On the External Validity of Average-case Analyses of Graph Algorithms

2.1 Heterogeneity

We define the heterogeneity of a graph as the logarithm (base 10) of the coefficient of variation of its degree distribution. To make this specific, let \(\mu = \frac{1}{n} \sum _{v \in V} \deg (v)\) be the average degree of \(G = (V, E)\), and let \(\sigma ^2 = \frac{1}{n} \sum _{v \in V} (\deg (v) - \mu)^2\) be the variance. Then, the coefficient of variation is \(\sigma / \mu\), i.e., the standard deviation relative to the mean. Thus, the heterogeneity is \(\log _{10}(\sigma / \mu)\). The resulting distribution of heterogeneity values is shown in Figure 1. The figure includes thresholds for extreme heterogeneity values that we use to filter real-world graphs in our visualizations; see Section 4 for details.

Fig. 1.

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2.2 Locality

We define locality as a combination of two different notions of locality on edges. The degree locality of an edge is high if its endpoints have many common neighbors. This is similar to the commonly known local clustering coefficient of vertices. Our second measure, the distance locality, captures the locality of edges that do not have common neighbors. In the following, we first introduce these parameters. The subsequent discussion helps to interpret them and justifies our choices. For the distribution of the different locality values over the networks, see Figure 1.

Degree Locality. For \(u, v \in V\), let \(\deg (u, v) = |N(u) \cap N(v)|\) be the common degree of u and v. For a non-bridge edge \(\lbrace u, v\rbrace \in E\), the degree locality is defined as \(\begin{equation*} {L}_{\deg }(\lbrace u, v\rbrace) = \frac{\deg (u, v)}{\min (\deg (u), \deg (v)) - 1}. \end{equation*}\) The \(-1\) in the denominator accounts for the fact that u is always in v’s neighborhood but never in the common neighborhood. With this, we get \({L}_{\deg }(\lbrace u, v\rbrace) \in [0, 1]\) and \({L}_{\deg }(\lbrace u, v\rbrace) = 0\) if and only if the neighborhoods of u and v are disjoint. Moreover, \({L}_{\deg }(\lbrace u, v\rbrace) = 1\) if and only if u is only connected to v and to neighbors of v or vice versa. Essentially, \({L}_{\deg }(\lbrace u, v\rbrace)\) measures in how many triangles \(\lbrace u, v\rbrace\) appears. Note that the denominator is never 0 as we require the edge to not be a bridge and thus u and v both have degree at least 2.

The degree locality \({L}_{\deg }(G)\) of G is the average degree locality over all non-bridge edges.

Distance Locality. For \(u, v \in V\), let \(\operatorname{dist}(u, v)\) be the distance between u and v in G. If \(\operatorname{dist}(u, v) = 1\), i.e., \(\lbrace u, v\rbrace\) is an edge, we are additionally interested in the detour we have to make when not allowing to use the direct edge \(\lbrace u, v\rbrace\). To this end, we define the detour distance \(\operatorname{dist}^+(u, v)\) to be the distance between u and v in \(G - \lbrace u, v\rbrace\).

For a set of vertex pairs \(P \subseteq ({V \atop 2})\), we define \(\operatorname{dist}(P)\) to be the average distance of P, i.e., \(\begin{equation*} \operatorname{dist}(P) = \frac{1}{|P|} \sum _{\lbrace u, v\rbrace \in P} \operatorname{dist}(u, v). \end{equation*}\) Analogously, we define \(\operatorname{dist}^+(P)\) to be the average detour distance of P.

Let \(\overline{E} = ({V \atop 2}) \setminus E\) be the non-edges of G and assume that \(\overline{E} \not= \emptyset\). Note that \(\operatorname{dist}(\overline{E}) \ge 2\), as non-adjacent vertex pairs have distance at least 2. Assume for now that \(\operatorname{dist}(\overline{E}) \gt 2\). For a non-bridge edge \(\lbrace u, v\rbrace \in E\), we define the distance locality as \(\begin{equation*} {L}_{\operatorname{dist}}(\lbrace u, v\rbrace) = 1 - \frac{\operatorname{dist}^+(u, v) - 2}{\operatorname{dist}(\overline{E}) - 2}. \end{equation*}\) Note that the numerator is 0 if and only if u and v have a common neighbor, which yields \({L}_{\operatorname{dist}}(\lbrace u, v\rbrace) = 1\). Moreover, we have \({L}_{\operatorname{dist}}(\lbrace u, v\rbrace) = 0\) if and only if the detour distance between u and v equals the average distance between non-adjacent vertex pairs in G. Finally, \({L}_{\operatorname{dist}}(\lbrace u, v\rbrace)\) can be negative if \(\lbrace u, v\rbrace\) connects a vertex pair that is otherwise more distant than one would expect for a non-adjacent vertex pair. Thus, the distance locality essentially measures how short the edge \(\lbrace u, v\rbrace\) is compared to the average distance in the graph. In the special case of \(\operatorname{dist}(\overline{E}) = 2\), we define \({L}_{\operatorname{dist}}(\lbrace u, v\rbrace) = 0\).

The distance locality \({L}_{\operatorname{dist}}(G)\) of G is the maximum of 0 and the average distance locality over all non-bridge edges.

Locality. The locality \({L}(G)\) of G is the average of the degree and the distance locality, i.e., \({L}(G) = ({L}_{\deg }(G) + {L}_{\operatorname{dist}}(G)) / 2\).

