Strong and Weak Random Walks on Signed Networks

TL;DR

This work addresses how to design random walks on signed networks to uncover multi‑community structure under weak balance, going beyond traditional two‑community (strong) models. It introduces a strong walk tied to the signed Laplacian and a novel weak walk that permits at most one negative edge per path, incorporating restart/teleportation to yield finite, interpretable kernels. Through synthetic SSBMs and real networks (e.g., Highland Tribes, Sampson’s Monastery), the weak walk consistently outperforms the strong walk when the network contains more than two communities or exhibits density asymmetries, and it remains competitive with the three‑parameter SRWR while being parameter‑free. These findings suggest that replacing strong‑balance‑biased walks with weak walks can improve a wide range of signed‑network algorithms, including unsupervised and semi‑supervised clustering, with broad practical impact for community detection and network embedding on signed graphs.

Abstract

Random walks play an important role in probing the structure of complex networks. On traditional networks, they can be used to extract community structure, understand node centrality, perform link prediction, or capture the similarity between nodes. On signed networks, where the edge weights can be either positive or negative, it is non-trivial to design a random walk which can be used to extract information about the signed structure of the network, in particular the ability to partition the graph into communities with positive edges inside and negative edges in between. Prior works on signed network random walks focus on the case where there are only two such communities (strong balance), which is rarely the case in empirical networks. In this paper, we propose a signed network random walk which can capture the structure of a network with more than two such communities (weak balance). The walk results in a similarity matrix which can be used to cluster the nodes into antagonistic communities. We compare the characteristics of the so-called strong and weak random walks, in terms of walk length and stationarity. We show through a series of experiments on synthetic and empirical networks that the similarity matrix based on weak walks can be used for both unsupervised and semi-supervised clustering, outperforming the same similarity matrix based on strong walks when the graph has more than two communities, or exhibits asymmetry in the density of links. These results suggest that other random-walk based algorithms for signed networks could be improved simply by running them with weak walks instead of strong walks.

Figures (9)

• Figure 1: When estimating the similarity between two nodes, here colored in black, strong walks may lead to conflictual signals when the network is composed of more than 2 clusters, here illustrated via a positive walk in blue and a negative walk in red (solid: positive; dashed: negative). Weak walks only consider walks with at most one negative edge and thus properly consider the two nodes as dissimilar.
• Figure 2: Distribution of the expected walk lengths starting from different nodes, both from the theoretical results in blue and Monte-Carlo simulations in orange on $3$ community symmetric SSBM with $100$ nodes in each community, $p=0.05$ and different levels of teleportation (left: $p_w=1$; middle: $p_w=0.8$; right: $p_w=0.5$, where the strong walk with $p_s=0.8, 0.5$ has expected walk length $5, 2$, respectively).
• Figure 3: (left) Adjacency matrix for the Highland Tribes network. (centre) Strong walk kernel resulting from a 500 step walk on the Highland Tribes network with teleportation parameter $p_{s} = 0.7$. (right) Weak walk kernel resulting from a 500 step walk with no teleportation $p_{w} = 1.0$. Positive and negative values have been set to +1 or -1 respectively for clearer visualization.
• Figure 4: (left) Adjacency matrix for the Sampson's Monastary network. (centre) Strong walk kernel resulting from a 500 step walk on the Highland Tribes network with teleportation parameter $p_s = 0.8$. (right) Weak walk kernel resulting from a 500 step walk with no teleportation. Positive and negative values have been set to +1 or -1 respectively for clearer visualization.
• Figure 5: ARI from the semi-supervised clustering for the weak walk, strong walk and SRWR on 2 community SSBMs (strong balance).