When describing a real-world system as a network, each element of the system is represented by a vertex or node, and relationships or interactions between elements are represented by edges between nodes. Two nodes are said to be neighbors if they are connected by an edge, and the degree k_i of node i is the number of neighbors it has. The minimum path length between two nodes is the minimum number of edges that must be traversed to get from one node to the other. The mean value of the minimum path length over all node pairs will be denoted by L.

A key concept in defining small-worlds networks is that of ‘clustering’ which measures the extent to which the neighbors of a node are also interconnected. Watts and Strogatz [3] defined the clustering coefficient of node i by(1)where E_i is the number of edges between the neighbors of i. The clustering coefficient of the network C^ws is then the mean of over all nodes. An alternative definition of network clustering in common use [14], based on transitivity, is expressed by(2)where a ‘triangle’ is a set of three nodes in which each contacts the other two. Both capture intuitive notions of clustering but, though often in good agreement, values for C^ws and C^Δ can differ by an order of magnitude for some networks. We consider mainly C^Δ here, but report where using C^ws leads to different results.

A network G with n nodes and m edges is a small-world network [3] if it has a similar path length but greater clustering of nodes than an equivalent Erdös-Rényi (E–R) random graph [15] with the same m and n (an E–R graph is constructed by uniquely assigning each edge to a node pair with uniform probability). More formally, let L_g be the mean shortest path length of G and its clustering coefficient using (2). Let L_rand and be the corresponding quantities for the corresponding E–R random graph. These ideas may be used to supply a semi-quantitative categorical definition of a small world network [3]

Put(3)and(4)We then define a quantitative metric of ‘small-world-ness’ S^Δ according to(5)In a similar way, putting(6)we define S^ws(7)The categorical definition of small-world network above implies λ_g≥1 and , which, in turn, gives S^Δ>1. We can, therefore, now make a quantitative categorical definition of a ‘small-world’ network

We first checked that the metric S^Δ behaves as required on the canonical Watts-Strogatz (WS) model of small-world generation [3]. The WS model begins with a ring of n nodes, each node connected to its nearest neighbors out to some range K. Each edge in turn is ‘re-wired’ to a new target node with probability p (Figure 1A). Values of p = 0 and p = 1 give regular and random networks, respectively, with intermediate p values resulting in ‘small-world’ networks that share properties of both provided that the network is connected and sparse — densely connected networks trivially have small mean path lengths and high clustering coefficients.

Figure 1 shows that small-world-ness captures the topology changes: it has a unique maximum at intermediate values of the re-wiring parameter p, indicating the maximum trade-off between high clustering and low path length (Figure 1B), and decays with increasing edge density for a fixed size of network, reflecting the requirement of sparseness (Figure 1C). We can see why this occurs for increasing density. The edge density of a network is given by(8)

As ξ→1 then both C^ws, C^Δ→1 and L→1 because all nodes become connected; and as this would apply for both a given real-world network and its E-R random graph equivalent, so S^Δ, S^ws→1 regardless of n: high edge density results in low small-world-ness.

We computed S^Δ and S^ws for a broad range of technological, biological, social, and information networks (33 networks in total; Table 1, see Materials and Methods). To our surprise, we found that both forms of small-world-ness scale linearly with the size of the network across all systems falling into the small-world class (Figure 2A,B), irrespective of their originating domain or their other topological properties (e.g. their degree distribution, degree correlation). For S^Δ, it was not possible to find or calculate C^Δ (and hence S^Δ) for 6 of the 33 networks. However, for the remaining 27, all had S^Δ>1 and were therefore deemed to be small-world in the new scheme (Definition 2). To ensure the robustness of the categorization, networks with borderline values 1≤S^Δ≤3 were tested for significance using Monte Carlo sampling of 1000 equivalent E–R random graphs for each network, estimating 99% confidence intervals using standard methods (see Materials and Methods). All such networks had small-world-ness scores significantly greater than an equivalent E–R random graph. For the 27 networks for which S^Δ>1, linear regression on log-transformed quantities (see Materials and Methods) allowed an estimate of the best power law fit: S^Δ = 0.023n^0.96 (r² = 0.78; p = 3×10⁻⁹). This is an essentially linear scaling of S^Δ with n.

