Counting and detecting small subgraphs via equations and matrix multiplication

SAHAD: Subgraph Analysis in Massive Networks Using Hadoop

“…Given an arbitrary template of size k and a graph with n nodes, the best known rigorous result for the subgraph isomorphism problem is obtained by Eisenbrand et al [10] with a running time of roughly O(n ωk/3 ) (which improves on the naive O(n k ) time), where ω denotes the exponent of the best possible matrix multiplication algorithm. If the template has an independent set of size s, Vassilevska et al [28] give an algorithm with an improved running time of O(2 s n k−s+3 k O (1) ); this is improved slightly by Kowaluk et al [16]. When the template is a tree or has a bounded treewidth, Alon et al [1] develop the color coding technique which is a randomized approximation algorithm with running time O(k|E|2 k e k log (1/δ) Here is an application of detecting subgraph isomorphism in financial networks (as in [7], [5]): G is an underlying graph whose nodes represent customers (c), suspicious (s) or banks (b).…”

Quad Census Computation: Simple, Efficient, and Orbit-Aware

“…While there is extensive work on determining the subgraph census for varying subgraph sizes [9,10,12] and also for directed graphs [4], the focus is almost always on the global distribution, i.e., say, the number of triangles a graph contains but not how often a given node is part of a triangle. However, for many properties describing nodes and edges, respectively, it is necessary to know the subgraph census on a node or edge level basis.…”

“…The general approach to determine a subgraph census on the global level is to solve a system of equations that relates the non-induced subgraph frequency of each non-isomorphic k-node subgraph with the number of occurrences in other k-node subgraphs [4,5,9,10,12]. It is known that the time needed to solve the system of equations for the 4-node subgraph census, which we refer to as the quad census, on a global level is O(a(G)m + i(G)) [12], where i(G) is the time needed to calculate the frequency of a given 4-node induced subgraph in G, and a(G) is the arboricity, i.e., the minimum number of spanning forest needed to cover E. Following the idea of relating non-induced and induced subgraph counts, Marcus and Shavitt [13] present a system of equations for the orbitaware connected quad census on a node level that runs in O(Δ(G)m + m 2 ) time with Δ(G) denoting the maximum degree of G. Because of the larger number of algorithms invoked by Marcus and Shavitt's approach, Hočevar and Demšar [6] present a different system of equations, again restricted to connected quads, that requires fewer counting algorithms and runs in O(Δ(G) 2 m) time, but does not determine the non-induced counts.…”

The complexity of the overlay network verification and its related problems

“…Many variations of these two problems have been investigated. For example in [5] Kowaluk, Lingas, and Lundell studied counting and detecting subgraph isomorphism and induced subgraph isomorphism of a fixed size graph H with an independent set of size s on a host graph G with arbitrary sizes. They used techniques involving equations and rectangular matrix multiplications and showed that, in the induced subgraph isomorphism problem, the size of graph H and the size of an independent set cannot make a problem easier than the original problem and, in the case of the subgraph isomorphism problem, they gave a better upper bound for a lager size of an independent set.…”