Recursive Shortest Path Algorithm with Application to Density-integration of Weighted Graphs

Graph theory is increasingly commonly utilised in genetics, proteomics and neuroimaging. In such fields, the data of interest generally constitute weighted graphs. Analysis of such weighted graphs often require the integration of topological metrics with respect to the density of the graph. Here, density refers to the proportion of the number of edges present in that graph. When topological metrics based on shortest paths are of interest, such densityintegration usually necessitates the iterative application of Dijkstra’s algorithm in order to compute the shortest path matrix at each density level. In this short note, we describe a recursive shortest path algorithm based on single edge updating, which replaces the need for the iterative use of Dijkstra’s algorithm. Our proposed procedure is based on pairs of breadthfirst searches around each of the vertices incident to the edge added at each recursion. An algorithmic analysis of the proposed technique is provided. When the graph of interest is coded as an adjacency list, our algorithm can be shown to be more efficient than an iterative use of Dijkstra’s algorithm. Introduction The last ten years has seen a surge of interest in graph theory among biologists, physicists and other natural scientists. This was primarily stimulated by the seminal papers of Watts and Strogatz (1998) and Barabasi and Albert (1999). In particular, a wide range of different data types are now analyzed through systematic calculations of various topological measures, such as the characteristic path length or clustering coefficient. In systems biology and neuroscience, subject-specific networks can be constructed in order to compare several populations of networks for testing putative differences between groups of subjects (see Bullmore and Sporns, 2009, for a review). (For convenience, the terms network and graph will here be used interchangeably, as this reflects some of the recent developments in the literature.) Such biological networks, however, tend to be weighted undirected graphs, which generally correspond to some standardized covariance matrices between a set of regions of interest. By contrast, most of the topological measures introduced by Watts and Strogatz (1998) and Barabasi and Albert (1999) pertain to unweighted networks. There is currently no general consensus on how to compute or compare the topology of weighted graphs. This is a particularly arduous problem, since it requires the use of real-valued mathematical tools on objects, which are essentially discrete. One of the possible solutions to this conundrum has been advanced by He et al. (2009), who suggested integrating the topological measures of interest with respect to the density of the network (see also Achard and Bullmore, 2007, Ginestet and Simmons, 2011). The density of a network is here defined as the proportion of the number of edges in a given graph. Such integration, however, is computationally expensive, and its complexity grows quadratically with the number of nodes. A Monte Carlo scheme has been proposed in the literature to address this issue and approximate the value of such an integral (Ginestet et al., Submitted). Such Monte Carlo methods, however, also necessitates large number of simulations in order to reduce the variability of the resulting estimates. Most of the topological metrics of interest to researchers in neuroscience and systems biology tend to involve the computation of the matrix of shortest paths, denoted D. This includes, for instance, the global and local efficiency measures proposed by Latora and Marchiori (2001) (see also Latora and Marchiori, 2003). The computation of D for a given network can be done Cedric E. Ginestet & Andrew Simmons 2 Recursive Shortest Path Algorithm efficiently using the celebrated Dijstra’s algorithm (Dijkstra, 1959). However, when considering weighted networks, Dijstra’s algorithm may need to be invoked as many times as the number of edges in the graph of interest. In this short note, we address this specific problem by proposing a recursive shortest path algorithm based on applying single edge updates to D. In this setup, we only work with the shortest path matrix and compute the value of the desired topological metric at every density level. Taken together, we therefore provide an efficient algorithm for the density-integration of the topological functions of weighted networks. Density-integration of Topological Metrics In this paper, our main focus will be on undirected weighted graphs, containing no graph loops or multiple edges. However, since we also need to refer to unweighted graphs, we introduce the following notation. A graph G is here defined as a triple (V, E ,W), where V(G) is the standard vertex set, E(G) is the edge set and W(G) is a multiset of real-valued weights. Our convention generalizes to directed graphs. In addition, this also includes undirected unweighted (simple) graphs as special cases, for which the elements of W belong to {0, 1}. Such a setup may, for instance, apply to the consideration of correlation matrices or other matrices of similarity measures with real-valued entries. In addition, we will make use of the following notation, NV := |V(G)|, NE := |E(G)|, and NI := NV (NV − 1) 2 , where := signifies that the left-hand side is defined as the right-hand side. We define NI as the number of shortest paths in G. Naturally, NI here takes this value because G is undirected. For a directed network, NI would be NV (NV − 1). For convenience, we will interchangeably use the following two sets of indices to label the elements of W, W(G) = {wv1v2 , . . . , wij , . . . , wNV −1,NV } = {w1, . . . , we, . . . , wNE}. (1) Albeit we will here restrict our attention to undirected graphs, an extension of our proposed technique to directed networks will be discussed in the conclusion. A range of topological metrics necessitating the computation of the shortest path matrix have been proposed in the literature. Two popular choices of topological measures are the global and local efficiency measures introduced by Latora and Marchiori (2001). Both of these quantities can be derived from the general definition of the efficiency, E(·), of a simple graph G = (V, E ,W), which is defined as follows, E(G) := 1 NI NV ∑