Abstract
Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the dkseries, a complete set of basic characteristics of the network structure, to study the statistical dependencies between different network properties. We consider six real networks—the Internet, US airport network, human protein interactions, technosocial web of trust, English word network, and an fMRI map of the human brain—and find that many important local and global structural properties of these networks are closely reproduced by dkrandom graphs whose degree distributions, degree correlations and clustering are as in the corresponding real network. We discuss important conceptual, methodological, and practical implications of this evaluation of network randomness, and release software to generate dkrandom graphs.
Introduction
Network science studies complex systems by representing them as networks^{1}. This approach has proven quite fruitful because in many cases the network representation achieves a practically useful balance between simplicity and realism: while always grand simplifications of real systems, networks often encode some crucial information about the system. Represented as a network, the system structure is fully specified by the network adjacency matrix, or the list of connections, perhaps enriched with some additional attributes. This (possibly weighted) matrix is then a starting point of research in network science.
One significant line of this research studies various (statistical) properties of adjacency matrices of real networks. The focus is often on properties that convey useful information about the global network structure that affects the dynamical processes in the system that this network represents^{2}. A common belief is that a selforganizing system should evolve to a network structure that makes these dynamical processes, or network functions, efficient^{3,4,5}. If this is the case, then given a real network, we may ‘reverse engineer’ it by showing that its structure optimizes its function. In that respect the problem of interdependency between different network properties becomes particularly important^{6,7,8,9,10}.
Indeed, suppose that the structure of some real network has property X—some statistically over or underrepresented subgraph, or motif^{11}, for example—that we believe is related to a particular network function. Suppose also that the same network has in addition property Y—some specific degree distribution or clustering, for example—and that all networks that have property Y necessarily have property X as a consequence. Property Y thus enforces or ‘explains’ property X, and attempts to ‘explain’ X by itself, ignoring Y, are misguided. For example, if a network has high density (property Y), such as the interarial cortical network in the primate brain where 66% of edges that could exist do exist^{12}, then it will necessarily have short path lengths and high clustering, meaning it is a smallworld network (property X). However, unlike social networks where the smallworld property is an independent feature of the network, in the brain this property is a simple consequence of high density.
The problem of interdependencies among network properties has been long understood^{13,14}. The standard way to address it, is to generate many graphs that have property Y and that are random in all other respects—let us call them Yrandom graphs—and then to check if property X is a typical property of these Yrandom graphs. In other words, this procedure checks if graphs that are sampled uniformly at random from the set of all graphs that have property Y, also have property X with high probability. For example, if graphs are sampled from the set of graphs with high enough edge density, then all sampled graphs will be small worlds. If this is the case, then X is not an interesting property of the considered network, because the fact that the network has property X is a statistical consequence of that it also has property Y. In this case we should attempt to explain Y rather than X. In case X is not a typical property of Yrandom graphs, one cannot really conclude that property X is interesting or important (for some network functions). The only conclusion one can make is that Y cannot explain X, which does not mean however that there is no other property Z from which X follows.
In view of this inherent and unavoidable relativism with respect to a null model, the problem of structure–function relationship requires an answer to the following question in the first place: what is the right base property or properties Y in the null model (Yrandom graphs) that we should choose to study the (statistical) significance of a given property X in a given network^{15}? For most properties X including motifs^{11}, the choice of Y is often just the degree distribution. That is, one usually checks if X is present in random graphs with the same degree distribution as in the real network. Given that scalefree degree distributions are indeed the striking and important features of many real networks^{1}, this null model choice seems natural, but there are no rigorous and successful attempts to justify it. The reason is simple: the choice cannot be rigorously justified because there is nothing special about the degree distribution—it is one of infinitely many ways to specify a null model.