To interpret this parameter, let \({L}(e) = \frac{1}{2}({L}_{\deg }(e) + {L}_{\operatorname{dist}}(e))\) be the edge locality of a non-bridge edge \(e = \lbrace u, v\rbrace \in E\). Note that \({L}(G)\) is basically⁷ the average of all edge localities. Observe that \({L}_{\deg }(e) \gt 0\) and \({L}_{\operatorname{dist}}(e) = 1\) if u and v have a common neighbor. Otherwise, \({L}_{\deg }(e) = 0\) and \({L}_{\operatorname{dist}}(e) \lt 1\). Thus, we in particular get the following regimes for \({L}(e)\):

Discussion and Comparison to the Clustering Coefficient. The degree locality is closely related to the commonly known local clustering coefficient. Note that the degree locality (like the clustering coefficient) only cares for triangles. Though this is desirable in some cases, it does not provide a good separation between graphs with few triangles. An extreme case is bipartite graphs that have no triangles and thus degree locality and clustering coefficient 0. However, we would regard, e.g., grids as highly local. The distance locality solves this issue by essentially defining a measure of locality that distinguishes between graphs of low degree locality. Figure 2 shows a comparison of our locality values to the local clustering coefficient. Note how, for high values, the locality behaves as the clustering coefficient (scaled by a factor of 2), while it provides additional separation for networks with clustering close to 0. See Section B.1 for an extended discussion.

Fig. 2.

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Limitations. We note that our locality definition has some limitations; e.g., it is undefined for trees as trees have no non-bridge edges. However, these limitations do not pose an issue in the context of this article; see Section B.2 for more details.

Computing the Locality. Computing the distance locality efficiently is not straightforward. In Section B.3, we show how to compute it by computing two values for a given graph: the average distance between all vertex pairs and the average detour distance over all non-bridge edges. The latter can be computed exactly in reasonable time using some insights from Section 4.1 (see Section B.3 for details).

For the average distance between all vertex pairs, we use an approximation. Specifically, we implemented the algorithm by Chechik et al. [12]. We observed that there is a simple way to significantly improve its approximation ratio by conditioning on random choices that have been made earlier in the algorithm. Additionally, we compared the algorithm with the straightforward method of computing the shortest path between uniformly sampled vertex pairs. Our experiments indicate that the preferable method depends on the locality and heterogeneity of a network. Though these results are interesting in their own right, they are somewhat beyond the scope of this article and thus deferred to Appendix C.

3.1 Real-world Networks

We use 2740 graphs from Network Repository [39], which we selected as follows. We started with all networks with at most \(1 \,{\rm M}\) edges and reduced each network to its largest connected component. We removed multi-edges and self-loops and ignored weights or edge directions. We tested the resulting connected, simple, undirected, and unweighted graphs for isomorphism and kept only one copy for each group of isomorphic networks. This resulted in 2977 networks. From this we removed 237 networks for different reasons.

3.2 Random Networks

We use three random graph models to generate networks: the Erdős–Rényi model [27] (non-local and homogeneous), the Chung–Lu model [15, 16] (non-local and varying heterogeneity), and the geometric inhomogeneous random graphs (GIRG) model [11] (varying locality and heterogeneity). For the latter, we use the efficient implementation in [8]. In the following we briefly define the models, discuss the model choice, and specify what networks we generate.

Network Model Definitions. Given n and m, the Erdős–Rényi model draws a graph uniformly at random among all graphs with n vertices and m edges.

Given n weights \(w_1, \dots , w_n\) with \(W = \sum w_v\), the Chung–Lu model generates a graph with n vertices by creating an edge between the uth and vth vertex with probability \(\begin{equation*} p_{u, v} = \min \left\lbrace \frac{w_u w_v}{W}, 1\right\rbrace\! . \end{equation*}\) The expected degree of v is then roughly proportional to \(w_v\). We use weights that follow a power law. Specifically, for a constant c and a power-law exponent \(\beta \gt 2\), we choose \(\begin{equation*} w_v = c\cdot v^{-\frac{1}{\beta - 1}}. \end{equation*}\) Thus, the parameters are the number of vertices n, the expected average degree (controlled via c), and the power-law exponent \(\beta\). The latter controls the heterogeneity of the network: heterogeneity is high for small \(\beta\), and for \(\beta \rightarrow \infty\) we get uniform weights.

The Geometric inhomogeneous random graphs (GIRG) model augments the Chung–Lu model with an underlying geometry. In a first step, each vertex v is mapped to a random position \(\boldsymbol {v}\) in some d-dimensional ground space. Let \(||{\boldsymbol {u}}-{\boldsymbol {v}}||\) denote the distance between vertices u and v in that space. Then, for a so-called temperature \(T \in (0, 1)\), the vertices u and v are connected with probability \(\begin{equation*} p_{u, v} = \min \left\lbrace \left(\frac{1}{||{\boldsymbol {u}}-{\boldsymbol {v}}||^d}\cdot \frac{w_u w_v}{W}\right)^{\frac{1}{T}}, 1\right\rbrace . \end{equation*}\) Additionally, for \(T=0\), a threshold variant is obtained, with \(p_{u, v} = 1\) if \(||{\boldsymbol {u}}-{\boldsymbol {v}}||^d \le w_u v_w / W\) and \(p_{u, v} = 0\) otherwise. For the weights, we use the same power-law weights as for the Chung–Lu model. As ground space, we usually use a two-dimensional torus \(\mathbb {T}^2 = [0, 1] \times [0, 1]\) with maximum norm, which is basically just a unit square with distances wrapping around in both dimensions. More precisely, for \(\boldsymbol {u} = (x_u, y_u) \in \mathbb {T}\) and \(\boldsymbol {v} = (x_v, y_v) \in \mathbb {T}\), we have \(||{\boldsymbol {u}}-{\boldsymbol {v}}|| = \max \lbrace \min \lbrace |x_u - x_v|, 1 - |x_u - x_v|\rbrace ,\min \lbrace |y_u - y_v|, 1 - |y_u - y_v|\rbrace \rbrace\). With this, the parameters that remain to be chosen for the GIRG model are the same as for the Chung–Lu model plus the temperature T, which mainly controls the locality.

