## Abstract

Borrowing from concepts in expander graphs, we study the expansion properties of real-world, complex networks (e.g., social networks, unstructured peer-to-peer, or P2P networks) and the extent to which these properties can be exploited to understand and address the problem of decentralized search. We first produce samples that concisely capture the overall expansion properties of an entire network, which we collectively refer to as the *expansion signature*. Using these signatures, we find a correspondence between the magnitude of maximum expansion and the extent to which a network can be efficiently searched. We further find evidence that standard graph-theoretic measures, such as average path length, fail to fully explain the level of “searchability” or ease of information diffusion and dissemination in a network. Finally, we demonstrate that this high expansion can be leveraged to facilitate decentralized search in networks and show that an expansion-based search strategy outperforms typical search methods.

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## Notes

- 1.
There is a large body of work studying expansion in theoretic computer science and graph theory. However, much of this work focuses on (1) synthetic graphs that do not normally arise in the real-world such as \(d\)-regular graphs and (2) the

*minimum*(not maximum) expansion in these graphs [16]. - 2.
- 3.
We employ the concept of

*vertex*expansion here. The*vertex expansion*of an entire graph is typically taken to mean \(\min _{S \subset V}\frac{|N(S)|}{|S|}\) [16]. - 4.
For directed networks that are very weakly connected, a sample with high maximum expansion (based on out-degree) may exist, but the nodes in the sample itself may not be reachable from substantial portions of the network. Samples such as this may shed little light on searchability. Possible approaches to addressing these scenarios include computing expansion signatures using the

*expected*maximum expansion of*connected*samples or examining both in-degree and out-degree expansion. In the present work, however, links are treated as undirected (or bidirectional). - 5.
As mentioned, one may compute

*expansion signatures*using either the*expansion quality*or the*discovery quotient*, as both are essentially normalized representations of the expansion. For the present work, we plot the*expansion quality*of samples, which is \(\mathcal X _Q(S) = \frac{|N(S)|}{|V-S|}\). - 6.
Alternatively, the approximation guarantee for Maximum Expansion can be shown through formulation as an optimization problem with a submodular objective function and the well-known result by Nemhauser et al. [33] regarding such submodular functions. Maximum Expansion can also simply be mapped to a Maximum Coverage instance via a reduction.

- 7.
In the context of P2P, we assume each node knows the

*identity*of its neighbors’ neighbors, but not necessarily the*files*stored by its neighbors’ neighbors. - 8.
The normalized cumulative nodes discovered (\(\frac{|N(S) \cup S|}{|V|}\)) is simply the

*discovery quotient*, as defined in Sect. 3.1. - 9.
See [20] for more information on conductance and its relation to the concept of expansion.

- 10.
- 11.
For the purposes of this experiment, in order to ensure every node in \(V\) is traversed, we disallowed all node revisits. That is, for all search strategies, if all neighbors of the current node have been visited, the next step is selected uniformly at random from among the nodes in \(N(S)\).

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Maiya, A.S., Berger-Wolf, T.Y. Expansion and decentralized search in complex networks.
*Knowl Inf Syst* **38, **469–490 (2014). https://doi.org/10.1007/s10115-012-0596-4

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### Keywords

- Expansion
- Decentralized search
- P2P
- Peer-to-peer networks
- Social network analysis
- Complex networks
- Graph mining
- Expander graphs