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arXiv2026

Graphlets as structural fingerprints of complex networks

Complex networks are often compared using selected graph-theoretical measures that capture a selected set of properties with effects ranging from local to global, such as degree, clustering or betweenness centrality. Here we introduce a structural fingerprinting framework based on graphlets: small rooted subgraphs whose distributions provide a systematic description of local-to-mesoscale topology. Across synthetic networks generated from several random graph models, graphlet fingerprints capture parameter-dependent structural differences, outperform standard graph-theoretical measures, and identify even subtle local patterns driving discrimination. We then apply the framework to empirical resting-state functional connectomes, documenting that while graphlets show superior sensitivity also to controlled topological perturbations of brain connectivity, specifically in schizophrenia-control classification they perform only comparably to classical graph-theoretical features. This is in line with the notion that schizophrenia-related alterations are dominated by spatially localized connectivity changes rather than general topological reorganization. Altogether, the generative modeling, targeted perturbations and real-world neuroimaging classification challenge position graphlets as flexible structural fingerprints of complex networks, while carefully outlining their strength and weaknesses compared to more classical graph theoretical features.

cs.SI↗

arXiv2021

On Aharoni's rainbow generalization of the Caccetta-Häggkvist conjecture

For a digraph $G$ and $v \in V(G)$, let $δ^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-Häggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $δ^+(v) \ge k$ for all $v \in V(G)$, then G contains a directed cycle of length at most $\lceil n/k \rceil$. In [2], Aharoni proposes a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. In this paper, we prove this conjecture if each color class has size $Ω(k \log k)$.

math.CO↗