Discussion of the Network Models. We note that there is a plethora of network models out there. So, why did we choose these particular models? To answer this, consider two different use-cases for network models: first, to explain why and how certain properties emerge in networks assuming some mechanism existing in the real world, and second, to draw conclusions from the existence of certain properties in a network.

An example of the first perspective is the Barabási–Albert model [3], where the simple and believable mechanism of preferential attachment yields a power-law degree distribution.

The second perspective is what we are concerned with in this article. We do not aim at explaining how locality or heterogeneity emerge in networks. Instead, we want to study the implications of these properties on algorithmic performance. For this purpose, we believe that it is crucial for the model to be maximally unbiased beyond the explicitly assumed properties. Ideally, one uses the maximum entropy model with respect to the given constraints.

With this in mind, note that the Erdős–Rényi model draws graphs uniformly at random, given the number of vertices and edges. This is the maximum entropy model and thus maximally unbiased with the constrains that the number of vertices and edges are given.

For graphs whose degree distribution follows a power law, one could use the above-mentioned Barabási–Albert model. However, this model is biased in all sorts of ways that go beyond assuming a power-law degree distribution (e.g., the resulting graphs are k-degenerate for average degree 2k). The maximum entropy model for graphs with power-law distribution is the (soft) configuration model [41]. The Chung–Lu model we use is close to the soft configuration model but much easier to work with. Thus, it is a good compromise between being unbiased beyond the assumptions we want to make and being able to efficiently generate networks or conduct theoretical analyses.

Finally, the GIRG model enhances the Chung–Lu model with locality by assuming an underlying geometry. Though the model is set up to be as unbiased as possible beyond this assumption, we are not aware of a formal proof for this. Moreover, it is up for debate whether a geometry is the best way to introduce locality. Instead, one could, e.g., assume a fixed set of communities like in the stochastic block model [32]. However, assigning vertices to geometric positions seems like the most straightforward way of introducing a smooth notion of similarity. Thus, at the current state of research, we believe that assuming an underlying geometry is the best way to achieve locality for our purpose. As an added bonus, the amount of locality can be controlled with just a single parameter (the temperature) and there is an efficient generator [8] readily available.

Generated Networks and Parameter Choices. For all models, we generate networks with \(n = 50 \,{\rm k}\) vertices and (expected) average degree 10. For the power-law exponent \(\beta\) in Chung–Lu graphs and GIRGs as well as for the temperature T in GIRGs, we chose the values shown in Figure 3, yielding a rather uniform distribution of heterogeneity and locality.

Fig. 3.

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Note that we have 10 different values for each of the parameters \(\beta\) and T, which results in 100 different parameter configurations for GIRGs, 10 configurations for Chung–Lu graphs, and a single configuration for Erdős–Rényi graphs. For each of these 111 configurations, we generated five networks. In the plots, we always use a single dot for each configuration representing the average over five individual networks.

As for the real-world networks, we reduce each generated graph to its largest connected component. With the above parameter choices, this does not change the size of the networks by too much; see Section D.1.

In Section 4.6, we additionally consider networks of average degree 20, using otherwise the same parameter settings.

We note that the power-law weights converge to uniform weights for \(\beta \rightarrow \infty\). Thus, in the limit, the probability distribution of the Chung–Lu model (almost)⁹ coincides with that of the Erdős–Rényi model. It is thus not surprising that the Erdős–Rényi graphs and the Chung–Lu graphs with \(\beta = 25\) occupy almost the same spot in Figure 3.

Geometric Ground Space. We usually use a torus with dimension \(d = 2\) as ground space of the GIRG model. For temperature \(T = 0\), we have a threshold model where two vertices are connected if and only if their distance is sufficiently small compared to the product of their weights. Thus, for \(\beta \rightarrow \infty\), this converges to the commonly known model of random geometric graphs, just on the torus \(\mathbb {T}\) instead of the more commonly used unit square in the Euclidean plane. Moreover, for smaller \(\beta\), we basically get hyperbolic random graphs [36], just in one more dimension than usual.

The reason for using a torus instead of a square in the GIRG model is that a square leads to special situations close to the boundary, while the torus wraps around and thus is completely symmetric. This usually simplifies theoretic analysis without changing too much otherwise. Interestingly, we observe in our experiments that the choice of torus vs. square makes a substantial difference for computing the diameter; see Section 4.2. To make this comparison, we additionally generated five GIRGs with a square as ground space for each of the parameter settings; see Section D.2 for details on how we generated these networks.

4.1 Bidirectional Search

We can compute a shortest path between two vertices s and t using a breadth-first search (BFS). The BFS explores the graph layer-wise, where the ith layer contains the vertices of distance i from s. By exploring the ith layer, we refer to the process of iterating over all edges with an endpoint in the ith layer, thereby finding layer \(i+1\). The search can stop once the current layer contains t.

