typo ''fist'' to ''first''

preliminaries/measuring-networks-random-graphs.md

@@ -104,7 +104,7 @@ $$
 where $$\bar k$$ is the average degree. From this, we can see that the clustering coefficient of $$G_{np}$$ is very small. If we generate bigger and bigger graphs with fixed average degree $$\bar k$$, then $$C$$ decreases with graph size $$n$$. $$\mathbb{E}[C_i] \to 0$$ as $$n \to \infty$$.
 
 ### The Path Length of $$G_{np}$$
-To discuss the path length of $$G_{np}$$, we fist introduce the concept of **expansion.** Graph $$G(V, E)$$ has expansion $$\alpha$$ if $$\forall S \subset V$$, the number of edges leaving $$S \geq \alpha \cdot \min (|S|, | V \setminus S|)$$. Expansion answers the question ''if we pick a random set of nodes, how many edges are going to leave the set?'' Expansion is a measure of robustness: to disconnect $$\ell$$ nodes, one must cut $$\geq \alpha \cdot \ell$$ edges.
+To discuss the path length of $$G_{np}$$, we first introduce the concept of **expansion.** Graph $$G(V, E)$$ has expansion $$\alpha$$ if $$\forall S \subset V$$, the number of edges leaving $$S \geq \alpha \cdot \min (|S|, | V \setminus S|)$$. Expansion answers the question ''if we pick a random set of nodes, how many edges are going to leave the set?'' Expansion is a measure of robustness: to disconnect $$\ell$$ nodes, one must cut $$\geq \alpha \cdot \ell$$ edges.
 
 Equivalently, we can say a graph $$G(V,E)$$ has an expansion $$\alpha$$ such that
 
@@ -230,7 +230,7 @@ Up to now, we have only considered $$K_1$$ initiator matrices with binary values
 
 To negate this effect, stochasticity is introduced by relaxing the assumption that the entries in the initiator matrix can only take binary values. Instead entries in $$\Theta_1$$ can take values on the interval $$[0,1]$$, and each represents the probability of that particular edge appearing. Then the matrix (and all the generated larger matrix products) represent the probability distribution over all possible graphs from that matrix.
 
-More concretely, for probaility matrix $$\Theta_1$$, we compute the $$k^{th}$$ Kronecker power $$\Theta_k$$ as the large stochastic adjacency matrix. Each entry $$p_{uv}$$ in $$\Theta_k$$ then represents the probability of edge $$(u,v)$$ appearing. 
+More concretely, for probability matrix $$\Theta_1$$, we compute the $$k^{th}$$ Kronecker power $$\Theta_k$$ as the large stochastic adjacency matrix. Each entry $$p_{uv}$$ in $$\Theta_k$$ then represents the probability of edge $$(u,v)$$ appearing. 
 
 {% include marginnote.html id='note-bipartite-folded' note='Note that the probabilities do not have to sum up to 1 as each the probability of each edge appearing is independent from other edges.' %}
 
@@ -278,4 +278,4 @@ In practice, the stochastic Kronecker graph model is able to generate graphs tha
 
 <br/>
 
-|[Index](../) | [Previous](./introduction-graph-structure) | [Next](../)|
+|[Index](../) | [Previous](./introduction-graph-structure) | [Next](./motifs-and-structral-roles_lecture)|

preliminaries/motifs-and-structral-roles_lecture.md

@@ -6,7 +6,7 @@ title: Motifs and Structral Rules in Network
 
 ## Subgraphs
 
-In this section, we will begin by introducing the defination of subgraphs. Subnetworks, or subgraphs, are the building blocks of networks which enable us to characterize and discriminate networks.
+In this section, we will begin by introducing the definition of subgraphs. Subnetworks, or subgraphs, are the building blocks of networks which enable us to characterize and discriminate networks.
 
 For example, in Figure 1 we show all the non-isomorphic directed subgraphs of size 3. These subgraphs differ from each other in the number of edges or direction of edges.