6 10 −2 10 −1 10 0 0 1 2 3 4 5 6 7 0 0.02 0.04 0.06 0 2 4 6 f ℓ$_{f}$ − ℓ$_{1}$ 10 × C$_{f}$ 100 × E$_{f}$ E$_{f}$ ℓ$_{f}$ − ℓ$_{1}$ 10 × C$_{f}$ ℓ$_{f}$ − ℓ$_{1}$ C$_{f}$ E$_{f}$ FIG. 5: (Color online) Watts-Strogatz small-world network: ℓ$_{f}$ − ℓ$_{1}$ (red circles), 10 × C$_{f}$ (open squares), and 100 × E$_{f}$ (blue triangles) as functions of rewiring fraction f . The inset shows ℓ$_{f}$ − ℓ$_{1}$ and C$_{f}$ as functions of E$_{f}$ for f ≥ 10 − $^{2}$. Observe the linear relation between E$_{f}$ and ℓ$_{f}$ − ℓ$_{1}$ , which suggests that ℓ$_{f}$ − ℓ$_{1}$ might be a good indicator of how well the bond-percolation process on a network can be approximated by tree-based theory. 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]. 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 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. 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) In order to assess the strength of a relation between the theory error E and some characteristic of the net- B. Real-World Networks and Additional Examples E = 1 M M ∑ i =1 | S$_{th}$ ( p$_{i}$ ) − S$_{num}$ ( p$_{i}$ ) | . (2)