The bidirectional BFS alternates the exploration of layers between a forward BFS from s and a backward BFS from t. The alternation strategy we study here greedily applies the cheaper of the two explorations in every step. The cost of exploring a layer is estimated via the sum of degrees of vertices in that layer. The search stops once the current layers of forward and backward search intersect. The resulting algorithm is called balanced bidirectional BFS [10], which we often abbreviate with just bidirectional BFS in the following.

The cost c for the bidirectional BFS is the average number of edge explorations over 100 random st-pairs. Note that \(c \le 2m\), as each edge can be explored at most twice, once from each side. Figure 4 shows the exponent x for \(c = m^x\) depending on heterogeneity and locality.

Impact of Locality and Heterogeneity. For the generated data, we see that networks with high locality and homogeneous degree distribution (top-left corner) have an exponent of around 1 (red). Thus, the cost of the bidirectional BFS is roughly m, which matches the worst-case bound. If the network is heterogeneous (right) or less local (bottom), we get significantly lower exponents of around \(0.5\), indicating a cost of roughly \(\sqrt {m}\). The cost is particularly low for very heterogeneous networks. Overall, we get a strict separation between hard (high locality, low heterogeneity) and easy (low locality or high heterogeneity) instances.

For real-world networks, we observe the same overall behavior that instances that are local and homogeneous tend to be hard, while all others tend to be easy. There are only few exceptions to this, indicating that the heterogeneity and locality are usually the crucial properties impacting the performance of the bidirectional BFS.

Discussion. Borassi and Natale [10] found that the bidirectional BFS is surprisingly efficient on many networks, which they used to efficiently compute the betweenness centrality. Additionally, they studied the bidirectional BFS theoretically on random network models where the edges are drawn independently, i.e., when there is no locality. They in particular prove that the search requires with high probability only \(O(\sqrt {n})\) time for degree distributions with bounded variance. This includes the Erdős–Rényi model and the Chung–Lu or the configuration model with power-law degree distribution with power-law exponent \(\beta \ge 3\). For \(\beta \in (2, 3)\) (unbounded variance), the bound is \(O(n^x)\) for \(x \in (0.5, 1)\)¹⁰.

For graphs with locality, it is known that the bidirectional BFS requires linear (\(x = 1\)) and sublinear (\(x \in (0.5, 1)\))¹⁰ time on geometric random graphs in Euclidean and hyperbolic space, respectively [7]. We note that geometric random graphs in Euclidean and hyperbolic space essentially correspond to the top-left and top-right corner of Figure 4, as they can be viewed as special cases of GIRGs without or with heterogeneity, respectively.

Overall, the theoretical results on network models cover all four corners of the space spanned by heterogeneity and locality. Moreover, the predictions on the models match the observations on real-world networks; i.e., for most networks, the practical runtime of the bidirectional BFS is not surprising but as expected. This indicates that locality and heterogeneity are the core deciding features for the performance of the bidirectional BFS.

4.2 Diameter

The eccentricity of \(s \in V\) is \(\max _{t\in V}\operatorname{dist}(s, t)\), i.e., the distance to the vertex farthest from s. The diameter of the graph G is the maximum eccentricity of its vertices. It can be computed using the iFUB algorithm [20]. It starts with a root \(r \in V\) from which it computes the BFS tree T. It then processes the vertices bottom up in T and computes their eccentricities using a BFS per vertex. This process can be pruned when the distance to r is sufficiently small compared to the largest eccentricity found so far. Pruning works well if r is a central vertex in the sense that it has low distance to many vertices and a shortest path between distant vertices gets close to r.

There are different strategies of choosing the central vertex r. The iFUB+hd algorithm simply chooses a vertex of highest degree as the starting vertex. A more sophisticated way of selecting r works as follows. A double sweep [38] starts with a vertex u, chooses a vertex v at maximum distance from u, and returns a vertex w from a middle layer of the BFS tree from v. A 4-sweep [20] consists of two double sweeps, starting the second sweep with the result w of the first sweep. The iFUB+4-sweephd algorithm chooses r by doing a 4-sweep from a vertex of maximum degree.

The cost c of iFUB+hd and iFUB+4-sweephd is the number of BFSs it performs (including the four initial BFSs of the 4-sweep for iFUB+4-sweephd). Note that \(c \le n\). Figure 5 shows the exponent x for \(c = n^x\) depending on heterogeneity and locality. For iFUB-hd and iFUB+4-sweephd, 17 and 16 real-world networks, respectively, are excluded from the plots as they exceeded the time limit of \(30 \,{\rm min}\). The GIRG model uses a square as ground space instead of the usual torus; see discussion below for details.

Fig. 5.

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Impact of Locality and Heterogeneity. The general dependence is the same for generated and real-world networks. For iFUB+hd (Figure 5(a)), an almost linear number of BFSs is required for networks lacking heterogeneity. On heterogeneous networks, i.e., if there are vertices of high degree, the number of BFSs is substantially sublinear. The more sophisticated iFUB+4-sweephd variant (Figure 5(b)) additionally performs well on homogeneous networks with locality. Observe that the picture for real-world networks shows some noise, which indicates that there are properties besides locality and heterogeneity that impact the performance. We note, however, that this observation agrees with the models, where we get highly varying exponents for individual parameter settings; e.g., for iFUB+4-sweephd on the five GIRGs with power-law exponent \(\beta = 3.35\) and temperature \(T = 0.62\), we get exponents ranging from \(0.18\) to \(0.77\) for the five generated instances.

