${\mathcal{G}}$, for $q$, includes all response items from each ${{\tau}_{q}} \in {\mathcal{T}_{q}}$, as vertices. Vertices are connected by taking into account the degree of relationship between their corresponding response items, and the degree of their relationships to $q$.

![Extraction of a fusion graph (adapted from [@Dourado:2019:FG]).[]{data-label="fig:graphFusionFromSample"}](graphFusionFromSample2){width=".8\linewidth"}

Embedding of Rank-Fusion Graphs {#sec:fusionGraphEmbedding}
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Let ${\mathbb{G}}=\{{\mathcal{G}}_i\}_{i=1}^n$ be the fusion graph set related to the response set of a given collection. Based on ${\mathbb{G}}$, an embedding function ${\mathcal{E}}$ defines a vector space model in order to project a fusion graph ${\mathcal{G}}$ into that space as a fusion vector ${\mathcal{V}}$, i.e. ${\mathcal{V}}={\mathcal{E}}({\mathcal{G}})$ for any ${\mathcal{G}}$.

We investigate how ${\mathcal{V}}$ can be adopted as a representation model of multi-ranked objects. It encodes the use of multiple rankers and allows the fusion of multiple modalities, being therefore suitable for prediction tasks. ${\mathcal{E}}$ can be defined by unsupervised or supervised approaches. We explore three approaches in this paper, preliminarly presented in [@Dourado:2019:FV] in the context of retrieval tasks. Different from that paper, here we focus its use on prediction tasks involving supervised learning.

This first one, ${\mathcal{E}}_V$, derives the vector space based on vertex analysis. Let $w(v)$ be the weight of the vertex $v$, if $v \in 