Since there exists no unique preferred null model, we have to consider a series of null models satisfying certain requirements. Here we consider a particular realization of such series—the dkseries^{16}, which provides a complete systematic basis for network structure analysis, bearing some conceptual similarities with a Fourier or Taylor series in mathematical analysis. The dkseries is a converging series of basic interdependent degree and subgraphbased properties that characterize the local network structure at an increasing level of detail, and define a corresponding series of null models or random graph ensembles. These random graphs have the same distribution of differently sized subgraphs as in a given real network. Importantly, the nodes in these subgraphs are labelled by node degrees in the real network. Therefore, this random graph series is a natural generalization of random graphs with fixed average degree, degree distribution, degree correlations, clustering and so on. Using dkseries we analyse six real networks, and find that they are essentially random as soon as we constrain their degree distributions, correlations, and clustering to the values observed in the real network (Y=degrees+correlations+clustering). In other words, these basic local structural characteristics almost fully define not only local but also global organization of the considered networks. These findings have important implications on research dealing with network structurefunction interplay in different disciplines where networks are used to represent complex natural or designed systems. We also find that some properties of some networks cannot be explained by just degrees, correlations, and clustering. The dkseries methodology thus allows one to detect which particular property in which particular network is nontrivial, cannot be reduced to basic local degree or subgraphbased characteristics, and may thus be potentially related to some network function.
Results
General requirements to a systematic series of properties
The introductory remarks above instruct one to look not for a single base property Y, which cannot be unique or universal, but for a systematic series of base properties Y_{0}, Y_{1},…. By ‘systematic’ we mean the following conditions: (1) inclusiveness, that is, the properties in the series should provide strictly more detailed information about the network structure, which is equivalent to requiring that networks that have property Y_{d} (Y_{d}random graphs), d>0, should also have properties Y_{d′} for all d′=0, 1,…, d−1; and (2) convergence, that is, there should exist property Y_{D} in the series that fully characterizes the adjacency matrix of any given network, which is equivalent to requiring that Y_{D}random graphs is only one graph—the given network itself. If these Yseries satisfy the conditions above, then whatever property X is deemed important now or later in whatever real network, we can always standardize the problem of explanation of X by reformulating it as the following question: what is the minimal value of d in the above Yseries such that property Y_{d} explains X? By convergence, such d should exist; and by inclusiveness, networks that have property Y_{d′} with any d′=d, d+1,…, D, also have property X. Assuming that properties Y_{d} are once explained, the described procedure provides an explanation of any other property of interest X.
The general philosophy outlined above is applicable to undirected and directed networks, and it is shared by different approaches, including motifs^{11}, graphlets^{17} and similar constructions^{18}, albeit they violate the inclusiveness condition as we show below. Yet one can still define many different Yseries satisfying both conditions above. Some further criteria are needed to focus on a particular one. One approach is to use degreebased tailored random graphs as null models for both undirected^{19,20,21} and directed^{22,23} networks. The criteria that we use to select a particular Yseries in this study are simplicity and the importance of subgraph and degreebased statistics in networks. Indeed, in the network representation of a system, subgraphs, their frequency and convergence are the most natural and basic building blocks of the system, among other things forming the basis of the rigorous theory of graph family limits known as graphons^{24}, while the degree is the most natural and basic property of individual nodes in the network. Combining the subgraph and degreebased characteristics leads to dkseries^{16}.
dkseries
In dkseries, properties Y_{d} are dkdistributions. For any given network G of size N, its dkdistribution is defined as a collection of distributions of G’s subgraphs of size d=0, 1,…, N in which nodes are labelled by their degrees in G. That is, two isomorphic subgraphs of G involving nodes of different degrees—for instance, edges (d=2) connecting nodes of degrees 1, 2 and 2, 2—are counted separately. The 0k‘distribution’ is defined as the average degree of G. Figure 1 illustrates the dkdistributions of a graph of size 4.
Thus defined the dkseries subsumes all the basic degreebased characteristics of networks of increasing detail. The zeroth element in the series, the 0k‘distribution’, is the coarsest characteristic, the average degree. The next element, the 1kdistribution, is the standard degree distribution, which is the number of nodes—subgraphs of size 1—of degree k in the network. The second element, the 2kdistribution, is the joint degree distribution, the number of subgraphs of size 2—edges—between nodes of degrees k_{1} and k_{2}. The 2kdistribution thus defines 2node degree correlations and network’s assortativity. For d=3, the two nonisomorphic subgraphs are triangles and wedges, composed of nodes of degrees k_{1}, k_{2} and k_{3}, which defines clustering, and so on. For arbitrary d the dkdistribution characterizes the ‘d’egree ‘k’orrelations in dsized subgraphs, thus including, on the one hand, the correlations of degrees of nodes located at hop distances below d, and, on the other hand, the statistics of dcliques in G. We will also consider dkdistributions with fractional d∈(2, 3) which in addition to specifying twonode degree correlations (d=2), fix some d=3 substatistics related to clustering.