Impact of the Ground Space: Torus vs. Square. As mentioned in Section 3.2, the default ground space for the GIRG model is a torus. Though this might seem less natural than using a unit square, using the torus often simplifies theoretical analysis while not making a big difference otherwise. However, in the particular case of the iFUB+4-sweephd algorithm, the ground space makes a significant difference. Figure 5(c) compares the efficiency of iFUB+4-sweephd between torus-GIRGs and square-GIRGs.

Observe that the difference mainly shows for homogeneous and local networks (top-left corner), where iFUB+4-sweephd requires an almost linear number of BFSs on torus-GIRGs, while being sublinear for square-GIRGs (as we have already observed in Figure 5(b)). This difference can be explained as follows. For square-GIRGs, the 4-sweep can find a vertex in the center of the ground space, which has relatively low distance to many vertices. As mentioned above, starting iFUB from such a vertex lets us prune the search early. In a torus, however, all points are identical and thus homogeneous torus-GIRGs do not have central vertices, rendering the iFUB approach ineffective.

As mentioned above, iFUB+4-sweephd performs well on most homogeneous and local real-world networks (Figure 5(b)). This indicates that square-GIRGs are a better representation for these networks than torus-GIRGs. There are, however, exceptions. Three such cases are the local and homogeneous networks man_5976, tube1, and barth4 with exponents of \(0.894\), \(0.873\), and \(0.89\), respectively. Interestingly, all three graphs exhibit torus-like structures in the sense that they “wrap around” in some way, which obstructs the existence of a central vertex; see Figure 6. Thus, also in this case, the predictions of the models match the real-world behavior.

Fig. 6.

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4-Sweep Lower Bound. Diameter approximation methods such as the double-sweep lower bound are known to perform well on real-world graphs [21, 38] and heterogeneous graph models [9]. As our experiments involve doing a 4-sweep anyway, we report the quality of the resulting 4-sweep lower bounds on our dataset of real-world networks. We know the exact diameter for 2726 real-world networks (no timeout in iFUB+hd or iFUB+4-sweephd). The 4-sweep lower bound matched the diameter for 2218 (\(81 \,\%\)) of these networks. For 2703 (\(99 \,\%\)) networks, the difference between the lower bound and the exact diameter is at most 2.

Discussion. The problem of computing the diameter of a graph is closely related to the all pairs shortest path (APSP) problem, i.e., computing the distance between all pairs of points. It is an open problem whether one can compute the diameter of arbitrary graphs faster than APSP [17]. The best-known algorithms for APSP run in time \(O(n^\omega)\) with \(\omega \lt 2.38\) [2] or \(O(nm)\). Both running times are infeasible for large real-world networks.

In practice, however, Crescenzi et al. [20] introduced the iFUB algorithm and demonstrated that it performs much better on most real-world networks than the worst case suggests. They note that it works particularly well on networks with high difference between radius and diameter. This corresponds to the existence of a central vertex with eccentricity close to half the diameter. Our experiments confirm this and provide the following more detailed picture. First, in heterogeneous networks, the high-degree nodes serve as central vertices. The results of Borassi et al. [9] provide a theoretical foundation for this phenomenon. They in particular show that heterogeneous Chung–Lu graphs allow the computation of the diameter in sub-quadratic time, indeed using a vertex of high degree as central vertex. Second, for homogeneous networks, locality facilitates the existence of a central vertex, unless the graph exhibits a toroidal structure.

Our results point to some practical as well as theoretical open questions. The highest potential for practical improvements can be achieved by focusing on homogeneous graphs without locality or with a toroidal underlying geometry. Erdős–Rényi graphs or random geometric graphs with a torus as underlying geometry may help guide the development of such an algorithm. Moreover, it would be interesting to strengthen the theoretical foundation by complementing the above results by Borassi et al. [9] on Chung–Lu graphs to network models exhibiting locality (homogeneous as well as heterogeneous).

4.3 Vertex Cover Domination

A vertex set \(S \subseteq V\) is a vertex cover if every edge has an endpoint in S, i.e., removing S from G leaves a set of isolated vertices. We are interested in finding a vertex cover of minimum size. For two adjacent vertices \(u, v \in V\), we say that udominates v if \(N[v] \subseteq N[u]\). The dominance rule states that there exists a minimum vertex cover that includes u. Thus, one can reduce the instance by including u in the vertex cover and removing it from the graph.

To evaluate the effectiveness of the dominance rule, we apply it exhaustively, i.e., until no dominant vertices are left. Moreover, we remove isolated vertices. We refer to the number c of vertices in the largest connected component of the remaining instances as the kernel size. Figure 7 shows the relative kernel size \(c / n\) with respect to locality and heterogeneity.

Fig. 7.

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Impact of Locality and Heterogeneity. We see a sharp separation for the generated networks. For low locality and heterogeneity (bottom left), the reduction rule cannot be applied. For high locality and heterogeneity (top right), the dominance rule completely solves the instance. For real-world networks, the separation is less sharp; i.e., there is a larger range of locality/heterogeneity values in the middle where the dominance rule is effective sometimes. Nonetheless, we see the same trend that the reduction rule is more likely to be effective the higher the locality and heterogeneity. In the extreme regimes (bottom left or top right), we observe the same behavior as for the generated networks with relative kernel sizes close to 1 and 0, respectively, for almost all networks. Moreover, there is dichotomy in the sense that many instances are either (almost) completely solved by the dominance rule or the rule is basically inapplicable. In fact, \(30.9 \,\%\) of the real-world networks are reduced to below \(5 \,\%\) of their original size, while \(16.3 \,\%\) are reduced by less than \(5 \,\%\).

