Eccentricities on small-world networks

Abstract

This paper focuses on the efficiency issue of computing and maintaining the eccentricity distribution on a large and perhaps dynamic small-world network. Eccentricity distribution evaluates the importance of each node in a graph, providing a node ranking for graph analytics; moreover, it is the key to the computation of two fundamental graph measurements, diameter, and radius. Existing eccentricity computation algorithms are not scalable enough to handle real large networks unless approximation is introduced. Such an approximation, however, leads to a prominent relative error on small-world networks whose diameters are notably short. Our solution optimizes existing eccentricity computation algorithms on their bottlenecks—one-node eccentricity computation and the upper/lower bounds update—based on a line of original insights; it also provides the first algorithm on maintaining the eccentricities of a dynamic graph without recomputing the eccentricity distribution upon each edge update. On real large small-world networks, our approach outperforms the state-of-the-art eccentricity computation approach by up to three orders of magnitude and our maintenance algorithm outperforms the recomputation baseline (recompute using our superior eccentricity computation approach) by up to two orders of magnitude, as demonstrated by our extensive evaluation.

Appendices

The proof of Lemma 4

Based on Definition 3, $pecc(x| V_{\le \lambda }^z) = \max _{u\in V_{\le \lambda }^z} dist(x,u).$$dist(x, u) \le dist(x,z) + dist(z,u) \le dist(x,z) + \lambda$, for $\forall u \in V_{\le \lambda }^z$. Therefore, $pecc(x| V_{\le \lambda }^z) = \max _{u\in V_{\le \lambda }^z} dist(x,u) \le \max _{u\in V_{\le \lambda }^z} (dist(x, z) + \lambda ) = dist(x, z) + \lambda .$

The proof of Lemma 5

According to the definition of a $\lambda$-partial set, $V' \cup V_{\le \lambda }^z = V$, and the definition of the eccentricity $ecc(x) = max_{u\in V} dist(u,x)$, $ecc(x) = \max \{pecc(x|V'), pecc(x| V_{\le \lambda }^z\}$.

The proof of Lemma 9

Let a shortest path from u to x be $\langle v_0, v_1, v_2, \ldots , v_k \rangle$ with $v_0=u$, $v_k=x$ and $k=dist(x,u)$. Since the new snapshot is taken on a stable state, consider edges $(v_0, v_1)$, $(v_1, v_2), \ldots , (v_{k-1}, v_k)$, we have $\overline{ecc}_{new}(u) = \overline{ecc}_{new}(v_0) \le \overline{ecc}_{new}(v_1) + 1 \le \overline{ecc}_{new}(v_2) + 2 \le \ldots \le \overline{ecc}_{new}(v_k) + k = \overline{ecc}_{new}(x) + k.$ Therefore, $\overline{ecc}_{new}(u) \le \min \{\overline{ecc}_{old}(u), \overline{ecc}_{new}(x) + dist (x,u)\}.$ Similar proof can be applied on showing that $\underline{ecc}_{new}(v_0) \ge \underline{ecc}_{new}(v_{1}) - 1 \ge \ldots \ge \underline{ecc}_{new}(v_k) - k.$ By plugging $v_0 = u$, $v_k = x$ and $k = dist(x,u)$ in the above inequality, we complete the proof.

The proof of Lemma 10

Observe that in Step 2) of the iterative update, $\overline{ecc}(l)$ of node l will be updated only if $\overline{ecc}(r)$ of its neighbor r is small enough such that $\overline{ecc}(l) > \overline{ecc}(r) + 1$. Once the update takes place, we conceptually associate with $\overline{ecc}(l)$ a source $\overline{ecc}(l).s \leftarrow r$ to record the source of the bound. Note that this source field may be overwritten upon a subsequent update; however, it will not be removed once created.

$\overline{ecc}_{new}(y).s$ exists since $\overline{ecc}(y)$ must have been updated to let $\overline{ecc}_{new}(y) <\overline{ecc}_{old}(y)$. Now, we trace from y via the source link of $\overline{ecc}_{new}(\cdot ).s$, generating a path $\langle v'_0, v'_1, \ldots , v_{k'}'\rangle$ with $v'_0 = y$, $v'_i = \overline{ecc}_{new}(v'_{i-1}).s$, for each $i\in [1, k']$ while $\overline{ecc}_{new}(v_{k'}')$ does not have a source. Note that in this sequence, we have $\overline{ecc}_{new}(v'_{i-1}) = \overline{ecc}_{new}(v'_i) + 1$ for all $i\in [1,k']$; thus, the sequence cannot contain a loop and thus $k'\le n$. We have $\overline{ecc}_{new}(y) = \overline{ecc}_{new}(v_0') = \overline{ecc}_{new}(v_{k'}') + k'.$

We prove that $v'_{k'}$ must be the trigger node x. If otherwise, $\overline{ecc}_{new}(v_{k'}')$ has no source, and thus, $\overline{ecc}_{new}(v_{k'}') = \overline{ecc}_{old}(v_{k'}')$. Therefore, $\overline{ecc}_{old}(y) > \overline{ecc}_{new}(y) = \overline{ecc}_{old}(v_{k'}') + k'$. According to pigeon principle, there must be $\exists j\in [1,k']$ such that $\overline{ecc}_{old}{v'_{j-1}} > \overline{ecc}_{old}{v'_j} + 1$—violating the assumption that the old snapshot is stable.

