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A self-loop is an edge whose two endpoints are the same vertex. In undirected total-degree counting it contributes two to the degree of a vertex, since both endpoints are incident to that vertex. Simple graphs exclude self-loops; the linearized chord diagram model retains them and starts from one self-loop.

There are genuine normalization inconsistencies in this PDF. In the standard linearized chord diagram model $G_n^{(1)}$, time $n$ gives $n$ edges and total vertex degree $2n$, with self-loops counted twice. The paper calls $2n$ the number of edges. Its right-endpoint hint also uses $\sqrt{i/(2n)}$ rather than $\sqrt{i/n}$. These cannot both describe the standard model; part (c)'s exponent is consequently inconsistent as well. We give the definition, prove the alternative construction and its limits, and explicitly identify the corrected conclusion rather than infer an invalid constant from the printed hint.

Start from the empty graph at time zero, or from one vertex with one self-loop at time one. To obtain $G_t$ from $G_{t-1}$, add vertex $t$ and an edge directed from $t$ to $I_t$, whereA self-choice creates a self-loop. The old vertex degrees sum to $2(t-1)$, so these probabilities sum to one. This is exact preferential attachment, including the outward half of the new edge as weight one for the new vertex; it is not a rule normalized by $2t$ or a process adding two edges at each step. The degree of a vertex here is its total incoming plus outgoing vertex degree. The construction in part (a) has exactly one chord per vertex and makes the edge-count discrepancy explicit.

Pair $2n$ linearly ordered endpoints uniformly. Each pair is a chord; its larger endpoint ends a vertex block, with all endpoints after the previous right endpoint merged into that vertex. Orient the chord from its right-endpoint block to its left-endpoint block. Removing the last endpoint and its mate leaves a uniform smaller pairing. Reinserting the mate into one of $2n-1$ equally likely gaps chooses an existing block proportionally to its endpoint count, or the final gap creates a self-loop. This proves equivalence with the linearized chord diagram model.

An equivalent continuous representation draws $n$ independent pairs of independent uniform points on $[0,1]$, orders the pairs by their maxima $R_1<\cdots <R_n$, and calls their minima $L_i$. The $R_i^2$ are uniform order statistics; conditional on all the $R_i$, the $L_i$ are independent uniform points on $[0,R_i]$.