For S^ws, 3 networks in our data-set were not small-worlds: relationships amongst students [20] S^ws = 0.27 (network #9); a freshwater food web [21], [1] S^ws = 0.74 (network #28); and the E-Coli reaction graph [22] S^ws = 0.67 (network #32). This demonstrates both that the small-world property is not robustly achieved for small networks, and that it is contingent on the particular measure of clustering used. Once again, networks with borderline values 1≤S^ws≤3 were tested using Monte Carlo methods for significant membership of the small-world category and were found to satisfy this criterion. For the 30 networks with S^ws>1, a similar regression to that used for S^Δ gave S^ws = 0.012n^1.11 (r² = 0.84; p = 1.3×10⁻¹¹). Thus, there is also a robust linear scaling of S^ws with n (see Text S1 for further details).

Linear scaling of small-world-ness with network size was unexpectedly shown by the canonical Watts-Strogatz model [3] of small-world network generation. We now show that this result can be explained analytically. In what follows, many of the relationships are held only approximately, but because these approximations are often very good we show them as equalities. Note that from here on we use subscript names to identify analytic quantities that pertain to a particular network model only.

If L_ws is the mean shortest path length in the WS model, then it is known [14] that(9)Similarly, it is known for E-R random graphs [23] that(10)where 〈k〉 is the expected value of the degree across the network. In the WS model, node degree and range are related by 〈k〉 = 2K, so that(11)Using (11), (9) and (4) the path length ratio λ_ws for the WS model is(12)The function f(x) in (9) has an upper asymptote of ln(2x)/4x if . Thus, assuming , (12) becomes(13)If is the clustering coefficient of the WS model (using (2) as the metric), then it is known [24] that(14)For E–R random graphs [23], to a good approximation, , so that using 〈k〉 = 2K(15)Therefore, using, (14), (15) and (3)(16)From (5), (16) and (13),(17)where h(K, p) is a function of K and p only. The term in the square brackets tends to 1 as n→∞ and so, for large enough n, S^Δ for the WS model scales with n. To quantify this approximation, we performed a linear regression on log-transformed quantities (just as for the real networks) over the typical range of n encountered in our sample of networks, 10²≤n≤10⁷, and found a linear fit, with r² within 10⁻⁵ of unity.

The WS network is often used as a generative model for real small-world networks [e.g. 8]–[13]. This is assumed to establish a ‘first-pass’ model of that system’s topology, which may be augmented by considering other factors such as degree sequence [23], degree correlation [25], modularity [26] and other properties.

In matching the WS parameters K, p, n to the target system, we know n, can measure 〈k〉 (giving K = 〈k〉/2), but estimating p has, hitherto, remained problematic. However, using our new metric of small-world-ness, it is possible to establish p in a principled way. Thus, if G is a real (target) network with measured small-world-ness , we identify it with the WS network with the same value of S^Δ. That is, form , where is given by the right hand side of (17), and minimise e with respect to p, keeping K, n at their measured values. We did this for our sample of real-world systems, omitting those for which 〈k〉≤2 since the expressions used in defining are inaccurate in these cases (we used Matlab routine fzero, initial value of p = 0.5). The resulting p values for the equivalent WS model are listed in Table 1

Given that the real-world networks showed S^Δ∝n, the WS networks derived from them under the procedure described here must do likewise (they have identical S^Δ values). However, the result in the previous section would suggest that this implies K, p are roughly constant for this set of WS networks.

To investigate the constancy of K we used the result that 〈k〉 = 2m/n (where m is the number of edges in the network). So, using K = 〈k〉/2, constant K is equivalent to establishing m∝n. Figure 2C shows the result of regressing m against n (using log-transformed quantities) for the real world networks. For networks with S^Δ>1, the best fit model was m = 2.46n^1.06 (27 networks, r² = 0.92, p = 4×10⁻¹⁵), implying a mean node degree of 〈k〉 = 2m/n≈5; for networks with S^ws>1, the best fit model was m = 3.16n^1.03 (30 networks, r² = 0.91, p = 2×10⁻¹⁵) implying 〈k〉≈6.32. Thus, the real-world networks fulfill the prediction of constant mean node degree. A similar result holds for p values; we found that all testable real-world systems fall into a very limited range of p for the equivalent WS model (0.64≤p≤0.95 and σ_p = 0.0806).

An alternative view of these results is as follows. We could start with the empirically observed approximate constancy of mean node degree 〈k〉 and calculated rewiring parameter p for the real world networks, and deduce a linear scaling of S^Δ for the WS models. Then, under the equivalence of S^Δ for both real-world networks and their WS counterparts, we could have predicted that S^Δ for the real-world networks would also scale linearly.