The dkseries is inclusive because the (d+1)kdistribution contains the same information about the network as the dkdistribution, plus some additional information. In the simplest d=0 case for example, the degree distribution P(k) (1kdistribution) defines the average (0kdistribution) via . The analogous expression for d=1, 2 are derived in Supplementary Note 1.
It is important to note that if we omit the degree information, and just count the number of dsized subgraphs in a given network regardless their node degrees, as in motifs^{11}, graphlets^{17} or similar constructions^{18}, then such degreekagnostic dseries (versus dkseries) would not be inclusive (Supplementary Discussion). Therefore, preserving the node degree (‘k’) information is necessary to make a subgraphbased (‘d’) series inclusive. The dkseries is clearly convergent because at d=N, where N is the network size, the Nkdistribution fully specifies the network adjacency matrix.
A sequence of dkdistributions then defines a sequence of random graph ensembles (null models). The dkgraphs are a set of all graphs with a given dkdistribution, for example, with the dkdistribution in a given real network. The dkrandom graphs are a maximumentropy ensemble of these graphs^{16}. This ensemble consists of all dkgraphs, and the probability measure is uniform (unbiased): each graph G in the ensemble is assigned the same probability , where is the number of dkgraphs. For d=0, 1, 2 these are well studied classical random graphs (ref. 25), configuration model^{26,27,28} and random graphs with a given joint degree distribution^{29}, respectively. Since a sequence of dkdistributions is increasingly more informative and thus constraining, the corresponding sequence of the sizes of dkrandom graph ensembles is nonincreasing and shrinking to 1, , Fig. 1. At low d=0, 1, 2 these numbers can be calculated either exactly or approximately^{30,31}.
We emphasize that in dkgraphs the dkdistribution constraints are sharp, that is, they hold exactly—all dkgraphs have exactly the same dkdistribution. An alternative description uses soft maximumentropy ensembles belonging to the general class of exponential random graph models^{32,33,34,35} in which these constraints hold only on average over the ensemble—the expected dkdistribution in the ensemble (not in any individual graph) is fixed to a given distribution. This ensemble consists of all possible graphs G of size N, and the probability measure P(G) is the one maximizing the ensemble entropy S=−∑_{G}P(G)lnP(G) under the dkdistribution constraints. Using analogy with statistical mechanics, sharp and soft ensemble are often called microcanonical and canonical, respectively.
As a consequence of the convergence and inclusiveness properties of dkseries, any network property X of any given network G is guaranteed to be reproduced with any desired accuracy by high enough d. At d=N all possible properties are reproduced exactly, but the Nkgraph ensemble trivially consists of only one graph, Gself, and has zero entropy. In the sense that the entropy of dkensembles is a nonincreasing function of d, the smaller the d, the more random the dkrandom graphs, which also agrees with the intuition that dkrandom graphs are ‘the less random and the more structured’, the higher the d. Therefore, the general problem of explaining a given property X reduces to the general problem of how random a graph ensemble must be so that X is statistically significant. In the dkseries context, this question becomes: how much local degree information, that is, information about concentrations of degreelabelled subgraphs of what minimal size d, is needed to reproduce a possibly global property X with a desired accuracy?
Below we answer this question for a set of popular and commonly used structural properties of some paradigmatic real networks. But to answer this question for any property in any network, we have to be able to sample graphs uniformly at random from the sets of dkgraphs—the problem that we discuss next.
dkrandom graph sampling
Soft dkensembles tend to be more amenable for analytic treatment, compared with sharp ensembles, but even in soft ensembles the exact analytic expressions for expected values are known only for simplest network properties in simplest ensembles^{36,37}. Therefore, we retreat to numeric experiments here. Given a real network G, there exist two ways to sample dkrandom graphs in such experiments: dkrandomize G generalizing the randomization algorithms in refs 38, 39, or construct random graphs with G’s dksequence from scratch^{16,40}, also called direct construction^{41,42,43,44}.
