 3 a square (3hyperlink), for d = 4 a pentagon (4hyperlink), and so on. At each time step, a structure of d − 1 new nodes forming an open ring with d − 2 edges (see the sketches in Fig. 4a–c) are added to the network, in order to close a d hyperedge with an existing link. Once again, the probability that a specific link (i,j) is chosen to close the hyperedge is
N_{ℓ}(t) being the total number of links at time t, and k_{ij,d}(t) the generalized degree of the (i,j) link, that is, the total number (at time t) of d uniform hyperedges which are incorporating the link ij. Figure 4 reports the node degree distributions P(k) (d–f) and the generalized degree distribution \(P(k_{\ell,d})\) (g–i), where \(k_{\ell,d}\) now characterizes the number of triangles (for d = 2), squares (for d = 3), or pentagons (for d = 4) supported by each link l = (i, j). The reported curves are obtained at different values of B, and the color code is visible in the legend of (a). It is possible to see that, in all cases, the grown structures display the same scaling behavior predicted by our mixed model for the case of SCs of order 2. The extension of our techniques to the growth of SC’s of higher order will be reported elsewhere.
In summary, complex networks encode the basic architecture of social, biological, and technological networks, touching upon the most crucial challenges of modern science, from the spread of epidemics in social networks^{20,49} to the resilience of our ecosystems and critical infrastructure^{50}.
In the context of pairwise interactions, the most natural measure of centrality is a node’s individual degree, capturing its potential dynamic impact on the system^{51}. The discovery that most realworld networks exhibit extreme levels of degree heterogeneity was disruptive—indicating that networks are highly centralized, with a potentially disproportionate role played by a small fraction of their components^{42,43}.
As we deepen our investigation into the interaction patterns of complex systems, it becomes increasingly clear, however, that higher order structures, beyond pairwise interactions, underlie much of the observed richness of realworld networks. Hence, we sought the fundamental rules that prescribe centrality, and govern its distribution, in an SC environment. In addition to this, the proposed models are expected to exhibit the smallworld property, simultaneously featuring short average distances which increase proportionally to the logarithm of the network size (see Fig. 5) and large modularity and clustering coefficients—as new nodes enter the network closing triads. A natural measure is the number of complexes, here triangles, that a link participates in. Indeed, an SC represents a potentially functional unit, such as a collaboration of a social team^{17}, or a trio of interacting biochemical agents^{20,21}. A component that is part of many such complexes is, therefore, likely central in the functionality of the system.
Several growing network models have been introduced and studied, which help exposing the roots of SC heterogeneity, shedding light on the emergence of centrality beyond the degree distribution. As we seek to understand the behavior of complex systems, their resilience and dynamic functionality, we hope that our comprehension into the microscopic processes of their formation can provide meaningful macroscopic insights.
Methods
In this final section, we furnish all the details of the analytical results regarding our models. The starting point is that, as the synthesized networks have transitivity coefficient T = 1 (i.e., no links exist which do not form part of at least a triangle), the degree of each vertex is the number of triangles containing that vertex multiplied by 2, and one has that N_{v}(k_{v}, t) = N(2k, t), where N_{v}(k_{v}, t) is the number of nodes participating in k_{v} triangles and N(2k, t) is the number of nodes having degree 2k at time t. As a consequence one has that P(k_{v}) = P(k), and is therefore entitled to concentrate on either one of such distributions, depending on which one finds the simpler analytical treatment.
The nonpreferential attachment case
Let N_{v}(k, k_{v}, t) be the number of nodes with k neighbors participating in k_{v} triangles at time t. The rate equation is
where the unwritten terms account for the formation of triangles from two or more linked edges. In the sequel, we assume that the chosen edges are not linked (none of the nodes of a selected edge is linked to any of the nodes of another selected edge, which is always the case for m_{tri} = 1). Hence, the number of edges and triangles related to a given node increases one by one. Only the new nodes entering the system have the number of edges (2m_{tri}) double than the number of triangles in which they are participating (m_{tri}). This way, for k big enough, one has
After summing over all values of k, one obtains an approximate equation for N_{v}(k_{v},t), the number of nodes participating in k_{v} triangles
One then can proceed as in the case of the degree distribution P(k) (see the main text), and seek a solution of the form N_{v}(k_{v},t) = tP(k_{v})
The solution for k_{v} ≥ m_{tri} is
Therefore, one has also that P(k) ~ k^{−3} which coincides with Eq. (3) in the main text.
