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where p$_{i}$ = i/M for i = 1 , 2 , . . . , M are uniformly-spaced values in the interval [0 , 1]. Taking the spacing 1 /M to be sufficiently fine (we use 1 /M = 10 − $^{3}$) implies that the er- ror measure E$_{f}$ approaches the average vertical distance between the S$_{th}$ ( p ) and S$_{f}$ ( p ) curves for p ∈ [0 , 1]. 

In Fig. 5, we plot the values of ℓ$_{f}$ − ℓ$_{1}$ , C$_{f}$ (scaled by a factor of 10 for ease of visualization), and E$_{f}$ (scaled by a factor of 100) as functions of the rewiring parame- ter f . For values of f greater than 10 − $^{2}$, the quantities ℓ$_{f}$ and E$_{f}$ exhibit similar behavior, whereas C$_{f}$ remains near its f = 0 value of 2 / 3 until f is much larger [54]. We highlight the similar scaling of ℓ$_{f}$ and E$_{f}$ in the inset of Fig. 5, in which we plot ℓ$_{f}$ − ℓ$_{1}$ directly as a function of E$_{f}$ for f ≥ 10 − $^{2}$. The approximately linear depen- dence that we observe contrasts to the clearly nonlinear relation between E$_{f}$ and the clustering C$_{f}$ that we show in the same inset. This strongly suggests that differences between theory and numerics are related more directly to the mean intervertex distance than to the clustering coefficient. 

## B. Real-World Networks and Additional Examples 

The above results for Watts-Strogatz small-world net- works motivate the examination of a range of real-world networks in order to seek a clear relationship between an error measure similar to (1) and some other characteris- tic of the network, such as clustering or mean intervertex distance. For each network, we calculate the inaccuracy of P ( k, k $^{′}$)-theory in terms of the error E , which measures the distance between the actual (numerically calculated) 

GCC size curve S$_{num}$ ( p ) and the theoretical prediction S$_{th}$ ( p ): 

Essentially, E gives the average distance between the numerics (black disks) and theory (solid blue curve) in Fig. 1. In Fig. 6(a), we show a scatter plot of log$_{10}$ E ver- sus log$_{10}$ C , where C is the clustering coefficient of each network. We use logarithmic coordinates in Fig. 6 in or- der to fully resolve the range of values for both variables, as they vary by one or more orders of magnitude. 

We also include synthetic examples, such as Watts- Strogatz small-world networks and clustered random net- works generated using the recent models described in Refs. [12, 13], which we now briefly recall. The fun- damental quantity defining the γ -theory networks of Ref. [13] is the joint probability distribution γ ( k, c ), which gives the probability that a randomly chosen node has degree k and is a member of a c -clique (an all-to-all connected subgraph of c nodes). With γ (3 , 3) = 1 (and zero for other values of k and c ), each node in such a network has degree 3 and is part of exactly one triangle. This is equivalent to the p$_{1}$$_{,}$$_{1}$ = 1 case in the clustered random graph model of Ref. [12], where p$_{s,t}$ is the prob- ability that a randomly chosen node is part of t different triangles and in addition has s single edges (which don’t belong to the triangles). In each synthetic network, we P -rewire a fraction f of links and show our results for f = { 10 − $^{3}$, 4 × 10 − $^{3}$, 0 . 04 , 0 . 1 , 0 . 4 } . 

In order to assess the strength of a relation between the theory error E and some characteristic of the net- 

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