{"action":"created","comment":{"url":"https://api.github.com/repos/pomonadb/graphstreams/pulls/comments/54521809","id":54521809,"diff_hunk":"@@ -13,48 +15,118 @@\n \\numberwithin{thm}{subsection}\n \n \\begin{document}\n-\n-\\begin{center} \\Large Shortest Path definitions\\end{center}\n+\\title{Temporal Graphs Research at Pomona College\\\\\n+  \\textit{Working Definitions and Vocabulary}\n+}\n+\\maketitle\n+\n+\\section{About}\n+\n+As we continue researching temporal graphs and their related properties there is\n+an increased need to establish fundamental definitions. Although many papers\n+have established definitions of the terms and concepts contained within this\n+paper, there are subtle details of each of these definitions that vary from\n+paper to paper making it difficult to efficiently communicate exact ideas.\n+\n+This document shall serve as an ongoing record of the definitions we will use\n+in our research. The definitions contained within should serve as the defaults\n+in conversations; if you intend to use an alternate definition to a concept\n+defined in this document, you should state and/or cite the alternate definition.\n+(In the future, we will look to incorporate alternate definitions in this\n+document as well.) It should be noted that these definitions may change as we\n+continue our research.\n \n \\section{Preliminary Definitions}\n \n-Last week we talked about several potential definitions of `temporal shortest\n-path' that have different semantic value. We will consider these definitions\n-and address their usefulness by examining algorithms and attempting to prove\n-different facts.\n+\\begin{defn}\n+  A \\textbf{temporal graph} or \\textbf{temporal network}\\footnote{We will treat\n+  the terms \\textit{graph} and \\textit{network} synonymously throughout this\n+  paper.} is defined as a tuple $G \\in V \\times E$ where $V$ is the set of\n+  vertices, and $E \\in V^2 \\times T^2$ is the set of edges such that\n+  $T = [0, \\infty)$:\n+\n+  $$G = (V, E)$$\n+\n+  Each edge $e \\in E$ is a 5-tuple $(v_i, v_j, t_0, t_f, w)$ such that\n+  $v_i, v_j \\in V$, $t_0, t_f \\in T$, and $t_0 \\leq t_f$. An edge is said to\n+  connect $v_i$ to $v_j$ (with the possibility of $v_i = v_j$) between the time\n+  $[t_0, t_f]$ with weight $w$. That is, all edges are to be treated as directed\n+  edges with some weight and time-window (possibly instantaneous) in which they\n+  are \\textit{active}.\n+\\end{defn}\n+\n+With this generalized definition of a temporal network, we will begin to explain\n+some common, specialized descriptors of these networks:\n+\n+\\begin{defn}\n+  We will describe a temporal graph as \\textbf{undirected} $\\forall e \\in E$\n+  if there exists $e' \\in E$ such that $e = (v_i, v_j, t_0, t_f,w)$ and $e' =\n+  (v_j, v_i, t_0, t_f, w)$.\\footnote{As we consider the real-life applications\n+  of these temporal networks, it may be convenient to define an edge as\n+  a 6-tuple with the last component $l$ being a unique label or edge\n+  differentiator. This would allow for the following scenario: if author\n+  \\texttt{A} and \\texttt{B} collaborate in window $[t_0, t_f]$ on multiple\n+  papers with the same weight, we could differentiate two edges connecting the\n+  two authors instead of condensing the two paper collaborations into one edge.\n+  In many cases, it may not make sense to do this, but it is valuable to have\n+  an umbrella definition that encompasses every type of graph we could create.}\n+\n+  Informally, if for every directed edge in the graph, there exists a\n+  corresponding edge in the opposite direction connecting the same two\n+  vertices and labeled with the exact same $t_i, t_f,$ and $w$, we will consider\n+  the temporal graph to be undirected.\n+\\end{defn}\n+\n+\\begin{defn}\n+  We will describe a temporal graph as \\textbf{unweighted} if every edge in $E$\n+  has equal weight $w$.\n+\\end{defn}\n+\n+\\begin{defn}\n+  \\textbf{Edge persistence} is the amount of time for which an edge is present.\n+  More formally, in a graph $(V,E)$ the persistence of an edge $(u,v,t_1,t_2,\n+  w)$ is defined as $t_2 - t_1$ or simply $\\delta t$.\n+\\end{defn}\n+\n+\\begin{defn}\n+  An edge $e \\in E$ is said to be \\textbf{infinitely persistent}\n+  if $t_f = \\infty$.\n+\\end{defn}\n \n \\begin{defn}\n-  An \\textbf{edge-varying temporal network} is a tuple $G \\in V \\times E$ where\n-  the set $V$ is the set of vertices, and $E \\in V^2 \\times T^2$, where $V$ is\n-  the set of vertices and $T = [0,\\infty)$.  The first vertex represents the\n-  A \\textbf{vertex-varying temporal network} is a tuple $G \\in V \\times E$ where\n-  the set $V$ is the set of vertices -- each labelled with a start and end time,\n-  and $E \\in V^2$ , a pair of vertices.