The preferential attachment case
In order to obtain P(k) and P(k_{ℓ}), one needs here to make use of the following lemma reported by Chung and Lu^{52}:
Lemma 1 Suppose that a sequence {a_{t}} satisfies a recurrence relation
where t_{0} and t_{1} are arbitrary, positive, fixed, values. Furthermore, suppose that \({\mathrm{lim}\,}_{t\to \infty }{b}_{t}=b\,> \, 0\) and \({\mathrm{lim}\,}_{t\to \infty }{c}_{t}=c\,> \,0\).
Then, \({\mathrm{lim}\,}_{t\to \infty }\frac{{a}_{t}}{t}\) exists and one has
For instance (and indicating by \({\mathbb{E}}(\cdot )\) the expectation value), P(k) for k = 2m_{tri} can be obtained by setting \({b}_{t}=b=\frac{2{m}_{{\rm{tri}}}}{3}\) and c_{t} = c = 1. Application of Lemma 1 ensures that the limit \({\mathrm{lim}\,}_{t\to \infty }\frac{{\mathbb{E}}(N(2{m}_{{\rm{tri}}},t))}{t}\) exists and is equal to \(\frac{3}{3+2{m}_{{\rm{tri}}}}\), leading to \(P(1)=\frac{3}{3+2{m}_{{\rm{tri}}}}\).
On the other hand, P(k) for k > 2m_{tri} is obtained assuming that P(k − 2) exists, and applying Lemma 1 again with \({b}_{t}=b=\frac{k}{3}\) and \({c}_{t}={\mathbb{E}}(N(k2,t))\frac{k2}{3t}\) (i.e., taking \(c=P(k2)\frac{k2}{3}\)). Such a choice, indeed, entitles one to write a recurrence relation, and to obtain an explicit formula for P(k)
which coincides with Eq. (9) of the main text. Here, Γ is the gamma function.
Furthermore, one can demonstrate that P(k) is sharp, by the use of the following theorem:
Theorem 1 For any fixed ε > 0 and δ > 0 and for any large enough t, the difference between the number of vertices with degree k at time t and P(k)t is smaller than εt with probability larger than 1 − δ.
The theorem can be proved by considering the martingale \({X}_{l}={\mathbb{E}}(N(k,t) {{\mathcal{F}}}_{l})\) (where \({{\mathcal{F}}}_{l}\) is the σalgebra generated by the probability space at time l). It is rather easy to show that ∣X_{l+1} − X_{l}∣ is bounded by 4, and therefore the theorem follows from the Azuma–Hoeffding inequality^{53,54}.
When trying to obtain P(k_{ℓ}), one can encounter a problem in the case in which the triangles added at step t have one or more common edges, so that the number of different edges added at time t might not be equal to 2m_{tri}.
Denoting by e(t, k_{ℓ}) the number of added edges of degree k_{ℓ} at time t, the more accurate recurrence relation for k_{ℓ} > 1 is the following:
and for k_{ℓ} = 1 one has
Using Theorem 1 (and some nontrivial math, of which we omit the technical details), one can prove that \({\mathrm{lim}\,}_{t\to \infty }e(t,1)=2{m}_{{\rm{tri}}}\) and \({\mathrm{lim}\,}_{t\to \infty }e(t,{k}_{{\rm{\ell }}})=0\) for k_{ℓ} > 1, so that one has \({\mathrm{lim}\,}_{t\to \infty }{\mathbb{E}}(e(t,1))=2{m}_{{\rm{tri}}}\) and \({\mathrm{lim}\,}_{t\to \infty }{\mathbb{E}}(e(t,{k}_{{\rm{\ell }}}))=0\) for k_{ℓ} > 1.
Furthermore, one can prove another theorem:
Theorem 2 The number of different edges divided by t converges always to 2m_{tri}.
