highercore

Highercore is a generalization of the core decomposition concept in graph theory, used in network science to identify robust substructures in multilayer or time-evolving networks. It extends the classic k-core by requiring connectivity thresholds to hold simultaneously across multiple layers or time steps. In a single-layer graph, the highercore reduces to the standard k-core when the threshold vector has a single component.

Consider a multilayer network with vertex set V and layers L1, ..., Lm. Each layer i defines a degree function di(v). For a threshold vector t = (t1, ..., tm) with ti integers, a highercore is a maximal subgraph H = (VH, EH) such that for every v in VH, di(v) >= ti for all i, after iteratively removing vertices that fail to meet any threshold.

Variants may replace degree with other measures, such as strength in weighted networks, or use adaptive thresholds. Highercore yields a hierarchy of subgraphs as thresholds vary, providing a multiscale view of network core structure.

Computationally, highercores are typically computed by iterative pruning across layers: initialize with all vertices, remove any vertex that fails any layer’s threshold, update degrees, and repeat until stable. The resulting cores form a nested sequence as thresholds increase, enabling analysis of core-periphery structure across the multilayer network.

Applications include identifying resilient cores in social, transportation, and biological networks, analyzing multilayer collaborations, and studying the evolution of networks over time. The concept draws on the k-core idea from graph theory and has been explored in multilayer network research as a generalization of core decomposition. See also k-core, core decomposition, multilayer networks.

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