Defensive Alliances in Signed Networks

TL;DR

Defensive Alliances in Signed Networks introduces a new combinatorial notion of defensive alliance for signed graphs and studies its algorithmic and structural properties. The authors prove NP-hardness for existence and related problems, provide polynomial-time algorithms for alliance-building on several graph classes, and develop a rich parameterized framework centered on the new snd parameter, including ILP formulations and FPT results. They also connect the DA concept to correlation clustering and demonstrate implications for real-world networks with coexistence of positive and negative ties. The work lays foundational theory and opens directions for further refinements, weighted models, and directed-sign variants with practical impact on social, economic, and geopolitical networks.

Abstract

The analysis of (social) networks and multi-agent systems is a central theme in Artificial Intelligence. Some line of research deals with finding groups of agents that could work together to achieve a certain goal. To this end, different notions of so-called clusters or communities have been introduced in the literature of graphs and networks. Among these, defensive alliance is a kind of quantitative group structure. However, all studies on the alliance so for have ignored one aspect that is central to the formation of alliances on a very intuitive level, assuming that the agents are preconditioned concerning their attitude towards other agents: they prefer to be in some group (alliance) together with the agents they like, so that they are happy to help each other towards their common aim, possibly then working against the agents outside of their group that they dislike. Signed networks were introduced in the psychology literature to model liking and disliking between agents, generalizing graphs in a natural way. Hence, we propose the novel notion of a defensive alliance in the context of signed networks. We then investigate several natural algorithmic questions related to this notion. These, and also combinatorial findings, connect our notion to that of correlation clustering, which is a well-established idea of finding groups of agents within a signed network. Also, we introduce a new structural parameter for signed graphs, signed neighborhood diversity snd, and exhibit a parameterized algorithm that finds a smallest defensive alliance in a signed graph.

Figures (4)

• Figure 1: The structure of relationships between different sub-tribes in some mountain area in Papua New Guinea; dashed lines indicate hostility and solid lines friendship. Two defensive alliances as observed in the country are colored blue and red. Other alliances, like $\{7,8\}$, have not been observed, possibly, as also $\{7\}$ or $\{8\}$ is already an alliance; in contrast, $\{3,4\}$ might be more interested in building an alliance together.
• Figure 2: Defensive alliances on unsigned versus signed graphs, where dashed or solid lines mean negative or positive edges, respectively.
• Figure 3: Reduction construction for \ref{['thm:def-alliable']}.
• Figure 4: Clause gadget: the squares represent the not-in-the-solution gadget.

Theorems & Definitions (46)

• Definition 1
• Definition 2
• Definition 3
• proof
• corollary 1
• Theorem 1
• proof
• corollary 2
• corollary 3
• corollary 4