Every testable (infinite) property of bounded-degree graphs contains an infinite hyperfinite subproperty

One of the most fundamental questions in graph property testing is to characterize the combinatorial structure of properties that are testable with a constant number of queries. We work towards an answer to this question for the bounded-degree graph model introduced in [GR02], where the input graphs have maximum degree bounded by a constant d. In this model, it is known (among other results) that every hyperfinite property is constant-query testable [NS13], where, informally, a graph property is hyperfinite, if for every δ > 0 every graph in the property can be partitioned into small connected components by removing δn edges. In this paper we show that hyperfiniteness plays a role in every testable property, i.e. we show that every testable property is either finite (which trivially implies hyperfiniteness and testability) or contains an infinite hyperfinite subproperty. A simple consequence of our result is that no infinite graph property that only consists of expander graphs is constant-query testable. Based on the above findings, one could ask if every infinite testable non-hyperfinite property might contain an infinite family of expander (or near-expander) graphs. We show that this is not true. Motivated by our counterexample we develop a theorem that shows that we can partition the set of vertices of every bounded degree graph into a constant number of subsets and a separator set, such that the separator set is small and the distribution of k-discs on every subset of a partition class, is roughly the same as that of the partition class if the subset has small expansion

Every Property of Outerplanar Graphs is Testable

A D-disc around a vertex v of a graph G=(V,E) is the subgraph induced by all vertices of distance at most D from v. We show that the structure of an outerplanar graph on n vertices is determined, up to modification (insertion or deletion) of at most epsilon n edges, by a set of D-discs around the vertices, for D=D(epsilon) that is independent of the size of the graph. Such a result was already known for planar graphs (and any hyperfinite graph class), in the limited case of bounded degree graphs (that is, their maximum degree is bounded by some fixed constant, independent of |V|). We prove this result with no assumption on the degree of the graph. A pure combinatorial consequence of this result is that two outerplanar graphs that share the same local views are close to be isomorphic. We also obtain the following property testing results in the sparse graph model: * graph isomorphism is testable for outerplanar graphs by poly(log n) queries. * every graph property is testable for outerplanar graphs by poly(log n) queries. We note that we can replace outerplanar graphs by a slightly more general family of k-edge-outerplanar graphs. The only previous general testing results, as above, where known for forests (Kusumoto and Yoshida), and for some power-law graphs that are extremely close to be bounded degree hyperfinite (by Ito)

Every property is testable on a natural class of scale-free multigraphs

In this paper, we introduce a natural class of multigraphs called hierarchical-scale-free (HSF) multigraphs, and consider constant-time testability on the class. We show that a very wide subclass, specifically, that in which the power-law exponent is greater than two, of HSF is hyperfinite. Based on this result, an algorithm for a deterministic partitioning oracle can be constructed. We conclude by showing that every property is constant-time testable on the above subclass of HSF. This algorithm utilizes findings by Newman and Sohler of STOC'11. However, their algorithm is based on the bounded-degree model, while it is known that actual scale-free networks usually include hubs, which have a very large degree. HSF is based on scale-free properties and includes such hubs. This is the first universal result of constant-time testability on the general graph model, and it has the potential to be applicable on a very wide range of scale-free networks.Comment: 13 pages, one figure. Difference from ver. 1: Definitions of HSF and SF become more general. Typos were fixe

Finite graphs and amenability

Hyperfiniteness or amenability of measurable equivalence relations and group actions has been studied for almost fifty years. Recently, unexpected applications of hyperfiniteness were found in computer science in the context of testability of graph properties. In this paper we propose a unified approach to hyperfiniteness. We establish some new results and give new proofs of theorems of Schramm, Lov\'asz, Newman-Sohler and Ornstein-Weiss

Relating two property testing models for bounded degree directed graphs

We study property testing algorithms in directed graphs (digraphs) with maximum indegree and maximum outdegree upper bounded by d. For directed graphs with bounded degree, there are two different models in property testing introduced by Bender and Ron (2002). In the bidirectional model, one can access both incoming and outgoing edges while in the unidirectional model one can only access outgoing edges. In our paper we provide a new relation between the two models: we prove that if a property can be tested with constant query complexity in the bidirectional model, then it can be tested with sublinear query complexity in the unidirectional model. A corollary of this result is that in the unidirectional model (the model allowing only queries to the outgoing neighbors), every property in hyperfinite digraphs is testable with sublinear query complexity

Approximating the Spectrum of a Graph

The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$, consists of the eigenvalues of the normalized Laplacian $L= I - D^{-1/2} A D^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure of the graph, including the extent to which the graph posses community structures at multiple scales. We study the problem of approximating the spectrum $\lambda = (\lambda_1,\dots,\lambda_{|V|})$, $0 \le \lambda_1,\le \dots, \le \lambda_{|V|}\le 2$ of $G$ in the regime where the graph is too large to explicitly calculate the spectrum. We present a sublinear time algorithm that, given the ability to query a random node in the graph and select a random neighbor of a given node, computes a succinct representation of an approximation $\widetilde \lambda = (\widetilde \lambda_1,\dots,\widetilde \lambda_{|V|})$, $0 \le \widetilde \lambda_1,\le \dots, \le \widetilde \lambda_{|V|}\le 2$ such that $\|\widetilde \lambda - \lambda\|_1 \le \epsilon |V|$. Our algorithm has query complexity and running time $exp(O(1/\epsilon))$, independent of the size of the graph, $|V|$. We demonstrate the practical viability of our algorithm on 15 different real-world graphs from the Stanford Large Network Dataset Collection, including social networks, academic collaboration graphs, and road networks. For the smallest of these graphs, we are able to validate the accuracy of our algorithm by explicitly calculating the true spectrum; for the larger graphs, such a calculation is computationally prohibitive. In addition we study the implications of our algorithm to property testing in the bounded degree graph model

A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor

Motivated by the problem of testing planarity and related properties, we study the problem of designing efficient {\em partition oracles}. A {\em partition oracle} is a procedure that, given access to the incidence lists representation of a bounded-degree graph $G= (V,E)$ and a parameter \eps, when queried on a vertex $v\in V$, returns the part (subset of vertices) which $v$ belongs to in a partition of all graph vertices. The partition should be such that all parts are small, each part is connected, and if the graph has certain properties, the total number of edges between parts is at most \eps |V|. In this work we give a partition oracle for graphs with excluded minors whose query complexity is quasi-polynomial in 1/\eps, thus improving on the result of Hassidim et al. ({\em Proceedings of FOCS 2009}) who gave a partition oracle with query complexity exponential in 1/\eps. This improvement implies corresponding improvements in the complexity of testing planarity and other properties that are characterized by excluded minors as well as sublinear-time approximation algorithms that work under the promise that the graph has an excluded minor.Comment: 13 pages, 1 figur