From 5033cba59c423afbc0a4059b487f16bd9de5ce9c Mon Sep 17 00:00:00 2001 From: Liz Guo Date: Thu, 17 Oct 2019 21:49:35 -0700 Subject: [PATCH 1/3] update latex in markdown --- preliminaries/motifs-and-structral-roles_lecture.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/preliminaries/motifs-and-structral-roles_lecture.md b/preliminaries/motifs-and-structral-roles_lecture.md index 8f6672a..67caa24 100755 --- a/preliminaries/motifs-and-structral-roles_lecture.md +++ b/preliminaries/motifs-and-structral-roles_lecture.md @@ -19,20 +19,20 @@ Network motifs are recurring, significant patterns of interconnections in the ne Recurrence of motif represents it occurs with high frequency. We allow overlapping of motifs. Significance of a motif means it is more frequent than expected. The key idea here is we say subgraphs that occur in a real network much more often than in a random network have functional significance. Significance can be measured using Z-score which is defined as: \begin{equation} Z_{i} = \frac{N_{i}^{real} - \overline N_{i}^{rand}}{std(N_{i}^{rand})} \end{equation}
-where $N_{i}^{real}$ is #(subgraphs of type i) in network $G^{real}$ and $N_{i}^{rand}$ is #(subgraphs of type i) in randomized network $G^{rand}$. +where $$N_{i}^{real}$$ is #(subgraphs of type i) in network $$G^{real}$$ and $$N_{i}^{rand}$$ is #(subgraphs of type i) in randomized network $$G^{rand}$$. Network significance profile (SP) is defined as: \begin{equation} SP_{i} = \frac{Z_{i}}{\sqrt{\sum_{j} {Z_j^{2}}}} \end{equation} where SP is a vector of normalized Z-scores. ### Configuration Model -Configuration model is a random graph with a given degree sequence $k_1$, $k_2$, ..., $k_N$ which can be used as a "null" model and then compared with real network. Configuration model can be generated in an easy way as shown in Figure 2. +Configuration model is a random graph with a given degree sequence $$k_1$$, $$k_2$$, ..., $$k_N$$ which can be used as a "null" model and then compared with real network. Configuration model can be generated in an easy way as shown in Figure 2. ![Figure 2](../assets/img/Configuration_Model.png?style=centerme) Another way for generation is as following: 1) start from a given graph G; -2) select a pair of edges A->B, C->D at random, exchange the endpoints to give A->D, C->B, repeat the switching step Q* $\vert$ E $\vert$ times. +2) select a pair of edges A->B, C->D at random, exchange the endpoints to give A->D, C->B, repeat the switching step Q* $$\vert$$ E $$\vert$$ times. By this way, we will get a randomly rewired graph with same node degrees and randomly rewired edges. @@ -49,7 +49,7 @@ Finding size-k motifs/graphlets requires us: Just knowing if a certain subgraph exists in a graph is a hard computational problem. Also, computation time grows exponentially as the size of the motif/graphlet increases. ### ESU Algorithm -Exact Subgraph Enumeration (ESU) Algorithm involves two sets, while $V_{subgraph}$ contains nodes in currently constructed subgraph, and $V_{extension}$ is a set of candidate nodes to extend the motif. The basic idea of ESU is firstly starting with a node v, then adding nodes u to $V_{extension}$ set when u's node id is larger than that of v, and u may only be neighbored to some newly added node w but not of any node already in $V_{subgraph}$. +Exact Subgraph Enumeration (ESU) Algorithm involves two sets, while $$V_{subgraph}$$ contains nodes in currently constructed subgraph, and $$V_{extension}$$ is a set of candidate nodes to extend the motif. The basic idea of ESU is firstly starting with a node v, then adding nodes u to $$V_{extension}$$ set when u's node id is larger than that of v, and u may only be neighbored to some newly added node w but not of any node already in $$V_{subgraph}$$. ESU is implemented as a recursive function, Figure 3 shows the pseudocode of this algorithm: ![Figure 3](../assets/img/Exact_Subgraph_Enumeration.png?style=centerme) From d63061a8eff64272e0da03f2d738c62dac736e6e Mon Sep 17 00:00:00 2001 From: microLizzy <32970198+microLizzy@users.noreply.github.com> Date: Thu, 17 Oct 2019 22:03:03 -0700 Subject: [PATCH 2/3] Update motifs-and-structral-roles_lecture.md --- .../motifs-and-structral-roles_lecture.md | 15 +++++++++++++-- 1 file changed, 13 insertions(+), 2 deletions(-) diff --git a/preliminaries/motifs-and-structral-roles_lecture.md b/preliminaries/motifs-and-structral-roles_lecture.md index 67caa24..fd435de 100755 --- a/preliminaries/motifs-and-structral-roles_lecture.md +++ b/preliminaries/motifs-and-structral-roles_lecture.md @@ -18,10 +18,21 @@ Network motifs are recurring, significant patterns of interconnections in the ne Recurrence of motif represents it occurs with high frequency. We allow overlapping of motifs. -Significance of a motif means it is more frequent than expected. The key idea here is we say subgraphs that occur in a real network much more often than in a random network have functional significance. Significance can be measured using Z-score which is defined as: \begin{equation} Z_{i} = \frac{N_{i}^{real} - \overline N_{i}^{rand}}{std(N_{i}^{rand})} \end{equation}
+Significance of a motif means it is more frequent than expected. The key idea here is we say subgraphs that occur in a real network much more often than in a random network have functional significance. Significance can be measured using Z-score which is defined as: + +$$ +Z_{i} = \frac{N_{i}^{real} - \overline N_{i}^{rand}}{std(N_{i}^{rand})} +$$ + where $$N_{i}^{real}$$ is #(subgraphs of type i) in network $$G^{real}$$ and $$N_{i}^{rand}$$ is #(subgraphs of type i) in randomized network $$G^{rand}$$. -Network significance profile (SP) is defined as: \begin{equation} SP_{i} = \frac{Z_{i}}{\sqrt{\sum_{j} {Z_j^{2}}}} \end{equation} where SP is a vector of normalized Z-scores. +Network significance profile (SP) is defined as: + +$$ +SP_{i} = \frac{Z_{i}}{\sqrt{\sum_{j} {Z_j^{2}}}} +$$ + +where SP is a vector of normalized Z-scores. ### Configuration Model From 1bb5cebf8885483f8d193680f4e18a1d49952c72 Mon Sep 17 00:00:00 2001 From: microLizzy <32970198+microLizzy@users.noreply.github.com> Date: Thu, 17 Oct 2019 22:07:02 -0700 Subject: [PATCH 3/3] Update motifs-and-structral-roles_lecture.md --- preliminaries/motifs-and-structral-roles_lecture.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/preliminaries/motifs-and-structral-roles_lecture.md b/preliminaries/motifs-and-structral-roles_lecture.md index fd435de..df82f09 100755 --- a/preliminaries/motifs-and-structral-roles_lecture.md +++ b/preliminaries/motifs-and-structral-roles_lecture.md @@ -43,7 +43,7 @@ Configuration model is a random graph with a given degree sequence $$k_1$$, $$k_ Another way for generation is as following: 1) start from a given graph G; -2) select a pair of edges A->B, C->D at random, exchange the endpoints to give A->D, C->B, repeat the switching step Q* $$\vert$$ E $$\vert$$ times. +2) select a pair of edges A->B, C->D at random, exchange the endpoints to give A->D, C->B, repeat the switching step Q* $$\vert E\vert$$ times. By this way, we will get a randomly rewired graph with same node degrees and randomly rewired edges.