Minimum spanning trees of weighted scale-free networks

– A complete characterization of real networks requires us to understand the consequences of the uneven interaction strengths between a system’s components. Here we use minimum spanning trees (MSTs) to explore the effect of correlations between link weights and network topology on scale-free networks. Solely by changing the nature of the correlations between weights and network topology, the structure of the MSTs can change from scale-free to exponential. Additionally, for some choices of weight correlations, the efficiency of the MSTs increases with increasing network size, a result with potential implications for the design and scalability of communication networks. Introduction. – The study of many complex systems has benefited from representing them as networks [1]. There is now extensive empirical evidence indicating that the degree (or connectivity) distribution of the nodes in many networks follows a power law, strongly influencing properties from network robustness [2] to disease spreading [3]. However, to fully characterize these systems, we need to acknowledge the fact that the links can differ in their strength and importance [4–7]. This is demonstrated, e.g., in social networks where the relationship between two long-time friends presumably differs from that between two casual business associates [8], and in ecological systems where the strength of a particular species pair-interaction is crucial for the population dynamics [9, 10]. The fact that links in complex networks are weighted rather than binary (present or absent) must be taken into account when considering dynamical network processes like multicast or broadcast, which have important applications in modern computer networks [11]. For instance, when trying to broadcast a message like a new routing table, one must factor in that different paths and links are characterized by varying degrees of time delay, bandwith, and transmission costs. Consequently, one would attempt to reach all nodes preferably by using as few connections as possible, while at the same time keeping the total weight (e.g., delay) of the traversed links minimal. Similarly, the effective spreading of a computer or biological virus to all nodes in a network would also opt for the low-weight paths. These broadcasting problems all boil down to finding and characterizing the minimum spanning tree (MST) of a (∗) E-mail: [email protected] c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10232-x P.J. Macdonald et al.: MST of weighted SF networks 309 network [12, 13], where the MST is defined as the connected, loopless subgraph consisting of (N − 1) links reaching all N nodes while minimizing the sum of the link weights. Additionally, MSTs have received attention as examples of optimal path spanning trees [14, 15], and recently spanning trees have been invoked to explain properties of traceroute measurements of the Internet [16]. At a more fundamental level, MSTs are closely connected to invasion bond percolation [17], which has been studied extensively on regular lattices with both random and correlated link weights [18], turning IBP into a key model of non-equilibrium statistical mechanics [19]. In this paper, we present the first results on how the structure and efficiency of MSTs change as a function of the correlations between the link weights and the network topology. We start by examining the correlations between the weights and the network structure in several real systems. We use the resulting insights to generate weighted scale-free networks, the MSTs of which are either scale-free or exponential depending on the nature of the linkweight correlations. In contrast, when correlations between weights and topology are removed, we find that the MST degree distribution follows a power law with a degree exponent close to that of the original network, independent of the weight distribution. Finally, we find that the exponential MSTs are increasingly more efficient with increasing network size N , while the efficiency of the scale-free MSTs quickly saturates at a finite value. Topology correlated weights. – To uncover the relationship between network topology and link weights, in fig. 1 we display the dependence of the weights on the node degrees for the E. coli metabolic network, where the link weights represent the optimal metabolic fluxes [20]; the US Airport Network (USAN) where the weights reflect the total number of passengers travelling between two airports between 1992 and 2002; and the link betweenness-centrality (BC), representing the number of shortest paths along a link for the Barabási-Albert (BA) scale-free 10 0 10 1 10 2 10 3 10 4 10 5 10 6 kikj 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 < w ij > / < w m ax > USAN E.coli BC (kikj) 0.5