Graph neural networks (GNNs) have become increasingly popular in modeling graph-structured data due to their ability to learn node representations by aggregating local structure information. However, it is widely acknowledged that the test graph structure may differ from the training graph structure, resulting in a structure shift. In this paper, we experimentally find that the performance of GNNs drops significantly when the structure shift happens, suggesting that the learned models may be biased towards specific structure patterns. To address this challenge, we propose the Cluster Information Transfer (\textbf{CIT}) mechanism, which can learn invariant representations for GNNs, thereby improving their generalization ability to various and unknown test graphs with structure shift. The CIT mechanism achieves this by combining different cluster information with the nodes while preserving their cluster-independent information. By generating nodes across different clusters, the mechanism significantly enhances the diversity of the nodes and helps GNNs learn the invariant representations. We provide a theoretical analysis of the CIT mechanism, showing that the impact of changing clusters during structure shift can be mitigated after transfer. Additionally, the proposed mechanism is a plug-in that can be easily used to improve existing GNNs. We comprehensively evaluate our proposed method on three typical structure shift scenarios, demonstrating its effectiveness in enhancing GNNs' performance.

We consider compact routing schemes in networks of low doubling dimension, where the doubling dimension is the least value α such that any ball in the network can be covered by at most 2 α balls of half radius. There are two variants of routing-scheme design: (i) labeled (name-dependent) routing, in which the designer is allowed to rename the nodes so that the names (labels) can contain additional routing information, for example, topological information; and (ii) name-independent routing, which works on top of the arbitrary original node names in the network, that is, the node names are independent of the routing scheme. In this article, given any constant ϵ ∈ (0, 1) and an n -node edge-weighted network of doubling dimension α ∈ O (loglog n ), we present —a (1 + ϵ)-stretch labeled compact routing scheme with ⌈log n ⌉-bit routing labels, O (log 2 n /loglog n )-bit packet headers, and ((1/ϵ) O (α) log 3 n )-bit routing information at each node; —a (9 + ϵ)-stretch name-independent compact routing scheme with O (log 2 n /loglog n )-bit packet headers, and ((1/ϵ) O (α) log 3 n )-bit routing information at each node. In addition, we prove a lower bound: any name-independent routing scheme with o ( n (ϵ/60) 2 ) bits of storage at each node has stretch no less than 9 − ϵ for any ϵ ∈ (0, 8). Therefore, our name-independent routing scheme achieves asymptotically optimal stretch with polylogarithmic storage at each node and packet headers. Note that both schemes are scale-free in the sense that their space requirements do not depend on the normalized diameter Δ of the network. We also present a simpler nonscale-free (9 + ϵ)-stretch name-independent compact routing scheme with improved space requirements if Δ is polynomial in n .

We study the compact routing problem in networks whose shortest path metrics are decomposable. Decomposable metrics are more general than doubling metrics, growth-bounded metrics, and metrics induced by graphs excluding Kr,r as a minor. In this work, we present both name-dependent and name-independent constant stretch compact routing schemes for bounded decomposable metrics with polylogarithmic storage requirements at each node and polylogarithmic packet headers. Our work is the first to design compact routing schemes with constant stretch for networks as general as decomposable metrics.

No abstract available.

We consider the problem of compact routing with slack in networks of low doubling dimension. Namely, we seek name-independent routing schemes with (1+ε) stretch and polylogarithmic storage at each node: since existing lower bound precludes such a scheme, we relax our guarantees to allow for (i) a small fraction of nodes to have large storage, say size of O(n log n) bits, or (ii) a small fraction of source-destination pairs to have larger, but still constant, stretch.

We consider the problem of name-independent routing in doubling metrics. A doubling metric is a metric space whose doubling dimension is a constant, where the doubling dimension of a metric space is the least value α such that any ball of radius r can be covered by at most 2 α balls of radius r/2. Given any δ> 0 and a weighted undirected network G whose shortest path metric d is a doubling metric with doubling dimension α, we present a name-independent routing scheme for G with (9+δ)-stretch, (2+ 1 δ)O(α) (log ∆) 2 (log n)bit routing information at each node, and packet headers of size O(log n), where ∆ is the ratio of the largest to the smallest shortest path distance in G. In addition, we prove that for any ǫ ∈ (0, 8), there is a doubling metric network G with n nodes, doubling dimension α ≤ 6 − log ǫ, and ∆ = O(2 1/ǫ n) such that any name-independent routing scheme on G with routing information at each node of size o(n (ǫ/60)2)-bits has stretch larger than 9 − ǫ. Therefore assuming that ∆ is bounded by a polynomial on n, our algorithm basically achieves optimal stretch for name-independent routing in doubling metrics with packet header size and routing information at each node both bounded by a polylogarithmic function of n.