\n-  When the vertex pair within the edges is ordered, call the graph\n-  \\textbf{directed}; otherwise call it \\textbf{undirected}.\n+  A \\textbf{instantaneous edge} or \\textbf{contact edge} is any edge $e \\in E$\n+  where $t_0 = t_f$.\n \\end{defn}\n \n-Specifically we will analyze the effect of these definitions on the coauthorship\n-and citation networks.\n+\\begin{defn}\n+  \\textbf{Interval networks} or \\textbf{windowed networks} shall be defined as","path":"paths.tex","position":114,"original_position":114,"commit_id":"ffdb7edd6c1e4014ac5f18841dec9be258dc1952","original_commit_id":"ffdb7edd6c1e4014ac5f18841dec9be258dc1952","user":{"login":"ericthewry","id":10351855,"avatar_url":"https://avatars.githubusercontent.com/u/10351855?v=3","gravatar_id":"","url":"https://api.github.com/users/ericthewry","html_url":"https://github.com/ericthewry","followers_url":"https://api.github.com/users/ericthewry/followers","following_url":"https://api.github.com/users/ericthewry/following{/other_user}","gists_url":"https://api.github.com/users/ericthewry/gists{/gist_id}","starred_url":"https://api.github.com/users/ericthewry/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/ericthewry/subscriptions","organizations_url":"https://api.github.com/users/ericthewry/orgs","repos_url":"https://api.github.com/users/ericthewry/repos","events_url":"https://api.github.com/users/ericthewry/events{/privacy}","received_events_url":"https://api.github.com/users/ericthewry/received_events","type":"User","site_admin":false},"body":"We need to be careful with terminology here.  An interval graph is a static structure that is often used in a very different way than we are describing (its something I've been intending to bring up but haven't since it hasn't been pressing, might be useful for our definitions of contemporaneity). Since you above said that network and graph are interchangable, we'll get into trouble when we want to introduce these interval graphs.","created_at":"2016-03-01T04:30:17Z","updated_at":"2016-03-01T04:30:17Z","html_url":"https://github.com/pomonadb/graphstreams/pull/4#discussion_r54521809","pull_request_url":"https://api.github.com/repos/pomonadb/graphstreams/pulls/4","_links":{"self":{"href":"https://api.github.com/repos/pomonadb/graphstreams/pulls/comments/54521809"},"html":{"href":"https://github.com/pomonadb/graphstreams/pull/4#discussion_r54521809"},"pull_request":{"href":"https://api.github.com/repos/pomonadb/graphstreams/pulls/4"}}},"pull_request":{"url":"https://api.github.com/repos/pomonadb/graphstreams/pulls/4","id":61025984,"html_url":"https://github.com/pomonadb/graphstreams/pull/4","diff_url":"https://github.com/pomonadb/graphstreams/pull/4.diff","patch_url":"https://github.com/pomonadb/graphstreams/pull/4.patch","issue_url":"https://api.github.com/repos/pomonadb/graphstreams/issues/4","number":4,"state":"open","locked":false,"title":"Generalized Temporal Graph Definition","user":{"login":"rwoll","id":11915034,"avatar_url":"https://avatars.githubusercontent.com/u/11915034?v=3","gravatar_id":"","url":"https://api.github.com/users/rwoll","html_url":"https://github.com/rwoll","followers_url":"https://api.github.com/users/rwoll/followers","following_url":"https://api.github.com/users/rwoll/following{/other_user}","gists_url":"https://api.github.com/users/rwoll/gists{/gist_id}","starred_url":"https://api.github.com/users/rwoll/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/rwoll/subscriptions","organizations_url":"https://api.github.com/users/rwoll/orgs","repos_url":"https://api.github.com/users/rwoll/repos","events_url":"https://api.github.com/users/rwoll/events{/privacy}","received_events_url":"https://api.github.com/users/rwoll/received_events","type":"User","site_admin":false},"body":"@ericthewry @MelanieWu \r\n\r\n1. I have added a `builds` directory in the root of the project so you can easily read the PDFs without\r\n    having to clone and compile on your machine. (I know this is not necessarily best practice since the\r\n    PDF could get out of sync with the `.tex` file, but I figured it was convenient.)\r\n\r\n2. As you will see in the diff below/above I modified the definitions to use a more generalize, all-\r\n    encompassing temporal graph definition which corresponds to our discussion on Friday with Prof.\r\n    Wu. I also added some general terms and their meaning since I figured it would be convenient to have\r\n    everything in one place and reworded some of Eric's definitions in terms of those.\r\n\r\nIt's simply a quick revision, so please feel free to comment with corrections/discussion points/reversions.\r\n\r\nBest,\r\n@rwoll 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