Fig. 27

Testing the accuracy of ${\mathsf {kBFSEcc}}$

Fig. 28

Testing the running time of ${\mathsf {kBFSEcc}}$ and ${\mathsf {ECC}}$

Table 4 Testing ${\mathsf {ECC}}$ on road networks

Table 5 Dataset description and the results of ${\mathsf {HybridEcc}}$

The fact that $v'_{k'} = x$ implies three important results:

• 1. For $v_j'$ with $j \in [0,k')$, $\overline{ecc}_{new}(v_j') < \overline{ecc}_{old}(v_j')$: if otherwise, the path would have stopped at j instead of $k'$.

• 2. $\overline{ecc}_{new}(x) = ub_x < \overline{ecc}_{old}(x)$. Since if $\overline{ecc}_{new}(x) = \overline{ecc}_{old}(x)$, there will be a violation to the assumption that the old snapshot is stable. Besides, $\overline{ecc}_{new}(x)$ has no source; thus, it has not been updated in Step 2) of the iterative update. Therefore, $\overline{ecc}_{new}(x) = \min \{\overline{ecc}_{old}(x), ub_x\} = ub_x < \overline{ecc}_{old}(x).$

• 3. $\langle v'_0, v'_1, \ldots , v_{k'}'\rangle$ is a shortest path from y to x. Since $k'$ is the length of a path from x to y, $k' \ge dist(x,y)$. Based on Lemma 9, that is, $\overline{ecc}_{new}(y) \le \overline{ecc}_{new}(x) + dist(x,y)$, it can be assured that $k' = dist(x,y)$ since $\overline{ecc}_{new}(y) = \overline{ecc}_{new}(x) + k'$.

From the above three results, we complete the proof.

The proof of Lemma 13

We only prove the lemma for the case of edge insertion since the case of edge deletion can be symmetrically proved. Let p (and $p'$, resp.) be a shortest path from v to w in G (and in $G'$, resp.) with length L(p) (or $L(p')$, resp.). If $p'$ does not include edge e, then both p and $p'$ are paths between v and w in both G and $G'$. Since p and $p'$ are shortest paths of G and $G'$, respectively, $dist(v,w) = L(p) = L'(p) = dist'(v,w)$, contradiction. Therefore, $p'$ must include e.

The proof of Lemma 14

We only consider the case of inserting the edge of e(a, b) to G since the case of edge deletion can be symmetrically proved. For two nodes $u,v \in V$, if $dist(u,v) \ne dist'(u,v)$, then all of the shortest paths from u to v on graph $G'$ must pass e (Lemma 13), i.e., either $dist'(u,a) = dist'(u, b) - 1$ or $dist'(u,b) = dist'(u, a) - 1$. When $dist'(u,a) = dist'(u, b) - 1$, since $dist'(b,v) = dist(b,v)$ and $dist'(u,a) = dist(u,a)$ (the shortest paths from u to a and from b to v on $G'$ do not include e), and $dist(u,v) \ne dist'(u,v)$, we have $dist(u,b) \ne dist'(u,b)$, and $dist(v,a) \ne dist'(v,a)$, $u \in C^b$ and $v \in C^a$. Similarly, when $dist(u,b) = dist(u, a) - 1$, $u \in C^a$ and $v \in C^b$.

Comparison with approximate methods

We show that computing exact eccentricities is necessary by evaluating the approximate algorithms of ${\mathsf {HybridEcc}}$ [27] and ${\mathsf {kBFSEcc}}$ [26] on four graphs: Twitter, Youtube, Lastfm, and Indochina. The algorithms of ${\mathsf {HybridEcc}}$ and ${\mathsf {kBFSEcc}}$ have been introduced in Sect. 5; the descriptions of the four real graphs and the computation time of ${\mathsf {ECC}}$ are given in Table 5. All algorithms were run with a single thread.

We measure an approximate algorithm with its accuracy—the percentage of nodes whose eccentricities are correctly estimated—and its running time.

Table 5 shows the running time and accuracy of $\mathsf {HybridEcc}$. In comparison with $\mathsf {ECC}$, $\mathsf {HybridEcc}$ shows a trade-off between the precision and efficiency. On Twitter, $\mathsf {HybridEcc}$ uses longer running time to achieve a high accuracy (higher than 96%). On Indochina, $\mathsf {HybridEcc}$ performs well both in terms of accuracy and running time. On Youtube and Lastfm, the running time of $\mathsf {HybridEcc}$ is significantly lower than that of $\mathsf {ECC}$ while the accuracy is below 66%. $\mathsf {HybridEcc}$ has its accuracy varying dramatically on different graphs and thus fails in providing stable and reliable estimations.