The first option, dkrandomization, is easier. It accounts for swapping random (pairs of) edges, starting from G, such that the dkdistribution is preserved at each swap, Fig. 2. There are many concerns with this prescription^{45}, two of which are particularly important. The first concern is if this process ‘ergodic’, meaning that if any two dkgraphs are connected by a chain of dkswaps. For d=1 the twoedge swap is ergodic^{38,39}, while for d=2 it is not ergodic. However, the socalled restricted twoedge swap, when at least one node attached to each edge has the same degree, Fig. 2, was proven to be ergodic^{46}. It is now commonly believed that there is no edgeswapping operation, of this or other type, that is ergodic for the 3kdistribution, although a definite proof is lacking at the moment. If there exists no ergodic 3kswapping, then we cannot really rely on it in sampling dkrandom graphs because our real network G can be trapped on a small island of atypical dkgraphs, which is not connected by any dkswap chain to the main land of many typical dkgraphs. Yet we note that in an unpublished work^{47} we showed that five out of six considered real networks were virtually indistinguishable from their 3krandomizations across all the considered network properties, although one network (power grid) was very different from its 3krandom counterparts.
The second concern with dkrandomization is about how close to uniform sampling the dkswap Markov chain is after its mixing time is reached—its mixing time is yet another concern that we do not discuss here, but according to many numerical experiments and some analytic estimates, it is O(M)^{16,29,38,39,40,46}. Even for d=1 the swap chain does not sample 1kgraphs uniformly at random, yet if the edgeswap process is done correctly, then the sampling is uniform^{20,21}.
A simple algorithm for the second dksampling option, constructing dkgraphs from scratch, is widely known for d=1: given G’s degree sequence {k_{i}}, build a 1krandom graph by attaching k_{i} halfedges (‘stubs’) to node i, and then connect random pairs of stubs to form edges^{27}. If during this process a selfloop (both stubs are connected to the same node) or doubleedge (two edges between the same pair of nodes) is formed, one has to restart the process from scratch since otherwise the graph sampling is not uniform^{48}. If the degree sequence is powerlaw distributed with exponent close to −2 as in many real networks, then the probability that the process must be restarted approaches 1 for large graphs^{49}, so that this construction process never succeeds. An alternative greedy algorithm is described in ref. 42, which always quickly succeeds and gives an efficient way of testing whether a given sequence of integers is graphical, that is, whether it can be realized as a degree sequence of a graph. The base sampling procedure does not sample graphs uniformly, but then an importance sampling procedure is used to account for the bias, which results in uniform sampling. Yet again, if the degree distribution is a power law, then one can show that even without importance sampling, the base sampling procedure is uniform, since the distribution of sampling weights that one can compute for this greedy algorithm approaches a delta function. Extensions of the naive 1kconstruction above to 2k are less known, but they exist^{16,29,44,50}. Most of these 2kconstructions do not sample 2kgraphs exactly uniformly either^{46}, but importance sampling^{20,44} can correct for the sampling biases.
Unfortunately, to the best of our knowledge, there currently exists no 3kconstruction algorithm that can be successfully used in practice to generate large 3kgraphs with 3kdistributions of real networks. The 3kdistribution is quite constraining and nonlocal, so that the dkconstruction methods described above for d=1, 2 cannot be readily extended to d=3 (ref. 16). There is yet another option—3ktargeting rewiring, Fig. 2. It is 2kpreserving rewiring in which each 2kswap is accepted not with probability 1, but with probability equal to min(1, exp(−βΔH)), where β is the inverse temperature of this simulatedannealinglike process, and ΔH is the change in the L^{1} distance between the 3kdistribution in the current graph and the target 3kdistribution before and after the swap. This probability favours and, respectively, suppresses 2kswaps that move the graph closer or farther from the target 3kdistribution. Unfortunately, we report that in agreement with^{40} this 2kpreserving 3ktargeting process never converged for any considered real network—regardless of how long we let the rewiring code run, after the initial rapid decrease, the 3kdistance, while continuing to slowly decrease, remained substantially large. The reason why this process never converges is that the 3kdistribution is extremely constraining, so that the number of 3kgraphs is infinitesimally small compared with the number of 2kgraphs , (refs 16, 30). Therefore, it is extremely difficult for the 3ktargeting Markov chain to find a rare path to the target 3kdistribution, and the process gets hopelessly trapped in abundant local minima in distance H.
