Enumeration of spanning trees in a pseudofractal scale-free web

TL;DR

This paper derives exact formulas for counting spanning trees in a pseudofractal scale-free network, revealing fewer spanning trees than regular lattices and challenging assumptions about network reliability.

Contribution

It provides the first exact expressions for spanning trees in a specific scale-free network, highlighting structural differences affecting network robustness.

Findings

Spanning tree entropy is less than 1 in the studied network.

Number of spanning trees is significantly less than in regular lattices.

Scale-free networks can be more robust despite fewer spanning trees.

Abstract

Spanning trees are an important quantity characterizing the reliability of a network, however, explicitly determining the number of spanning trees in networks is a theoretical challenge. In this paper, we study the number of spanning trees in a small-world scale-free network and obtain the exact expressions. We find that the entropy of spanning trees in the studied network is less than 1, which is in sharp contrast to previous result for the regular lattice with the same average degree, the entropy of which is higher than 1. Thus, the number of spanning trees in the scale-free network is much less than that of the corresponding regular lattice. We present that this difference lies in disparate structure of the two networks. Since scale-free networks are more robust than regular networks under random attack, our result can lead to the counterintuitive conclusion that a network with more spanning trees may be relatively unreliable.