Is the relationship S∝n inevitable for all systems? (The subsequent argument holds for S based on either definition of clustering coefficient and so superscripts Δ, ws are dropped). To investigate this we note that it is always possible to write S_i = α_in_i for the ith system, for some value α_i; in the case of linear scaling, α_i is constant. To proceed further, we now express α_i in terms of other system parameters. Using the definition of S and (10) for random graphs,(18)where C_i, L_i, 〈k〉_i are the clustering coefficient, mean shortest path length, and mean node degree of system i respectively. While we do not know exactly how L_i depends on n, we note that the mean shortest path length for small-world networks is usually assumed to scale logarithmically like random graphs: from (11), L_rand = [1/ln(〈k〉)]ln(n); and for the WS model, using (9) with large n, L_ws = (1/4K²p)ln(n). Both relations are of the form L = βln(n) where β is independent of n. We therefore write L_i = β_iln(n_i), where β_i is the factor that ensures the equality to be true (i.e it plays a similar role in this respect as α_i).

This gives(19)In general, there is no a priori reason to suppose that the variables C_i, 〈k〉_i and β_i are either all constant, or co-vary in a way commensurate with constancy for α_i. However, for the sample of networks used here, as noted above, the mean node degree 〈k〉_i is approximately constant. It is now instructive to see how much co-variation is required between the remaining two variables in order to ensure a significantly different power law holds between S and n.

Thus, suppose that we fit a model S = μn^1.5 so that we expect . For the range of n encountered here – approximately four orders of magnitude – α_i would therefore have to range over 2 orders of magnitude. For this to occur, there must be sufficient variation in C_i and β_i, and these two quantities should correlate well with n. The ranges of the two variables are reasonably large in the data-set – using C^Δ, 0.209≤β_i≤2.52 and 0.005≤C_i≤0.72 – and could plausibly generate the required 100-fold variation. However, the correlation coefficients with n are very small: for β_i, r² = 0.028 and for C_i, r² = 0.025. This would therefore appear to preclude a nonlinear relationship between S^Δ and n for the networks studied here.

To study the effect of a lack of correlation between n and network parameters like C_i on linear scaling between S^Δ and n, we ran a Monte Carlo simulation (see Materials and Methods). Each one of 1000 experiments consisted of sampling 27 randomly drawn C values for networks with constant β, and with a spread of n over 4 orders of magnitude. For each network its small-world-ness was computed and a linear regression of S against n performed. This resulted in a mean of r² = 0.89±0.06 s.d. across the 1000 experiments, showing the strong tendency for linearity in this case.

The linear relationship is, however, sensitive to deviations from the approximation that 〈k〉 is constant. That is, networks that deviate furthest from the linear m∝n model in turn deviate furthest from the linear S∝n model (Figure 3). This was shown using a novel regress-delete-regress procedure outlined in Materials and Methods (we were able to directly test the sensitivity to 〈k〉, rather than using a Monte Carlo approach as above, because the strong correlation of m with n provided a baseline from which we could quantify deviation of 〈k〉 from constancy). Further, Figure 3 shows that if we delete a random set of networks from the data-set, then the average effect is to not change the fit to the linear S∝n model: the linear scaling is robust, and does not depend on a specific network set.

The sensitivity of small-world-ness linearity with n to degree 〈k〉 suggests that introduction of networks with very high edge density into our sample would destroy the linear scaling. We can rewrite (8) using 〈k〉 = 2m/n(20)and see that mean degree scales linearly with edge density. Thus, a network with high edge density implies high mean degree, which in turn would fall far from the linear S = αn model, as we have just shown.

One exemplar of a real system with high edge density is the network of individual neurons within a single vertebrate brain region. Detailed network data for these are not available because of the great technical difficulties in reliably reconstructing even small networks such as the 302 neuron C. Elegans nervous system [27]. Indeed, high edge density itself may be the primary cause of technical problems in reconstructing complete systems from many domains, resulting in their absence from the network literature. Nonetheless, approximate reconstructions can be attempted. Quantitative anatomical models of individual brain regions suggest that each of the hundreds of thousands or millions of neurons receive many thousands of connections, and each themselves connect to similar numbers of target neurons [16], [28]. Such networks of neurons can have very low small-world-ness values for their size [16], and thus fall far from the linear S∝n model discovered here.

We conclude here that the linear relationship between small-world-ness and system size does not hold for an arbitrary collection of networks, but is highly likely if all such networks have a similar mean node degree.

Having established that S scales linearly with n, it is also instructive to look at how its component ratios scale with n. We find, as expected, that most networks falling into the small-world class have approximately the same mean shortest path length as their equivalent E–R random graphs, and so λ≈1. Given this, it is unsurprising that both γ^Δ and γ^ws then scale linearly with n (see Figure S1). We did find that three networks in our data-set — email messages (#7), software packages (#21), and software classes (#22) — had λ≈0.1, indicating that their mean shortest path length was an order of magnitude smaller than the equivalent E–R random graph. These networks are thus ultra-small [29], and indeed both email message (#7) and software package (#21) networks fall further from the linear model than any others.