Discussion. Though vertex cover is NP-hard [34], it is rather approachable: it can be solved in \(1.1996^nn^{O(1)}\) time [46] and there is a multitude of FPT algorithms with respect to the solution size k [22], the fastest one running in \(O(1.2738^k + kn)\) [13]. This basis was improved by Harris and Narayanaswamy in a recent preprint [31] stating a running time of \(O^*(1.25400^k),\) where \(O^*\) suppresses polynomial factors. Moreover, there are good practical algorithms [1, 45]. Both algorithms apply a suite of reduction rules, including the dominance rule or a generalization. We note that the dominance rule is closely related to Weihe’s reduction rules for hitting set [45]. Our previous experiments for Weihe’s reduction rules match our results: they work well if the instances are local and heterogeneous [6]. Concerning theoretic analysis on models, we know that on hyperbolic random graphs, the dominance rule is sufficiently effective to yield a polynomial time algorithm [5]. Thus, it is not surprising that the top-right corner in Figure 7 is mostly green.

We see two main directions for future research. First, concerning the dominance rule, we have seen that heterogeneity and locality are good predictors for the effectiveness in the extreme regimes. However, there is a regime of moderate heterogeneity and locality where we see a mix of high and low effectiveness. This indicates that other properties are the deciding factor for these instances and it would be interesting to figure out these properties. The second direction is to investigate the effectiveness of other techniques for solving vertex cover depending on the network properties. This includes the study of techniques used in existing solvers [1, 45] but also the development of new techniques that are tailored toward less local or less heterogeneous instances. Specifically, the algorithm of Akiba and Iwata [1] performs well on social networks (high locality and heterogeneity) but fails on road networks (low heterogeneity). Thus, techniques tailored toward solving homogeneous instances could lead to an algorithm that is more efficient on a wider range of instances.

4.4 The Louvain Algorithm for Graph Clustering

Let \(V_1 \cup \dots \cup V_k = V\) be a clustering where each vertex set \(V_i\) is a cluster. One is usually interested in finding clusterings with dense clusters and few edges between clusters, which is formalized using some quality measure, e.g., the so-called modularity. A common subroutine in clustering algorithms is to apply the following local search. Start with every vertex in its own cluster. Then, check for each vertex \(v \in V\) whether moving v into a neighboring cluster improves the clustering. If so, v is moved into the cluster yielding the biggest improvement. This is iterated until no improvement can be achieved. Doing this with the modularity as the quality measure (and subsequently repeating it after contracting clusters) yields the well-known Louvain algorithm [4].

The runtime of the Louvain algorithm is dominated by the number of iterations of the initial local search. Figure 8 shows this number of iterations. It excludes four real-world networks with more than \(1 \,{\rm k}\) iterations (1,403, 1,403, 5,003, and 19,983).

Fig. 8.

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Impact of Locality and Heterogeneity. For the generated instances, the number of iterations is generally low but increasing when decreasing the locality. To discuss this in more detail, recall that each dot in the left plot represents the average of five networks randomly generated with the same set of parameters. Moreover, recall that the bottom-left dot represents Erdős–Rényi graphs, the rest of the bottom row represents Chung–Lu graphs, and all other points represent GIRGs.

For GIRGs, the average number of iterations ranges from \(9.4\) to \(55.4\) for the different parameter configurations. Moreover, the number of iterations starts to rise only for low localities. For \(83 \,\%\) of the configurations, the average number of iterations lies below 20. For the Erdős–Rényi graphs we obtain an average of \(196.4\) iterations, and for Chung–Lu graphs it goes up to \(443.40\) for one configuration. However, these average values over five generated instances have to be taken with a grain of salt as there is a rather high variance; e.g., the number of iterations for the five Chung–Lu graphs with power-law exponent 25 ranges from 104 up to 333.

For the real-world networks, there is no clear trend depending on locality or heterogeneity. In general, the number of iterations is rather low except for some outliers. While the strongest outlier requires almost \(20 \,{\rm k}\) iterations, \(98.6 \,\%\) of the networks have at most 100 iterations.

Discussion. The Louvain algorithm has been introduced by Blondel et al. [4]. Discussing the plethora of work building on the Louvain algorithm is beyond the scope of this article. Concerning its running time, the worst-case number of iterations of the Louvain algorithm can be upper bounded by \(O(m^2)\) due to the fact that the modularity is bounded and each vertex move improves it by at least \(\Omega (1 / m^2)\). Moreover, there exists a graph family that requires \(\Omega (n)\) iterations for the first local search [33, Proposition 3.1]. Note that a linear number of iterations leads to a quadratic running time, which is prohibitive for larger networks.

Our experiments indicate that locality or heterogeneity are not the properties that discriminate between easy and hard instances. For generated instances, there is the trend that low locality increases the number of iterations, which does not transfer to the real-world networks (or is at least less clear). However, the general picture that most instances require few iterations while there are some outliers coincides for generated and real-world networks.

We believe that the observed behavior can be explained as follows. There are two factors that increase the number of iterations. First, one can craft very specific situations in which individual vertices move into a cluster, trigger a change there, and then continue into another cluster. These specific structures are unlikely to appear in practice as well as in random instances, which is why we observe the overall low number of iterations in our experiments. The second factor increasing the number of iterations is a lack of clear community structure. A vertex that fits more or less equally well to different communities is likely to swap its cluster more often when the clustering slightly changes during the local search. This is why we see an increased number of iterations on the generated networks with low locality. This interpretation is also supported by the fact that Louvain performs provably well on the stochastic block model [18], a network model with a very clear community structure of non-overlapping clusters.