Therefore, on one hand, even though 3krandomized versions of many real networks are indistinguishable from the original networks across many metrics^{47}, we cannot use this fact to claim that at d=3 these metrics are not statistically significant in those networks, because the 3krandomization Markov chain may be nonergodic. On the other hand, we cannot generate the corresponding 3krandom graphs from scratch in a feasible amount of compute time. The 3krandom graph ensemble is not analytically tractable either. Given that d=2 is not enough to guarantee the statistical insignificance of some important properties of some real networks, see ref. 47 and below, we, as in ref. 40, retreat to numeric investigations of 2krandom graphs in which in addition to the 2kdistribution, some substatistics of the 3kdistribution is fixed. Since strong clustering is a ubiquitous feature of many real networks^{1}, one of the most interesting such substatistics is clustering.
Specifically we study 2.1krandom graphs, defined as 2krandom graphs with a given value of average clustering , and 2.5krandom graphs, defined as 2krandom graphs with given values of average clustering (k) of nodes of degree k (ref. 40). The 3kdistribution fully defines both 2.1k and 2.5kstatistics, while 2.5k defines 2.1k. Therefore, 2kgraphs are a superset of 2.1kgraphs, which are a superset of 2.5kgraphs, which in turn contain all the 3kgraphs, . Therefore if a particular property is not statistically significant in 2.5krandom graphs, for example, then it is not statistically significant in 3krandom graphs either, while the converse is not generally true.
We thus generate 20 dkrandom graphs with d=0, 1, 2, 2.1, 2.5 for each considered real network. For d=0,1,2 we use the standard dkrandomizing swapping, Fig. 2. We do not use its modifications to guarantee exactly uniform sampling^{20,21}, because: (1) even without these modifications the swapping is close to uniform in powerlaw graphs, (2) these modifications are nontrivial to efficiently implement, and (3) we could not extend these modifications to the 2.1k and 2.5k cases. As a consequence, our sampling is not exactly uniform, but we believe it is close to uniform for the reasons discussed above. To generate dkrandom graphs with d=2.1, 2.5, we start with a 2krandom graph, and apply to it the standard 2kpreserving 2.xktargeting (x=1, 5) rewiring process, Fig. 2. The algorithms that do that, as described in ref. 40, did not converge on some networks, so that we modified the algorithm in ref. 10 to ensure the convergence in all cases. The details of these modifications are in Supplementary Methods (the parameters used are listed in Supplementary Table 4), along with the details of the software package implementing these algorithms that we release to public^{51}.
Real versus dkrandom networks
We performed an extensive set of numeric experiments with six real networks—the US air transportation network, an fMRI map of the human brain, the Internet at the level of autonomous systems, a technosocial web of trust among users of the distributed Pretty Good Privacy (PGP) cryptosystem, a human protein interaction map, and an English word adjacency network (Supplementary Note 2 and Supplementary Table 3 present the analysed networks). For each network we compute its average degree, degree distribution, degree correlations, average clustering, averaging clustering of nodes of degree k and based on these dkstatistics generate a number of dkrandom graphs as described above for each d=0, 1, 2, 2.1, 2.5. Then for each sample we compute a variety of network properties, and report their means and deviations for each combination of the real network, d, and the property. Figures 3, 4, 5, 6 present the results for the PGP network; Supplementary Note 3, Supplementary Figs 1–10, and Supplementary Tables 1–2 provide the complete set of results for all the considered real networks. The reason why we choose the PGP network as our main example is that this network appears to be ‘least random’ among the considered real networks, in the sense that the PGP network requires higher values of d to reproduce its considered properties. The only exception is the brain network. Some of its properties are not reproduced even by d=2.5.
Figure 2 visualizes the PGP network and its dkrandomizations. The figure illustrates the convergence of dkseries applied to this network. While the 0krandom graph has very little in common with the real network, the 1krandom one is somewhat more similar, even more so for 2k, and there is very little visual difference between the real PGP network and its 2.5krandom counterpart. This figure is only an illustration though, and to have a better understanding of how similar the network is to its randomization, we compare their properties.