Given the existence of the linear scaling with n, the scaling of small-world-ness with some other topological properties is completely determined. We can directly determine from (20) how edge density behaves in our data-set (values for ξ are given in Table 1). Taking our fitted linear model S = αn, we can substitute n = S/α in (20) and find that(21)Substituting our found values of 〈k〉 and α for the fits to either S^ws or S^Δ confirms that this is a good approximation. Therefore, because small-world-ness linearly scales with network size, and degree is approximately constant, then S also has a simple inverse linear scaling with edge density.

We can show that the specific scaling coefficient α in the relationship S = αn for the real-world networks studied here does not maximize small-world-ness for a particular size of network. First, we show that the WS model predicts an approximately constant amount of rewiring p that maximizes S^Δ, independent of network size. To do this, given the above analytic expressions (13) for λ_ws and (16) for (and again assuming ), we found , and set . Solving this equality for p would then give us the value of p that maximized , if one existed — see Text S1 for details of the solution.

We did this over the range with K = 〈k〉/2 = 2.5, since this is implied by the result of Figure 2C. If p^* is the value of p giving maximal , we found that for small networks (n = 10³) p^* = 0.222 and, as n→∞, p^*→0.246, so that the range of p^* is very small. The constant K and very small range of p^* imply that the associated maximum values should scale linearly with n. It transpires that the theoretical maximum depends almost exactly in a linear way on n with slope 0.181 (and plotted in Figure 2A). Thus, S^Δ is not maximized by the real-world networks.

We have established and explained many simple properties of real-world networks and of their equivalence class in the WS model. We now show how the specific, sub-maximal, linear scaling of S = αn could have been generated. The models we examine here are intended as informative examples of the generation and limits on S scaling, not an exhaustive list of those which could generate the specific linear scaling we found — that remains the subject of future work.

Many of the real-world systems share common generative principals despite their widely differing origins. Most systems have a growth process, showing some form of preferential attachment [30] that is limited by the cost of adding new edges and by the capacity to maintain them (as might be induced by aging) [31]. Simple models of this process result in ‘scale-free’ networks with power-law or truncated power-law degree distributions [30], [31], a property that is also common to many real-world systems considered here [32] (but see [33] for an alternative view of some biological networks). However, networks generated by these models are not ‘small-world’ by either Definition 1 or 2. Their clustering coefficient is inversely proportional to n, going to zero as n grows large [34]. Thus, they cannot show linear scaling of S: it is at best constant and at worst goes to zero with increasing n.

A noisy, limited growth process can generate the specific linear S = αn relationships we report. A generalized form of the Klemm-Eguiluz model (GKE)[34], [35] encapsulates this process, and has the unique property of creating networks that are both small-world (short path length, high clustering) and ‘scale-free’ (having a truncated power-law degree distribution) as found for many real-world systems considered here. (To the best of our knowledge, all known real-world systems with power-law-like degree distributions also fall into the broad ‘small-world’ class we discussed in the Introduction; it is only the scale-free networks formed by the simple models that form a distinct set of ‘scale-free-only’ networks). By using the GKE model, we therefore also show that linear scaling of S can occur whether or not the real-world systems have ‘scale-free’ properties.

The GKE model begins with an active set of M nodes. At every time-step a new node is added, connecting d edges: one edge added to a random inactive node with probability ρ, adding noise to the process; all remaining edges connect to randomly chosen active nodes. One of the active nodes is deactivated with a probability proportional to the active nodes’ degrees; finally, the new node is activated. The sequence repeats until the desired size of network is obtained.

We found that specific values for M and ρ could generate GKE networks with the same linear scaling relationships between network size and S^ws and S^Δ that we observed for the real-world networks (Figure 4; see Materials and Methods, and Figure S2). Therefore, a possible general mechanism for particular linear scaling rates of small-world-ness is a common size of both active node set and quantity of noise during creation of the real-world systems.

We collated a database of real-world networks' topological properties, combining published results with our own analyses of available data-sets. These are presented in Table 1, extending the previous considerable effort of collating topological properties by Mark Newman [1]. All networks are treated as undirected. We list 33 real-world systems in total: we could compute S^ws for all systems and S^Δ for 27 systems — C^Δ could not be found or computed for those systems.