For future research, we believe it is highly interesting to support our above interpretation with theoretical evidence. In particular, it would be interesting to understand how the local search behaves on a model with underlying geometry, which facilitates structures of overlapping communities.

4.5 Maximal Cliques

A clique is a subset of vertices \(C \subseteq V\) such that C induces a complete graph; i.e., any pair of vertices in C is connected. A clique is maximal if it is not contained in any other clique. In the following, we are never interested in non-maximal cliques. Thus, whenever we use the term clique, it implicitly refers to a maximal clique. To enumerate all cliques, we used the algorithm by Eppstein et al. [24], using their implementation [25, 26].

As the cliques of a network can be enumerated in polynomial time per clique [40], the number of maximal cliques is a good indicator for the hardness of an instance. To our surprise, the number of cliques does not exceed the number of edges m for all generated and most real-world networks. Out of the 2,740 real-world networks, 2,552 (\(93 \,\%\)) have at most m and 187 have more than m cliques. For the remaining one network, the timeout of \(30 \,{\rm min}\) was exceeded. Figure 9 shows the number of cliques relative to m for all networks with at most m cliques.

Fig. 9.

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Impact of Locality and Heterogeneity. One can see that the networks (generated and real-world) with low locality have roughly m cliques, while the number of cliques decreases for increasing locality. Moreover, among networks with locality, there is the slight trend that networks with higher heterogeneity have more cliques.

It is not surprising that networks with low locality have roughly m cliques, as graphs without triangles have exactly m cliques. For graphs with higher locality, there are two effects counteracting each other. On the one hand, multiple edges combine into larger cliques, which decreases the number of cliques. On the other hand, each edge can be contained in multiple cliques, which increases the number of cliques. Our experiments show that the former effect usually supersedes the latter; i.e., the size of the cliques increases more than the number of cliques each edge is contained in.

Discussion. There are many results on enumerating cliques and on the complementary problem of enumerating independent sets; discussing them all is beyond the scope of this article. Here, we focus on discussing two results that are closely related. They are based on the degeneracy and the so-called (weak) closure¹¹ of a network.

The degeneracy is a robust measure for the sparsity of a network; i.e., the degeneracy is low only if the graph does not include a dense subgraph. Eppstein et al. [25] give an algorithm for enumerating all cliques that runs in \(O(dn3^{d/3})\) time, where d is the degeneracy. We use this algorithm for our experiments as it performs very well in practice.

Fox et al. [28] introduced the closure as a measure that captures the tendency that common neighbors facilitate direct connections, i.e., as a measure for locality. Additionally, they introduced the weak closure as a more robust measure. Fox et al. show that weakly c-closed graphs have at most \(3^{(c - 1)/3} n^2\) cliques. For c-closed graphs they give the additional upper bound of \(4^{(c + 4)(c - 1) / 2} n^{2- 2^{1 - c}}\). Together with an algorithm that takes polynomial time per clique [40], this shows that enumerating all cliques is fixed-parameter tractable with respect to the (weak) closure.

Comparing this to our empirical observations, the number of cliques is at most m for \(93 \,\%\) of real-world networks, while the theoretical results give exponential bounds on how much bigger than m the number cliques can be. Nonetheless, qualitatively speaking, the bounds for degeneracy and closure show that sparsity and locality lead to a low number of cliques. This matches our observations on real-world networks, i.e., most of them are sparse and have few cliques and the number of cliques decreases with increasing locality. Unfortunately, this interpretation does not withstand a closer look. To study how indicative the degeneracy and (weak) closure actually are for the number of cliques, we also computed¹² these parameters for the networks in our dataset. We then study how the number of cliques depends on these parameters as well as the relation between the parameters. As this is somewhat beyond the core focus of the article, we only mention the main insights here. The detailed results can be found in Appendix E.

For the \(93 \,\%\) of networks with at most m cliques, the closure does not correlate with the number of cliques. Even worse, the weak closure and the degeneracy have a slight negative correlation; i.e., a lower parameter corresponds to a higher number of cliques. On the remaining \(7 \,\%\) of networks, the picture is somewhat different. While the closure is still a bad indicator for the number of cliques, the number of cliques is highly correlated with the degeneracy and the weak closure. Thus, degeneracy and weak closure provide a good measure of hardness at least for the few somewhat hard instances with more than m cliques. However, we note that degeneracy and weak closure are almost identical on these networks. Thus, weak closure is mostly a measure of sparsity rather than of locality.

To summarize, although the existing theoretical results provide valuable insights, they cannot explain the observed behavior on the majority of the networks. While providing strong upper bounds, these parameterized results suffer from the same issue that a worst-case analysis often has. The worst case is usually harder than the typical case, and this does not change by restricting the set of considered instances to those with small degeneracy or closure. In contrast, our experiments show that the network models yield surprisingly accurate predictions. They predict that one should generally expect sparse networks to have at most m cliques and that the number of cliques is lower for networks with high locality. Moreover, even the slight increase with increasing heterogeneity is given for the generated as well as for the real-world networks. For future research, we believe that it is highly interesting to complement our empirical findings theoretically by proving that the expected number of cliques is below m. Moreover, our findings reopen the question of finding deterministic parameters that allow to give strong bounds on the number of cliques while capturing most real-world networks.