It follows that \(P({k}_{{\rm{\ell }}})={\mathrm{lim}\,}_{t\to \infty }\frac{{\mathbb{E}}({N}_{{\rm{e}}}({k}_{{\rm{\ell }}},t))}{2{m}_{{\rm{tri}}}t}\).
Finally, one can apply Lemma 1 with \({b}_{t}=b=\frac{1}{3}\) and \(c=\mathrm{lim}\,1+O\left(\frac{1}{{t}^{2}}\right)=1\) to get that \(P(1)={\mathrm{lim}\,}_{t\to \infty }\frac{{\mathbb{E}}({N}_{{\rm{e}}}(1,t))}{2{m}_{{\rm{tri}}}t}\) exists and is equal to \(\frac{3}{4}\). For k_{ℓ} > 1, one assumes that \(P({k}_{{\rm{\ell }}}1)={\mathrm{lim}\,}_{t\to \infty }\frac{{\mathbb{E}}({N}_{{\rm{e}}}({k}_{{\rm{\ell }}}1,t))}{2{m}_{{\rm{tri}}}t}\) exists, and applies again Lemma 1 with \({b}_{t}=b=\frac{{k}_{{\rm{\ell }}}}{3}\) and \(c={\mathrm{lim}}\,{c}_{t}=P({k}_{{\rm{\ell }}}1)\frac{({k}_{{\rm{\ell }}}1)}{3t}\). Hence, \(P({k}_{{\rm{\ell }}})={\mathrm{lim}\,}_{t\to \infty }\frac{{\mathbb{E}}({N}_{{\rm{e}}}({k}_{{\rm{\ell }}},t))}{2{m}_{{\rm{tri}}}t}\) exists and is equal to \(P({k}_{{\rm{\ell }}}1)\frac{{k}_{{\rm{\ell }}}1}{{k}_{{\rm{\ell }}}+3}\). From such a recurrence relation, one finally obtains the explicit formula for P(k_{ℓ})
which is identical to Eq. (4) of the main text.
Finally, since the degree of the vertex is the number of triangles containing this vertex multiplied by 2, one has N_{v}(k_{v},t) = N(2k,t) and as a consequence one obtains
The mixed case
The recurrence relation for N_{v}(k_{v},t) (the number of vertices participating in k_{v} triangles at time t) is
For t > 0 and k_{v} = m_{tri} one has
Furthermore, the use of Lemma 1 gives
and
Therefore, one has
or
where \({\gamma }_{v}=1+\frac{1}{A+B}\).
In addition, one can consider \(N_{\ell}(k_{\ell}, t)\), the number of links with degree k_{ℓ} at moment t. Its recurrence equation is
For k_{ℓ} = 1 one has
One can apply again Lemma 1, and obtain
Therefore, one gets
Let us consider separately the cases B = 0 (the fully nonpreferential case) and B > 0.
For B = 0, one has
Since \(A=\frac{1}{2}\), one obtains
which fully coincide with the exponential scaling derived in the main text for the nonpreferential case.
On the other hand, when instead B > 0, one has
where \({\gamma }_{\ell}=1+\frac{2}{B}\).
Data availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Code availability
All the codes used in this study can be found in Supplementary Software file.
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Acknowledgements
I.S.N. acknowledges support from the Ministerio de Economía, Industria y Competitividad, Spain, under project FIS201784151P. The work of A.M.R. and D.M. was supported by the Russian Federation Government (Grant number 0751520191926).
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K.K., I.S.N., N.K., S.B., and B.B. equally contributed to the manuscript. S.B., B.B., I.S.N., and D.M. conceived the research study. K.K. and N.K. carried out all the analytical treatments. I.S.N. wrote the codes and conducted the simulations and the analysis of the data. A.M.R. proved Theorem 2. K.K., I.S.N., N.K., A.D., D.M., A.M.R., K.A.B., B.B., and S.B. discussed the results, drew the conclusions, and jointly wrote the manuscript.
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Kovalenko, K., SendiñaNadal, I., Khalil, N. et al. Growing scalefree simplices. Commun Phys 4, 43 (2021). https://doi.org/10.1038/s4200502100538y
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DOI: https://doi.org/10.1038/s4200502100538y
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