We emphasise that the networks were not chosen for their ability to fit the linear model of S∝n. The majority of the data-set (21 of 33) were obtained from a previous collation [1]: networks were only omitted from that prior data-set if neither C^ws or C^Δ were available for them (and hence were of no practical use to us here). Many of the additional networks we added filled sub-domains missing from the prior data-set, for example: the dolphin network [41] is an example of an animal social network; the cortical area connectivity map [42] is an example of large-scale neural connectivity. In addition, the regress-delete-regress sequence we used in the main text (and see below) shows two properties. First, that we could have applied that method to the data-set in Table 1 before further analysis, pruning the data-set down to those networks that showed the best fit to the linear model (by choosing the ‘most-deviant’ networks to omit), but did not. Second, that the linear scaling property is robust across randomly chosen sub-sets of the network data-set: on average, randomly deleting networks from the data-set did not significantly reduce the fit to the linear model.

We assess the significance of borderline small-world-ness scores S1 using Monte Carlo methods. The null hypothesis for the Watts-Strogatz definition of small-world networks is that the system is an Erdös-Rényi (E–R) random graph. We thus constructed N = 1000 E–R networks with the same number of nodes n and edges m for each tested real-world system, computing and for the ith E–R network. The 99% confidence limits for the null hypothesis were then defined for each system. We first found the central 99% interval [a*, b*], that is [43](22)and similarly for S^ws. The 99% confidence interval for the system is then(23)The upper 99% confidence limit is then CL^0.01  =  1 + CI (where by definition S^ws, S^Δ = 1 for an E–R random graph). A network with S>CL^0.01 was therefore considered to significantly differ from a random network. We note that adopting a quantitative definition (Definition 2) of small-world-ness has led us to a procedure for a general statistical test for the presence of small-world structure, as defined by Watts and Strogatz [3], which is particularly useful for establishing meaningful departures from randomness in small networks.

Least-squares regressions on small-world-ness S and number of edges m against size of system n were performed on log₁₀-transformed data to normalize magnitude of errors across range of n. Best fit linear model log₁₀(x) = a+blog₁₀(n) back-transformed to a linear basis, giving x = αn^β, where α = 10^a and β = b. MATLAB (Mathworks) function regress was used to perform the regressions. The validity of r² significance values was established by confirming that the residuals of each regression had a normal distribution at p = 0.01 using the Anderson-Darling test [44].

We test the effect of real-world networks deviating from the constant 〈k〉 assumption using the following iterative procedure:

We do this for 3 selection cases in step 2. First, we tested removing the network with the largest deviation from m∝n linear model in each iteration, hypothesizing that this should lead to an overall increase in fit to a linear model (increased r²) for S = f(n) if the WS model behaviour reflected that of real-world systems. Second, we tested removing the network with the smallest deviation from m∝n linear model at each iteration, hypothesizing that this should lead to an overall decrease in fit to a linear model (decreased r²) for S = f(n) if the WS model behaviour reflected that of real-world systems. Third, we tested random deletion, where a random network was deleted at each iteration, irrespective of the regression outcome, to establish the baseline effect of removing systems from the data-set. The first and second cases are unique sequences of deleted networks; the third case we repeated 1000 times.

We tested the dominance of linear S∝n scaling given an approximately constant mean node degree 〈k〉 and path length scaling β. For each Monte Carlo simulation, we drew 27 random C values from a uniform distribution in [0,1], computing α_i for each from Eq. (19) with constant β_i = 1 and 〈k〉_i = 6. We then computed S_i = α_in_i for each, using a logarithmic spread of 27 network sizes to closely match the spread of the real-world system sizes. Linear regression (as detailed above, including the log₁₀-transform) was then performed on the set of simulated 27 S values and the r² recorded. We repeated this procedure 1000 times.

We wished to determine if the generalized Klemm-Eguiluz model (GKE)[31], [32] model could explain the particular scaling relationships we found for the real-world systems:(24)(25)

We explored the (M, ρ) parameter space, searching over in steps of 1, and in steps of 0.1. The lower limit on M is set by the number of edges added per new vertex, and here we set d = 3 to give a mean degree of 〈k〉≈6 for a GKE model network, approximately the same degree that was implied by the linear m∝n relationship for the real-world systems. For each (M, ρ) pair, we constructed 5 GKE model networks for each value of and computed their S^ws and S^Δ scores. We took the mean of these 5 scores for each n, giving sets and . The fit of the GKE model networks was then assessed by computing the root mean square error (RMSE) between these mean values and those given by the scaling relationships (24) and (25) for the tested sizes of network n. The parameters that minimised RMSE are given in Figure 4; the error landscapes are shown in Figure S2.