4.6 Chromatic Number

The chromatic number \(\chi\) of G is the smallest number of colors one can use to color the vertices of G without assigning adjacent vertices the same color. The size of any clique in G is a lower bound on the chromatic number. Thus, for the clique number \(\omega\) of G, which is the size of the maximum clique in G, we have \(\chi \ge \omega\). To compute \(\chi\), it is safe to reduce G to its \(\omega\)-core, i.e., to iteratively remove all vertices with degree less than \(\omega\). After coloring the resulting kernel optimally, one can always color the previously removed vertices with at most \(\omega\) colors [42].

The size of the kernel after applying this reduction is a good indicator for the hardness of an instance. For each network in our dataset, we first compute the clique number \(\omega\) by enumerating all maximal cliques as described in Section 4.5. We then compute the \(\omega\)-core to obtain the kernel. Let c be the number of vertices of the kernel. Figure 10(a) shows the exponent x for the kernel size \(c = n^x\), excluding one network where the clique computation exceeded the timeout of 30 min. As we observed that the average degree of the generated instances has a substantial impact on the kernel size, we additionally consider generated instances with average degree 20 (instead of the usual average degree of 10).

Fig. 10.

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Impact of Locality and Heterogeneity. For the generated networks, one can make three main observations. First, graphs with higher locality have smaller kernels. Second, for highly heterogeneous networks, the kernel size is always moderate independent of the locality. Third, the average degree has a strong impact on the applicability of the reduction rule, with much smaller kernels for lower average degrees.

Many real-world instances are either heavily reduced or the reduction rule is almost inapplicable, yielding a very noisy picture. The noisiness indicates that locality and heterogeneity are not the only relevant factors for the kernel size. However, the overall trend is similar to the generated instances. Moreover, the noisiness is also present in the network models: the kernel size varies strongly with the average degree, but even for fixed average degree we observe high variance; e.g., the exponents we get for the five generated GIRGs with average degree 20, power-law exponent \(5.1\), and temperature \(0.76\) are \(0.00\), \(0.00\), \(0.99\), \(0.99\), and \(0.99\).

In the following, we first discuss the impact of the average degree in more detail and then offer a potential explanation for the observed variance in kernel size.

Average Degree of Generated Instances. As mentioned above, the comparison of Figures 10(a) and 10(b) shows that a higher average degree leads to larger kernels and thus harder instances. This dependence on the average degree can be seen in more detail in Figure 10(c). It shows the exponent (averaged over 30 sampled instances) for GIRGs with three parameter settings depending on the average degree. For GIRGs with high temperature (low locality), we see a threshold behavior going from small kernels for low average degree to large kernels for high average degree. For lower temperatures (high locality), the transition happens later and less steeply. This fits to the previously observed separation between difficult non-local and easy local instances, which moves up to higher locality values for an increasing average degree.

Variance in the Kernel Size: Clique Number and Degeneracy. Studying the clique number in comparison to the degeneracy yields a potential explanation for the observed variance in the kernel size; see Figure 11.

Fig. 11.

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Recall that we use the clique number \(\omega\) as the lower bound for the chromatic number. Thus, a larger \(\omega\) yields a better lower bound, implying that the reduction rule can remove vertices of higher degree. However, the size of the kernel still depends on the structure of the remaining graph, i.e., on whether there are sufficiently many low-degree vertices that can be removed. This property is captured by the degeneracy d of the graph. In particular, note that the degeneracy is lower-bounded by \(\omega - 1\). If the degeneracy is exactly \(\omega - 1\), then the reduction rule completely solves the instance. If the degeneracy is higher, the kernel can be very large.

In Figure 11, we can make multiple observations. Locality and heterogeneity seem to be the core factors impacting the clique size and the degeneracy, and the dependence is similar for generated and real-world networks. However, the dependence is almost identical for clique size and degeneracy. Thus, whether \(d = \omega - 1\), which yields an empty kernel, or \(d \gt \omega - 1\), yielding a non-empty kernel, often comes down to a property other than locality and heterogeneity, which explains the high variance observed above. Also observe how the difference between clique size and degeneracy is slightly higher for less local graphs, which fits to the previous observation that non-local graphs tend to have larger kernels.

Discussion. The reduction rule we considered here has been described by Verma et al. [42], who demonstrated that the clique number is a good lower bound for the chromatic number on real-world instances. Another lower-bound method based on degree-bounded independent sets performing very well in practice has been introduced by Lin et al. [37]. Concerning the clique number, Walteros and Buchanan [43] have shown that in many real-world networks, the clique-core gap \(g = d+1-\omega\) is small and provide an \(O(m^{1.5})\) algorithm for the clique number if this gap is constant. However, they leave a potential parameterization for the chromatic number based on the clique-core gap as an open problem.

Concerning the impact of locality and heterogeneity, our results reveal a highly interesting effect. We have two characteristics (clique number and degeneracy), where locality and heterogeneity are the crucial factors. However, for an algorithm whose efficiency is essentially based on the comparison of the two characteristics, properties besides locality and heterogeneity have to tip the scale for whether the algorithm performs well or not. This is an interesting insight as this effect might pose a threat to external validity when considering more complicated algorithms based on multiple structural properties. In this specific case, however, the behavior observed on real-world instances is nonetheless quite similar to that on generated instances, including the high variance due to the above effect.

For future research, it would be interesting to study the reduction rule based on maximum cliques theoretically on the different network models. We believe that this can give a better understanding of what properties tip the scale in case the degeneracy is higher than the clique number, potentially leading to improved algorithms. This can also lead to interesting technical contributions like, e.g., understanding the potential threshold behavior when it comes to the dependence on the average degree.