[Paper Review] Return probability and k-step measures

Nicholas Dronen, Qin Lv|arXiv (Cornell University)|May 23, 2011

Complex Network Analysis Techniques2 references3 citations

TL;DR

This paper introduces the Pólya power index (PPI), a computationally efficient k-step measure based on return probability in random walks on undirected graphs. It computes node centrality by tracking the probability a random walk returns to its origin within k steps, achieving O(n + m) time complexity—significantly faster than beta centrality or subgraph centrality—while maintaining equivalent discriminatory power for identifying influential nodes in networks.

ABSTRACT

The notion of return probability -- explored most famously by George Pólya on d-dimensional lattices -- has potential as a measure for the analysis of networks. We present an efficient method for finding return probability distributions for connected undirected graphs. We argue that return probability has the same discriminatory power as existing k-step measures -- in particular, beta centrality (with negative beta), the graph-theoretical power index (GPI), and subgraph centrality. We compare the running time of our algorithm to beta centrality and subgraph centrality and find that it is significantly faster. When return probability is used to measure the same phenomena as beta centrality, it runs in linear time -- O(n+m), where n and m are the number of nodes and edges, respectively -- which takes much less time than either the matrix inversion or the sequence of matrix multiplications required for calculating the exact or approximate forms of beta centrality, respectively. We call this form of return probability the Pólya power index (PPI). Computing subgraph centrality requires an expensive eigendecomposition of the adjacency matrix; return probability runs in half the time of the eigendecomposition on a 2000-node network. These performance improvements are important because computationally efficient measures are necessary in order to analyze large networks.

Motivation & Objective

• To develop a computationally efficient alternative to existing k-step centrality measures like beta centrality and subgraph centrality.
• To demonstrate that return probability, as a probabilistic measure, offers equivalent discriminatory power for identifying influential nodes in networks.
• To show that return probability can be computed in linear time, making it suitable for large-scale network analysis.
• To establish the Pólya power index (PPI) as a viable, efficient alternative to existing centrality measures with minimal normalization and intuitive interpretation.

Proposed method

• The method uses a stochastic transition matrix P derived from the adjacency matrix A to model random walks on undirected, connected graphs.
• It computes the k-step return probability for each node i as the (i,i) entry of P^k, tracking the probability of returning to i after k steps.
• For k=1, return probability is zero; for k=2, it is the sum over neighbors j of 1/deg(j), reflecting the chance a neighbor randomly selects i as a partner.
• The algorithm avoids expensive matrix operations by using iterative matrix-vector multiplication, achieving O(n + m) time for the PPI (k=1,2) and O(k × nnz_k) for higher k.
• The Pólya power index (PPI) is defined as the return probability at k=2, which correlates strongly with exclusionary power measures like beta centrality (negative β) and GPI.
• The method enables exact control over walk length k and ensures values are always in [0,1], eliminating the need for normalization.

Experimental results

Research questions

• RQ1 Can return probability serve as a computationally efficient alternative to established k-step centrality measures like beta centrality and subgraph centrality?
• RQ2 Does return probability exhibit the same discriminatory power as beta centrality, GPI, and subgraph centrality in identifying influential nodes?
• RQ3 Can return probability be computed in linear time, and how does its performance scale with network size?
• RQ4 How does the Pólya power index (PPI) compare to other measures in capturing structural exclusionary power in networks?

Key findings

• The Pólya power index (PPI), based on 2-step return probability, runs in O(n + m) time, significantly faster than the matrix inversion or repeated multiplication required by exact or approximate beta centrality.
• For a 2000-node network, computing return probability takes half the time of eigendecomposition required for subgraph centrality.
• Return probability achieves the same discriminatory power as beta centrality (negative β), GPI, and subgraph centrality, despite being a probabilistic measure with no normalization.
• The algorithm’s time complexity scales linearly with network size for k=1 and k=2, and as O(k × nnz_k) for higher k, making it practical for large-scale networks.
• Return probability provides exact control over walk length k and naturally bounds values in [0,1], enabling direct comparison across different networks without normalization.