The performance of ${\mathsf {kBFSEcc}}$ is highly dependent on a key parameter of k, and we show the accuracy of ${\mathsf {kBFSEcc}}$ in Fig. 27 and running time in Fig. 28 with k varying from 1 to $2^{14} = 16{,}384$. Note that, in order to eliminate the impact of randomness, we used the same random seed for different values of k. This means that the sampled node set S of Phase 1 (see Sect. 5) of a smaller k is a subset of that of a larger k.

Figure 27 shows the accuracy of ${\mathsf {kBFSEcc}}$. On 2 out of 4 graphs, ${\mathsf {kBFSEcc}}$ achieves 100% accuracy with a small k: $k = 2^4$ on Youtube, $k = 2^8$ on Indochina. In contrast, on Twitter, it requires a large $k = 2^{14}$ to reach 100% accuracy while it never achieves 100% accuracy on Lastfm. A weird fluctuation of the accuracy of ${\mathsf {kBFSEcc}}$ has been observed. Note that this phenomena is random-seed independent—similar phenomena can be observed under other random seeds. On all four graphs, an increase of k can dramatically reduce the accuracy. For example, the accuracy drops from $95.59\%$ to $39.01\%$ on Lastfm when k is increased from $2^3$ to $2^4$. In this sense, ${\mathsf {kBFSEcc}}$ still needs analysis on the relationship between k and the accuracy to establish the reliability of the estimation.

Figure 28 shows the running time of ${\mathsf {kBFSEcc}}$ that increases linearly with k. In general, $\mathsf {kBFSEcc}$ reaches a high accuracy within $10\%$ of the running time of $\mathsf {ECC}$. In particular, on Indochina, $\mathsf {kBFSEcc}$ reached 100% accuracy $17.5\times$ times faster than $\mathsf {ECC}$. However, this superiority is not guaranteed: when $\mathsf {ECC}$ completed its computation on Twitter ($k = 2^8$), ${\mathsf {kBFSEcc}}$ can only achieve an accuracy of $93.10\%$. In conclusion, in comparison with $\mathsf {ECC}$, $\mathsf {kBFSEcc}$ provides a faster yet less reliable estimation on the eccentricity distribution.

Corner case on road networks

We show the applicability of our algorithm by conducting experiments on road networks whose diameters are large. Three road networks Luxembourg, Usroads, and RoadNetPA were used (downloaded from Network Repository⁷). Tables 4, 5 show the comparison of ${\mathsf {ECC}}$ and ${\mathsf {BoundEcc}}$ on road networks. ${\mathsf {ECC}}$ is slower than ${\mathsf {BoundEcc}}$ in all three graphs; ${\mathsf {ECC}}$ cannot finish the computation within one day on RoadNetPA. Since our algorithm is highly dependent on the features of small-world networks, it is not applicable to graphs with large diameters.

Fig. 29

Testing ${\mathsf {ECC}}$ (varying # reference nodes)

Fig. 30

Testing ${\mathsf {ECC}}\text {-}{\mathsf {LS}}$ (processing time for ${\mathsf {Eccentricity}}$)

Fig. 31

Testing ${\mathsf {ECC}}\text {-}{\mathsf {LS}}$ (# $\mathsf {PLL}$ queries)

Detailed explanation of $\mathsf {PLL}$

Algorithm 7 illustrates the main steps of the ${\mathsf {PLL}}$ approach. In iteration i, node $v_i$ performs pruned BFS (Lines 1–4). When $v_i$ visits a node u (Line 5), we first check whether the current labels S can answer the distance between $v_i$ and u. If so, u is pruned and we stop the traversal from u (Lines 6–7). Otherwise, $(v_i,dist(v_i,u))$ is inserted into S(u) (Line 8). Moreover, we continue BFS from u by adding the neighbors of u into the queue Q (Lines 9–12). Finally, the total set $S = \{S(v)|v \in V\}$ is returned as the answer ${\mathsf {PLL}}$ (Line 13).

Fig. 32

Testing distribution of distance to reference nodes

Fig. 33

Testing average distance to reference nodes

Fig. 34

Testing the average number of nodes in each step of ${\mathsf {ECC}}\text {-}{\mathsf {DY}}$ for edge insertion

Additional experimental results

This section shows the results on the graphs apart from Slashdot, Twitter, DBLP, and Wiki-talk in the following four experiments.

• Exp-2: Testing ${\mathsf {ECC}}$. Figure 29 shows the performance of ${\mathsf {ECC}}$ under a varying number k of reference nodes.

• Exp-3: Testing ${\mathsf {ECC}}\text {-}{\mathsf {LS}}$. Figure 30 shows the processing time when varying the number k of reference nodes. Figure 31 shows the number of $\mathsf {PLL}$ queries for ${\mathsf {ECC}}$ and ${\mathsf {ECC}}\text {-}{\mathsf {LS}}$.

• Exp-5: Testing distance to reference nodes Figure 32 shows the distribution of the distance $\lambda _0(u)$ from a node u to its reference node z. Figure 33 illustrates the average distance of a node to its nearest reference node.

• Exp-8: Rule-effectiveness for edge insertion. Figure 34 shows the number of nodes pruned by each of Lemmas 19–21 and the number of nodes in $C^{b'}$.