Consistent Strong Triadic Closure in Multilayer Networks

TL;DR

This work adapts the definitions of the STC and STC+ for multilayer networks and provides ILP formulations to solve the problems exactly and provides an efficient 2-approximation for the STC and a 6-approximation for the STC+ minimization variants.

Abstract

Social network users are commonly connected to hundreds or even thousands of other users. However, these ties are not all of equal strength; for example, we often are connected to good friends or family members as well as acquaintances. Inferring the tie strengths is an essential task in social network analysis. Common approaches classify the ties into strong and weak edges based on the network topology using the strong triadic closure (STC). The STC states that if for three nodes, $\textit{A}$, $\textit{B}$, and $\textit{C}$, there are strong ties between $\textit{A}$ and $\textit{B}$, as well as $\textit{A}$ and $\textit{C}$, there has to be a (weak or strong) tie between $\textit{B}$ and $\textit{C}$. Moreover, a variant of the STC called STC+ allows adding new weak edges to obtain improved solutions. Recently, the focus of social network analysis has been shifting from single-layer to multilayer networks due to their ability to represent complex systems with multiple types of interactions or relationships in multiple social network platforms like Facebook, LinkedIn, or X (formerly Twitter). However, straightforwardly applying the STC separately to each layer of multilayer networks usually leads to inconsistent labelings between layers. Avoiding such inconsistencies is essential as they contradict the idea that tie strengths represent underlying, consistent truths about the relationships between users. Therefore, we adapt the definitions of the STC and STC+ for multilayer networks and provide ILP formulations to solve the problems exactly. Solving the ILPs is computationally costly; hence, we additionally provide an efficient 2-approximation for the STC and a 6-approximation for the STC+ minimization variants. The experiments show that, unlike standard approaches, our new highly efficient algorithms lead to consistent strong/weak labelings of the multilayer network edges.

Figures (5)

• Figure 1: A toy multilayer network with four nodes in two layers. The strong edges are colored blue and weak edges red. In (a) the labeling is inconsistent in the two layers, e.g., the edges between $A$ and $B$ have different labels in the two layers. (b) shows a consistent labeling, i.e., there are no two nodes that are connected in both layers but with differently labeled edges.
• Figure 2: A multilayer graph with three layers. The blue edges are strong and the red are weak. (a) shows a non-optimal solution for MaxMultiLayerSTC by solving MaxSTC optimally in each layer. (b) shows an optimal solution for MaxMultiLayerSTC. The value of the solution in (a) is seven, the optimal value in (b) is eight.
• Figure 3: (a) shows a multilayer graph with two layers. In each layer, the wedge graph is shown with its nodes as squares and edges colored green. (b) shows the combined node-weighted wedge graph as computed in \ref{['alg:klayerstc']} and the aggregated multilayer graph in the background.
• Figure 4: A multilayer graph with two layers and an optimal solution for MinMultiLayerSTC. The blue (red) edges are strong (weak, resp.). Adding the edge $\{C,B\}$ in layer one reduces the number of weak edges while adding the same edge in layer two does not improve the labeling.
• Figure 5: The empirical approximation ratios (*using lower bound of optimal solution after 12h time limit).

Theorems & Definitions (12)

• Lemma 1: sintos2014using
• Definition 1
• Lemma 2
• proof
• Theorem 1
• proof
• Lemma 3
• proof
• Lemma 4
• proof