Analysing social networks is challenging. Key features of relational data require the use of non-standard statistical methods such as developing system-specific null, or reference, models that randomize one or more components of the observed data. Here we review a variety of randomization procedures that generate reference models for social network analysis. Reference models provide an expectation for hypothesis testing when analysing network data. We outline the key stages in producing an effective reference model and detail four approaches for generating reference distributions: permutation, resampling, sampling from a distribution, and generative models. We highlight when each type of approach would be appropriate and note potential pitfalls for researchers to avoid. Throughout, we illustrate our points with examples from a simulated social system. Our aim is to provide social network researchers with a deeper understanding of analytical approaches to enhance their confidence when tailoring reference models to specific research questions.

A set S of vertices is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The size of a smallest determining set for G is called its determining number, Det(G). A graph G is said to be d-distinguishable if there is a coloring of the vertices with d colors so that only the trivial automorphism preserves the color classes. The smallest d for which G is d-distinguishable is its distinguishing number, Dist(G). If Dist(G) = 2, the cost of 2-distinguishing, ρ(G), is the size of a smallest color class over all 2-distinguishing colorings of G. The Mycielskian, µ(G), of a graph G is constructed by adding a shadow master vertex w, and for each vertex vi of G adding a shadow vertex ui, with edges so that the neighborhood of ui in µ(G) is the same as the neighborhood of vi in G with the addition of w. That is, N (ui) = NG(vi) ∪ {w}. The generalized Mycielskian µ (t) (G) of a graph G is a Mycielskian graph with t layers of shadow vertices, each with edges to layers above and below, and the shadow master only adjacent to the top layer of shadow vertices. A graph is twin-free if it has no pair of vertices with the same set of neighbors. This paper examines the determining number and, when relevant, the cost of 2-distinguishing for Mycielskians and generalized Mycielskians of simple graphs with no isolated vertices. In particular, if G = K2 is twin-free with no isolated vertices, then Det(µ (t) For G with twins, we develop a framework using quotient graphs with respect to equivalence classes of twin vertices to give bounds on the determining number of Mycielskians. Moreover, we identify classes of graphs with twins for which Det(µ (t) (G)) = (t+1) Det(G).

Any finite simple graph $G = (V,E)$ can be represented by a collection $\mathscr{C}$ of subsets of $V$ such that $uv\in E$ if and only if $u$ and $v$ appear together in an odd number of sets in $\mathscr{C}$. Let $c_2(G)$ denote the minimum cardinality of such a collection. This invariant is equivalent to the minimum dimension of a faithful orthogonal representation of $G$ over $\mathbb{F}_2$ and is closely connected to the minimum rank of $G$. We show that $c_2 (G) = \operatorname{mr}(G,\mathbb{F}_2)$ when $\operatorname{mr}(G,\mathbb{F}_2)$ is odd, or when $G$ is a forest. Otherwise, $\operatorname{mr}(G,\mathbb{F}_2)\leqslant c_2 (G)\leqslant \operatorname{mr}(G,\mathbb{F}_2)+1$. Furthermore, we show that the following are equivalent for any graph $G$ with at least one edge: i. $c_2(G)=\operatorname{mr}(G,\mathbb{F}_2)+1$; ii. the adjacency matrix of $G$ is the unique matrix of rank $\operatorname{mr}(G,\mathbb{F}_2)$ which fits $G$ over $\mathbb{F}_2$; iii. there is a minimum collection $\mathscr{C}$ as described in which every vertex appears an even number of times; and iv. for every component $G'$ of $G$, $c_2(G') = \operatorname{mr}(G',\mathbb{F}_2) + 1$. We also show that, for these graphs, $\operatorname{mr}(G,\mathbb{F}_2)$ is twice the minimum number of tricliques whose symmetric difference of edge sets is $E$. Additionally, we provide a set of upper bounds on $c_2(G)$ in terms of the order, size, and vertex cover number of $G$. Finally, we show that the class of graphs with $c_2(G)\leqslant k$ is hereditary and finitely defined. For odd $k$, the sets of minimal forbidden induced subgraphs are the same as those for the property $\operatorname{mr}(G,\mathbb{F}_2)\leq k$, and we exhibit this set for $c_2(G) \leqslant 2$.

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