Abstract
Biased random walk has been studied extensively over the past decade especially in the transport and communication networks communities. The mean first passage time (MFPT) of a biased random walk is an important performance indicator in those domains. While the fundamental matrix approach gives precise solution to MFPT, the computation is expensive and the solution lacks interpretability. Other approaches based on the Mean Field Theory relate MFPT to the node degree alone. However, nodes with the same degree may have very different local weight distribution, which may result in vastly different MFPT. We derive an approximate bound to the MFPT of biased random walk with short relaxation time on complex network where the biases are controlled by arbitrarily assigned node weights. We show that the MFPT of a node in this general case is closely related to not only its node degree, but also its local weight distribution. The MFPTs obtained from computer simulations also agree with the new theoretical analysis. Our result enables fast estimation of MFPT, which is useful especially to differentiate between nodes that have very different local node weight distribution even though they share the same node degrees.
Highlights
Scale-free node degree distribution, small network diameter, large clustering coefficients – these are common properties found to be present in complex networks arising from seemingly disparate fields such as biology, computer science, cosmology, etc. [1,2]
Quantities such as stationary distribution and mean first passage time (MFPT) are important as they can be used to answer the questions about the performance of networks as mentioned above
While the focus of [16] is to show the general trend of MFPT with regard to node degree and spectral dimension, we focus on explaining the differences in MFPT for nodes with same degree when the underlying random walk falls into the non-compact exploration regime
Summary
Scale-free node degree distribution, small network diameter, large clustering coefficients – these are common properties found to be present in complex networks arising from seemingly disparate fields such as biology, computer science, cosmology, etc. [1,2]. We generalize the FPT analysis as discussed in [14] for a class of random walks with short relaxation time where the nodes have arbitrary weights. Outline we will show the detailed analysis for approximating the FPT decay rate and MFPT of random walks with short relaxation time.
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