We split the properties that we compare into the following categories. The microscopic properties are local properties of individual nodes and subgraphs of small size. These properties can be further subdivided into those that are defined by the dkdistributions—the degree distribution, average neighbour degree, clustering, Fig. 3—and those that are not fixed by the dkdistributions—the concentrations of subgraphs of size 3 and 4, Fig. 4. The mesoscopic properties—kcoreness and kdensity (the latter is also known as mcoreness or edge multiplicity, Supplementary Note 1), Fig. 5—depend both on local and global aspects of network organization. Finally, the macroscopic properties are truly global ones—betweenness, the distribution of hop lengths of shortest paths, and spectral properties, Fig. 6. In Supplementary Note 3 we also report some extremal properties, such as the graph diameter (the length of the longest shortest path), and Kolmogorov–Smirnov distances between the distributions of all the considered properties in real networks and their corresponding dkrandom graphs. The detailed definitions of all the properties that we consider can be found in Supplementary Note 1.
In most cases—henceforth by ‘case’ we mean a combination of a real network and one of its considered property—we observe a nice convergence of properties as d increases. In many cases there is no statistically significant difference between the property in the real network and in its 2.5krandom graphs. In that sense these graphs, that is, random graphs whose degree distribution, degree correlations, and degreedependent clustering (k) are as in the original network, capture many other important properties of the real network.
Some properties always converge. This is certainly true for the microscopic properties in Fig. 3, simply confirming that our dksampling algorithm operates correctly. But many properties that are not fixed by the dkdistributions converge as well. Neither the concentration of subgraphs of size 3 nor the distribution of the number of neighbours common to a pair of nodes are fully fixed by dkdistributions with any d<3 by definition, yet 2.5krandom graphs reproduce them well in all the considered networks. Most subgraphs of size 4 are also captured at d=2.5 in most networks, even though d=3 would not be enough to exactly reproduce the statistics of these subgraphs. We note that the improvement in subgraph concentrations at d=2.5 compared with d=2.1 is particularly striking, Fig. 4. The mesoscopic and especially macroscopic properties converge more slowly as expected. Nevertheless, quite surprisingly, both mesoscopic properties (kcoreness and kdensity) and some macroscopic properties converge nicely in most cases. The kcoreness, kdensity, and the spectral properties, for instance, converge at d=2.5 in all the considered cases other than Internet’s Fiedler value. In some cases a property, even global one, converges for d <2.5. Betweenness, for example, a global property, converges at d=1 for the Internet and English word network.
Finally, there are ‘outlier’ networks and properties of poor or no dkconvergence. Many properties of the brain network, for example, exhibit slow or no convergence. We have also experimented with community structure inferred by different algorithms, and in most cases the convergence is either slow or nonexistent as one could expect.
Discussion
In general, we should not expect nonlocal properties of networks to be exactly or even closely reproduced by random graphs with local constraints. The considered brain network is a good example of that this expectation is quite reasonable. The human brain consists of two relatively weakly connected parts, and no dkrandomization with low d is expected to reproduce this peculiar global feature, which likely has an impact on other global properties. And indeed we observe in Supplementary Note 3 that its two global properties, the shortest path distance and betweenness distributions, differ drastically between the brain and its dkrandomizations.
Another good example is community structure, which is not robust with respect to dkrandomizations in all the considered networks. In other words, dkrandomizations destroy the original peculiar cluster organization in real networks, which is not surprising, as clusters have too many complex nonlocal features such as variable densities of internal links, boundaries and so on, which dkrandomizations, even with high d, are expected to affect considerably.
Surprisingly, what happens for the brain and community structure does not appear representative for many other considered combinations of real networks and their properties. As a possible explanation, one can think of constraintbased modelling as a satisfiability (SAT) problem: find the elements of the adjacency matrix (1/0, True/False) such that all the given constraints in terms of the functions of the marginals (degrees) of this matrix are obeyed. We then expect that the 3kconstraints already correspond to an NPhard SAT problem, such as 3SAT, with hardness coming from the global nature of the constraints in the problem. However, many realworld networks evolve based mostly on local dynamical rules and thus we would expect them to contain correlations with d<3, that is, below the NPhard barrier. The primate brain, however, has likely evolved through global constraints, as indicated by the dense connectivity across all functional areas and the existence of a strong coreperiphery structure in which the core heavily concentrates on areas within the associative cortex, with connections to and from all the primary input and subcortical areas^{12}.
However, in most cases, the considered networks are dkrandom with d≤2.5, that is, d≤2.5 is enough to reproduce not only basic microscopic (local) properties but also mesoscopic and even macroscopic (global) network properties^{6,7,8,9,10}. This finding means that these more sophisticated properties are effectively random in the considered networks, or more precisely, that the observed values of these properties are effective consequences of particular degree distributions and, optionally, degree correlations and clustering that the networks have. This further implies that attempts to find explanations for these complex but effectively random properties should probably be abandoned, and redirected to explanations of why and how degree distributions, correlations and clustering emerge in real networks, for which there already exists a multitude of approaches^{52,53,54,55,56,57}. On the other hand, the features that we found nonrandom do require separate explanations, or perhaps a different system of null models.
We reiterate that the dkrandomization system makes it clear that there is no a priori preferred null model for network randomization. To tell how statistically significant a particular feature is, it is necessary to compare this feature in the real network against the same feature in an ensemble of random graphs, a null model. But one is free to choose any random graph model. In particular, any d defines a random graph ensemble, and we find that many properties, most notably the frequencies of small subgraphs that define motifs^{11}, strongly depend on d for many considered networks. Therefore, choosing any specific value of d, or more generally, any specific null model to study the statistical significance of a particular structural network feature, requires some nontrivial justification before this feature can be claimed important for any network function.
Yet another implication of our results is that if one looks for network topology generators that would veraciously reproduce certain properties of a given real network—a task that often comes up in as diverse disciplines as biology^{58} and computer science^{59}—one should first check how dkrandom these properties are. If they are dkrandom with low d, then one may not need any sophisticated missionspecific topology generators. The dkrandom graphgeneration algorithms discussed here can be used for that purpose in this case. We note that there exists an extension of a subset of these algorithm for networks with arbitrary annotations of links and nodes^{60}—directed or coloured (multilayer) networks, for instance.
The main caveat of our approach is that we have no proof that our dkrandom graph generation algorithms for d=2.1 and d=2.5 sample graphs uniformly at random from the ensemble. The randomgraph ensembles and edgerewiring processes employed here are known to suffer from problems such as degeneracy and hysteresis^{35,61,62}. Ideally, we would wish to calculate analytically the exact expected value of a given property in an ensemble. This is currently possible only for very simple properties in soft ensembles with d=0, 1, 2 (refs 36, 37). Some mathematically rigorous results are available for d=0, 1 and for some exponential randomgraph models^{28,34}. Many of these results rely on graphons^{24} that are applicable to dense graphs only, while virtually all real networks are sparse^{49}. Some rigorous approaches to sparse networks are beginning to emerge^{63,64}, but the rigorous treatment of global properties, which tend to be highly nontrivial functions of adjacency matrices, in random graph ensembles with d>2 constraints, appear to be well beyond the reach in the near future. Yet if we ever want to fully understand the relationship between the structure, function and dynamics of real networks, this future research direction appears to be of a paramount importance.
Additional information
How to cite this article: Orsini, C. et al. Quantifying randomness in real networks. Nat. Commun. 6:8627 doi: 10.1038/ncomms9627 (2015).
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Acknowledgements
We acknowledge financial support by NSF Grants No. CNS1039646, CNS1345286, CNS0722070, CNS0964236, CNS1441828, CNS1344289, CNS1442999, CCF1212778, and DMR1206839; by AFOSR and DARPA Grants No. HR00111210012 and FA95501210405; by DTRA Grant No. HDTRA10910039; by Cisco Systems; by the Ministry of Education, Science, and Technological Development of the Republic of Serbia under Project No. ON171017; by the ICREA Academia Prize, funded by the Generalitat de Catalunya; by the Spanish MINECO Project No. FIS201347282C21P; by the Generalitat de Catalunya Grant No. 2014SGR608; and by European Commission Multiplex FP7 Project No. 317532.
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All authors contributed to the development and/or implementation of the concept, discussed and analysed the results. C.O., M.M.D., and P.C.S. implemented the software for generating dkgraphs and analysed their properties. D.K. wrote the manuscript, incorporating comments and contributions from all authors.
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Supplementary Figures 110, Supplementary Tables 15, Supplementary Notes 13, Supplementary Discussion, Supplementary Methods and Supplementary References (PDF 760 kb)
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Orsini, C., Dankulov, M., ColomerdeSimón, P. et al. Quantifying randomness in real networks. Nat Commun 6, 8627 (2015). https://doi.org/10.1038/ncomms9627
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